Pearson Complete Guide to NDA/NA Examination, 2/e, The [2nd ed] 9788131793602, 9789332520325, 8131793605, 9332520321

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The Pearson Guide to the

NDA/NA Examination Second Edition

Edgar Thorpe Showick Thorpe

Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd. Licensees of Pearson Education in South Asia No part of this eBook may be used or reproduced in any manner whatsoever without the publisher’s prior written consent. This eBook may or may not include all assets that were part of the print version. The publisher reserves the right to remove any material in this eBook at any time. ISBN 9788131793602 eISBN 9789332520325 Head Office: A-8(A), Sector 62, Knowledge Boulevard, 7th Floor, NOIDA 201 309, India Registered Office: 11 Local Shopping Centre, Panchsheel Park, New Delhi 110 017, India

contents Preface to the Second Edition

v

Preface

vi SECTION–I

IntroductIon Armed Forces—Pride of the Nation

1.3

How to Crack the NDA/NA Examination

1.7 SECTION–II

PArt—A EnGLISH Chapter–1

Synonyms

2.1–2.12

Chapter–2

Antonyms

2.13–2.20

Chapter–3

Error Identification

2.21–2.32

Chapter–4

Sentence Completion

2.33–2.44

Chapter–5

Jumbled Words in a Sentence

2.45–2.68

Chapter–6

Jumbled Sentences in a Paragraph

2.69–2.78

Chapter–7

Sentence Improvement

2.79–2.86

Chapter–8

Reading Comprehension

2.87–2.108

PArt—B GEnErAL KnoWLEdGE Chapter–1

Indian History

2.109–2.150

Chapter–2

Geography

2.151–2.208

Chapter–3

Indian Polity

2.209–2.234

Chapter–4

Indian Economy

2.235–2.288

Chapter–5

General Science

2.289–2.382

Chapter–6

Basic General Knowledge

2.383–2.404

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▌ Contents

SECTION–III

MAtHEMAtIcS Chapter–1

Algebra

3.1–3.52

Chapter–2

Matrices and Determinants

Chapter–3

Trigonometry

Chapter–4

Analytical Geometry in 2D and 3D

3.111–3.144

Chapter–5

Differential Calculus

3.145–3.168

Chapter–6

Integral Calculus and Differential Equations

3.169–3.192

Chapter–7

Vector Algebra

3.193–3.208

Chapter–8

Statistics and Probability

3.209–3.228

3.53–3.74 3.75–3.110

•  Additional Support Material

3.229–3.354

SECTION–IV

PrEVIouS YEArS’ PAPErS •  Test 1 (Apr 2011)

4.1–4.24

•  Test 2 (Aug 2011)

4.25–4.44

•  Test 3 (Apr 2012)

4.45–4.64

•  Test 4 (Aug 2012)

4.65–4.84

•  Test 5 (Apr 2013)

4.85–4.102

Preface to the Second Edition It has been a few seasons since the first edition of The Pearson Guide to the NDA Examination was published. One of the reasons for having such a long gap between the first edition and the second one is the static nature of the format of the written test that has not witnessed much change at the macro level. However, there have been a few changes in some domains at micro level, especially in the general knowledge section, that have been the prime mover for bringing in the revised edition. Overall, the structure of the book affirms its extensiveness for the NDA examination aspirants. While working on the second edition, the focus has been on refurbishing the internal content and tuning it to suit the requirements of the aspirants. Special care has been taken to incorporate more practice questions of the existing type, add new topics/types of questions, and classify the topics properly to facilitate better understanding. You will observe a lot of pruning been done to reduce the extent of certain portions which have either become extinct or turned obsolete in the general knowledge section. Among the major changes are the new question banks for various chapters, additional study material and some new features such as NDA Boxes. Such changes take the usability of this book for test preparation to the next level. Along with these, the latest original question papers have been introduced for students to work on and test their preparation level against the type of questions that generally appear in the original tests. As a part of the constant endeavor to impart qualitative inputs to readers preparing for their target examinations, the book offers online support on general awareness and some additional practice material. Please visit www.thorpeseducation.com for extensive web resource. Finally, a special note of thanks to the valued readers for their constant encouragement and feedback. This new edition has been meticulously designed to help them achieve their goal. All the best. Showick Thorpe

Preface The Pearson Guide to the NDA Examination is structured to be a comprehensive tool for aspirants preparing for the NDA Examination. It not only prepares the readers on different areas of study which comprise the NDA examination but also assesses their preparation level. The book covers all important topics as per the UPSC syllabus. Each chapter begins with a presentation of the simplest and most basic information and methods, progressing gradually to the more advanced topics. It follows an easy-to-understand, pattern-oriented format, keeping a close watch on the trend of previous years’ original question papers. The subject matter covered in this book requires a 10+2 level of understanding in Mathematics, along with basic English and General Knowledge. These three topics constitute two papers of the examination, the book treats each of these in sufficient depth to facilitate real understanding and provides a considerable degree of confidence to the aspirants. In addition to the brief introduction to the various question types asked in the paper, the book has comprehensive practice exercises which provide numerous multiple-choice questions (MCQs) that involve students in an advanced learning process and increase their accuracy levels. A careful scrutiny of previous years’ question papers was also done for preparing this book. To encourage you to read and study actively, we have included some special features in this manual. A quick glance through the pages shows importance given to lists and tables. As mentioned above, a significant part of the book is devoted to the MCQs. With an objective to familiarize candidates with the current trends in examination and types of questions asked therein, we have provided a few recent years’ original papers at the end. These test papers will help students while assessing their level of preparation. In our constant endeavour to impart qualitative inputs to readers preparing for competitive examinations, we have provided online support on general awareness and some additional practice material. Please visit us at www.thorpeseducation.com for extensive web resources. Crafting this preparation manual has been a team effort. It is impossible to mention everyone who has contributed to the development of this book, but I would be sorely remiss if I did not express my gratitude to many students who have approached me for consultations and guidance. They have been my teachers as well as students, and this book is an expression of my gratitude to them. I was fortunate to have been assisted by a talented team of researchers who went through the proofs meticulously in order to weed out errors. Some errors, however, might have crept in at some stage despite our best efforts. I hope the reader will appreciate the book. Suggestions for improving the text are most welcome. Finally, I am merely continuing with the endeavour that was initiated by my father, the late Edgar Thorpe. Therefore, I am thankful to my publishers for including his name on the cover of this book. All the best. Showick Thorpe

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INTRODUCTION Armed Forces—Pride of the Nation How to Crack the NDA/NA Examination

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armed Forces–Pride of the nation Career in armed ForCes Choosing the right career is one of the most difficult decision a person has to make. There are professions which Indian parents generally want their children to go for such as being a doctor, an engineer, a teacher or an IT professional. A career in army is also one of the noble professions which one can feel proud of. The armed force officers are known for their honesty, discipline, self-reliance, courage and an ability to lead and inspire people around them in the most difficult circumstances. The Indian Armed Forces have a reputation of being one of the best in the world. The individuals in the armed forces are imparted training and made capable of handling complex and difficult situations. The army serves the nation both in time of war and peace. Army keeps the boundary safe from external aggression and helps in case internal crises arise or during natural calamities like floods and cyclones. The Army, Navy and Air Force are the three services of armed force in India. All the services safeguard the country from external threats, they are often involved in missions like the UN Peacekeeping forces and in disaster management. In case of earthquakes, floods or avalanches, the Army, Navy and Air Force are involved in search and rescue operations. One of the advantages of joining the Armed Forces is that one starts a career early in life. In some countries compulsory enlistment in army is there but in most of the democratic countries there is no such compulsion. The biggest advantage of joining the army as a teenager is that after a decade or so of service one is still young enough to

choose a career outside the Armed forces. For instance a naval officer can make a transit to Merchant Navy where as Air Force pilots have the option to switch to commercial airlines later in their career. Military training lays great emphasis on physical fitness and discipline. And ex service men are definitely fitter than most civilians because of the training they receive and their habit to lead a disciplined life even after retirement. Sailing and mountaineering, horse-riding, rifle shooting and other games and sports are a part of training in army. In the later years of service there is an option of taking the sport up seriously as a profession. The silver medalist in shooting in Olympics 2003, Major Raghvendra Singh Rathore is an Army officer. Army life is full of challenges and people who like to live life on the edge can be sure of adrenaline pumping moments in the army. Armed forces teach an individual to be selfless, to rise above the differences on the basis of caste, creed, colour etc. They foster the feeling of friendship and brotherhood. The army personals have a very dynamic life. They get posted in interesting in remote areas where they might never have gone if they had chosen some other career. They get to see the world. Besides that the Army has some of the finest clubs, hospitals, golf courses and schools in the country. Unlike the hassles of getting into clubs and schools and golf courses army personnel do not have to pay hefty amounts to get admission or membership. A career in the Armed Forces makes for a wholesome personality, and indeed, a wholesome life.

1.4

▌ Introduction

national deFenCe aCademy (nda) Anyone can be heroic from time to time, but a gentleman is something which you have to be all the time. Which isn’t easy.

Luigi Pirandello, Italian dramatist

about nda The National Defence Academy (NDA) at Khadakwasla, Pune can rightfully boast of raising year-after-year what other training institutes merely dream of: heroes and gentlemen. The raw talent is forged in the furnace of discipline, patriotism and courage under the watchful guidance of a dedicated team of educators

and trainers. The motto ‘Service Before Self” has been evident in the actions and deeds of the offi cers who have passed out from NDA over the years. The credit for the leadership of the Army can be given to the alumni of the NDA from where most of its offi cers have passed.

brieF History The foundation stone of the academy at Khadakwasla, Pune was laid by the first Prome Minister of India, Pt Jawaharlal Nehru on October 6, 1949. Located south-west of Pune City and north-west of Khadakwasla Lake on 7015 acres of land the National Defence

Academy was chosen while keeping factors like climatic conditions and suitability of terrain for training and proximity to Arbian sea.

tHe aim oF training The NDA transforms young men into courageous, dynamic and cultured and educated young officers of integrity. The training at the NDA aims to inculcate in the trainees or cadets moral values, leadership qualities, mental and physical strength, an adventurous spirit and a determination to win. The objectives of the training are achieved by focusing on character building, discipline, military and academic subjects, physical fitness and adventurous activities. The purpose of activity at the Academy is to promote the feeling of brotherhood, friendship, team spirit and a commitment to achieve excellence. The

curriculum and syllabi keep the cadets’ up-to-date with current theoretical and technological developments. The educational training aims at the overall development of cadets by providing a broad academic base for their future professional development. Each separate course contributes in the augmenting the potential of the cadet in the areas of analysis, reasoning, problem solving, decision making and expression. The training for future officers at the NDA ensures a gradual development of a cadet from a civilian to an effective commander.

about indian army The Army is the largest and the oldest branch of the Indian armed forces. The army fights on land and consists of various divisions including mainly the infantry, the artillery, and the armoured corps. The main responsibility of the army is to protect the boundaries of the country from external invasions and to invade other countries on behalf of their country. The army training teaches both offense and defense.

The army also functions in the time of peace in relief and rescue operations during natural calamities like floods, earthquakes and cyclones and in the maintenance of other essential services. The Army also assists the civil administration during internal security disturbances and in the maintenance of law and order. To achieve these objectives, the Army has to be constantly modernised, suitably structured and be equipped and trained.

Armed Forces—Pride of the Nation ▌ 1.5

about indian air ForCe The Indian Air Force Act established the “Royal Indian Air Force’’ on October 8, 1932. IAF’s first Squadron came into being on April 1, 1933. The Indian Air Force has been serving the nation now for more than sixty years. The IAF has continued to modernize and move along with the demands of contemporary advancements and today stands as a credible air power.

In addition to the traditional wartime roles of the IAF of conducting air strikes and supporting combat operations, surface, the Air Force also helps civil authorities during natural calamities like the cyclones and the earthquakes.

about indian navy The origins of the modern Indian navy are traced to a maritime force established by the East India Company in the seventeenth century. The Indian Navy force had a variety of names like the Indian Navy, Bombay Marine, and the Indian Marine. India has thriving seaborne trade linking it with other nations. The country’s economic well being is thus very closely

related to the ability to keep sea-lanes clear and open at all times. Besides protecting the country during war time the Indian Navy also helps in the time of natural calamity like floods or cyclones.

road to ‘an oFFiCer and a gentleman’ Army has a history of turning neighbourhood boys into officers and gentlemen. In fact, it can be rightly said that it is a road to an officer and a gentleman. Every year thousands of candidates appear in exams for joining the army and there are only a handful of them who get selected. The selected candidates become cadets and their journey towards being an officer and a gentleman begins from there. The cadets are trained to be disciplined, courageous, self-reliant and the ability to take complex situations into their stride is imbibed into them. The cadets on completion of their training become officers, they are no longer boys, they are men. They have to undergo strenuous training which makes them physically and mentally tough. Focus is also on character building, academic subjects, and adventurous activities. The feeling of tolerance towards other castes, creeds, races and the feeling of brotherhood, friendship and team spirit is promoted amongst the gentlemen cadets. The efficiency and success of the country’s Armed Forces — the Army, the Navy and the Air Force — depends largely on the quality of the officers who lead the men who comprise their ranks.

Becoming an officer is just the beginning but its not sufficient enough to make the officer a gentleman as well. The officer has also to put in efforts on his own. Besides honesty and professionalism, a gentleman has to be moral, courteous and chivalerous. Having good manners and etiquettes is important not only to be socially acceptable; it is the quality of a gentleman. An army officer is expected to be better behaving than a civilian. And also to be honest and direct in all their dealings. The advantage of being an officer and gentleman is confined not only to being in service but after ending of the period of commission also. For instance naval officers make a transit to Merchant navy, Air Force pilots also have the option of becoming commercial pilots. The training received in army goes a long way, the natural ability to lead helps in corporate organizations at managerial level positions, the mind developed by critical analysis and reasoning is a powerful tool in business. It is relatively easier for most ex-army officers to impress the people around with their good manners, communication skills, honesty and fair play in their dealings.

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How to Crack the nda/na examination overview oF tHe nda examination The National Defence Academy (NDA) Examination comprises (a) The written test, and, (b) Interview for intelligence and personality

test: of such candidates as may be called for interview at one of the Services selection centres.

seleCtion ProCedure Details pertaining to objective type tests including sample questions will be supplied to candidates along with the Admission Certificate in the “Candidates Information Manual”. However, some of the standard practices in the NDA examination are: 1. The candidates are not permitted to use calculators for answering objective type papers (test booklets). They should not, therefore, bring the same inside the examination hall. 2. The papers in all the subjects will consist of objective type questions only. 3. The question papers (test booklets) will be set in English only. 4. The commission has the discretion to alter the format of the written examination.

5. In the question papers, wherever necessary, questions involving only the metric system of weights and measures will be set. 6. Candidates must write the papers in their own hand. In no circumstances will they be allowed the help of a scribe to write answers for them. 7. The commission has discretion to fix the qualifying marks in any or all the subjects of the examination. 8. The standard of the paper in Elementary Mathematics will be of the matriculation examination standard. 9. The standard of papers in other subjects will approximately be such as may be expected of a graduate of an Indian university. 10. Please refer to the “Candidates Information Manual” for any recent changes in the scheme of the examination.

ComPosite time The composite time (time allotted for both) has to be budgeted in order to be able to attempt each paper of the written test. You may attempt the papers/questions in any order. However, it is advisable not to spend too much time on any one particular paper. To qualify in the written examination, each paper has

to be attempted separately and it is also important to obtain a sufficiently high rank in the order of merit. However, it may be noted that, in case, there is no clear time limit per paper, you are advised to adhere to self-discipline acquired through prior practice.

1.8

▌ Introduction

interview For intelligenCe and Personality test In addition to the interview, the candidates will be put through intelligence tests, both verbal and non-verbal, designed to assess their basic intelligence. They will also be put through group tests such as group discussions, group planning, outdoor group tasks,

and asked to give brief lectures on specified subjects. All these tests are intended to judge the mental calibre of the candidate. In broad terms, this is really an assessment of not only his intellectual qualities but also his social traits and interest in current affairs.

SYLLABUS FOR NDA (Source: www.upsc.gov.in)

PAPER-I

Mathematics (Maximum Marks: 300)

algebra Concept of a set, operations on sets, Venn diagrams, De Morgan laws, Cartesian product, relation, equivalence relation. Representation of real numbers on a line. Complex numbers — basic properties, modulus, argument, cube roots of unity. Binary system of numbers. Conversion of a number in decimal

system to binary system and vice-versa. Arithmetic, Geometric and Harmonic progressions. Quadratic equations with real coefficients. Solution of linear in equations of two variables by graphs. Permutation and Combination. Binomial theorem and its application. Logarithms and their applications.

matriCes and determinants Types of matrices, operations on matrices Determinant of a matrix, basic properties of determinant. Adjoint and inverse of a square

matrix, Applications — Solution of a system of linear equations in two or three unknowns by Cramer’s rule and by Matrix Method.

trigonometry Angles and their measures in degrees and in radians. Trigonometrical ratios. Trigonometric identities Sum and difference formulae.

Multiple and Sub-multiple angles. Inverse trigonometric functions. Applications — Height and distance, properties of triangles.

analytiCal geometry oF two and tHree dimensions Rectangular Cartesian Coordinate system. Distance formula. Equation of a line in various forms. Angle between two lines. Distance of a point from a line. Equation of a circle in standard and in general form. Standard forms of parabola, ellipse and hyperbola. Eccentricity and axis of a conic.

Point in a three dimensional space, distance between two points. Direction Cosines and direction ratios. Equation of a plane and a line in various forms. Angle between two lines and angle between two planes. Equation of a sphere.

1.10

▌ Introduction

diFFerential CalCulus Concept of a real valued function — domain, range and graph of a function. Composite functions, one to one, onto and inverse functions. Notion of limit, Standard limits, Continuity of functions — examples, algebraic operations on continuous functions. Derivative of a function at a point, geometrical and physical interpreatation of a

derivative — applications. Derivatives of sum, product and quotient of functions, derivative of a function with respect of another function, derivative of a composite function. Second order derivatives. Increasing and decreasing functions. Application of derivatives in problems of maxima and minima.

integral CalCulus and diFFerential equations Integration as inverse of differentiation, integration by substitution and by parts, standard integrals involving algebraic expressions, trigonometric, exponential and hyperbolic functions. Evaluation of definite integrals — determination of areas of plane regions bounded by curves — applications. Definition of order and degree

of a differential equation, formation of a differential equation by examples. General and particular solution of a differential equation, solution of first order and first degree differential equations of various types. Application in problems of growth and decay.

veCtor algebra Vectors in two and three dimensions, magnitude and direction of a vector. Unit and null vectors, addition of vectors, scalar multiplication of vector, scalar product or dot product of two-vectors. Vector product

and cross product of two vectors. Applications — work done by a force and moment of a force, and in geometrical problems.

statistiCs and Probability Statistics: Classification of data, Frequency distribution, cumulative frequency distribution-examples Graphical representation — Histogram, Pie Chart, Frequency Polygonexamples. Measures of Central tendency — mean, median and mode. Variance and standard deviation — determination and comparison. Correlation and regression. Probability: Random experiment, outcomes and associated sample space, events, mutually exclusive and exhaustive events,

impossible and certain events. Union and Intersection of events. Complementary, elementary and composite events. Definition of probability — classical and statistical- examples. Elementary theorems on probability-simple problems. Conditional probability, Bayes’ theorem-simple problems. Random variable as function on a sample space. Binomial distribution, examples of random experiments giving rise to Binominal distribution.

PAPER-I

general Ability Test (Maximum Marks: 600)

Part a

ENgLISh The question paper in English will be designed to test the candidate’s understanding of English and workman like use of words. The syllabus covers various aspects like: Grammar and

(Maximum Marks: 200)

usage, vocabulary, comprehension and cohesion in extended text to test the candidate’s proficiency in English.

Part b

gENERAL KNOwLEDgE The question paper on General Knowledge will broadly cover the subjects: Physics, Chemistry, General Science, Social Studies, Geography and Current Events. The syllabus given below is designed to indicate the scope of these subjects included in this paper. The topics mentioned are

(Maximum Marks: 400)

not to be regarded as exhaustive and questions on topics of similar nature not specifically mentioned in the syllabus may also be asked. Candidates’ answers are expected to show their knowledge and understanding of the subject.

PHysiCs Physical Properties and States of Matter, Mass, Weight, Volume, Density and Specific Gravity, Principle of Archimedes, Pressure Barometer. Motion of objects, Velocity and Acceleration, Newton’s Laws of Motion, Force and Momentum, Parallelogram of Forces, Stability and Equilibrium of bodies, Gravitation, elementary ideas of work, Power and Energy. Effects of Heat, Measurement of temperature and heat, change of State and Latent Heat, Modes of transference of Heat.

Sound waves and their properties, Simple musical instruments. Rectilinear propagation of Light, Reflection and refraction. Spherical mirrors and Lenses. Human Eye. Natural and Artificial Magnets, Properties of a Magnet, Earth as a Magnet. Static and Current Electricity, conductors and Non-conductors, Ohm’s Law, Simple Electrical Circuits, Heating, Lighting and Magnetic effects of Current, Measurement of Electrical Power, Primary and Secondary Cells, Use of X-Rays.

1.12

▌ Introduction

general PrinCiPles in tHe working oF tHe Following Simple Pendulum, Simple Pulleys, Siphon, Levers, Balloon, Pumps, Hydrometer, Pressure Cooker, Thermos Flask, Gramophone,

Telegraphs, Telephone, Periscope, Telescope, Microscope, Mariner’s Compass; Lightening Conductors, Safety Fuses.

CHemistry Physical and Chemical changes. Elements, Mixtures and Compounds, Symbols, Formulae and simple Chemical Equations, Law of Chemical Combination (excluding problems). Properties of Air and Water. Preparation and Properties of Hydrogen, Oxygen, Nitrogen and Carbondioxide, Oxidation and Reduction. Acids, bases and salts.

Carbon — different forms Fertilizers — Natural and Artificial Material used in the preparation of substances like soap, Glass, Ink, Paper, Cement, Paints, Safety Matches, and Gun-Powder. Elementary ideas about the Structure of Atom, Atomic, Equivalent and Molecular Weights, Valency.

general sCienCe Difference between the living and non-living. Basis of Life-Cells, Protoplasms and Tissues. Growth and Reproduction in Plants and Animals. Elementary knowledge of human Body and its important organs. Common Epidemics, their causes and prevention. Food — Source of Energy for man. Constituents of food, Balanced Diet. The Solar System — Meteors and Comets, Eclipses. Achievements of Eminent Scientists. Section ‘D’ (History, Freedom Movement etc.) A broad survey of Indian History, with emphasis on Culture and Civilisation. Freedom Movement in India.

Elementary study of Indian Constitution and Administration. Elementary knowledge of Five Year Plans of India. Panchayati Raj, Co-operatives and Community Development. Bhoodan, Sarvodaya, National Integration and Welfare State, Basic Teachings of Mahatma Gandhi. Forces shaping the modern world; Renaissance, Exploration and Discovery; War of American Independence. French Revolution, Industrial Revolution and Russian Revolution. Impact of Science and Technology on Society. Concept of one World, United Nations, Panchsheel, Democracy. Socialism and Communism. Role of India in the present world.

geograPHy The Earth, its shape and size. Lattitudes and Longitudes, Concept of time. International Date Line. Movements of Earth and their effects.

Origin of Earth. Rocks and their classification; WeatheringMechanical and Chemical, Earthquakes and volcanoes.

oCean Currents and tides Atmosphere and its composition; Temperature and Atmospheric Pressure, Planetary Winds, cyclones and Anti-cyclones; Humidity; Condensation and Precipitation; Types of Climate. Major Natural regions of the World.

Regional Geography of India — Climate, Natural vegetation. Mineral and Power resources; location and distribution of agricultural and industrial activities. Important Sea ports and main sea, land and air routes of India. Main items of Imports and Exports of India.

Current events Knowledge of Important events that have happened in India in the recent years.

Current important world events.

Prominent personalities — both Indian and International including those connected with cultural activities and sports.

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ENGLISH CHAPTER 1. SYNONYMS CHAPTER 2. ANTONYMS CHAPTER 3. ERROR IDENTIFICATION CHAPTER 4. SENTENCE COMPLETION CHAPTER 5. JUMBLED WORDS

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SENTENCE

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CHAPTER 7. SENTENCE IMPROVEMENT CHAPTER 8. READING COMPREHENSION

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synonyms Format oF the Questions A synonym is a word or expression accepted as a figurative or symbolic substitute for another word or expression. In other words, we can say that a synonym is a word which has the same, or almost the same meaning as that of another word in the same language. For example: mix, blend and mingle are synonyms as they convey more or less the same meanings. English has the largest vocabulary and, therefore, has more synonyms than any other language in the world. It may, however, be noted that, except in rare cases, no two words are exact synonyms that are interchangeable in all the contexts in which either might appear. However, within a given context, there are often close synonyms. For example: teach, educate and instruct, convey more or less same meaning and can be called they slightly differ from one another, principally in degrees of abstraction. Hence, words which are synonymous in one of their meanings may differ considerably in their other meanings.

Direction: In each of the following questions, a word is given in capital letters followed by four alternative words marked a-d. Select from the alternatives, the word nearest in meaning to the word given in capital letters: 1. DISTASTEFUL (a) not delicious (c) unpleasant

(b) tasteless (d) useless

2. ACQUIESCE (a) something liquid (c) consent

(b) watery (d) to know someone

3. ESCALATE (a) retard (c) hamper

(b) step up (d) oppose

2. ACCOMPLICE (a) friend (c) criminal

(b) abetter (d) thief

3. ALLEGIANCE (a) disaffection (c) loyalty

(b) satisfactory (d) enemity

answer Key 1. (c)

2. (c)

3. (b)

test oF synonyms In the following questions, a word is given in capital letters followed by four alternative words marked a–d. Select, from the alternatives, the word that conveys the same meaning as the word given in capital letters: 1. ABASH (a) acquit (c) blush

(b) confuse (d) condemn

2.4

▌ English

4. BIASED (a) careful (c) expert

(b) partial (d) unskilled

20. ABASE (a) low (c) humble

(b) peaceful (d) cruel

5. INTENSIFY (a) boiling (c) aggravate

(b) satisfy (d) destroyed

21. FORSAKE (a) forgive (c) abandon

(b) punish (d) benefit

6. REPULSIVE (a) frustrating (c) admirable

(b) repellent (d) pleasant

22. GRUDGE (a) hard (c) dangerous

(b) very hot (d) envy

7. PROHIBIT (a) banish (c) forbid

(b) punish (d) exhibit

23. PRUDENT (a) skilled (c) cautious

(b) experienced (d) criminal

8. CALLOUS (a) large (c) thick

(b) hardened in feeling (d) dangerous

24. VOTARY (a) devotee (c) religious

(b) conscious (d) amiable

9. COLLOSSAL (a) dangerous (c) fatal

(b) gigantic (d) honourable

25. TETE-A-TETE (a) delightful (c) piercing

(b) penetrating (d) conversation

10. DELUSION (a) actuality (c) clarity

(b) illusion (d) emulating

26. ABET (a) force (c) assist

(b) discourage (d) neglect

11. AMICABLE (a) poisonous (c) satisfying

(b) friendly (d) heartening

27. ABRIDGE (a) desire (c) graceful

(b) to condense (d) disconnect

12. BRUTAL (a) humane (c) ruthless

(b) adamant (d) criminal

28. IMPOSTER (a) imposer (c) cheat

(b) lazy (d) painful

13. EXQUISITE (a) clumsy (c) graceful

(b) rough (d) unrefined

29. SHUN (a) destroy (c) avoid

(b) weaken (d) include

14. CONCISENESS (a) correctness (c) brevity

(b) frankness (d) refiness

30. COWARDICE (a) timidity (c) criminal

(b) gallantry (d) humanity

15. CANDID (a) sweet (c) frank

(b) pleasing (d) dishonest

31. TEMPERATE (a) cool (c) favourable

(b) moderate (d) dangerous

16. BOISTEROUS (a) boiling (c) noisy

(b) hot (d) dangerous

32. VIRGINITY (a) decency (c) beauty

(b) maidenhood (d) attractiveness

17. EXTEMPORE (a) praiseworth (c) extraordinary

(b) without preparation (d) essential

33. ANTIQUE (a) old-fashioned (c) modern

(b) rare (d) unique

18. ACUMEN (a) beauty (c) insight

(b) appearance (d) distaste

34. INCITE (a) discourage (c) internal

(b) encourage (d) crime

19. ABSTAIN (a) influence (c) retain

(b) refrain (d) retard

35. TWAIN (a) discarded (c) couple

(b) worthless (d) strong

Synonyms

▌ 2.5

36. TRIVIAL (a) economical (c) rare

(b) uneconomical (d) commonplace

52. APPLAUD (a) rebuke (c) praise

(b) slam (d) condemn

37. ZENITH (a) sharp (c) bravery

(b) bright (d) climax

53. Denounce (a) support (c) maintain

(b) sustain (d) condemn

38. STRUMPET (a) surprise (c) noisy

(b) prostitute (d) peaceful

54. Corroborate (a) verify (c) disprove

(b) refute (d) disapprove

39. SLANDER (a) circular (c) fragile

(b) spherical d) damage

55. Serenity (a) agitation (c) stir

(b) protest (d) calmness

40. REVERBERATE (a) reflect (c) revitalise

(b) accelerate (d) gibberish

56. Anxiety (a) worry (c) faith

(b) comfort (d) reassurance

41. REVITALISE (a) accelerate (c) minimise

(b) vigorise (d) maximise

57. Accord (a) agreement (c) dispute

(b) disagreement (d) disparity

42. WEARISOME (a) beautiful (c) handsome

(b) attractive (d) tiresome

58. LETHAL (a) deadly (c) smooth

(b) sluggish (d) unlawful

43. UNDAUNTED (a) careless (c) worthless

(b) not discouraged (d) prudent

59. LIMPID (a) moist (c) transparent

(b) dear (d) dark

44. LETHAL (a) sharp (c) deadly

(b) brutal (d) sickly

60. MACHINATION (a) labour saving (c) factory work

(b) evil plot (d) engine part

45. LEVITATE (a) evaporate (c) to rise

(b) to rape (d) subsidized

61. MALINGERING (a) creating ill will (c) defaming

(b) feigning illness (d) being habitually lazy

46. REVAMP (a) prostitute (c) decamp

(b) dangerous (d) reconstruct

62. MELEE (a) kindness (c) simple song

(b) brawl (d) primitive dance

47. TENACIOUS (a) tough (c) uncomfortable

(b) poisonous (d) piercing

63. MOLECULE (a) little heap of earth (c) enormous

(b) birth mark (d) particle

48. THESPIAN (a) alcoholic (c) inexperienced

(b) actor (d) ancient

64. NAUTICAL (a) perverse (c) naval

(b) disgusting (d) unaffected

49. VENIAL (a) contagious (c) worthless

(b) filthy (d) excusable

65. NOSTALGIA (a) aroma (c) home sickness

(b) sea sickness (d) cure-all

50. STUBBORN (a) fearless (c) loyal

(b) shameless (d) determined

66. OBTUSE (a) difficult (c) blunt

(b) interfering (d) concealed

51. SHREWD (a) naive (c) raw

(b) clever (d) youthful

67. OMNIPOTENT (a) all knowing (c) everlasting

(b) all powerful (d) all merciful

2.6

▌ English

(b) puzzling (d) illusive

83. GRAPHIC (a) pictorial (c) geometrical

(b) squarish (d) mathematical

69. ORIENTATION (a) eastward migration (c) adjustment to facts

(b) likeness to rules (d) teaching theories

84. HANDSOME (a) polite (c) gentle

(b) noble (d) good-looking

70. OSTENSIBLY (a) actually (c) apparently

(b) conspicuously (d) quietly

85. VOCATION (a) employment (c) occupation

(b) hobby (d) past

86. DYNAMIC (a) active (c) forceful

(b) robust (d) vigorous

87. PLEASURE (a) disappointment (c) grief

(b) happiness (d) anxiety

88. RIVAL (a) friend (c) associate

(b) partner (d) opponent

89. GENERATE (a) prefer (c) command

(b) race (d) produce

90. ACUTE (a) rice (c) severe

(b) accidental (d) curious

91. LATENT (a) hard (c) visible

(b) concealed (d) display

92. Robust (a) wear (c) forceful

(b) useless (d) stupid

93. LENIENT (a) cruel (c) kind

(b) rough (d) harsh

94. VENERATE (a) defame (c) respect

(b) abuse (d) accuse

95. SHALLOW (a) not deep (c) hidden

(b) low (d) high

96. REWARD (a) retribution (c) forfeiture

(b) penalty (d) demotion

97. FOREIGNER (a) alien (c) local

(b) native (d) national

98. GATHER (a) spend (c) amass

(b) scatter (d) separate

68. ENIGMATIC (a) pithy (c) complicated

71. OSTENTATION (a) protruding (c) decorative

(b) wealthy (d) showy

72. PANORAMA (a) broad scene (c) light weight hat

(b) deck of cards (d) holiday

73. PENITENCE (a) retribution (c) confinement

(b) submission (d) repentance

74. MYSTICAL (a) imaginary (c) prophetic

(b) vague (d) spiritually symbolic

75. SINGULAR (a) extraordinary (c) to the point

(b) simple (d) representative

76. SAVOURING (a) eating (c) liking

(b) enjoying (d) loving

77. CASTIGATE (a) deligate (c) criticise

(b) evaluate (d) enforce

78. SKIRMISH (a) fight (c) enimity

(b) contact (d) relations

79. DOMAIN (a) area (c) marketing

(b) main (d) fielding

80. EXCEEDINGLY (a) politely (c) freely

(b) extremely (d) outwardly

81. UNANIMOUSLY (a) friendly (c) unquestionably

(b) forcefully (d) democratically

82. STUPID (a) disobedient (b) incapable (c) insane (d) silly

Synonyms 99. CONCEALED (a) disclosed (c) camouflaged

100. INNOCENT (a) guilty (c) deadly

(b) opened (d) released

▌ 2.7

(b) clean (d) sinful

answer Key 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

(b) (b) (c) (b) (c) (b) (c) (b) (b) (b) (b) (c) (c)

14. 15. 16. 17. 18. 19.

(c) (c) (c) (d) (c) (b)

27. 28. 29. 30. 31. 32.

(b) (c) (c) (a) (b) (b)

40. 41. 42. 43. 44. 45.

(a) (b) (d) (b) (c) (c)

53. 54. 55. 56. 57. 58.

(d) (a) (d) (a) (a) (a)

66. 67. 68. 69. 70. 71.

(c) (a) (b) (c) (c) (d)

79. 80. 81. 82. 83. 84.

(a) (b) (c) (d) (a) (d)

20. 21. 22. 23. 24. 25. 26.

(c) (c) (d) (c) (a) (d) (c)

33. 34. 35. 36. 37. 38. 39.

(a) (b) (c) (d) (d) (b) (d)

46. 47. 48. 49. 50. 51. 52.

(d) (a) (b) (d) (d) (b) (c)

59. 60. 61. 62. 63. 64. 65.

(c) (a) (b) (b) (d) (c) (c)

72. 73. 74. 75. 76. 77. 78.

(a) (d) (d) (a) (b) (c) (a)

85. 86. 87. 88. 89. 90. 91.

(c) (a) (b) (d) (d) (c) (b)

PraCtiCe eXerCise Direction: In each of the sentences, a word is printed in capital. For each of the capital word, four words are listed below. Choose the word nearest in meaning to it.

8. IMPROVIDENT (a) thriftiness (c) unhappy

(b) incautious (d) shabby

9. PENITENCE (a) confinement (c) submission

(b) retribution (d) repentance

1. QUAY (a) peninsula (c) target

(b) wharf (d) island

2. PERNICKETY (a) spiteful (c) fussy

(b) careless (d) ignorant

10. APPELLATION (a) anonymity (c) submission

(b) regard (d) repentance

3. QUAINT (a) old (c) haunted

(b) quiet (d) usual

11. RAUCOUS (a) flavourful (c) aggressive

(b) boisterous (d) evil

4. ALBEIT (a) although (c) because

(b) further more (d) not only

12. QUACK (a) pyromaniac (c) dilemma

(b) charlatan (d) victim

5. LETHARGY (a) dynamism (c) cool-headed

(b) laziness (d) lavish

13. VISAGE (a) appearance (c) hermitage

(b) doubt (d) armour

6. BUMPTIOUS (a) hurry (c) modest

(b) haste (d) assertive

14. MASTICATE (a) swallow (c) meander

(b) organise (d) reduce

7. EBULLIENT (a) exorbitant (c) obscure

(b) exuberant (d) still

15. MAWKISH (a) certain (c) sentimental

(b) devious (d) carefree

92. 93. 94. 95. 96. 97.

(c) (c) (c) (a) (a) (a)

98. (c) 99. (c) 100. (b)

2.8

▌ English

16. MILITANT (a) dramatic (c) combative

(b) religious (d) quaint

17. OSTENSIBLE (a) apparent (c) traditional

(b) elaborate (d) showy

18. MACABRE (a) innocent (c) gruesome

(b) tarried (d) pleasing

32. ENTICE (a) cheat (c) flight

(b) lure (d) love

33. PENSIVE (a) habitually tardy (b) confident (c) given to quiet reflection (d) affectionate 34. SLOTHFUL (a) fat (c) ignorant

(b) stubborn (d) lazy

35. EMANCIPATE (a) set free (c) correct morally

(b) exist (d) restrain

36. OBTUSE (a) stupid (c) moderate

(b) diligent (d) champ

37. PARLOUS (a) tempting (c) dangerous

(b) obsolete (d) concrete

38. ADMONITION (a) thrash (c) warning

(b) hindrance (d) rebuke

39. SURREPTITIOUS (a) deceiving (c) tricky

(b) secret (d) imaginary

19. DENOUEMENT (a) outcome (c) action

(b) eschew (d) character

20. MANACLE (a) fetter (c) spectacles

(b) ornament (d) order

21. FEIGN (a) hesitate (c) deserve

(b) pretend (d) attend

22. ASSAY (a) attack (c) case

(b) accept (d) evaluate

23. GIST (a) contribution (c) prestige

(b) substance (d) accessory

24. SOLICITIOUS (a) nonchalant (c) reverential

(b) firm (d) worried

25. HOMELY (a) refined (c) reliable

(b) plain (d) amiable

40. MOLLIFY (a) appease (c) tarnish

(b) justify (d) appreciate

26. LOQUACIOUS (a) verbose (c) rational

(b) taciturn (d) alluring

41. Oracular (a) false (c) absurd

(b) mysterious (d) predicting

27. PACIFY (a) placate (c) harass

(b) rouse (d) rejoice

42. WAIVE (a) restrict (c) permit

(b) relax (d) admit

28. INFIRM (a) dull (c) slim

(b) sickly (d) unyielding

43. ELAN (a) flair (c) inspiration

(b) spiritual (d) boredom

29. ACCREDITED (a) authorised (c) unauthorised

(b) disallowed (d) not debited

44. GAUCHE (a) vain (c) polished

(b) rich (d) tactless

30. RESOLUTE (a) stout (c) strong

(b) determined (d) feeble

45. PREPOSTEROUS (a) formal (c) ridiculous

(b) judicious (d) ceremonious

31. LOATH (a) tired (c) sickly

(b) unwilling (d) spirited

46. DOMINION (a) recreation (c) issue

(b) separation (d) realm

Synonyms 47. ATTEST (a) allure (c) assign

(b) corroborate (d) enlarge

49. ANNOTATION (a) prologue (c) translation

(b) explanatory note (d) quip

48. QUALM (a) crisis (c) altercation

(b) scruple (d) attribute

50. EQUIPOISE (a) patience (c) balance

(b) modesty (d) representation

▌ 2.9

answer Key 1. (b) 2. (c) 3. (c)

8. (b) 9. (d) 10. (a)

15. (c) 16. (c) 17. (a)

22. (d) 23. (b) 24. (d)

29. (a) 30. (b) 31. (b)

36. (a) 37. (c) 38. (c)

43. (a) 44. (d) 45. (c)

4. 5. 6. 7.

11. 12. 13. 14.

18. 19. 20. 21.

25. 26. 27. 28.

32. 33. 34. 35.

39. 40. 41. 42.

46. 47. 48. 49.

(a) (b) (d) (b)

(c) (b) (a) (d)

(c) (a) (a) (b)

(b) (a) (a) (b)

(b) (b) (d) (a)

(b) (a) (d) (b)

(d) (b) (b) (b)

nDa BoX Directions: In this section, you will find a number of sentences, parts of which are underlined. For each underlined part, four words/phrases are listed below. Choose the word phrase nearest in meaning to the underlined part. 1. He was not at all abashed by her open admiration. (a) delighted (b) piqued (c) embarrassed (d) livid 2. Rahul was amazed at how affable his new employer was. (a) demanding (b) polite (c) repulsive (d) quality-conscious 3. Since our plans are amorphous we shall send you the detailed programme at a later date. (a) impractical (b) prohibitive (c) inimical (d) formless 4. Preeti’s arduous efforts had sapped her energy. (a) over-ambitious (b) strenuous (c) sterile (d) apocryphal 5. The manager’s articulate presentation of the advertising campaign impressed his employers. (a) well-prepared (b) effective (c) superficial (d) banal 6. I do not wish to be beholden to anyone in this office. (a) dependent (b) opposed (c) obligated (d) sycophant Directions (For the following 7 items): Each of the following seven items consists of a word in capital letters, followed by four words or groups of words. Select the word or group of words that is most similar in meaning to the word in capital letters.

7. PREPOSTEROUS (a) Careful (c) Absurd

(b) Casual (d) Deterrent

8. DELECTATION (a) Caution (b) Entertainment (c) Reward (d) Composure 9. TO EULOGIZE (a) To criticize (b) To praise (c) To work very hard (d) To spend recklessly 10. SLOTH (a) Laziness (c) Sludge

(b) Anger (d) Malady

11. TO OBFUSCATE (a) To rebuke (b) To make fun (c) To make something difficult to understand (d) To escape 12. PRECURSOR (a) Forerunner (b) Prerequisite (c) Prerogative (d) Vibrations 13. PARSIMONIOUS (a) Thrifty (c) Extravagant

(b) Cunning (d) Upright

50. (c)

2.10

▌ English

Directions (For the following 10 items): Each of the following ten items consists of a word in capital letters, followed by four words or groups of words. Select the word or group of words that is most similar in meaning to the word in capital letters. 14. DESPOT (a) Pragmatic person (b) Cruel ruler (c) Knowledgeable historian (d) Leading industrialist 15. MUSTY (a) Tiring (c) Swanky

(b) Deserted (d) Obsolete

16. PERNICKETY (a) Highly talented (c) Fussy

(b) Futuristic (d) Extravagant

17. ROSETTE (a) Badge (c) Exhibition

(b) Wild jungle (d) Palatial building

18. ORNATE (a) Dimension (c) Richly decorated

(b) Ancient (d) Vigour

19. DIAPHANOUS (a) Cylindrical (c) Complicated

(b) Almost transparent (d) Steep climb

20. COME-UPPANCE (a) Retribution (c) Leisure

(b) Lenience (d) Competitive nature

21. UNSCATHED (a) Unstinting (c) Unspeakable

(b) Unsociable (d) Unharmed

22. EPAULETTE (a) Ammunition (c) Equipment

(b) Shoulder Ornament (d) Cockpit

23. EXPIATION (a) Expansion (c) Explanation

(b) Atonement (d) Addition

Directions (For the next 9 items): Each of the following nine items consists of a word or a group of words in capital letters, followed by four words or groups of words. Select the word or group of words that is most similar in meaning to the word in capital letters. 24. DEBONAIR (a) Bed-ridden (c) Elegant

(b) Candid (d) Thrifty

25. TO ASSUAGE (a) To assume (c) To resolve

(b) To forget (d) To mitigate

26. TO BROWBEAT (a) To bully (c) To give a hint

(b) To chase (d) To revive old friendship

27. BEE IN ONE’S BONNET (a) Sickness (c) Misery

(b) Obsession (d) Freshness

28. THE WORSE FOR WEAR (a) Worrisome (c) Hostile

(b) Financially poor (d) Shabby from use

29. TO JETTISON (a) To go on trekking (c) To abandon

(b) To sail (d) To fire a gun

30. VICARIOUS (a) Ambitious (b) Not experienced personally (c) Nostalgic (d) Vindictive 31. FORTY WINKS (a) A person beyond 40 years of age (b) A studious person (c) Ordeals of life (d) A short nap 32. CRAVEN (a) Greedy (b) Cowardly (c) Flattering (d) Restless Directions (for the 8 items which follow): Each of the following eight items consists of a word or group of words in capital letters, followed by four words or groups of words. Select the word or group of words that is most similar in meaning to the word or group of the words in capital letters. 33. ENSCONCE (a) To surround (c) To honour

(b) To promote (d) To settle comfortably

34. INGRATIATE (a) To regret (b) To provoke (c) To place oneself in the favourable position (d) To feel delighted 35. AUGURY (a) Dispute (c) Place of refuge

(b) Altar (d) Omen

36. TRANSGESS (a) To take loan (b) To go beyond the limit (c) To discuss at the length (d) To show a clean pair of heels

Synonyms 37. FERRET OUT (a) To search (c) To hide

39. MERCURIAL (a) Mechanical (c) Clownish

(b) To trap (d) To Flee

▌ 2.11

(b) Heavy (d) Quick-changing

40. ALLEVIATE (a) To release (b) To lesson (c) To deprive (d) To deceive

38. USURP (a) To be lazy (b) To climb (c) To seize power or position illegally (d) To yield

answer Key 1. 2. 3. 4. 5.

(c) (b) (d) (b) (a)

6. 7. 8. 9. 10.

(c) (c) (b) (b) (a)

11. 12. 13. 14. 15.

(c) (a) (a) (b) (d)

16. 17. 18. 19. 20.

(c) (a) (c) (b) (d)

21. 22. 23. 24. 25.

(d) (b) (b) (c) (d)

26. 27. 28. 29. 30.

(a) (b) (d) (c) (b)

31. 32. 33. 34. 35.

(d) (b) (d) (c) (d)

36. 37. 38. 39. 40.

(b) (a) (c) (d) (b)

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Antonyms FormAt oF the questions An antonym is a word having an opposite meaning to another word or nearly so. For example: ‘incite’ means to arouse to action or ‘exhort’. Its opposite word or its antonym can be ‘hinder’ or ‘discourage’. Similarly, ‘disparate’ which means unlike or unequal is an antonym of ‘similar’, which means ‘alike’. Direction: Select from amongst the four alternatives, marked a–d, the word that is most nearly opposite in meaning to the word given in capital letters: 1. ABSTAIN (a) refuse (c) permit

2. CANDID (a) useless (c) worthless

(b) misleading (d) legal

3. IMPLICIT (a) satire (c) unexplained

(b) baseless (d) definite

(b) oppose (d) run away

Answer Key 1. (c)

2. (b)

3. (d)

test oF Antonyms Direction: Each of the following questions consists of a word printed in capital letters, followed by four words or phrases. Select the word or phrase that is most closely opposite in meaning to the capitalized word. 1. REFRACTORY (a) refreshing (c) privileged

(b) burdensome (d) manageable

2. ADROIT (a) deterred (c) foolish

(b) skillful (d) awkward

3. PALLIATE (a) apologize (c) wait impatiently 4. VILIFY (a) sing the praises of (b) show satisfaction with (c) regard with distrust (d) welcome with glee 5. IRASCIBLE (a) placid (c) shameless

(b) hesitate (d) worsen

(b) fortuitous (d) entrancing

2.14

▌ English

6. GELID (a) chilly (c) mature

(b) solid (d) boiling

22. MANUMIT (a) throw (c) promise

(b) lock (d) enslave

7. CONDIGN (a) unavoidable (c) guarded

(b) satisfactory (d) undeserved

23. PIQUANT (a) factitious (c) insipid

(b) vain (d) colorful

8. PUNCTILIOUS (a) tardy (c) careless

(b) correct (d) apathetic

24. OPPORTUNE (a) surprisingly agreeable (c) badly shaped

(b) closely berthed (d) poorly timed

9. FECKLESS (a) spotted (c) fawning

(b) fatuous (d) strong

25. PETULANT (a) pliable (c) uncouth

(b) equable (d) abnormal

10. INSOLENT (a) sullen (c) determined

(b) rich (d) affable

26. SAVORY (a) sad (c) lost

(b) hidden (d) unpalatable

11. SERENDIPITOUS (a) calm (c) flat

(b) planned (d) evil

27. FULFILLED (a) satirical (c) gorgeous

(b) dry (d) needy

12. FETID (a) in an embryonic state (c) acclaimed by peers

(b) easily enraged (d) having a pleasant odour

28. RECLUSIVE (a) joined (c) gregarious

(b) obscure (d) urban

13. ILLUSORY (a) nimble (c) powerful

(b) realistic (d) underrated

29. COURTEOUS (a) flaccid (c) insolent

(b) emollient (d) scrupulous

14. DOUR (a) gay (c) wealthy

(b) sweet (d) responsive

30. USURP (a) rise rapidly (c) produce quickly

(b) use fully (d) own rightfully

15. MENDACIOUS (a) broken (c) truthful

(b) efficacious (d) destructive

31. ACRIMONIOUS (a) legal (c) cursive

(b) severe (d) harmonious

16. ENERVATE (a) debilitate (c) introduce

(b) fortify (d) conclude

32. SKEPTIC (a) cryptic (c) support

(b) believer (d) eminent

17. DISCRETE (a) loud (c) loose

(b) combined (d) circle

33. INDUBITABLE (a) wavering (c) unmitigated

(b) aesthetic (d) questionable

18. PRIMITIVE (a) polite (c) weak

(b) naive (d) sophisticated

34. DELETERIOUS (a) impulsive (c) pathetic

(b) salubrious (d) inclusive

19. PARTITION (a) solidify (c) parse

(b) unify (d) enjoin

35. PUISSANCE (a) ignorance (c) impotence

(b) approbation (d) repudiation

20. CLANDESTINE (a) above ground (c) outside

(b) public (d) burnt out

36. SYCOPHANCY (a) speak harmoniously (c) push forcefully

(b) shout harshly (d) advise candidly

21. PHLEGMATIC (a) hoarse (c) oral

(b) voluntary (d) effusive

37. ABERRATION (a) typical behaviour (c) straight aim

(b) correct manners (d) full truthfulness

Antonyms

▌ 2.15

38. ANOMALOUS (a) capacious (c) connected

(b) vicious (d) usual

54. SEDULOUS (a) vociferous (c) concomitant

(b) derelict (d) itinerant

39. COGNISANCE (a) idiom (c) abeyance

(b) ignorance (d) anecdote

55. IMPERTURBABLE (a) militant (c) conical

(b) cynical (d) agitated

40. QUIESCENT (a) restless (c) malignant

(b) exempt (d) mendicant

56. PROFUSION (a) travesty (c) scarcity

(b) validity (d) retraction

41. ESCHEW (a) traduce (c) use

(b) invite (d) emanate

57. TEMPERATE (a) aged (c) truncated

(b) unlimited (d) inebriated

42. TACITURN (a) dubious (c) strategic

(b) garrulous (d) pleasant

58. MITIGATION (a) aggravation (c) obscenity

(b) verdancy (d) restriction

43. RECONDITE (a) miniature (c) arable

(b) philosopher (d) obvious

59. INIQUITY (a) fairness (c) peace

(b) rectitude (d) apostasy

44. REDUNDANT (a) dilatory (c) astute

(b) apocryphal (d) necessary

60. PROTUBERANCE (a) cadence (c) indentation

(b) habitation (d) attachment

45. RESTITUTION (a) inflation (c) deprivation

(b) cataclysm (d) constitution

61. EFFULGENCE (a) murky (c) mercenary

(b) harshness (d) quiet

46. PARSIMONY (a) closely held (c) acting apishly

(b) free spending (d) poorly expressed

62. AMELIORATE (a) increase (c) clasp

(b) worsen (d) dissemble

47. PERSPICACITY (a) homelike ambiance (c) vague memory

(b) precise meaning (d) partial fulfillment

63. BENIGN (a) sick (c) damaged

(b) poor (d) evil

48. PREPOSTEROUS (a) complaisant (c) apologetic

(b) conceited (d) credible

64. SALUTARY (a) noxious (c) moderate

(b) objectionable (d) farewell

49. SANCTIMONIOUS (a) proud (c) wealthy

(b) stubborn (d) devout

65. ALIENATE (a) go native (c) make friends

(b) say clearly (d) give freely

50. EXTIRPATE (a) preserve (c) ingratiate

(b) inseminate (d) enter

66. DROLLERY (a) firm warning (c) incredible threat

(b) serious statement (d) witty aside

51. CAPRICIOUS (a) redoubtable (c) phlegmatic

(b) constant (d) solitary

67. RETALIATE (a) maintain serenity (c) turn the other cheek

(b) stand tall (d) improve relations with

52. CASUISTARY (a) resultant (c) sediment

(b) interior (d) verity

68. ANALYSIS (a) dialysis (c) parenthesis

(b) electrolysis (d) synthesis

53. CONTUMELY (a) willingness (c) praise

(b) sporadically (d) augmented

69. PEREMPTORY (a) humble (c) weak

(b) resistant (d) deferential

2.16

▌ English

70. CONDUCIVE (a) expensive (c) empty

(b) wholesome (d) religious

86. OVERPOWER (a) succumb (c) oust

(b) twinge (d) betray

71. INSOLVENT (a) physically pure (c) metaphysically correct

(b) financially stable (d) chemically active

87. PALACE (a) manse (c) skyscrapper

(b) dwelling (d) hovel

72. HOMOGENEOUS (a) parsimonious (c) variegated

(b) consciousness (d) loquacious

88. SUSPEND (a) turn on (c) switch

(b) resume (d) pend

(b) procrastinate (d) segregate

89. MUTTER (a) please oneself (c) speak distinctly

(b) resolve conflict (d) digress randomly

74. TEMERITY (a) imbroglio (c) cognomen

(b) diffidence (d) effervescence

90. TRANSPARENT (a) indelicate (c) opaque

(b) neutral (d) somber

75. MUTATION (a) constancy (c) adolescent

(b) decency (d) clangorous

91. ENSEMBLE (a) complement (c) coordination

(b) cacophony (d) solo

76. SYBARITIC (a) foolish (c) consistent

(b) obdurate (d) austere

92. RETAIN (a) allocate (c) relegate

(b) distract (d) discard

77. PROSAIC (a) fulsome (c) extraordinary

(b) mundane (d) certain

93. RADIATE (a) approach (c) absorb

(b) cool (d) tranish

78. TENDER (a) difficult (c) tepid

(b) leonine (d) hardened

79. INADEQUATE (a) glossy (c) aspiring

(b) rapid (d) sufficient

80. NON-STRIATED (a) marked (c) plagued

(b) striped (d) stippled

81. SCRUTINISE (a) question (c) sensitise

(b) leer (d) disregard

96. AMORTISE (a) loosen (b) denounce (c) suddenly increase one’s indebtedness (d) wisely cause to flourish

82. LOFTY (a) assessed (c) general

(b) debased (d) intrusive

97. EMACIATION (a) invigoration (c) amelioration

(b) glorification (d) inundation

83. COLOURLESS (a) red (c) flamboyant

(b) black (d) vicious

98. UNALLOYED (a) destabilised (c) assimilated

(b) unregulated (d) adulterated

84. CELERITY (a) depression (c) slowness

(b) postponement (d) choler

99. MINATORY (a) reassuring (c) creative

(b) genuine (d) obvious

85. FURIOUS (a) medium (c) ticklish

(b) restrained (d) disjointed

73. AMALGAMATE (a) recriminate (c) scintillate

94. EPICURE (a) a person ignorant about art (b) a person dedicated to a cause (c) a person motivated by greed (d) a person indifferent to food 95. PREVARICATION (a) tact (c) veracity

100. FLUCTUATE (a) work for (c) follow from

(b) consistency (d) silence

(b) flow over (d) remain steady

Antonyms

▌ 2.17

Answer Key 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

(d) (d) (d) (a) (a) (d) (d) (c) (d) (d) (b) (d) (b)

14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.

(a) (c) (b) (b) (d) (b) (b) (d) (d) (c) (d) (b) (d)

27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39.

(d) (c) (c) (d) (d) (b) (d) (b) (c) (d) (a) (d) (b)

40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52.

(a) (c) (b) (d) (b) (b) (b) (d) (d) (d) (a) (b) (d)

53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65.

(c) (b) (b) (c) (d) (a) (b) (c) (a) (b) (d) (a) (c)

66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78.

(b) (c) (d) (d) (b) (b) (c) (d) (b) (a) (d) (c) (d)

79. 80. 81. 82. 83. 84. 85. 86. 87. 88. 89. 90. 91.

(d) (b) (d) (b) (c) (c) (b) (a) (d)

92. 93. 94. 95. 96. 97. 98. 99. 100.

(b) (c) (c) (d)

PrACtiCe eXerCise Direction: In the following questions, a word is given in capital letters followed by four alternative words, marked a-d. You are required to select a word that is opposite in meaning to the word given in capital letters. 1. APPLAUSE (a) to make noise (c) abuse

(b) acclaim (d) prosper

2. AMIABLE (a) friendly (c) agreeable

(b) not agreeable (d) amicable

3. AMATEUR (a) arrogant (c) lazy

(b) humble (d) expert

4. APPEASE (a) sympathise (c) peaceful

(b) irritate (d) apart

5. ABSURD (a) awkward (c) reasonable

(b) unreasonable (d) worthless

6. BARREN (a) dry (c) distant

(b) fruitful (d) unfertile

7. BIZARRE (a) costly (b) attractive (c) normal (d) abnormal

8. COWARDLY (a) courageously (c) secretly

(b) silently (d) timidity

9. BENEVOLENCE (a) kindness (c) unhumanely

(b) humanely (d) malice

10. BRUTALITY (a) criminal (c) adamant

(b) humanely (d) without fear

11. BIASED (a) disinterested (c) cured

(b) distasteful (d) curved

12. BLEAK (a) cheering (c) stout

(b) cheerful (d) fine

13. PAUCITY (a) sincerity (c) adversity

(b) difficulty (d) abundance

14. EXAGGERATE (a) minimise (c) cessation

(b) maximise (d) difficult

15. PUNGENT (a) clear (c) sharp

(b) turbid (d) harmless

16. TURBULENT (a) boiling (c) disturbing

(b) tranquil (d) comfortable

(d) (c) (d) (c) (c) (a) (d) (a) (d)

2.18

▌ English

17. RIDICULE (a) hurried (c) baseless

(b) reasonable (d) faulty

33. VOCIFEROUS (a) laudable (c) dangerous

(b) quiet (d) powerful

18. VIGOROUS (a) strongly (c) weak

(b) dangerous (d) ruthless

34. CURSORY (a) flimsy (c) degrading

(b) careful (d) torment

19. INCITE (a) exhort (c) discourage

(b) to cut (d) encourage

35. CLUMSY (a) mercy (c) bright

(b) dull (d) dextrous

20. RECEDE (a) proceed (c) advance

(b) restore (d) refuse

36. STOUT (a) skilled (c) unskilled

(b) weak (d) strong

21. REPULSIVE (a) inducive (c) attractive

(b) dull (d) attentive

37. MALICE (a) affection (c) egoism

(b) benevolence (d) filthy

22. VINDICTIVE (a) praise (c) useless

(b) forgiving (d) piercing

38. CATASTROPHE (a) calamity (c) blessing

(b) disaster (d) storm

23. GIGANTIC (a) huge (c) dangerous

(b) small (d) fantastic

39. EXCULPATE (a) acquit (c) murder

(b) absolve (d) accuse

24. TORTUOUS (a) straight (c) disturbing

(b) adventurous (d) wavering

40. ASSERT (a) accept (c) contradict

(b) agree (d) affirm

25. ENDURE (a) sensor (c) discontinue

(b) censor (d) continue

41. PARTISAN (a) partial (c) disinterested

(b) subjective (d) biased

26. TRANQUIL (a) peaceful (c) painful

(b) disturbed (d) punishing

42. DESOLATE (a) abandon (c) cheerful

(b) dislocate (d) accelerate

27. INTRUDE (a) to tear away (c) force

(b) keep away (d) compelled

43. VAGUE (a) foggy (c) clear

(b) cloudy (d) rare

28. TIMIDITY (a) Gallantry (c) weakness

(b) fearing (d) guilty

44. ACQUIT (a) sentence (c) criminal

(b) convict (d) thief

29. MEAGRE (a) villager (c) danger

(b) ample (d) simple

45. EXONERATE (a) implicate (c) desperate

(b) turn out (d) punish

30. BLUSTERing (a) calm (c) boiling

(b) dangerous (d) risky

46. RIDICULOUS (a) Dangerous (c) hazardous

(b) reasonable (d) poisoner

31. HOSTILE (a) friendly (c) costly

(b) sudden (d) unfair

47. IMPLICATE (a) involve (c) corrupt

(b) exonerate (d) accuse

32. MALEVOLENCE (a) tenderness (c) benefiting

(b) kindness (d) indulgence

48. AMICABLE (a) dangerous (c) unfriendly

(b) criminal (d) compensatory

Antonyms 49. FALLACIOUS (a) true (c) wrong

50. IMPLODE (a) extract (c) burst

(b) faulty (d) dirty

▌ 2.19

(b) explode (d) erupt

Answer Key 1. 2. 3. 4. 5. 6. 7.

(c) (b) (d) (b) (c) (b) (c)

8. 9. 10. 11. 12. 13. 14.

(a) (d) (b) (a) (b) (d) (a)

15. 16. 17. 18. 19. 20. 21.

(d) (b) (b) (c) (c) (c) (c)

22. 23. 24. 25. 26. 27. 28.

(b) (b) (a) (c) (b) (b) (a)

29. 30. 31. 32. 33. 34. 35.

(b) (a) (a) (b) (b) (b) (d)

36. 37. 38. 39. 40. 41. 42.

(b) (b) (c) (d) (c) (c) (c)

43. 44. 45. 46. 47. 48. 49.

(c)

50. (b)

(b) (a) (b) (b) (c) (a)

nDA BoX Directions: In this section, each item consists of a word or a phrase which is underlined in the sentence given. It is followed by four words or phrases. Select the word or phrase which is closest to the opposite in meaning of the underlined word or phrase. 1. He showed a marked antipathy to foreigners. (a) Profundity (b) fondness (c) objection (d) willingness 2. The authorities took the corrective action with celerity. (a) reluctance (b) delay (c) promptness (d) lack of judgment 3. It seems churlish to refuse such a generous offer. (a) wise (b) sensible (c) polite (d) immature 4. A conscientious editor, Manjula checked every definition for its accuracy. (a) novice (b) careless (c) unscientific (d) biased 5. Sharma’s craven refusal to join the protest was criticized by his comrades. (a) strategic (b) bold (c) diplomatic (d) well-thought Directions: (For the following 7 items): Each of the following seven items consists of a word in capital letters, followed by four words. Select the word that is most nearly opposite in meaning to the word in capital letters.

8. IMPROMPTU (a) Rehearsed (c) Appropriate

(b) Foolish (d) Vast

9. LOQUACIOUS (a) Sentimental (c) Soothing

(b) Taciturn (d) Soporific

10. TRACTABLE (a) Unreal (c) Incapable

(b) Irreligious (d) Unmanageable

11. TREPIDATION (a) Fearlessness (c) Death

(b) Slowness (d) Adroitness

12. PROLIFIC (a) Controversial (c) Wise

(b) Barren (d) Unfocussed

Directions (For the following 10 items): Each of the following ten items consists of a word in capital letters, followed by four words or group of words. Select the word that is most nearly opposite in meaning to the word in capital letters. 13. CONSUMMATE (a) Sluggish (c) Melancholy

(b) Imperfect (d) Dull

14. REPROOF (a) Brawl (c) Ambiguity

(b) Omission (d) Approval

6. DEBILITATE (a) Bedevil (c) Strengthen

(b) Delude (d) Repress

15. FELICITOUS (a) Jealous (c) Demanding

(b) Inapt (d) Inaccessible

7. EQUIVOCAL (a) Universal (c) Mistaken

(b) Clear (d) Quaint

16. CHARLATAN (a) Knowledgeable doctor (c) Comedian

(b) Trickster (d) Senior government official

2.20

▌ English

17. FECUND (a) Underweight (c) Infertile

(b) Infidel (d) Uncritical

29. BLITHESOME (a) Graceful (c) Adventurous

(b) Sullen (d) Mammoth

18. OUTLANDISH (a) Migrant (c) Boastful

(b) Opulence (d) Conventional

30. ADULATION (a) Confusion (c) Consolidation

(b) Inertia (d) Condemnation

19. COMMON (a) Clam (c) Pungent

(b) Scarcity (d) Motivation

20. CYNOSURE (a) Abomination (c) Mundane

31. PREPOSSESSING (a) Incredible (c) Unattractive

(b) Arousing envy (d) Recent

(b) Picturesque (d) Fertile

21. PUERILE (a) Upright (c) Matured

(b) Peripheral (d) Steadfast

22. DECREPIT (a) Research-oriented (c) Intelligent

(b) Sprawling (d) Energetic

Directions (For the 8 items which follow): Each of the following eight items consists of a word in capital letters, followed by four words or groups of words. Select the word or group of words that is furthest in meaning to the word in capital letters. 32. PROTRACT (a) Not to display (b) To indulge in extravagance (c) Not to be careful about future (d) To cut short

Directions (For the next 9 items): Each of the following nine items consists of a word or a group of words in capital letters, followed by four words or groups of words. Select the word or group of words that is most nearly opposite in meaning to the word in capital letters. 23. INORDINATE (a) Distant (c) Moderate

(b) Facile (d) Attractive

24. REDOUBTABLE (a) Truthful (c) Widely travelled

(b) Crafty (d) Ordinary

25. CONSANGUINE (a) Of different kind (c) Urbane

(b) Invisible (d) Untenanted

26. ON THE BLINK (a) Apologetically (c) Legally

(b) Blindly (d) In working order

27. PREDILECTION (a) Analogy (c) Antipathy

(b) Anticlimax (d) Argument

28. SENILE (a) Suspicious (c) Corrupt

(b) Mentally alert (d) Affluent

33 DEBILITATE (a) To argue (c) To guess

(b) To strengthen (d) To conspire

34. PERTINACIOUS (a) Irretrievable (c) Irresolute

(b) Insipid (d) Reproof

35. IMECUNIOUSNESS (a) Smoothness (c) Affl uence

(b) Carefree nature (d) Stability

36. INMICAL (a) Supportive (c) Lack-lustre

(b) Inquisitive (d) Coarse

37. SCINTILLA (a) Unscientific (c) Mealy-mouthed

(b) Huge quantity (d) Unpromising

38. KNAVE (a) Honest (c) Timid

(b) Ignorant (d) Belligerent

39. APOCRYPAL (a) Confusing (c) True

(b) Modern (d) Unsophisticated

Answer Key 1. 2. 3. 4. 5.

(b) (b) (d) (b) (b)

6. 7. 8. 9. 10.

(c) (d) (a) (b) (d)

11. 12. 13. 14. 15.

(a) (b) (b) (d) (b)

16. 17. 18. 19. 20.

(a) (c) (d) (b) (a)

21. 22. 23. 24. 25.

(c) (d) (c) (d) (a)

26. 27. 28. 29. 30.

(d) (c) (b) (b) (d)

31. 32. 33. 34. 35.

(c) (d) (b) (c) (c)

36. 37. 38. 39.

(a) (b) (a) (c)

C

H

3

A

P

T

E

R

Error Identification This section of the examination has 10–12 sentences with grammatical/word usage and/or spelling errors, which the candidate is expected to delete and rectify. For attempting such questions,

familiarity with the basic grammar rules, including spellings and the correct usage of words, is essential. Some sentences may not have any error, in which case ‘No error’ is marked as your answer.

Format oF the Questions Questions for spotting errors may be in various formats, the most common of which is illustrated below: Instead of underlining the parts of a sentence, here you will find that the sentence is split into four parts and each part is separated by a bar(/) and is either numbered 1– 4 or marked a–d. Your task is to detect the part that contains any error. Illustration: Each of the following sentences is split into four parts. Detect the part containing an error. If there is no error in any of the parts, mark your answer ‘5’ or ‘e’ (i.e., ‘No error’):

1. This was one of the reason (a) why the industry flourished (b) in such rich countries (c) as Japan and Britain (d) No error (e) 2. This advertising has had (a) an adverse effect on our (b) present business and it may (c) damage our reputation in future (d) No error (e) 3. The number of orders (a) for that item have increased (b) greatly; several of them (c) have come from Egypt (d) No error (e).

answer Key 1. (a)

2. (b)

3. (b)

error identiFication Directions: Each of the following sentences are divided into parts, which are marked A, B, C and D. One of these parts may contain an error or may not be acceptable in standard written communication. Mark that part as your answer.

1. We threw out some old furniture(a)/so that the new television set(b)/has enough space.(c)/No error(d) 2. This small table(a)/will collapse(b)/if you will stand on it.(c)/ No error(d)

2.22

▌ English

3. Children often(a)/quarrel on(b)/petty issue.(c)/ No error(d) 4. To transport goods(a)/by sea is cheaper(b)/than land.(c)/No error(d) 5. Meatless Days (a)/have been made(b)/in to a film.(c)/No error(d) 6. He did not eat the apple(a)/because it tasted bitterly(b)/and was slightly rotten.(c)/No error(d) 7. When she arrived(a)/, I was pretty fed up(b)/because it was been a horrible weather(c)/since eight O’clock in the morning. No error(d) 8. He went on committing crime after crime(a)/, but in spite of my best efforts(b)/I could not prevent him to do so.(c)/No error(d) 9. Of the two thesis submitted for the degree of Ph.D in English(a)/ neither was found suitable(b)/by the examiners for the award of the degree.(c)/No error(d) 10. The doctor advised us(a)/not to told(b)/the patient about his disease.(c)/No error(d) 11. As sooner did he(a)/return from school(b)/than he was told to go to the shop.(c)/No error(d) 12. All teachers agree(a)/that Paresh is the(b)/more intelligent of all the boys is his class.(c)/No error(d) 13. But for some(a)/anxious moments, our expedition(b)/in the Himalayas was enjoyable.(c)/No error(d) 14. Javed had not forgotten(a)/the incident and could(b)/clear remember all the details.(c)/No error(d) 15. Whether you agree(a)/or not(b)/I have to send you at Bombay. (c)/No error(d) 16. How close the ball(a)/will come depends(b)/on how forcefully it was hit.(c)/No error(d)

26. As there are only one taxi available(a)/I have no other alternative(b)/but to pay the fare demanded by the driver.(c)/ No error(d) 27. Prayag and Parag used(a)/to work for almost twelve hours in the factory(b)/is were working earlier.(c)/No error(d) 28. Whenever they go out(a)/for shopping(b)/they take their pet dog with them.(c)/No error(d) 29. Monisha was to about(a)/tell her teacher the(b)/truth when her friend interrupted her.(c)/No error(d) 30. The children were(a)/playing with a ball(b)/and run around when the accident occurred.(c)/No error(d) 31. Avinash is not so(a)/good a batsman to(b)/be selected for the national cricket team.(c)/No error(d) 32. He ran so fastly(a)/that he reached(b)/the destination in just two minutes.(c)/No error(d) 33. The fireman could not(a)/succeed in rescue the(b)/child although they could put out the fire.(c)/No error(d) 34. Anjali’s performance in the(a)/drama was best than(b)/ Deepali’s but not as good as Vaishali’s.(c)/No error(d) 35. Although he is usually(a)/rude with everyone(b)/he behaved nice with all of us today.(c)/No error(d) 36. The policeman started(a)/firing the crowd(b)/when the striking workers got violent.(c)/No error(d) 37. Of all the friends(a)/I have had, he is the most helpful(b)/and less arrogant.(c)/No error(d) 38. Supposing if you(a)/are arrested(b)/what will you do?(c)/No error(d) 39. Although we are free(a)/for the last forty-five years or so(b)/yet we continue to be economically backward.(c)/No error(d)

17. In spite of the help(a)/offered to him(b)/he refuses to accept and I decided to leave him alone.(c)/ No error(d)

40. Satyajit Ray, who conceived, co-authored(a)/, and directed a number of good films, was(b)/one of India’s most talented film maker.(c)/No error(d)

18. If my father(a)/will approve(b)/I will go to Bombay.(c)/No error(d)

41. Neither the famine(a)/or the subsequent fire(b)/was able to destroy the spirit of the people.(c)/No error(d)

19. Bread and butter(a)/is(b)/wholesome food.(c)/No error(d)

42. The lawyer told his client(a)/that he would represent him(b)/ only if he pays up his fee.(c)/No error(d)

20. It is high time(a)/that we send(b)/the answer.(c)/No error(d) 21. The angry man ran hurried(a)/into the crowded room(b)/and shouted loud at the guests.(c)/No error(d) 22. Hardly he had arrived(a)/when the house caught fire(b)/and everything was reduced to ashes.(c)/No error(d) 23. Interviews for(a)/the posts of lecturer will(b)/begin from Monday.(c)/No error(d) 24. Good night.(a)/I am glad(b)/to see you of.(c)/No error(d) 25. We all(a)/take vegetable(b)/and fruit.(c)/No error(d)

43. The signpost at the gate(a)/of the garden read(b)/: Trespassers will be prosecuted.(c)/No error(d) 44. One of the drawbacks(a)/of modern education are(b)/that it does not encourage original thinking.(c)/No error(d) 45. He was(a)/acquitted from(b)/the charges.(c)/No error(d) 46. This watch is(a)/more superior and(b)/more expensive than that.(c)/No error(d) 47. Some people(a)/consider wealth(b)/superior than wisdom.(c)/ No error(d)

Error Identification ▌ 2.23 48. Judging by what he tells me.(a)/I don’t think he has made a good impression(b)/to the manager of the company where he works.(c)/No error(d) 49. He told me that he couldn’t buy(a)/the paintings which he wants(b)/very much because he didn’t have enough money. (c)/No error(d) 50. There was great difference(a)/of opinion between(b)/the members of the Board.(c)/No error(d) 51. I can speak Gujarati(a)/a little but not well(b)/as you do.(c)/No error(d) 52. He hesitated to accept the past(a)/as he did not think the salary would not be(b)/enough for a man with a family of ten.(c)/No error(d)

69. Alexander was(a)/the first European invasion(b)/ to set foot on the Indian soil.(c)/No error(d) 70. Brazil is(a)/the larger(b)/country in South America.(c)/No error(d) 71. Mercury is(a)/the best conductors(b)/of heat and is used in thermometers.(c)/No error(d) 72. We are(a)/fortunate in being able(b)/to visit Puri because there is many who cannot go there.(c)/No error(d) 73. If a man joins a post and do(a)/not work he is(b)/asked to resign the post.(c)/No error(d) 74. After Independence many(a)/people have say(b)/many things about the national language.(c)/No error(d)

53. Finally, after long years of hardship(a)/he succeeded(b)/which completely altered his outlook on life.(c)/No error(d)

75. Us(a)/may stay(b)/in Darjeeling until the monsoon starts.(c)/ No error(d)

54. While I was travelling(a)/in a train(b)/piece of luggage fell on my head.(c)/No error(d)

76. There will be(a)/increased emphasis on(b)/heavy industry in planning.(c)/No error(d)

55. Will the bank loan(a)/me money(b)/if I pledge my land?(c)/No error(d)

77. Due to the explosion the(a)walls bursted apart(b)/and the roof was blown off.(c)/No error(d)

56. No sooner did the teacher(a)/enter the class(b)/then the students stand up.(c)/No error(d)

78. The selection Board will(a)/call only those(b)/candidates whom have the proper qualifications.(c)/ No error(d)

57. Although my car is expensive(a)/I have had(b)/a lot of trouble with it.(c)/No error(d)

79. In the high school, girls often do(a)/as good as boys(b)/if not better than the boys.(c)/No error(d)

58. He asked Ramu(a)/ why your father(b)/ had lost his temper.(c)/ No error(d)

80. The authorities states(a)/that the students are(b)/responsible for the lack of discipline in the colleges.(c)/No error(d)

59. What to speak of milk(a)/, even water(b)/ was not available there.(c)/No error(d)

81. No employee are(a)/permitted to act on(b)/behalf of the company in financial matters.(c)/No error(d)

60. My father asked me(a)/ to go to the fair(b)/ and enjoy there.(c)/ No error(d)

82. She denied that(a)/she did not(b)commit the crime.(c)/No error(d)

61. Although winter in Kashmir is cold(a)/but it is pleasant(b)/in many ways.(c)/No error(d)

83. I prefer(a)/punishment(b)/than insult.(c)/No error(d)

62. Hari was unhappy(a)/that he would(b)/not attend the wedding of his friend yesterday.(c)/No error(d) 63. Rama could not(a)/go to the picnic(b)/for his mother was not well.(c)/No error(d)

84. My friend always prefers(a)/reading short stories(b)/than voluminous novels.(c)/No error(d) 85. It has been unbearable hot(a)/for(b)/the last two months.(c)/ No error(d)

64. Sarita was popular(a)/with her classmates that(b)/she always had someone or the other coming to her house.(c)/No error(d)

86. On the occasion,(a)/a tribal chieftain boldly asserted that any one of his boys could kill a tiger(b)/with his bare hands.(c)/No error(d)

65. The programme which(a)/came on television(b)/these days in the evenings is very interesting.(c)/No error(d)

87. It’s time(a)/we do something to check(b)/the incidence of this infectious disease.(c)/No error(d)

66. Prakash was leading(a)/a happy and leisurely(b)/life after his retirement from service.(c)/No error(d)

88. Neither John nor Jim was able to tell me(a)/who had visited our place(b)/in the evening in my absence.(c)/No error(d)

67. Elections to American(a)/President take place(b)/every fourth year in November.(c)/No error(d)

89. Neither of the contestant was(a)/in a proper physical condition(b)/to do his best work.(c)/No error(d)

68. Guru Ram Das was(a)/the fourth Guru of the Sikhs(b)/between 1674 to 1681.(c)/No error(d)

90. No everything(a)/that a daily newspaper prints(b)/is intended to be read for its educational value.(c)/No error(d)

2.24

▌ English

91. It was now six(a)/and he was tired(b)/because he has been working since dawn.(c)/No error(d)

96. Being a very hot day(a)/I stayed at home(b)/and prepared for my examination.(c)/No error(d)

92. A report of the meetings,(a)/and a list of the new officers(b)/ are printed on the third page.(c)/No error(d)

97. He told me(a)/the same old story(b)/in great details.(c)/No error(d)

93. Why you worry me(a)/when this problem of yours(b)/can be solved by your brother?(c)/No error(d)

98. The two brothers have never been(a)/on good terms(b)/to each other.(c)/No error(d)

94. When you have written your essay(a)/, read it over carefully(b)/ to correct possible mistakes.(c)/No error(d)

99. The man disappeared(a)/after he has committed(b)/a murder in the running train.(c)/No error(d)

95. If you will let me try(a)/I will make this engine go(b)/but you had better get out of the way first.(c)/No error(d)

100. This carpenter is the most skilled(a)/of all other carpenters(b)/ in our workshop.(c)/No error(d)

answer Key 1. (c) ‘has’ should be replaced with ‘had’

27. (c) ‘is’ should be replaced with ‘they’

2. (c) Remove ‘will’ after ‘you’

28. (d) This sentence is correct

3. (b) ‘on’ should be replaced with ‘over’

29. (a) ‘to about’ should be replaced with ‘about to’

4. (c) ‘land’ should be replaced with ‘the land’

30. (c) ‘run’ should be replaced with ‘running’

5. (b) ‘have’ should be replaced with ‘has’

31. (b) ‘to’ should be replaced with ‘as to’

6. (b) ‘bitterly’ should be replaced with ‘bitter’

32. (a) ‘fastly’ should be replaced with ‘fast’

7. (b) ‘has been’ should be used

33. (b) ‘rescue’ should be replaced with ‘rescuing’

8. (c) ‘to do so’ should be replaced with ‘from doing so’

34. (b) ‘best’ should be replaced with ‘better’

9. (a) ‘thesis’ should be replaced with ‘theses’

35. (c) ‘nice’ should be replaced with ‘nicely’

10. (b) ‘told’ should be replaced with ‘tell’

36. (b) ‘firing’ should be replaced with ‘firing at’

11. (a) ‘as’ should be replaced with ‘no’

37. (c) ‘less’ should be replaced with ‘the least’

12. (c) ‘more’ should be replaced with ‘most’

38. (a) ‘Supposing if’ should be replaced with ‘Suppose’

13. (d) This sentence is correct

39. (a) ‘are free’ should be replaced with ‘have been free’

14. (c) ‘clear’ should be replaced with ‘clearly’

40. (c) ‘maker’ should be replaced with ‘makers’

15. (c) ‘at’ should be replaced with ‘to’

41. (b) ‘or’ should be replaced with ‘nor’

16. (c) ‘was’ should be replaced with ‘has been’

42. (c) ‘pays’ should be replaced with ‘paid’

17. (c) ‘I decided’ should be replaced with ‘I have decided’

43. (d) This sentence is correct

18. (b) ‘will approve’ should be replaced with ‘approves’

44. (b) ‘are’ should be replaced with ‘is’

19. (d) The sentence is correct

45. (b) ‘from’ should be replaced with ‘of’

20. (c) ‘the’ should be replaced with ‘an’

46. (b) ‘more superior’ should be replaced with ‘superior to’

21. (c) ‘loud’ should be replaced with ‘loudly’

47. (c) ‘than’ should be replaced with ‘to’

22. (d) The sentence is correct

48. (c) ‘to’ should be replaced with ‘on’

23. (b) ‘posts’ should be replaced with ‘post’

49. (b) ‘wants’ should be replaced with ‘wanted’

24. (c) ‘see you of’ should be replaced with ‘see you off’

50. (b) ‘between’ should be replaced with ‘among’

25. (d) The sentence is correct

51. (b) ‘well’ should be replaced with ‘as well’

26. (a) ‘are’ should be replaced with ‘is’

52. (b) Delete ‘not’ after ‘would’

Error Identification ▌ 2.25 53. (d) This sentence is correct

77. (b) ‘bursted’ should be replaced with ‘burst’

54. (c) ‘piece’ should be replaced with ‘a piece’

78. (c) ‘whom’ should be replaced with ‘who’

55. (a) ‘loan’ should be replaced with ‘lend’

79. (d) This sentence is correct.

56. (c) ‘then’ should be replaced with ‘than’

80. (a) ‘states’ should be replaced with ‘state’

57. (b) ‘I’ should be replaced with ‘yet I’

81. (a) ‘are’ should be replaced with ‘is’

58. (b) ‘your’ should be replaced with ‘his’

82. (b) Delete ‘not’ after ‘did’

59. (a) ‘What’ should be replaced with ‘Not’

83. (c) ‘than’ should be replaced with ‘to’

60. (a) ‘asked’ should be replaced with ‘told’

84. (c) ‘than’ should be replaced with ‘to’

61. (b) ‘but’ should be replaced with ‘yet’

85. (a) ‘unbearable’ should be replaced with ‘unbearably’

62. (b) ‘would’ should be replaced with ‘could’

86. (d) This sentence is correct.

63. (d) This sentence is correct.

87. (b) ‘do’ should be replaced with ‘did’

64. (a) ‘was’ should be replaced with ‘was so’

88. (a) ‘was’ should be replaced with ‘were’

65. (b) ‘came’ should be replaced with ‘comes’

89. (c) ‘do his’ should be replaced with ‘perform’. Delete ‘work’

66. (d) The sentence is correct.

90. (a) ‘No’ should be replaced with ‘Not’

67. (b) ‘president’ should be replaced with ‘presidency’

91. (c) ‘has’ should be replaced with ‘had’

68. (c) ‘between’ should be replaced with ‘from’

92. (c) ‘are’ should be replaced with ‘is’

69. (b) ‘invasion’ should be replaced with ‘invader’

93. (a) ‘you’ should be replaced with ‘do you’

70. (b) ‘larger’ should be replaced with ‘largest’

94. (c) Delete ‘possible’ after ‘correct’

71. (b) ‘conductors’ should be replaced with ‘conductor’

95. (a) Delete ‘will’ after ‘you’

72. (c) ‘is’ should be replaced by ‘are’

96. (a) ‘Being’ should be replaced with ‘It being’

73. (a) ‘do’ should be replaced with ‘does’

97. (c) ‘details’ should be replaced with ‘detail’

74. (b) ‘say’ should be replaced with ‘said’

98. (c) ‘to’ should be replaced with ‘with’

75. (a) ‘Us’ should be replaced with ‘We’

99. (b) ‘has should be replaced with ‘had’

76. (b) ‘increased’ should be replaced with ‘an increased’

100. (b) Delete ‘other’ after ‘all’

Practice eXercise Directions: Each of the following sentences are divided into parts, which are marked a, b, c, and d. One of these parts may contain an error or may not be acceptable in standard written communication. Mark the part containing the error as your answer. If the sentence contains no error, mark ‘d’ as your answer.

4. Our has always been a multi-lingual country(a)/and people have always found their own strategies to (b)/deal with this. (c)/No error(d)

1. While the linking roads(a)/leading to a village can put many state highways to shame(b)/ the village has been fitted with solar street lights.(c)/No error(d)

6. We(a)/returned back to the guest house impressed with (b)/ what we had seen.(c)/No error(d)

2. I fail(a)/to understand why(b)/he deliberately absented from the meeting.(c)/No error(d)

7. The older one gets(a)/the more foolish(b)/and more wiser one becomes.(c)/No error(d)

3. Sometimes(a)/even words cannot(b)/describe about the beauty of the scene.(c)/no error(d)

8. What makes matters(a)/worse is that many a times(b)/the public address is not clearly audible.(c)/ No error(d)

5. Several hundred volunteers(a)/were detained of whom many(b)/ were remanded in to judicial custody.(c)/No error(d)

2.26

▌ English

9. Civil servants with a technical background(A )/are as good as if not better(b)/than those with humanities background.(c)/No error(d)

31. The principal along with the teachers(a)/were seen boarding a bus(b)/to go to a Picnic on a national holiday.(c)/No error(d)

10. There was a time(a)/when the universities took pride(b)/in the fact that its examinations were held in time.(c)/No error(d)

32. With the introduction of the new syllabus(a)/the number of colleges reporting(b)/high results were Decreasing year after year.(c)/No error(d)

11. The two friends walked(a)/besides each other(b)/in silence. (c)/No error(d) 12. It was so lovely weather(a)/that we spent(b)/the whole day in the garden.(c)/No error(d)

33. The management warned the employees(a)/that if they persist(b)/in their obstructionist attitude, they would be suspended.(c)/No error(d)

13. He suggested us(a)/that we should try to improve our condition(b)/by beginning to work hard.(c)/No error(d)

34. A morning bath(a)/is good not only for body(b)/but also for the mind.(c)/No error(d)

14. A man jumped(a)/in the river(b)/in a state of depression.(c)/ No error(d)

35. He has taken his(a)/degree examination last year(b)/,but failed. (c)/No error(d)

15. He said he was an employee(a)/of the gas company(b)/and has come to inspect the gas connect-ion.(c)/No error(d)

36. The police were(a)/on the alert(b)/to see that nothing goes wrong.(c)/No error(d)

16. Over three hundred farmers(a)/marched through the capital(b)/ to protest against the new tax.(c)/No error(d)

37. Such candidates who have not(a)/cleared the written test(b)/ will not be called for the interview.(c)/ No error(d)

17. The teacher(a)/kept encouraging us(b)/to work harder.(c)/No error(d)

38. Many precious lives were(a)/lose in a collision(b)/between a truck and a bus.(c)/No error(d)

18. He goes(a)/to college(b)/by foot everyday.(c)/No error(d)

39. He has been going to the office(a)/for a year now(b)/and he even can’t understand its working.(c)/No error(d)

19. Vipis is a big(a)/industrious town with(b)/a number of factories. (c)/No error(d) 20. The average age(a)/at which people begin to need eyeglasses(b)/vary considerably.(c)/No error(d) 21. The public(a)/are deadly against(b)/the imposition of any punitive tax.(c)/No error(d)

40. On second thoughts(a)/the employee has withdrawn(b)/his resignation from his job a week ago.(c)/ No error(d) 41. He would have lent me some money(a)/if he was knowing(b)/ that I had lost everything.(c)/No error(d)

22. He always acts the martyr(a)/when he has to do(b)/some domestic chores.(c)/No error(d)

42. The dark and tranquil atmosphere(a)/was occasionally disturbed(b)/by the thunder and lightning in the sky.(c)/No error(d)

23. Being the best football player(a)/he was invited by the local sports club(b)/to play a friendly match on Sunday.(c)/No error(d)

43. It surprises me to infer from whatever you say(a)/that he always intended to achieve his aim(b)/by hook or crook.(c)/No error(d)

24. No boy in his son’s class(a)/is as bright(b)/as his son.(c)/No error(d)

44. The manager of his bank(a)/is a close friend(b)/of me.(c)/No error(d)

25. Contrary to earlier reports(a)/I found him(b)/ brighter than during the encounter.(c)/No error(d)

45. He ought to start at one(a)/lest he may not miss the train(b)/and fail to appear for the interview.(c)/No error(d)

26. The teacher asked the students(a)/if each of them(b)/were ready to take practical class everyday.(c)/No error(d)

46. He refused to recognise(a)/that the goldmine there(b)/was of poor quality.(c)/No error(d)

27. There is no doubt(a)/that a majority of workers(b)/is in favour of increasing wages.(c)/No error(d) 28. No sooner he left the office(a)/than it started raining heavily(b)/ enough to make him completely wet.(c)/No error(d) 29. One of my first friend(a)/is both a novelist(b)/ and a poet of repute.(c)/No error(d) 30. Even after hearing the leader for a long time(a)/the followers could not make out(b)/ which he was talking about.(c)/No error(d)

47. They swam all night(a)/in the pond(b)/isn’t it?(c)/No error(d) 48. If you will work hard(a)/you will surely get a seat(b)/in any medical college in India.(c)/No error(d) 49. If a man runs after money he is greedy(a)/, if he keeps it he’s miser(b)/, and if he spends it he is extravagant.(c)/No error(d) 50. It passes my understanding(a)/as to how can you doubt the truth of(b)/what the venerable old man has said.(c)/No error(d)

Error Identification ▌ 2.27

answer Key 1. (a) While the link roads

26. (c) ‘were’ should be replaced with ‘was’

2. (c) he deliberately absented himself

27. (c) ‘is’ should be replaced with ‘are’

3. (c) cannot describe

28. (a) ‘No sooner he left’ should be replaced with ‘no sooner had he left’

4. (a) ours has always been 5. (c) were remanded to judicial custody 6. (b) returned to the guest house 7. (c) and wiser 8. (d) No error 9. (b) Delete ‘as’ after ‘good’ 10. (c) ‘were held on time’ 11. (b) ‘besides’ should be replaced with ‘beside’ 12. (a) ‘so’ should be replaced with ‘such a’ 13. (a) ‘us’ should be replaced with ‘to us’ 14. (b) ‘in’ should be replaced with ‘into’ 15. (a) ‘he’ should be replaced with ‘that he’ 16. (c) ‘to’ should be replaced with ‘in’ 17. (c) ‘harder’ should be replaced by ‘hard’ 18. (c) ‘by’ should be replaced by ‘on’ 19. (b) ‘industrious’ should be replaced by ‘industrial’

29. (a) ‘My first friend’ should be replaced with ‘my best friends’ 30. (c) ‘which he was’ should be replaced with ‘what he was’ 31. (b) ‘were seen’ should be replaced with ‘was seen’ 32. (c) ‘results are’ should be replaced with ‘results were’ 33. (b) ‘if they persist’ should be replaced with ‘if they persisted’ 34. (b) ‘only for body’ should be replaced with ‘only for the body’ 35. (a) ‘has taken’ should be replaced with ‘took’ 36. (c) ‘goes wrong’ should be repalced with ‘went wrong’ 37. (a) ‘such candidates’ should be replaced with ‘those candidates’ 38. (b) ‘lose in’ should be replaced with ‘lost in’ 39. (c) ‘he even’ should be replaced with ‘yet’ 40. (b) ‘has withdrawn’ should be replaced with ‘withdrew’ 41. (b) ‘was knowing’ should be replaced with ‘had known’ 42. (d) This sentence is correct 43. (c) ‘crook’ should be replaced with ‘by crook’ 44. (c) ‘of me’ should be replaced with ‘of mine’

20. (c) ‘vary’ should be replaced by ‘varies’

45. (b) Delete ‘not’

21. (b) ‘are deadly’ should be replaced by ‘is dead’

46. (c) ‘was of poor’ should be replaced with ‘is of poor’

22. (d) This sentence is correct

47. (c) ‘isn’t it’ should be replaced with ‘didn’t they’

23. (a) ‘Being the best’ should be replaced with ‘As the best’

48. (a) ‘you will work’ should be replaced with ‘you work hard’

24. (d) This sentence is correct

49. (d) This sentence is correct

25. (c) ‘brighter than’ should be replaced with ‘bright’

50. (b) Delete ‘the truth of’

nda BoX Directions: (i) In this section a number of sentences are given. The sentences are underlined in three separate parts and each one is labelled (a), (b), (c). Read each sentence to find out whether there is an error in any underlined part. No sentence has more than one error. When you find an error in any one of the underlined parts (a), (b) or (c), indicate your response on the separate answer sheet at the appropriate space. You may feel that there is no error in a sentence. In that case letter (d) will signify a ‘No error’ response. (ii) You are to indicate only one response for each item in your answer sheet. (If you indicate more than one response, your answer will be considered wrong.) Errors may be in grammar, word usage or idioms.

There may be a word missing or there may be a word which should be removed. (iii) You are not required to correct the error. You are required only to indicate your response on the answer sheet. 1. Amit's severe bout of flu (a)

debilitated him so much ( b)

that he was too tired to go for work for a week (c)

No error. (d)

2. This is the crux of the entire problem; evverything centres on (a) ( b) it being resolved. (c)

No error. (d)

2.28

▌ English

3. One of the major aims of the Air Force was thhe complete demolition of all means of (a) trransportation ( b) by the bombing of rail lines and terminals. (c) No error. (d) 4. His strong voice cut over (a)

the hum of conveersation ( b) No error. (d)

like a knife through butter (c)

5. Even though they weren't expecting us they (aa) managed to knock-up ( b) a marvellous meal (c)

surrounded by ussual crowd ( b)

of lackeys and hanger-ons (c)

No error. (d)

Directions (For the following 7 items): (i) In this section, a number of sentences are underlined in three separate parts and each one is labelled (a), (b), (c). Read each sentence to find out whether there is an error in any underlined part. No sentence has more than one error. When you find an error in anyone of the underlined parts (a), (b) or (c), indicate. your response on the separate answer sheet at the appropriate space. You may feel that there is no error in a sentence. In that case letter (d) will signify a ‘No error’ response. (ii) You are to indicate only one response for each item in your answer sheet. (If you indicate more than one response, your answer will be considered wrong.) Errors may be in grammar, word usage or idioms. There may be a word missing or there may be a word which should be removed. (iii) You are not required to correct the error. You are required only to indicate your response on the answer sheet. 7. I pretended as if (a)

interested in the converrsation ( b)

but really it was very boring. (c)

buut last summer we spent a few days ( b) at a very nice hotel by sea. (c) 9. I will try to be on time (a) when I am late. (c)

No error. (d)

No error. (d)

but do not worry ( b)

No error. (d)

10. The teacher remarked that they all had (a) very badly (c)

11. The train is supposed (a)

12. Ritu will be surprised (a) the news. (c)

donne it ( b)

No error. (d)

but it was an hour late (c)

No error. (d)

6. The celebrated singer was (a)

8. I don't usually like staying at hotels. (a)

to arrive at 10 a.m.. yesterday ( b) No error. (d) as she hears ( b)

No error. (d)

13. I asked two persons (a)

the way to the stationn ( b)

but none of them knew it (c)

No error. (d)

Directions (For the following 7 items): (i) In this section, a number of sentences are given. The sentences are underlined in their separate parts and each one is labelled (a), (b), (c). Read each sentence to find out whether there is an error in any underlined part. No sentence has more than one error. When you find an error in anyone of the underlined parts (a), (b), or (c), indicate your response on the separate answer sheet at the appropriate space. You may feel that there is no error in a sentence. In that case letter (d) will signify ‘No error’ response. (ii) You are to indicate only one response for each item in your answer sheet. (If you indicate more than one response, your answer will be considered wrong). Errors may be in grammar, word usage or idioms. There may be a word missing or there may be a word which should be removed. (iii) You are not required to correct the error. You are required only to indicate your response on the answer sheet.

Error Identification ▌ 2.29 14. If we really set to (a)

we can get the whole house ( b)

cleaned in an afternoon (c)

No error. (d)

15. Pieces of rock plummeted (a) in the ground below (c)

down the mountainnside ( b)

No error. (d)

16. Since the two parties each won (a)

the same nuumber of seats ( b)

the minority party holds the balance of power. (c)

(i) In this section a number of sentences are given. The sentences are underlined in three separate parts and each one is labelled (a), (b) and (c). Read each sentence to find out whether there is an error in any underlined part. No sentence hasmore than one error. When you find an error in any of the underlined parts’(a), (b) or (c), indicate your response on the separate Answer Sheet at the appropriate space. You may feel that there is no error in a sentence. In that case letter (d) will signify a ‘No error’ response. (ii) You are to indicate only one response for each item in your Answer Sheet. (If you indicate more than one response, your answer will be considered wrong.) Errors may be in grammar; word usage or idioms. There may be a word missing or there may be a word which should be removed. (iii) You are not required to correct the error. You are required only to indicate your response on the Answer Sheet. Examples ‘P’ and ‘Q’ have been solved for you.

No error. (d) 17. It's arrogant for you (a) win every time (c)

Directions (For the next 10 items):

P. The young child (a)

to assume you'll ( b)

No error. (d)

a very sweet song. (c)

No error. (d)

Q. We worked very hard throughout the season No error. (a) ( b) (c) (d)

18. We' ve paid for our travel and accommodationn (a) so we need only to take ( b) some pocket-money with us. (c)

singed ( b)

Explanation: In item P, the word ‘singed’ is wrong. The letter under this part is (b); so (b) is the correct answer. Similarly, for item Q, (d) is the correct answer, as the sentence does not contain any error. 21. An unit is an abstract idea, (a)

No error. (d)

defined eitheer by reference to ( b)

19. There's no evidence to show (a)

a randomly chosen material standard or

that informatiion technology secrets are more ( b) vulnerable in India than Britain or the U S. (c)

to a natural phenomenon (c)) No error. (d)

20. It is shameful that hunting (a)

No error. (d) 22.

Microwaves are the principle carriers (a)

is still consiidered sport ( b)

of television, telephone and data transmissionns ( b)

by some unscrupulous people in the civilized world. (c)

between stations on earth and between the earth and satellites. (c)

No error. (d)

No error. (d)

2.30

▌ English

23. Mahavira was an advocate of nonviolence and vegetarianism. Directions (For the 8 items which follow): (a) (i) In this section a number of sentences are given. The senwho revived and reorganizzed the Jain doctrine tences are underlined in three separate parts each one is ( b) labelled (a), (b) and (c). Read each sentence to find out and established rules for their monastic order whether there is an error in any underlined part. No sentence has more than one error. When you find an error in (c) any one of the underlined parts (a), (b) and (c), indicate No error. your response on the separate Answer sheet at appropri(d)) ate space. You may feel that there is no error in the sentence. In that case letter (d) will signify a ‘No error’ Amit has been deceiving Mona for many year r s but 24. response. (a) ( b) she has not still tumbled to it. (c)

No error. (d)

25. A major contribution of Mathura sculptors off that (a) period were the creation and popullarization of the ( b) Buddha's image in human form. No error. (c) (d) 26. I will try to put over some feelers to gauuge (a) ( b) people's reactions to our proposal. No error. (c) (d) 27. The modem office block sticks out like a sore thumb (a) ( b) among the old buildings in the area. No error. (c) (d) 28. He estimated his income tax bill (a) by extrappolation over figures ( b) submitted in previous years. (c) No error. (d) 29. The two books are the same (a) this has an answer in the back. (c)

(iii) You are not required to correct the error. You are required only to indicate your response on the Answer Sheet. Examples ‘P’ and ‘Q’ have been solved for you P. The young child (a)

singed ( b)

a very sweet song. (c)

Explanation: In item P, the word ‘singed’ is wrong. The letter under this parts is (b); so (b) is the correct answer. Similarly, for item Q, (d) is the correct answer, as the sentence does not contain any error. to the expense of takingg ( b)

music lessons if you never practice. (c) 32. You will find it difficult (a)

except for the fact that ( b) No error. (d)

thhe election but he has still to prove ( b) No error.. (d)

No error. (d)

Q. We worked very hard throughout the season No error . (a) ( b) (c) (d)

31. It's stupid to go (a)

30. The Prime Minister's good looks won him (a)

that he's not a just pretty face. (c)

(ii) You are to indicate only one response for each item in your Answer Sheet. (If you indicate more than one response, your answer will be considered wrong.) Error may be in grammar, word usage or idioms, There may be a word missing or there may be a word which should be removed.

to explain of your use ( b)

of such offensive language. (c) 33. Because of the (a)

No error. (d)

No error. (d)

extenuating circumstances (bb)

the court acquitted him out of the crime. (c) 34. The carpet was badly stained, (a)

No error. (d)

to such an extent that ( b)

you couldn't tell its original colour (c)

No error. (d)

Error Identification ▌ 2.31

35.

37.

It is greatly to Amit's credit (a)

exiled of his couuntry because of ( b)

ve back the money he found; that he gav ( b)

his part in the plot against the government. (c)

his honesty does for him credit. (c)

No error. (d)

No error. (d) 38. 36. A terrific hue and cry (a)

was raised ( b)

at the new tax proposals. (c)

The formers generals was (a)

The company has (a) set off itself some stiff production ( b)

No error. (d)

goals for this year. (c) No error. (d)

answer Key 1. 2. 3. 4. 5.

(b) () (c) (a) (b)

6. 7. 8. 9. 10.

(a) (c) (a) (c) (d)

11. 12. 13. 14. 15.

(a) (b) (d) (d) (c)

16. 17. 18. 19. 20.

(a) (a) (b) (c) (b)

21. 22. 23. 24. 25.

(a) (a) (b) (c) (b)

26. 27. 28. 29. 30.

(a) (d) (b) (c) (c)

31. 32. 33. 34. 35.

(b) (b) (c) (a) (c)

36. (a) 37. (b) 38. (b)

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Sentence Completion Format oF the QueStionS The ‘fill in the blanks’ or ‘sentence completion’ questions are meant to test your ability in several ways. First, the question tests your ability to grasp the context of a sentence even with some of the words missing. Second, the question tests your vocabulary as demonstrated by your ability to select the correct words to complete the given sentence so that it becomes a sound, logical, meaningful, verbal statement of the sort that a person with communication skill would be expected to make. Third, the question seeks to check your ability to recognize the finer relationships between various parts of a sentence. There may be various formats of these questions as illustrated below:

Format i

for each question. Choose the correct alternative out of the four choices to complete the sentence: 1. Gandhiji was a humble man known for his ____. (a) sweetness (b) pleasantness (c) modesty (d) kindness 2. The accused ________ having made any statement. (a) rejected (b) denied (c) refused (d) declined 3. If I ________ you, I would have told him the truth. (a) am (b) were (c) was (d) and

In the following questions, sentences are given with blanks to be filled in with an appropriate word. Four alternatives are suggested

anSwer Key 1. (c)

2. (b)

3. (b)

Format ii Sometimes, sentences with two blanks are given and two sets of words have to be selected from the answer choices to complete the sentence. 1. Although he had shown himself to be ________ in the most dangerous of situations in both war and peace, he revealed

himself as ________ in the social atmosphere of the femaledominated drawing room. (a) reliable/self-assured (b) interpid/craven (c) calm/cavalier (d) None of these

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▌ English

2. The opposition had ________ such strong arguments against his position on the sugar muddle, that even his most loyal adherents deserted him, and he was forced to ________. (a) fabricated/resist (b) marshalled /acquiesce (c) tendered /concentrate (d) tendered /resignation

3. Economic goods are those goods which are ________ in quantity as compared to ________ demand. (a) expensive/it (b) limited/ their (c) stable/its (d) available/ those

anSwer Key 1. (b)

2. (b)

3. (b)

SentenCe Completion Pick out the most effective word(s) from the given options to make the sentences meaningful. 1. The accelerated growth of public employment ________ the dramatic expansion of budgets and programmes. (a) parallels (b) contains (c) revolves (d) escapes 2. Not every ________ mansion, church, battle site, theatre, or other public hall can be preserved. (a) noble (b) structured (c) comparative (d) venerable 3. Man is still a ________ in the labour market. (a) glut (b) possibility (c) commodity (d) provision 4. As we moved on to Agra shortly after noon on Saturday, the clear air and the rolling ________ enthralled us. (a) stones (b) hovels (c) skyline (d) terrain 5. The book will be ________ by every Western student of the USSR, and it will be a thrilling adventure for any reader. (a) skimmed (b) persued (c) rejected (d) borrowed 6. There is a notion abroad that history has gotten away from us; that our lives are beyond control; that there are no points of ________ which mean anything any more. (a) conference (b) inference (c) prudence (d) reference

9. No one can say for sure how ________ the awards have been. (a) determined (b) effective (c) reducible (d) effervescent 10. For fifty years, only such women and the few intellectuals who shared his ________ scorn read Stendhal. (a) reverberating (b) explicit (c) sensational (d) sensuous 11. On the ground, liquid hydrogen must be stored in large stainless steel tanks with double walls filled with ________ and evacuated to a high vacuum. (a) velocity (b) visibility (c) sufficiency (d) insulation 12. Onthology is the word now used in place of metaphysics, on the grounds that it is less ________ and supernatural. (a) theological (b) applicable (c) reliable (d) approximate 13. All countries in the region, who are also UN members, are ________ to join. (a) embraced (b) eligible (c) prohibited (d) tantamount 14. An Australian woman ________ has discovered a way to boost the seed yield of pastureland. (a) agronomist (b) psychoanalyst (c) phrenologist (d) bioscopist

7. I cannot honestly number myself among the pious and I have frequently had the experience of being ________ amongst to being the unholy. (a) regenerated (b) deteriorated (c) complied (d) consigned

15. Basic research provides the ________ fund of scientific knowledge on which the applied researchers draw to give society a rich rate of interest. (a) depleted (b) endowed (c) capital (d) deterred

8. Today, we who read Latin return far more often to the exuberance of Apuleius than to the carefully moulded ________ of Cicero. (a) literature (b) redundancies (c) objects (d) piracies

16. He ________ his speech heavily with jargon of the trade. (a) retards (b) brakes (c) inflates (d) lards

Sentence Completion ▌ 2.35 17. The growing of cereals on a large scale was the first stage in a revolution that was to replace a food gathering ________ existence by an urban civilisation based on agriculture. (a) paternal (b) herbivorous (c) sedulous (d) nomadic

27. The teacher …… his students …… being late to school. (a) shouted …… at (b) reprimanded … for (c) reminded …… with (d) narrated … of

18. But a ________ wind built up during the race, and shortly most of the summertime boats turned to and went home. (a) infamous (b) helpless (c) snarling (d) ravishing

28. The dimensions of the …… are known and the solution is …… on. (a) problem …… agreed (b) theory …… dealt (c) measures …… decided (d) risks …… tempered

19. The sales of Jules Verne’s books continue to ________ like a runaway balloon. (a) flit (b) advance (c) revive (d) soar 20. The … manner of the officer …… all his fears. (a) haughty …… aggravated (b) officious …… levelled (c) amiable …… concentrated (d) genial …… dispelled 21. To ... his arguments, he… several references. (a) vindicate …… refer (b) press …… announced (c) substantiate …… cited (d) reveal …… declared 22. As censorship was …… people were …… the furious battle being fought. (a) clammed …… oblivious of (b) revoked …… indifferent to (c) imposed …… unaware of (d) dispelled …… ignorant of 23. He was … by letter that he had … selected. (a) informed …… been (b) told …… not (c) stated …… finally (d) offered …… been 24. Giving unfair …… to one’s relatives in the matter of appointments and such other benefits is …… . (a) punishment …… parochialism (b) advantage …… nepotism (c) benefits …… pragmatism (d) preferences …… chauvinism 25. The Vidhan Sabha voted to …… the ban …… drinking. (a) invoke …… for (b) revoke …… of (c) prohibit …… prevented (d) repeal …… on 26. Our army is …… …… the border of the neighbouring country. (a) posted …… with (b) accumulated …… for (c) assembled …… at (d) deployed …… along

29. There is so much …… between the two sisters that it is …… to know one from the other. (a) resemblance …… difficult (b) identity …… impossible (c) equality …… easy (d) disparity …… undesirable 30. With this realization, the people suddenly found themselves left with ______ moral values and little ethical _______. (a) obsolete … perspective (b) established … grasp (c) protentous … insinuation (d) extreme … judgement 31. These avant garde thinkers believe that the major peace movements are ineffective because the thinking that underlies these movements is old-fashioned, confused, ________ and out-of-step with the findings of ________ science. (a) stimulating … natural (b) delusionary … behavioral (c) loaded … true (d) uncertain … physical 32. The fact that a business has ________ does not create an ________ on it to give away its prosperity. (a) proposed … imperative (b) halted … insensitivity (c) incorporated … indecision (d) accumulated … aspect 33. It is usually a good thing when a discussion is taken firmly by the hand and led down from the heights of ________ to the level ground of hard ________. (a) ridiculousness … sublimity (b) audacity … sincerity (c) fantasy … fact (d) speculation … reality 34. Electronic eavesdropping technology has become so ________ that the comparatively little law on the subject has become as ________ as the horse and buggy. (a) repulsive … fictitious (b) omnivorous … ridiculous (c) sophisticated … outmoded (d) clandestine … entangled

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▌ English

35. It really looked as if the outclassed Portuguese were about to make as ________ an exit from the ________ as had the Italians. (a) ignominious … competition (b) differential … forum (c) emphatic … cavern (d) surreptitious … vista

48. You must bear ________ his rudeness. (a) to (b) with (c) out (d) of 49. You must dispense ________ his services. (a) of (b) in (c) at (d) with

36. Gandhiji was a humble man known for his ____. (a) sweetness (b) pleasantness (c) modesty (d) kindness

50. You must attend __________ the lecture. (a) on (b) with (c) in (d) to

37. The accused ________ having made any statement. (a) rejected (b) denied (c) refused (d) declined

51. He deals ________ vegetables these days. (a) out (b) to (c) for (d) in

38. If I ________ you, I would have told him the truth. (a) am (b) were (c) was (d) and

52. The blind man was feeling _______ the table. (a) in (b) for (c) by (d) out

39. The father divided his property among his sons lest they ________ quarrel afterwards. (a) should (b) may (c) would (d) might

53. Mr Rohit, their cashier, is working _________ a small salary. (a) for (b) at (c) on (d) in

40. Indians need sincere leaders, not people of ________ integrity. (a) doubting (b) doubtful (c) doubtless (d) double 41. ________ the corner of the street stood a policeman _________ an umbrella, shivering in the rain. (a) On, with (b) Near, under (c) By, in (d) At, under 42. Put the milk ________ the table and cover it ________ cloth. (a) on, with (b) near, by (c) on, by (d) in, with 43. I must hold a discussion ________ you _________ that matter shortly. (a) about, for (b) with, for (c) about, in (d) with, about 44. ________ cover of darkness, the enemy crept ________ the hill. (a) In, over (b) In, along (c) Under, up (d) Through, upon 45. There was no agreement _________ the great powers ________ a treaty to ban nuclear weapons. (a) with, about (b) among, on (c) between, about (d) in, for 46. _________ two weeks, he did not venture ________ the safety of his house, even once. (a) Within, for (b) For, beyond (c) For, over (d) Over, for 47. _________ the house tops the kite flew until it got lost _______ the clouds. (a) By, in (b) From, in (c) On, among (d) Over, among

54. He does not work with his hands, he works ________ the machine. (a) with (b) by (c) at (d) on 55. At the time of difficulty turn ________ your friend for help. (a) into (b) upon (c) up (d) down 56. Do not intrude, they are talking _________ a confidential matter. (a) on (b) for (c) over (d) in 57. We see ________ a box at the opera. (a) into (b) on (c) in (d) under 58. I slept after lunch ________ armed chair. (a) on (b) into (c) in (d) over 59. They planned to link the various cities one ________ another. (a) over (b) with (c) under (d) into 60. They were talking ________ the UN. (a) on (b) about (c) with (d) in 61. According to the conditions of my scholarship, after finishing my degree, _______. (a) my education will be employed by the university (b) employment will be given to me by the university (c) the university will employ me (d) I will be employed by the university

Sentence Completion ▌ 2.37 62. If it _______ more humid in the deserts the hot temperature would be unbearable. (a) be (b) is (c) was (d) were 63. _______ small specimen of the embryonic fluid is removed from a foetus, it will be possible to determine whether the baby will be born with birth defects. (a) A (b) That a (c) If a (d) When it is a 64. Prices for bikes can run ________ Rs 35,000. (a) as high as (b) as high to (c) so high to (d) so high as 65. Ancient civilizations such as Phoenicoans and the Mesopotamians’ ______ goods rather than use money. (a) use to trade (b) is used to trade (c) used to trade (d) was used to trade 66. Most executives do not object ________ them by their first names. (a) that I call (b) to my calling (c) for calling (d) that I do call 67. Travellers _______ their reservations well in advance if they want to travel during the Diwali holidays. (a) had better to get (b) had to get better (c) had better get (d) had better got 68. To generate income, magazine publishers must decide whether to increase the subscription price or _________. (a) to sell advertising (b) if they should sell advertising (c) selling advertising (d) sold advertising 69. All of the people at this conference are ________. (a) mathematic teachers (b) mathematics teachers (c) mathematics teacher (d) mathematic’s teacher 70. ______ Java man, who lived before the first Ice age, is the first man-like animal. (a) It is generally believed that (b) Generally believed it is (c) Believed generally is (d) That is a generally believed

73. An eighteen-year-old is ________ to vote in the election as per the Constitution of India. (a) old enough (b) as old enough (c) enough old (d) enough old as 74. The observation deck at the World Trade Center ______ in New Delhi. (a) is highest than any other (b) is higher than any other (c) is highest that any other (d) is higher that any other 75. The United States is _______ that there are five time zones. (a) much big (b) too big (c) so big (d) very big 76. Upon hatching ; _________. (a) young ducks know how to swim (b) swimming is known by young ducks (c) the knowledge of swimming is in young ducks (d) how to swim is known in young ducks 77. _________ 1000 species of fish have been identified. (a) As many as (b) As many (c) As much as (d) Much as 78. Not until a monkey is several years old _________ to exhibit signs of independence from its mother. (a) it begins (b) does it begin (c) and begin (d) beginning 79. ________ is necessary for the development of strong bones and teeth. (a) It is calcium (b) That calcium (c) Calcium (d) Alongwith calcium 80. Almost everyone fails ________ on the first try. (a) in passing his driver’s test (b) to pass his driver’s test (c) to have passed his driver’s test (d) passing his driver’s test 81. Spielburg is going to be nominated to receive the Academy Award for the best director, _______? (a) won’t he (b) didn’t he (c) doesn’t he (d) isn’t he

71. Since Vandana’s father never approved of ________ Sumit, the couple eloped to Italy where they lived and wrote. (a) her to marry (b) her marrying (c) she marrying (d) she to marry

82. ________ to watch football everyday? (a) Do people in your country like (b) People in your country like (c) May people in your country like (d) Have people in your country like

72. Only after food has been dried _________ . (a) that is should be stored for later consumption (b) should be stored for later consumption (c) should it be stored for later consumption (d) it should be stored for later consumption

83. The Ford Theater where Lincoln was shot _________. (a) must restore (b) must be restoring (c) must have been restored (d) must restored

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▌ English

84. The speaker is _________. (a) very well acquainted with the subject (b) recognized as an authority who knows, a great deal in terms of the subject (c) someone who knows well enough about the subject which he has undertaken to do the speaking (d) a person who has close awareness of the subject that he speaks about so much 85. After seeing the movie “The Day of the Jackal”, ________. (a) the book was read by many people (b) the book made many people want to read it (c) many people wanted to read the book (d) the reading of the book interested many people 86. ________ the formation of the sun, the planets, and other stars begin with the condensation of an interstellar cloud. (a) Except that (b) Accepted that (c) It is accepted that (d) That is accepted 87. In many ways, riding a bicycle is similar to _________. (a) the driving of a car (b) when you drive a car (c) driving a car (d) when driving a car 88. Fast-food restaurants have become popular because many working people want _________. (a) to eat quickly and cheaply (b) eating quickly and cheaply (c) eat quickly and cheaply (d) the eat quickly and cheaply 89. Although the weather in Kodaikanal is not ________ to have a year round tourist season, it has become a favourite summer resort. (a) goodly enough (b) good enough (c) good as enough (d) enough good 90. Although most adopted children want the right to know who their natural parents are, some who have found them wish that they ________ the experience of meeting. (a) hadn’t (b) didn’t have had (c) hadn’t had (d) hadn’t have 91. The consumer price index lists __________. (a) how a much cost every scooter (b) how much does every car cost (c) how much every car costs (d) how much are every car cost

92. Canada does not require that US citizens obtain passports to enter the country, and ________. (a) Mexico does neither (b) Mexico doesn’t either (c) neither Mexico does (d) either does Mexico 93. As soon as _______ with an arial salt, water is formed. (a) a base will react (b) a base reacts (c) a base is reacting (d) the reaction of a base 94. Several of these washes and dyers are not in order and ________. (a) need to be repairing (b) repairing is required of them (c) require that they be repaired (d) need to be repaired 95. Thirty-eight national sites are known as parks, another eightytwo as monuments and _________. (a) the another one hundred seventy-eight as historical sites (b) the other one hundred seventy-eight as historical sites (c) seventy-eight plus one hundred more as historical sites (d) as historical sites one hundred seventy-eight 96. A telephone recording tells callers _________. (a) what time the movie starts (b) what time starts the movie (c) what time does the movie starts (d) the movie starts what time 97. To answer accurately is more important than _________. (a) a quick finish (b) finishing quickly (c) to finish quickly (d) you finish quickly 98. Having been selected to represent the Association of Indian Managers at the International Convention, ____________. (a) the members applauded him (b) he gave a short acceptance speech (c) a speech had to be given by him (d) the members congratulated him 99. Not until a student has mastered algebra ________ the principals of geometry, trigonometry and physics. (a) he can begin to understand (b) can he begin to understand (c) he begins to understand (d) begins to understand 100. It costs about thirty rupees to have a tooth _________. (a) filling (b) filled (c) to fill (d) fill

Sentence Completion ▌ 2.39

anSwer Key 1. (a)

14. (a)

27. (b)

40. (b)

53. (a)

66. (b)

79. (c)

92. (b)

2. (d)

15. (c)

28. (a)

41. (a)

54. (d)

67. (c)

80. (b)

93. (b)

3. (c)

16. (d)

29. (a)

42. (a)

55. (d)

68. (a)

81. (d)

94. (d)

4. (c)

17. (d)

30. (b)

43. (d)

56. (c)

69. (b)

82. (a)

95. (b)

5. (d)

18. (c)

31. (b)

44. (c)

57. (c)

70. (a)

83. (c)

96. (a)

6. (a)

19. (d)

32. (a)

45. (b)

58. (c)

71. (b)

84. (a)

97. (b)

7. (d)

20. (d)

33. (d)

46. (b)

59. (b)

72. (c)

85. (c)

98. (b)

8. (d)

21. (c)

34. (c)

47. (b)

60. (b)

73. (a)

86. (c)

99. (b)

9. (b)

22. (c)

35. (a)

48. (b)

61. (d)

74. (b)

87. (c)

100. (b)

10. (d)

23. (a)

36. (c)

49. (d)

62. (d)

75. (c)

88. (a)

11. (d)

24. (b)

37. (b)

50. (d)

63. (c)

76. (a)

89. (b)

12. (a)

25. (d)

38. (b)

51. (d)

64. (a)

77. (a)

90. (c)

13. (b)

26. (d)

39. (a)

52. (b)

65. (c)

78. (b)

91. (c)

praCtiCe eXerCiSe Pick out the most effective word(s) from the given pair to make the sentences meaningfully complete. 1. Professionals focus their ……………… on fulfilling their responsibilities and achieving results, not on ………… a particular image. (a) planning …… devising (b) leadership …… attributing (c) abilities …… contributing (d) energies …… portraying 2. ‘Patriotism is the last refuge of a scoundrel,” says Johnson. In the modern world where the cunning selfish people …… and the hard working conscientious people …… the quotation holds good. (a) dominate …… suppress (b) thrive …… suffer (c) enjoy …… mutilate (d) empower …… subjected 3. We must be rest assured that our sorrows, if any, are shortlived and a period of joy … us. Such, thinking will always maintain our …… and peace of mind. (a) begets …… dream (b) befits …… body (c) bemoans …… skills (d) merits …… life 4. When you are living with your …… values and principles, you can be straightforward, honest and ……

(a) (b) (c) (d)

core ..... up-front inherited …… distinct innate … durable cultural … perceptive

5. In the role of a counsellor, you are an authority figure whose objective is to …… attentively and sensitively to employees who …… you with their feelings. (a) projects …… focus (b) manage …… direct (c) listen …… trust (d) concentrate …… believe 6. If a junior executive neglects his professional development and …… education, he can easily and quickly become obsolete in a world changing at …… rates. (a) management …… voluminous (b) higher …… vulnerable (c) better …… supreme (d) continuing …… dizzying 7. Harassment is just plain …… There is never a valid reason or a good …… for it. (a) wrong …… excuse (b) murder …… command (c) falsehood …… remedy (d) killing …… magic 8. Man is …… He likes to know how things work. The search for understanding is …… in its own right. (a) evolving …… prophetic (b) inquisitive …… prophetic (c) appreciative …… fundamental (d) curious …… philosophical

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9. Leadership is one of the world’s oldest ……… The understanding of leadership has figured strongly in the …… for knowledge. (a) cultures …… desire (b) institutions …… passion (c) preoccupations …… quest (d) subjects …… preference 10. Our …… to understand the process of learning underlying behaviour change are …… by the fact that any given behaviour is determined jointly by many processes. (a) nature …… determined (b) scope …… preceded (c) implications …… followed (d) attempt …… complicated

17. The fear …… when feeling incompetent is the fear of being humiliated, embarrassed and …… (a) endowed …… criticised (b) afflicted …… downtrodden (c) consistent …… damaged (d) imbued …… exposed 18. Gopal was frustrated with Sunder who would not …… himself to a deadline. Sunder claimed he was working well without a deadline, but Gopal …… and finally prevailed. (a) encourage …… enforced (b) inculcate …… ordered (c) cooperate …… stipulated (d) commit …… persisted

11. When interpersonal problems …… but are dealt with, the organization’s productivity inevitably …… (a) surface …… develops (b) focus …… increases (c) establish …… projects (d) develop …… exhibits

19. I am not easily …… by pressures that would interfere with accomplishing the goals of my unit. I stick with my …… (a) pessimistic …… views (b) swayed …… convictions (c) discouraged …… achievements (d) empowered …… organisation

12. Participatory management, in which everyone …… into a decision that a leader then makes, is a mechanism for …… employees (a) share …… protecting (b) value …… thwarting (c) motivation …… involving (d) contributes …… stimulating

20. Coercion sometimes leads to the …… of short-term goals, but its drawbacks far …… its advantages. (a) realization …… damage (b) appreciation …… percolate (c) accomplishment …… outweigh (d) achievement …… crumble

13. Lack of …… is the basis of good teamwork but our ability to work with others depends on our …… (a) rigidity …… compatibility (b) dogmatism …… motivation (c) professionalism ……vulnerability (d) positivism …… flexibility 14. Complete and constant openness is a notion that can be …… to absurdity. Am I …… to stop everyone on the street and tell them my reaction to their appearance? (a) consigned …… communicated (b) reduced …… required (c) attributed …… requested (d) projected …… destined 15. When organisations …… creativity and risk taking, the usual method of maintaining order and …… are indeed shaken. (a) encourage …… decorum (b) exhibit …… durability (c) propose …… humility (d) propagate … production 16. When I am an autocrat, I am …… in the extreme. My direct method is to …… power and control. (a) perfectionist …… explore (b) autocratic …… engulf (c) dominating …… seek (d) possessive …… reject

21. The teacher must …… the unique style of a learner in order to …… it to the desired knowledge. (a) advocate …… direct (b) perpetuate …… develop (c) appreciate …… focus (d) discover … harness 22. Not all counties benefit …… from liberalization, the benefits tend to …… first to the advantaged and to those with the right education to be able to benefit from the opportunities presented. (a) equally …… generate (b) richly …… downgrade (c) suitably …… ascribe (d) uniformally …… percolate 23. He has …… sense of words. Therefore, the sentence he constructs are always …… with rich meaning. (a) profound …… pregnant (b) distinguished …… loaded (c) terrific …… tempted (d) meaningful …… full 24. He was an …… musician, had been awarded the George Medal during the second world war and …… with the title of Rai Bahadur. (a) outstanding …… popularized (b) underestimated …… declared (c) accomplished …… honoured (d) impressive …… assigned

Sentence Completion ▌ 2.41 25. Whether it be shallow or not, commitment is the ……, the bedrock of any …… loving relationship. (a) expression …… perfunctory (b) foundation …… genuinely (c) manifestation …… deep (d) key …… alarmingly 26. Many people take their spirituality very seriously and …… about those who do not worrying about them and …… them to believe. (a) think …… criticising (b) pride …… appraising (c) rationalize …… enabling (d) wonder …… prodding 27. Unless new reserves are found soon, the world’s supply of coal is being …… in such a way that with demand continuing to grow at present rates reserves will be …… by the year 2050. (a) consumed …… completed (b) depleted …… exhausted (c) reduced …… argument (d) burnt …… destroyed 28. If you are …… you tend to respond to stressful situations, in a calm, secure, steady and …… way. (a) resilient …… rational (b) obdurate …… manageable (c) propitious …… stable (d) delectable …… flexible 29. Management can be defined as the process of …… organisational goals by working with and through human and non-human resources to …… improve value added to the world. (a) getting …… deliberately (b) managing …… purposefully (c) targeting …… critically (d) realising …… dialectically 30. If you are an introvert, you …… to prefer working alone and if possible, will …… towards projects where you can work by yourself or with as few people as possible. (a) like …… depart (b) advocate …… move (c) tend …… gravitate (d) express …… attract 31. Despite …… knowledgeable, he remained …… all through. (a) having …… ignorant (b) of …… doubtful (c) owing …… through (d) being …… poor 32. He used to …… the point that victory in any field needs …… courage. (a) emphasize …… little (b) stress …… exemplary (c) refute …… no (d) dismiss …… formidable

33. We will call off the agitation if the Chairman gives an …… that there will be no …… of workers. (a) linking …… harrassment (b) assurance …… need (c) explanation …… demand (d) undertaking …… retrenchment 34. It is a noble quality to …… the good and …… the bad. (a) retain …… preserve (b) imitate …… pretend (c) appreciate …… condemn (d) criticize …… ape 35. Using …… designs to …… the importance of others is not an healthy act. (a) nefarious …… undermine (b) architectural …… elevate (c) fictitious …… enhance (d) brittle …… activate 36. The so-called civilised human race has …… and ill-treated small and large animals in an attempt to prove its …… (a) abused …… supremacy (b) misuse …… power (c) cruelty …… altruism (d) advocated …… worthlessness 37. With large classes, it is difficult for teachers to …… regular essay type questions for homework because …… long answers would take too much time. (a) consider …… writing (b) revalue …… concise (c) pursue …… feeling (d) evaluate …… marking 38. India’s …… over the past half century since independence has been unique and …… in many ways. (a) thinking …… feeling (b) development …… commendable (c) victory …… celebrating (d) crash …… overbearing 39. Despite being the …… partner in the relationship, the franchiser doesn’t always have all the …… . (a) sincere …… limitations (b) vulnerable …… powers (c) active …… losses (d) dominant …… advantages 40. Ours is a democracy and any …… or use of force is out of question. Methods of …… and education are best suited to a democratic regime. (a) attempt …… coercion (b) compulsion …… persuasion (c) judgement …… prayer (d) inhuman …… apprehension

2.42

▌ English

41. Although he had shown himself to be ________ in the most dangerous of situations in both war and peace, he revealed himself as ________ in the social atmosphere of the femaledominated drawing room. (a) reliable/self-assured (b) interpid/craven (c) calm/cavalier (d) None of these 42. The opposition had ________ such strong arguments against his position on the sugar muddle, that Even his most loyal adherents deserted him, and he was forced to ________. (a) fabricated/resist (b) marshalled /acquiesce (c) tendered /concentrate (d) tendered /resignation 43. Economic goods are those goods which are ________ in quantity as compared to ________ demand. (a) expensive/it (b) limited/ their (c) stable/its (d) available/ those 44. Lockout is the closure of a factory by ________ when there is labour dispute and fear ________ damage to machinery and property.

(a) employees/for (c) government/with

(b) employer/in (d) employer/of

45. The rebels of the LTTE ________ outright an invitation by the Sri Lankan Prime Minister, to send ________ proposals to end the decade old-ethnic conflict. (a) have/their (b) rejected/ for (c) rejected/their (d) accepted/her 46. I could not refrain ________ shedding tears. (a) by (b) to (c) from (d) with 47. He is an amateur ________ stenography. (a) for (b) in (c) to (d) of 48. Children are the delight _________ the house. (a) for (b) of (c) to (d) with 49. No one has control ________ his tongue. (a) for (b) to (c) on (d) with 50. She was conscious _______ her weakness. (a) in (b) for (c) of (d) by

anSwer Key 1. 2. 3. 4. 5. 6. 7.

(d) (b) (d) (b) (c) (d) (a)

8. 9. 10. 11. 12. 13. 14.

(b) (b) (d) (a) (d) (b) (b)

15. 16. 17. 18. 19. 20. 21.

(d) (c) (d) (d) (b) (a) (d)

22. 23. 24. 25. 26. 27. 28.

(d) (a) (c) (b) (d) (b) (a)

29. 30. 31. 32. 33. 34. 35.

(d) (c) (d) (b) (d) (c) (a)

36. 37. 38. 39. 40. 41. 42.

(a) (d) (b) (d) (b) (b) (b)

43. 44. 45. 46. 47. 48. 49.

(b)

50. (d)

(d) (c) (c) (c) (b) (b)

nDa BoX Directions: In the following passage at certain points, you are given a choice of three words/phrases, underlined and marked (a), (b), (c). Choose the best word/phrase out of the three and indicate your choice in the relevant column of your answer sheet. 1. Andhra Pradesh Chief Minister (a) assures (b) ensures (c) insures that every other Head of State or Government visiting India comes to his capital city. So the mandarins promoting tourism in his government have decided to 2. (a) lie out (b) lay out (c)spread out the welcome by switching on city lights. After the success of

3. (a) lighting up (b) lightening up (c) brightening up the city’s iconic Charminar in different colours, a similar dynamic system now brightens up the Legislative Assembly building and the Saidanama Tomb facing the 4. (a) delightful (b) heart-warming (c) picturesque Hussainsagar Lake. The computerised lighting system throws up a 5. (a) frightening (b) staggering

(c) scray

sixteen million colour variations but only 150 are visible to the human eye.

Sentence Completion ▌ 2.43

Fill in the Blanks Directions: (for the following 6 items): Each of the following six sentences has a blank space and four words or groups of words are given after the sentence. Out of these four choices, select the word or group of words which you consider the most appropriate for the blank space and markyou choice on the answer sheet. 6. When the President is ill, his duties ..... the Vice-President. (a) evolve upon (b) devolve upon (c) rotate (d) revolve upon 7. He is ..... opponent; you must respect and fear him at all times. (a) a redoubtable (b) a craven (c) a disingenuous (d) an inveterate

8. They fired upon the enemy from behind trees, walls, and any other ........ point they could find. (a) exposed (b) vintage (c) vantage (d) definitive 9. The child was so much spoiled by her indulgent parents that she became ....... when she did not receive all their attention. (a) suspicious (b) sullen (c) discreet (d) elated

10. The doctor was overcome with ..... when he came to know that the patient had died due to negligence on his part. (a) emotion (b) remorse (c) humiliation (d) reality

11. I ..... with him about his rudeness. (a) remonstrated (b) demonstrated. (c) shouted (d) reiterated

anSwer Key 1. (b) 2. (c)

3. (c) 4. (a)

5. (b) 6. (b)

7. (a) 8. (c)

9. (b) 10. (b)

11. (a)

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C

H

5

A

P

T

E

R

Jumbled Words in a Sentence Format oF the QueStionS There are certain other type of questions asked in competitive examinations, wherein the format generally varies and the knowledge of grammar, vocabulary, spellings etc., attained so far would be sufficient to help you tackle these questions. Direction: In each of the questions, there is a sentence of which some parts have been jumbled up. You are required to rearrange these parts, which are labelled P,Q,R and S, to produce the correct sentence. Choose the proper sequence and mark in your answer sheet accordingly. 1. There was (P) needed for its everyday life (Q) a time when each family (R) for itself most of the things it (S) actually produced (a) QRSP (b) RQPS (c) RSPQ (d) QSRP

2. We have to (P) as we see it (Q) speak the truth (R) there is falsehood and darkness (S) even if all around us (a) RQSP (b) QRPS (c) RSQP (d) QPSR 3. The majestic mahogany table (P) belongs to an old prince (Q) which had one leg missing (R) who is now impoverished (S) but not without some pride (a) PQSR (b) QRSP (c) PRSQ (d) QPRS

anSWer Key 1. (d)

2. (d)

3. (d)

Jumbled WordS in a Sentence Directions: In the following items, some parts of the sentence have been jumbled uP : You are required to re arrange these parts which are labelled P, Q, R and S to produce the correct sentence. Choose the proper sequence and mark in your Answer Sheet accordingly.

1. She was born at 10am in india P Q

in the year 1978 18th Augusst . R S

Which one of the following is the correct sequence? (a) PQRS (b) QPSR (c) RSQR (d) SRPQ

2.46

▌ English

2. The highwayman behind the treetops P

when the sun disappearedd Q

rode to the cottage R

where his beloved livved . S

Which one of the following is the correct sequence? (a) PQRS (b) SPRQ (c) QRSP (d) RSQP 3. She danced at the town hall P

last night Q

beautifully R

in the concert . S

Which one of the following is the correct sequence? (a) PQRS (b) RSPQ (c) QSPR (d) RQPS 4. My father Shuts out the word who by closing his eyes is a loner . P Q R S Which one of the following is the correct sequence? (a) PQRS (b) SQPR (c) SPQR (d) QSPR 5. Because he detested the food available locallly P

the refugee Q

walking along the deserted

thought of the delicaciees

sttreets of the town R

of his country S

Which one of the following is the correct sequence? (a) RQSP (b) PQRS (c) RPQS (d) QPRS 6. I believe that Tendulkar's century P

will always remember Q

Criicket lovers on Tuesday . R S Which one of the following is the correct sequence? (a) SRQP (b) RQPS (c) PSRQ (d) PRQS 7. It is better to go the people for P if these kinds of accusations go on Q and if the goverrnment wants to do something good R even thouugh people do not want another election S

Which one of the following is the correct sequence? (a) SPQR (b) QSRP (c) QRPS (d) RPQS 8. The newspaper reported that can never reproduuce such a wonder P

the rain forest Q

is naturee's way of telling us that man R

for all his ability to create S

Which one of the following is the correct sequence? (a) QRSP (b) QRPS (c) SQPR (d) QPRS 9. The judge declared P will bee jailed R

that those people Q

Who cannot remain silent in the coourt S

Which one of the following is the correct sequence? (a) PQSR (b) QPSR (c) SPQR (d) QRPS 10. The first priority before us is irrespective of their

that people learn to geet on

backgrounds P

with one another as Q understand each other and

fellow human beings R

forrce bonds of friendship S

Which one of the following is the correct sequence? (a) QSPR (b) QRPS (c) QPRS (d) RPQS 11. we have come the to know that to who committe always ever came first P Q awarded R

a beautifull silver medal . S

The correct sequence should be (a) P-Q-R-S (b) R-D-P-Q (c) P-R-S-Q (d) Q-P-R-S 12. Sherlock Holmes is Charcter in fiction P more peopple R

than any other Q

perhaps known to S

The correct sequence should be (a) S-R-Q-R (b) Q-S-P-Q (c) Q-S-P-R (d) R-S-Q-P

Jumbled Words in a Sentence ▌ 2.47 13. The visitor said that he saw some people P in the street R

who shouted"Long live labbour unity'' Q

when he started from home . S

The correct sequence should be (a) P-Q-R-S (b) P-SQ-R (c) S-P-R-Q (d) S-R-P-Q 14. The principal shortcoming of the interpretatiion of culture P to encom mpass the social

lies in its failure Q so for as theory and

methodology is concerned nature of culture . R S The correct sequence should be (a) P-S-Q-R (b) Q-S-R-P (c) S-R-Q-P (d) S-Q-R-P 15. The whole project by drawing about

students, youth, teacher

60000 volunteers P persons would be

sand retired Q from different

completed R

groups such as . S

The correct sequence should be (a) R-S-P-Q (b) R-P-S-Q (c) P-S-Q-S (d) Q-R-S-P 16. It is unity P

which will generate dissensions Q

rather than disunity R

that is the need of the hour . S

The correct sequence should be (a) P-Q-R-S (b) P-S-R-Q (c) R-Q-P-S (d) Q-P-R-S 17. Deworming medicine P and were given R

suspected to have worms Q

all the animals were . S

The correct sequence should be (a) S-P-R-Q (b) S-Q-R-P (c) P-R-S-Q (d) P-Q-R-S

18. It is obvious that P

an open mind is a good thhing Q

but not outside of them R

within certainn limits . S

The correct sequence should be (a) Q-P-S-R (b) P-Q-S-R (c) R-P-Q-S (d) P-R-Q-S 19. National integration cannot be achieved P in the country R

wihtooth harnessing Q the communal forces . S

The correct sequence should be (a) R-S-P-Q (b) Q-P-R-S (c) P-Q-R-S (d) P-Q-S-R 20. He promised in the office P

for a suitable post Q

to consideer me R

when a vacancy arose . S

The correct sequence should be (a) S-R-Q-P (b) R-Q-P-S (c) Q-R-P-S (d) Q-R-S-P 21. Lived in the lake District Which lies P Q in thee North of England Wordsworth the great nattrue poet . R S The correct sequence should be (a) P-Q-R-S (b) P-S-Q-R (c) S-P-Q-R (d) Q-R-P-S 22. The poems and stories have been taken P from a sourcess R

for this book Q that have been selected . S

The correct sequence should be (a) S-Q-P-R (b) Q-P-S-R (c) R-P-Q-S (d) P-Q-R-S 23. In india (P)

of all the magazines (Q)

widely reaad (R)

Readers Digest is the most . (S)

The correct sequence should be (a) Q-P-S-R (b) Q-P-R-S (c) S-P-Q-P (d) R-P-S-Q

2.48

▌ English

24. was arrested (P)

he (Q)

for stealing (R)

by the police. S

The correct sequence should be (a) Q-P-S-R (b) P-S-Q-R (c) R-S-Q-P (d) R-P-S-Q 25. at are markably early age (P)

aggressive behavviour Q

children can show . R S The correct sequence should be (a) P-Q-R-S (b) P-S-Q-R (c) R-S-Q-P (d) R-P-S-Q 26. in the richness and variety (P) of its wild life (R)

in the world Q) (Q India is unique . (S)

The correct sequence should be (a) P-Q-R-S (b) Q-S-P-R (c) Q-R-P-S (d) S-Q-P-R 27. neither (P)

he was (Q)

intelligent (R)

nor hardworrking. (S)

The correct sequence should be (a) P-Q-R-S (b) Q-S-P-R (c) Q-P-R-S (d) R-S-P-Q 28. With rise P : like Bngladesh would experience severe flooding and erosion of land Q : in sea levels now R : island countries like the Maldives and low-lying coastal countries. S : an unequivocal certainly. Which one of the following is the correct sequence? (a) QPRS (b) RSQP (c) QSRP (d) RPQS 29. In medicine P : Mind, emotions, social environment and spirit Q : often ignored a holistic perspective, that is the body’s interaction with R : the emphasis has tended S : to focus on the human body, but it has Which one of the following is the correct sequence? (a) QPSR (b) RSQP (c) RPSQ (d) QSPR 30. Climatologists are P : the earth as a whole is becoming warmer Q : Antarctica would seem to be bucking the trend R : and largely due to increased human activity and yet as a continent S : however, unanimous in their opinion that regional variations not with standing

Which one of the following is the correct sequence? (a) RPSQ (b) SQRP (c) RQSP (d) SPRQ 31. Although P : there is contradictory evidence about Q : whether long-term drinking has permanent effects R : extreme alcohol consumption kills brain cells S : on congnitive abilities such as reasoning and memory Which one of the following is the correct sequence? (a) RPQS (b) SQPR (c) RQPS (d) SPQR 32. In a globalized world P : to re-examine the definition of ‘self-sufficiency’ Q : we need to be confident enough R : and ensure it has not become another word for poor use of available resources S : Where India has emerged as a major economic player Which one of the following is the correct sequence? (a) SRPQ (b) PQSR (c) SQPR (d) PRSQ 33. At first sight P : is as firmly entrenched in our system Q : it looks as though Panchayati Raj R : the lower layer of federalism in our Polity S : as is the older and higher layer comprising the Union Government and the States Which one of the following is the correct sequence? (a) PSQR (b) QRPS (c) PRQS (d) QSPR 34. The psychology of P : allowing us to experience other worlds Q : is a bit like play?–acting R : reading for pleasure, scholars point out S : and roles in our imagination Which one of the following is the correct sequence? (a) PQRS (b) RSPQ (c) PSRQ (d) RQPS 35. Whenever I have been to India P : of those they replace, and perform at a high level Q : seamlessly move into the Places R : bench both strong and ready so that players S : people have asked me how the Australian cricket keeps its Which one of the following is the correct sequence? (a) PQRS (b) SRQP (c) PRQS (d) SQRP 36. When you add P : your equation with him will change forever Q : and your approach in dealing R : the elements of respect to the relationship with your child S : With your child will mature and become positive Which one of the following is the correct sequence?

Jumbled Words in a Sentence ▌ 2.49 (a) RSQP (c) RPQS

(b) QPRS (d) QSRP

37. As in India P : press freedom once it begins to roll Q : would find it difficult to halt the juggernaut of R : when satellite TV spawned a whole genre of 24 hour new channels S : those in power in the Arab countries Which one of the following is the correct sequence? (a) QSRP (b) RPQS (c) QPRS (d) RSQP 38. Ideally situated at P : as a regional hub for international education. Q : the crossroads of Asia and R : Thailand is fast gaining an enviable reputation S : With state-of-the-art communications and transportation solutions. Which one of the following sequences is correct? (a) Q-P-S-R (b) R-S-P-Q (c) P-R-S-Q (d) Q-R-S-P 39. Despite the clear P : downward slope of a life-cycle bell curve Q : benefits of focusing and exploiting old discoveries R : enterprises that fail to recognise the need S : to explore for new opportunities typically find themselves on the Which one of the following sequence is correct? (a) P-S-R-Q (b) Q-S-R-P (c) P-R-S-Q (d) Q-R-S-P 40. The success story P : since, some of the most influential people in the west Q : of the Indian Diaspora should be viewed R : as a type of reverse colonization S : are Indians or of Indian origin. Which one of the following sequences is correct? (a) P-S-Q-R (b) Q-R-P-S (c) P-R-Q-S (d) Q-S-P-R 41. Rama won P : by employing highly mobile archers Q : described as monkeys in the epic Ramayana R : and a quick-footed infantry S : the battle against Ravana’s chariot-heavy army Which one of the following sequences is correct? (a) R-P-S-Q (b) S-Q-R-P (c) R-Q-S-P (d) S-P-R-Q 42. Perhaps the most significant factor P : is a failure of planning and Q : in the growth of all metropolitan crimes R : governance in the urban spraw1 S : including crimes against the elderly Which one of the following sequences is correct? (a) Q-S-P-R (b) P-R-Q-S (c) Q-R-P-S (d) P-S-Q-R

43. But, Kuala Lumpur P : Where modern Malay executives Q : but will never miss Friday prayers R : might have a cellular phone in hand, S : is a city firmly rooted in tradition. Which one of the following sequences is correct? (a) R-S-Q-P (b) S-P-R-Q (c) R-P-S-Q (d) S-Q-R-P 44. But nelson Mandela P : modern country in a modern way Q : and could run a new R : has the remedy S : so they could heal the racial conflict Which one of the following sequences is correct ? (a) R-S-Q-P (b) Q-P-R-S (c) R-P-Q-S (d) Q-S-R-P 45. SO, now India can P : loss out at first base because it’s too busy chasing an ephemeral dream Q : either put aside its hopes of getting the veto power and R : The security Council is enlarged or Which one of the following sequences is correct? (a) Q-P-S-R (b) S-R-Q-P (c) Q-R-S-P (d) S-P-Q-R 46. From a picnic table P : through the playground Q : while we unpacked a basket R : we watched them laugh and leap S : bulging with sandwiches and cookies. The correct sequence should be (a) RSQP (b) QPRS (c) RPQS (d) QSRP 47. As P : Hallucinations began Q : I hobbled over the rocks and gravel R : I thought they were going to burst S : My feet so excruciatingly painful The correct sequence should be (a) QSRP (b) RPQS (c) QPRS (d) RSQP 48. The completion P : It enables employees to feel a sense of accomplishment Q : and enables them to take pride in their work; no one enjoys producing junk R : of high quality products S : also enhances employee satisfaction because The correct sequence should be (a) PQRS (b) RQPS (c) PSRQ (d) RSPQ

2.50

▌ English

49. But now, P : not only on trade and war, but also on ancient civilizations Q : in the Arabian sea and the Bay of Bengal could give vital clues R : as many archaeologists feel that the wrecks of sea going vessels S : they wish to explore further The correct sequence should be (a) PQRS (b) SRQP (c) PRQS (d) SQRP 50. I estimate the value of your life P : as in the fact that whole sentiment of mankind will be affected by the principles which you have laid down Q : not so much by the wealth you have acquired R : and which you are putting into practice S : or even the distribution of it The correct sequence should be (a) RPSQ (b) QSPR (c) RSPQ (d) QPSR 51. Since life has now become intensely competitive (P))

The proper sequence should be (a) Q P R S (b) P S Q R (c) Q R P S (d) R S Q P 54. I can guarantee as medical representative (P)

without any doubbt (Q)

for the post for which

that this candidate

he has applied R) (R

will be successful (S)

The proper sequence should be (a) S P R Q (b) S R Q P (c) P S R Q (d) Q S P R 55. The broker said that a person (P)

should shop carefully (Q)

who buys a house (R)

for the lowest interest rate (S)

The proper sequence should be (a) S P R Q (b) P R Q S (c) S P Q R (d) P S R Q 56. Most of our small farmers

not only by the grades the obtains (Q)

do not get a fair return on the sale of thheir farm-products (P) (Q)

but also by his skill of observation and logicaal reasoning (R)

on their investments annd high cost of seeds

we judge a person's intellecctual worth. (S) The proper sequence should be (a) S Q R P (b) P S R Q (c) P Q R S (d) P S Q R 52. It is known that the world's greatest men (P) witth a view of (R)

become rich (Q)

have not laboured (S)

The proper sequence should be (a) P R Q S (b) R Q P S (c) P S R Q (d) R Q S P 53. By the time known to everyone (P)

we appeared on the scenee (Q)

he had already become

who had ever staryyed

an institution (R)

at Anand Bhawan (S)

in back breadking annd fertilizers (S) (R) The proper sequence should be (a) P Q R S (b) P Q S R (c) P R Q S (d) P R S Q 57. When the mother bird came home (P)

the storm had suubsided (Q)

and on not finding her bound ones (S)

cried piteously (R)

The proper sequence should be (a) Q S P R (b) P S R Q (c) P R S Q (d) Q P S R 58. The guide said that nowhere in the world (P) you will find (R)

a fairer building (Q)

than the Taj Mahal (S)

The proper sequence should be (a) R Q P S (b) P R Q S (c) P S Q R (d) R Q S P

Jumbled Words in a Sentence ▌ 2.51 59. The students knew that to control unruly students (P) our new principal (R)

in the college adminstration (Q)

took a number sequence should be (S)

The proper sequence should be (a) R S P Q (b) R S Q P (c) P Q R S (d) Q R P S 60. According to the standard big-bang theory, in a moment of

the universe came

infinite tempperature into existence (P) (Q) and density (S)

are not only (Q)

a number of discounts and (Q) conccessional tours (R)

the lowest sequence shoulld be (S)

The proper sequence should be (a) S R Q P (b) Q S P R (c) Q R P S (d) P S R Q 62. The theves fled away from those who were purssuing them (P)

with a dense shrubs and was (Q)

very favouurable to into a forest that was covered (R)

concealment (S)

The proper sequence should be (a) P R Q S (b) R P Q S (c) Q S R P (d) R Q S P

to cloud the othher fact (R)

The proper sequence should be (a) P S Q R (b) R S Q P (c) P Q R S (d) R Q S P 65. Only mountaineering (P) peoplle (R)

should take up (Q) who love adventure (S)

66. When the sun rose (P)

reach his destination (Q)

med the traveller resum his journey (R)

in time (S)

The proper sequence should be (a) R S Q P (b) P S R Q (c) P R Q S (d) S R Q P 67. The proposal that Indian Ocean should be decllared (P)

has been supported (Q)

as a zone of peace (R)

must not be allowed (Q) that man fas reached the moon (S))

The proper sequence should be (a) P Q R S (b) R P Q S (c) S Q R P (d) S P R Q

by all states (S)

The proper sequence should be (a) P S R Q (b) Q R S P (c) P R Q S (d) Q P S R 68. While I was going to the college, due to an accident ahead; (P) So I made a detour via pragati Maidan (Q) I found the road closed R) (R

63. The fact that man is ignorant of his own selff (P)

are more sppecifically of a nation (S)

as his objective was to

The proper sequence should be (a) P S R Q (b) Q P S R (c) R P Q S (d) Q S P R 61. Indian Airlines fares but also offer (P)

like India (R)

is lived (Q)

The proper sequence should be (a) P S Q R (b) R S Q P (c) P Q R S (d) R Q S P

some ten to fiffteen billion years ago (R)

64. The life of a nation principally in the villlages (P)

and reached in time (S) The proper sequence should be (a) R P Q S (b) R Q P S (c) Q R S P (d) P Q R S

2.52

▌ English

69. I who worked (P)

for the uplife of the poor (Q))

have always known him (R)

to be an honest maan (S)

The proper sequence should be (a) P Q R S (b) P S R Q (c) R S P Q (d) R P Q S 70. Ashok Kumar, whose name has almost become a legend (P) in his private life (Q) the evergreeen hero of the silver screen (R) is essentiially a quiet and sedate person (S) The proper sequence should be (a) R Q S P (b) Q S R P (c) S P R Q (d) R P S Q 71. Each is slightly different (P) and should be treated (R)

as an individual isssue (Q) industrial nooise control problem (S)

The proper sequence should be (a) S P R Q (b) R Q S P (c) S Q R P (d) R P S Q 72. The generation

of polllution because (R)

74. Undoubtedly and that too by shhowcasing at its traditional best (P)

Indian hockey (Q)

winning the tournaament (R)

is an achievement (S)

The proper sequence should be (a) P S R Q (b) R Q P S (c) P Q R S (d) R S P Q 75. The banks using more technology (P)

have already changedd (Q)

and selling techniques (R)

the old ways of doing businesss (S)

The proper sequence should be (a) P S Q R (b) Q R P S (c) Q S P R (d) P R Q S 76. The new directors use new technology embellish their productioon (P) improve on their acting and (Q) continuoously hone their skills (R) are rewritting reecord bookds (S)

within the environement is regarted as a form m (P)

The proper sequence should be (a) P S R Q (b) Q P S R (c) P S Q R (d) Q P R S

it lowers the quality of life (Q) of unreasonable noise (S)

The proper sequence should be (a) S P Q R (b) P R S Q (c) S P R Q (d) P R Q S 73. In anticipation Kazakhastan oficials have studied the (P) of the oil bonanza (Q) experience of other counttries that suddenly (R) found themselves richh in natural resources (S)

The proper sequence should be (a) S R Q P (b) P Q R S (c) S Q R P (d) P R Q S 77. In New York or a man visiting a playground without a chiild (P) carrying an open bottle of water on a bus (Q) are fineable offences (R) which is beelieved to be cash-strapped (S) The proper sequence should be (a) S Q R P (b) Q P R S (c) S Q P R (d) Q S P R

Jumbled Words in a Sentence ▌ 2.53 78. But people affected by the raising which testiffy to the fact that of the dam height to 90 metre the rehabilitation of (P) (Q)) has not been completed there are innumerrable cases (R) (S) The proper sequence should be (a) S Q P R (b) P R Q S (c) S R Q P (d) P Q R S 79. Characteristic proposed by both sides (P)

was the very hesitaant progress (Q)

of the confidence building measures (R)

of the cautious approach (S)

The proper sequence should be (a) S P R Q (b) R Q S P (c) S Q R P (d) R P S Q 80. The battle of all call centres outside the US (P) of comppromise of personal information by the prolliferation (Q)

The proper sequence should be (a) Q R S P (b) S P Q R (c) Q P S R (d) D R Q P 81. The Portuguese, as also for choosing thhe right place to build (Q) who arrived in Goa 400 years ago. (S)

Which one of the following is the correct sequence? (a) R – P – S – Q (b) S – Q – R – P (c) R – Q – S – P (d) S – P – R – Q 82. In life some rules are (P)

prehisttoric times the phases (Q) formed the basis of the earliest calendars (R) of the Moon haave been (S) Which one of the following is the correct sequence? (a) Q R P S (b) P S Q R (c) Q S P R (d) P R Q S 84. Today, in the country (P)

offers the best connectivitty, both (Q)

Kerala, among all States (R)

in teerms of telecom and datacom (S)

Which one of the following is the correct sequence? (a) R – P – Q – S (b) S – Q – P – R (c) R – Q – P – S (d) S – P – Q – R

the parties to a joint venture agreeement (Q) the chances of success (R) should joiintly prepare a detailed business plan (S) Which one of the following is the correct sequence? (a) P – Q – S – R (b) R – S – Q – P (c) R – Q – S – P (d) P – S – Q – R 86. In recent years, primarily because purchasing managers now (P)) has grown more intense (Q) exert much more influence over suppliers (R)

as in business (Q)

they seem almost instinctive (R)

used as a measure of time and have (P)

even before signing the joint venture agreem ment (P)

had taken a further twist with. (S)

for building housees (R)

83. Right from

85. To maximize

critics now raising the issue (R)

had a great talent (P)

Which one of the following is the correct sequence? (a) R S P Q (b) Q P S R (c) R P S Q (d) Q S P R

learnt so early that (S)

price competitioon in most industries (S)

2.54

▌ English Which one of the following is the correct sequence? (a) P – R – S – Q (b) S – Q – P – R (c) P – Q – S – R (d) S – R – P – Q

87. Another feature of nature where even (P)

juniors team functioningg beside (Q)

91. Making had formed from the receding tide (P) I noticeed a man fishing in (Q)

are encouraged to

cordiality_among seniors,

may way to a favouritte fishing spot (R)

speak their minnds (R)

is its democcratic (S)

a very small pool of wateer that (S)

Which one of the following is the correct sequence? (a) P – S – Q – R (b) Q – R – P – S (c) P – R – Q – S (d) Q – S – P – R 88. Affluent families in return for doing some small job (P) monetary help and helped R) (R

them coome up in life (Q)

took in those in need of (S)

Which one of the following is the correct sequence? (a) S – R – Q – P (b) P – Q – R – S (c) S – Q – R – P (d) P – Q – R – S 89. Summer vacation that their children learn something worth whhile (P) who are able to provide services forr anxious parents (Q)

has become a big business for those (S) Which one of the following is the correct sequence? (a) P – R – Q – S (b) S – Q – R – P (c) P – Q – R – S (d) S – R – Q – P 90. Each culture

draws strength from its (R)

92. Around the world painful terminal diseases (P)

the question of humane death (Q)

people were wrestling with (R)

especially in the face of (S)

Which one of the following is the correct sequence? (a) R – S – Q – P (b) P – Q – S – R (c) R – Q – S – P (d) P – S – Q – R 93. For thousands of years one of the most important fruits the strannge fruit of this bush (P) (Q) of the Mediterraneean region has been regarded as (R) (S) Which one of the following is the correct sequence? (a) P – Q – R – S (b) Q – S – P – R (c) P – S – Q – R (d) Q – R – P – S 94. Everyone knows that

who want to make sure R) (R

flourishes when it comes (P)

Which one of the following is the correct sequence? (a) S – P – Q – R (b) R – Q – S – P (c) S – Q – R – P (d) R – P – S – Q

own tradition andd yet (Q) into conntact with others (S)

Which one of the following is the correct sequence? (a) R – Q – P – S (b) S – P – Q – R (c) R – P – Q – S (d) S – Q – P – R

it is difficult (P)

to speak more politely (Q)

to go the manager (R)

and tell him that he ouught (S)

Which one of the following is the correct sequence? (a) Q – S – R – P (b) P – R – S – Q (c) Q – R – S – P (d) P – S – R – Q 95. The Great Powers are engaged in spite of common knowledge (P) in arming theemselves to the teeth (Q) total destruction of the human race (R) that another war will mean (S)

Jumbled Words in a Sentence ▌ 2.55 Which one of the following is the correct sequence? (a) Q – R – S – P (b) S – P – Q – R (c) Q – P – S – R (d) S – R – Q – P 96. It is a process of global production in so far as itt (P) well accepted fact that (Q)

98. Strong the making to curb the in various governmeent departments (P) (Q) paractices of nepotism measures are in (S) (R)) The correct sequence should be (a) RPSQ (b) SQRP (c) RQSP (d) SPRQ 99. The

brings togethher the various factors of production (R)

pulcked up courage to (P)

reflect global realiities (Q)

intternational transportation is crucial to thhe whole (S)

government has at last (R)

increase domestic fuel prices (S)

The correct sequence should be (a) QSPR (b) PRQS (c) QRPS (d) PSQR

The correct sequence should be (a) RPSQ (b) QPSR (c) RPSQ (d) QSPR

97. With

100. The

the new Bill in the Parliament, (P) the passage of (R)

exports arre expected (Q)

to get a filllip (S)

The correct sequence should be (a) QSRP (b) RPQS (c) QPRS (d) RSQP

spureme excellence (P) having to fight them (R)

of your enemies withoutt even (Q) is to subduue the armies (S)

The correct sequence should be (a) PSQR (b) QRPS (c) PRQS (d) QSPR

Practice eXerciSe 1. Those granted upon easier terms commonly spend life in one pursuit (P) (Q) for exvellence is nott often who attain any excellence (R) (S) The correct sequence should be (a) RPSQ (b) SQRP (c) RQSP (d) SPRQ 2. The bulk of people,

The correct sequence should be (a) SQRP (b) PRQS (c) SRQP (d) PQRS 3. Biotechnology describes, some specific properties (P) by means of which h life forms can be manipulated and altered at the most (Q)

for enforcement of law and order (P)

a number of powerful techniqques (R)

remain geenerally passive spectators (Q)

many of them only a few years old (S))

of the war beetween terrorists and the state organs (R) thhough peaceful and well intentioned (S)

The correct sequence should be (a) RQSP (b) PSQR (c) RSQP (d) PQSR

2.56

▌ English 8. Ironically,

4. These developments of which devotional music was the next to fllourish (P)

at global forums against the (P)

paved the way for the emergence Q) (Q

for a countryy that has been crying itself hoarse (Q)

marketed cassette based music (R)

Inddia apperar to be least prepared for a quotta less world (R)

of other varieties of commercially (S)

unfair quota regime imposedd on developing countries (S)

The correct sequence should be (a) RSQP (b) QPRS (c) RPQS (d) QSRP

The correct sequence should be (a) QPSR (b) SRQP (c) QRSP (d) SPQR 9. For a truly,

5. All the same,

customer-centred (P)

the kind of pressure it was (P)

even the competitorr

in the next foour years (Q) never subject to in its relatioons with the US (R) India is certain to come under (S) The correct sequence should be (a) QSPR (b) RPSQ (c) QPSR (d) RSPQ

greater than is acknowledged (P)

analyses have to be (R)

and macro-economic (S)

The correct sequence should be (a) PRSQ (b) QSRP (c) PSRQ (d) QRSP 10. We have even if he consistently underperforms (P)

6. Personal chemistry, with other coountries (Q)

plays a role in America's relatiions (R)

to a degree that is far (S)

The correct sequence should be (a) PSQR (b) RQSP (c) PQSR (d) RSQP 7. Bandwidth,

his place for more than 25 matches (Q) performannces ensure that a player (R) created a systeem where a few good (S) The correct sequence should be (a) SPQR (b) QRSP (c) SRQP (d) QPSR 11. Every today

of handheld devices come down (P) to a large extent on (R)

customer-based strategy Q) (Q

demand will also depend (Q)

how quickly prices (S)

The correct sequence should be (a) PRSQ (b) QSRP (c) PSRQ (d) QRSP

line to save the forests (P) in the firing (R)

it is the committted minority (Q)

of forest offficers who stand (S)

Which one of the following is the correct sequence? (a) PSRQ (b) QRSP (c) PRSQ (d) QSRP

Jumbled Words in a Sentence ▌ 2.57 16. It does seem funny

12. Our society has got used (P)

and is unable to (Q)

toolerate even

to sweepinng issues

statements of fact (R)

under the fact (S)

Which one of the following is the correct sequence? (a) QRPS (b) PSQR (c) QSPR (d) PRQS 13. Not only in India against the patriarchal order (P)

women had too struggle (Q)

that new laws on curbing indirect (Q)) influences of smoking are being formulatedd (R) have been effectively implemented (S) Which one of the following is the correct sequence? (a) QSPR (b) PRQS (c) QRPS (d) PSQR 17. The huge expenditure

to get into parliament in signiificant numbers (R)

when the existing ones to tackle public smokking (P)

but elsewhere in the worlld too (S)

Which one of the following is the correct sequence? (a) SPQR (b) RQPS (c) SQPR (d) RPQS 14. One of the central

into building state-of-the-art laboratoriess (P) one research and development has mostlyy gone (Q) but India is still lagging behind when it comes to innovation (R)

issues taken up by the G8 Summit (P)

buying new gadgetry, updating tools research (S)

In the coontext of climate change (Q)

Which one of the following is the correct sequence? (a) QPSR (b) RSPQ (c) QSPR (d) RPSQ

at Gleneagles iss the challenge of reducing green house gas emissions (R) and most contentious (S) Which one of the following is the correct sequence? (a) SPRQ (b) QRPS (c) SRPQ (d) QPRS 15. Technology helps unskilled to skilled work and thereby (P) to achieve a paradigm shift from (Q) move large numbers of the rural poor (R)

18. ISRO’s upcoming direct to most Indian homes (P) is expected too be the spur for beaming TV channels (Q) to be launched in mid-December (R) power-packedd satellite INSAT-4A (S) Which one of the following is the correct sequence? (a) PRQS (b) SQRP (c) PQRS (d) SRQP 19. There is

from the primaary to the secondary and tertiary

the regulatory framework laws increase acttivity in India (P) (Q)

sectors off the economic activity (S)

and services to help allso the need to improve (S) (R)

Which one of the following is the correct sequence? (a) SRPQ (b) QPRS (c) SPRQ (d) QRPS

Which one of the following is the correct sequence? (a) SPRQ (b) RQSP (c) SQRP (d) RPSQ

2.58

▌ English

20. The new technology of industrialised countries in scientific (P)) may also weaken the diminance (Q) and technoological innovations in favour of (R) the lesss wealthy but technology savvy nations (S) Which one of the following is the correct sequence? (a) RPQS (b) QSRP (c) RSQP (d) QPRS 21. Clearly system in India thrives on learning by rote (P) culture runs deep because the education (Q) and down plays questioning (R) lack of inttellectual property (S) Which one of the following is the correct sequence? (a) RPQS (b) SQPR (c) RQPS (d) SPQR 22. What is missing that has increasingly led to a drift of (P) stream to engineering and technology (Q) taleent from the pure science (R) though, is scieentific temper (S) Which one of the following is the correct sequence? (a) SRPQ (b) QPRS (c) SPRQ (d) QRPS 23. In the psychologists psychiatrists and social workeers (P) in marriage counselling in India (Q) trained in counselling act as marriage counnsellors (R) absence of any certification (S)

Which one of the following is the correct sequence? (a) PRSQ (b) SQPR (c) POSR (d) SRQP 24. When the year ago, the extent of damage it (P) Tsunami hit the coastline of Nagapattinam (Q) causedd to agricultural land was unimaginable (R) in Tamil Nadu more than a (S) Which one of the following is the correct sequence? (a) RSQP (b) QPRS (c) RPQS (d) QSPR 25. The fall in forest Department find hard to live down (P) the number of tigers in two key reserves (Q)) the State Government and its (R) in Rajasthaan is a scandal (S) Which one of the following is the correct sequence? (a) QSRP (b) PRSQ (c) QRSP (d) PSRQ 26. An apolitical of conviction, K.R. Narayanan has left andd no controversies (P) (Q) President with the couurage behind a virtuous life (R) (S) Which one of the following is the correct sequence? (a) RQSP (b) SPRQ (c) RPSQ (d) SQRP

Jumbled Words in a Sentence ▌ 2.59 27. Parents should which is making their children adults beforee their time (P) and will have repervussions later in life (Q)

(a) (b) (c) (d)

RSQP SRQP RSPQ SRPQ

31. A decade

availing various beauty trreatments at a tender age (R)

admits it has achieved only 30 percent of itts planned target. (P)

take steps to discourage the trend of children (S)

the Defence Research and Development Organization (Q)

(a) (b) (c) (d)

SQPR PRSQ SRPQ PQSR

after it sett an ambitious target of maximizing self-suufficiency (R) in defence by achiveing 70 perrcent indigenization (S)

28. Some frustrated that the bulk of career management advice (P)) or organization rather than finding a new one altogether (Q) focuses on doing better in your current field (R) community members have pointed out (S) (a) S P R Q (c) S P Q R

(b) P S Q R (d) P S R Q

29. This is a but written by someone who had privileged (P)) and exclusive insights into one of the (Q) personalized memoir partly in diary form (R)) most significant events in modern history (S) (a) S P Q R (c) S Q P R

(b) R Q P R (d) R P S Q

30. Private investments

(a) (b) (c) (d)

PSQR RQSP RSQP PQSR

32. Part of of the power generagted is lost as (P) corruptt politicians look the other way (Q) the probblem is that over a third (R) while ingeniouss industrialists steal it from under their noses (S) (a) (b) (c) (d)

PRQS RPSQ PRSQ RPQS

33. Why can’t fund and build one of (P) in an act of great philanthropy (Q) the leaders of our Silicon Valley come together and (R)

it is at the mercy of a lumbering decision making process (P)

the finest technnological museums in the

and no matter how nimble the private sector may be (Q)

world for our childdren to see (S)

realization thaat the government is the only buyer (R)

(a) (b) (c) (d)

howevver are taking place with eneasy (S)

RQPS QRPS RQSP QRSP

2.60

▌ English

34. It’s that to tell the government exactly how

(a) R S Q P (c) R S P Q

(b) S R P Q (d) S R Q P

38. Mobility of capital

much they earned in the year (P)

not only to seek low paid, informal wage empplyees

queue up in the swelttering heat (Q)

across national boundaries (P)

time of the year again (R)

and placce large number of workers in insecure

whenn taxpayers across the country (S) (a) Q P R S (c) Q S R P

(b) R S Q P (d) R P Q S

35. There’s no without ever fumbling or faltering (P) better juggler than a woman (Q) between her work, family and home (R) who has to strike a perfeect balance (S) (a) S Q P R (c) S Q R P

(b) Q S P R (d) Q S R P

36. When we can judgements and prejucides and come

but the threat of caapital flight can also serve to drive down wages (R) has given an unprecedented laveragee to companies (S) (a) (b) (c) (d)

QPRS SRPQ QRPS SPRQ

39. In the post independence era, and with the vanishing of land revenue as ann important source of revenue as an importaant source of revenue for the states (P) the IAS too, presented with opportunities in the economic developement sector (Q) began to regard tax collection and law and order

to a placee of peace inside (P) become more fully awaree of our inner battles (Q) we can easily let go of ego (R) and stop projecting them outsiide of ourselves (S) (a) Q S R P (c) Q P R S

and irregular employment (Q)

(b) R P Q S (d) R S Q P

37. The focus

jobbs as not being promising enough (R) from jobb satisfaction or carrer points of veiw (S) (a) (b) (c) (d)

QPRS PQSR QPSR PQRS

40. Many of but plants, and that we do not (P)

and subsequently expand our trade with the P) (P

eat meat siince that amounts to harming animals (Q)

outside and develop the co-operation (Q)

us humans pride ourselves (R)

iss to increase agricultural and industrical production (R)

by declearing thatt we eat nothing (S)

of our economic construction (S)

(a) R Q S P (c) R S P Q

(b) P S Q R (d) P Q S R

Jumbled Words in a Sentence ▌ 2.61 45. How strange that a refugee

41. It is difficult from major news channels without a deep sensse (P)

should fall in love (P)

or experience the image and comments Q) (Q

when he had got to Ameerica (Q)

not just of fear for the future, but of shame and embarassment (R)

with a girl less than half his age (R)

today to pick up a newspaper (S)

who had by the skin of his teeth escapedd death in Germany (S)

(a) Q S P R (c) Q S R P

(b) S Q R P (d) S Q P R

42. The old lady

46. I was

wanted to sell her precious table (P) having the capacity to pay a huge amount as its prrice (Q) to someone (R) with gold-coated legs and glossy surface (S) Which one of the following is the correct sequence? (a) Q S R P (b) P R S Q (c) Q R S P (d) P S R Q 43. Reuter reports that soldier fought each other in the streets of the Capital today (P)

just in time (Q)

with an egg in its mouth (R)

scurrying across the graass (S)

Which one of the following is the correct sequence? (a) S R Q P (b) Q P S R (c) S P Q R (d) Q R S P 47. From my plane flying over Namibia stretching 320 kilometers away (P) a shore of pink and silver sand (Q)

edged by a line of smokinng surf (S)

of armies loyal to rival political factions (R) following the weekend military (S) Which one of the following is the correct sequence? (a) R P S Q (b) Q S P R (c) R S P Q (d) Q P S R 44. I think it should be necessary for eggs as soon as they are laid (Q)

to be stamped with the date (R)

to see a mongoose (P)

I looked down at a haunting sight (R)

crackdown on demonstrators (Q)

by the poulterer (P)

Which one of the following is the correct sequence? (a) R Q P S (b) S P R Q (c) R P Q S (d) S Q P R

by the henn (S)

Which one of the following is the correct sequence? (a) Q S P R (b) S Q P R (c) S Q R P (d) Q S R P

Which one of the following is the correct sequence? (a) Q R P S (b) R Q S P (c) Q R S P (d) R Q P S 48. His favorite subject the secrets of life and of nature (P) because Science reveals (Q) happens to be Science (R)) which men did not know in the past (S) Which one of the following is the correct sequence? (a) P S R Q (b) R Q P S (c) P Q R S (d) R S P Q

2.62

▌ English

49. Physical training

50. Technology transfer

the character of (P)

from one country to another (P)

who are going to be respoonsible citizens (Q)

either thorughh a government policy (Q)

can shape well (R)

or via private channnels of communications (R)

young boys and girls (S)

implies the transsfer of technical knowledge (S)

Which one of the following is the correct sequence? (a) R P S Q (b) Q S P R (c) R S P Q (d) Q P S R

Which one of the following is the correct sequence? (a) S P R Q (b) P S Q R (c) S P Q R (d) P S R Q

anSWer Key 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

(d) (d) (b) (b) (a) (b) (c) (a) (a) (c) (c) (a) (b) (a) (b) (c) (b) (b) (d)

20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38.

(c) (c) (a) (d) (a) (c) (d) (c) (c) (b) (d) (a) (c) (b) (d) (b) (c) (d) (a)

39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57.

(d) (b) (d) (a) (b) (a) (c) (d) (a) (d) (c) (b) (d) (b) (c) (a) (b) (a) (d)

58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76.

(b) (a) (b) (b) (a) (a) (c) (b) (c) (c) (a) (c) (d) (a) (c) (d) (b) (c) (d)

77. 78. 79. 80. 81. 82. 83. 84. 85. 86. 87. 88. 89. 90. 91. 92. 93. 94. 95.

(b) (a) (c) (d) (d) (b) (c) (a) (c) (b) (d) (a) (b) (a) (d) (b) (c) (a) (c)

96. 97. 98. 99. 100. 101. 102. 103. 104. 105. 106. 107. 108. 109. 110. 111. 112. 113. 114.

(a) (b) (d) (c) (a) (b) (a) (c) (d) (a) (b) (d) (a) (b) (c) (d) (b) (c) (a)

115. 116. 117. 118. 119. 120. 121. 122. 123. 124. 125. 126. 127. 128. 129. 130. 131. 132. 133.

(b) (c) (a) (d) (a) (d) (b) (c) (b) (d) (a) (c) (c) (a) (d) (b) (c) (d) (a)

nda boX Directions: In the following items some parts of the sentence have been jumbled up. You are required to rearrange these parts which are labelled P, Q, R and S to make the correct sentence. Choose the proper sequence and mark in your answer sheet accordingly. 1. Although he’ tried to act normal as if of a trip to (P) Udaipur as his classmates were Q) (Q

Varun was as thrilled with the idea (R) he hadd been there a hundred of (S) The proper sequence should be (a) R S P Q (b) P Q R S (c) R Q P S (d) P S R Q

134. 135. 136. 137. 138. 139. 140. 141. 142. 143. 144. 145. 146. 147. 148. 149. 150.

(b) (d) (c) (b) (b) (a) (c) (d) (d) (a) (d) (b) (c) (d) (b) (a) (c)

Jumbled Words in a Sentence ▌ 2.63 2. With the US economy in limbo, makes

(a) S Q R P (c) S P R Q

makes it indispens (P) able to investors efficcient manufacturing base and (Q) burgeoning market the importance of China as a (R) low cost, and Japan and Europe stagnating. (S) The proper sequence should be (a) Q P R S (b) S R Q P (c) Q R P S (d) S Q R P 3. The capable and intelligent woman and appearing inpublic (P) in a man’s (R)

The proper sequence should be

further offended thhe conservatives (Q )

outfit by throwing away the veil. (S)

The proper sequence should be (a) Q S P R (b) R P S Q (c) Q P S R (d) R S P Q 4. Conscious of given them a stake in its own (P) prosperity have of its growing power, China is (Q ) deeper economic ties with team andties with teaam (R) and the trepidations most South East Asian countries (S) The proper sequence should be (a) P Q R S (b) S R Q P (c) P R Q S (d) S Q R P 5. To a considerable degree, of problems which (P) beset a person and to whhich the dream finds none (Q ) to which he knows no solution dreams are the result (R) of inner pressures which have found no relieff. (S)

(b) R P S Q (d) R Q S P

6. Normally for as long as four hours (P) becoming ever moore severe, (Q) but under certain conditions the storm may persist, (R) an individual thuunderstorm lasts about 45 minutes (S) The proper sequence should be (a) P Q R S (b) S R Q P (c) P R Q S (d) S Q R P Directions (For the following 7 items): In the following items some parts of the sentence have been jumbled up. You are required to rearrange these parts which are labeled P, Q, R and S to produce the correct sentence. Choose the proper sequence and mark in your answer sheet accordingly. 7. I am not an expert, and in the process, jeop pardized his inheritance of the great empirre (P) where Prince Salim’s love for a commonner (Q) so I do not know whether a moment like this had (R) m to revolt against his risen in history impelled him (S) The correct sequence should be: (a) S Q R P (b) R Q S P (e) S P R Q (d) R P S Q 8. Having been brought up at times without a whhimper (P) physical and mental bettering lyinng down (Q ) in a liberal society where I can always speak my mind (R) it baffles me how a woman could take (S) The correct sequence should be: (a) R S Q P (c) R Q S P

(b) P Q S R (d) P S Q R

2.64

▌ English

9. Beyond the India an India that is techno-savvvy, (P) of religious hate and riots, there is another India. (Q) and an emerging player in the global power structure (R) an India of untapped market. (S) The correct sequence should be: (a) P S Q R (b) Q R P S (c) P R Q S (d) Q S P R 10. First, the policy-makers large sums of money could be raised for the State coffers (P) wner ship, asssets from government control to private ow (Q ) economic efficiencies could be achieved and, not incidentally (R) were not convinced that by shifting (S) The correct sequence should be: (a) S P R Q (b) R Q S P (c) S Q R P (d) R P S Q 11. As transported me (P) the intensity of the mellody (Q) I stood there and listened (R) into another world (S) The correct sequence should be: (a) R P Q S (b) S Q P R (c) R Q P S (d) S P Q R 12. Non-resident Indians, and roads in their natiive villages in Andhra Pradesh (P) are spendiing crores of rupees (Q ) who feel duty-bound to give back to their home towns, (R) on impproving education and health facilities. (S))

The correct sequence should be: (a) R P S Q (c) P S Q R (b) R Q S P (d) S P Q R 13. No trip to Islamabad or Rawalpindi just 22 miles north off the twin cities (P) can be complete (Q) once the nerve-centre of the Buddhist learning (R) without paying a visit to Taxila. (S) The correct sequence should be: (a) R S Q P (b) Q P R S (c) R P Q S (d) Q S R P Directions: (For the following 9 items): In the following items, some parts of the sentence have been jumbled up. You are required to re-arrange these parts which are labelled P, Q, R and S to produce the correct sentence. Choose the proper sequence and mark in your Answer Sheet accordingly. 14. Polluted

on aqua life while Kerala rivers thhreatens the (P)

indigenous varieties (Q ) rivers are wreaking havoc (R) the introduction off exotic fish in the (S) Which one of the following is the correct sequence? (a) R-P-S-Q (b) Q-S-P-R (c) R-S-P-Q (d) Q-P-S-R 15. Typically despite the overwhelming evidence of thenegative role (P) or for the destructioon of natural habitats, (Q ) played by commercial logging and mining interests (R) it is local communities who are blamed for deforeestation (S) Which one of the following is the correct sequence? (a) P-R-S-Q (b) S-Q-P-R (c) P-Q-S-R (d) S-R-P-Q

Jumbled Words in a Sentence ▌ 2.65 16. If the idyllic and rehabilitation work beingg

20. It is possible between Finland and Sweden,

carried out across serval nations (P) only a cursory mention in the massive restoratioon (Q) island of Car Nicobar has found (R) part of the reason could be in accessibility of these islands (S)

whhich was an international accord (P) to find a solution to the Kashmir issue (Q ) settling a territorial dispute on the basis of staatus quo (R) on the lines of the agreement onn the Aland Islands (S)

Which one of the following is the correct sequence? (a) R-Q-P-S (b) S-P-Q-R (c) R-P-Q-S (d) S-Q-P-R

Which one of the following is the correct sequence? (a) Q-R-P-S (b) P-S-Q-R (c) Q-S-P-R (d) P-R-Q-S

17. The real founder and Humayun’s son Akbar the greatest ruler P) (P of the Mughal Empire was Babur’s grandsonn (Q ) Asoka a millennium and half earlier (R) to sit on an Indian throne after (S) Which one of the following is the correct sequence? (a) S-P-Q-R (b) Q-R-S-P (c) S-R-Q-P (d) Q-P-S-R 18. In the neighbourhood of the equator there is little need of clothhes or fire (P) stretched out on the bare groound beneath the shade of a tree (Q ) and no danger to health, to pass the live long dayy (R) and it is possible with perfect comfortt (S) Which one of the following is the correct sequence? (a) P-R-S-Q (b) Q-S-R-P (c) P-S-R-Q (d) Q-R-S-P 19. Hospitality to commendation on ths account P) (P is a virtue for which the natives of the east (Q ) and the people of Egypt are well entitled (R) in general are highly and deserveedly admired (S) Which one of the following is the correct sequence? (a) Q-S-R-P (b) P-R-S-Q (c) Q-R-S-P (d) P-S-R-Q

21. Once an Shanghai has metamorphosed into (P) in China’ss economic development strategy (Q ) a major international city which is a key element R) (R obscure fishing town (S) Which one of the following is the correct sequence? (a) R-P-S-Q (b) S-Q-R-P (c) R-Q-S-P (d) S-P-R-Q 22. When factors to the growth of biotechnology corne together (P) it heralds the arrival off a new venture (Q) and a government that is dedicated (R) like rich biodiversity, a cond ducive climate, a network of established ressearch institutes, educated and skilled mannpower (S) Which one of the following is the correct sequence? (a) Q-P-R-S (c) Q-R-P-S (b) S-R-P-Q (d) S-P-R-Q Directions (For the next 9 items): In the following items, some parts of the sentence have been jumbled up. You are required to re-arrange these parts which are labelled P, Q, R and S to produce the correct sentence. Choose the proper sequence and mark in your Answer Sheet accordingly. Example ‘Z’ has been solved for you. Z. It is well-known that the effect (P)

is very bad (Q )

on children (R)

of Cinema (S)

Which one of the following is the correct sequence? (a) P-S-R-Q (b) S-P-Q-R (c) S-R-P-Q (d) Q-S-R-P

2.66

▌ English

Explanation: The proper way of writing the sentence is “It is well-known that the effect of cinema on children is very bad”. This is indicated by the sequence P-S-R-Q and so (a) is the correct answer. 23. Many way to fuel growth (P) economists argue that (Q Q) and alleviate poverty (R) free trade is a magic bullet the quickest (S) Which one of the following is the correct sequence? (a) Q-P-S-R (b) R-S-P-Q (c) Q-S-P-R (d) R-P-S-Q 24. As a maestro appeared to be enjoying every bit off it (P) and followed every composition (Q ) the thunderous applause from (R) an appreciativve audience preceded (S) Which one of the following is the correct sequence? (a) P-Q-S-R (b) R-S-Q-P (c) P-S-Q-R (d) R-Q-S-P 25. Keeping farmers to smoke their fields during in view w (P) the prevailing weather conditions (Q ) agricultural experts have advised (R) the nightt to protect vegetables from cold (S) Which one of the following is the correct sequence? (a) S-R-P-Q (b) Q-P-R-S (c) S-P-R-Q (d) Q-R-P-S 26. It is stressful or joyful (P) with the belief in thee evanes-cence of life itself (Q ) nessessary to rise above the situations (R) and in the philosophical of the purpose of life (S)

Which one of the following is the correct sequence? (a) R-P-Q-S (b) Q-S-R-P (c) R-S-Q-P (d) Q-P-R-S 27. The difference and development on the other affects (P) in thhe relationship between death and birth ratees on the one hand (Q) but the age structure of the population (R) not just the rate of population growth (S) Which one of the following is the correct sequence? (a) S-R-Q-P (b) Q-P-S-R (c) S-P-Q-R (d) Q-R-S-P 28. Here another supposed discovery of 29-inch foot-pprints (P) claim to have found in Kerala (Q ) we go again with yet (R) which a group of amatteur anthropologists (S) Which one of the following is the correct sequence? (a) R-Q-S-P (b) S-P-R-Q (c) R-P-S-Q (d) S-Q-R-P 29. Creative world of reality writers and artists, throough their imagination (P) (Q ) transform the details of the into the world of (R) (S) Which one of the following is the correct sequence? (a) S-P-R-Q (b) Q-R-P-S (c) S-R-P-Q (d) Q-F-R-S 30. Thus against the state through the courts (P) of onne’s privacy against arbitrary intrusion byy the police (Q ) the court emphasized that the security is basic to a free (R) society annd enforceable (S)

Jumbled Words in a Sentence ▌ 2.67 Which one of the following is the correct sequence? (a) P-Q-S-R (c) P-S-Q-R

(b) R-S-Q-P (d) R-Q-S-P

Which one of the following is the correct sequence? (a) R P Q S (b) S Q P R (c) R Q P S (d) S P Q R 33. As things stand.

31. The preference responsibilities of looking after parents inn their old age (P)

But a majority still does not have access too English (P)

despite the fact that in a growing number of families (Q )

linguistic edge they are equippped with (Q)

at least in urban India, daughters are taking on the (R)

after globally because of the (R)

for the male child continues (S)

Indian professionals are much sought (S)

Which one of the following is the correct sequence? (a) S-Q-R-P (b) R-P-S-Q (c) S-P-R-Q (d) R-Q-S-P Directions (For the 8 items which follow): In the following items, some parts of the sentence have been jumpled up. You are required to rearrange there parts. Which are labelled P, Q, R and S to produce the correct sentence? Choose the proper sequence and mark in your Answer Sheet accordingly. Example ‘Z’ has been solved for you. Z. It is well-known that the effect (P)

is very bad (Q )

34. Which advocates of its provision with the (P) concern about a possible clash of some (Q ) of social reform have generally hailed the new 1egislation (R)

on children (R)

of Cinema (S)

Which one of the following is the correct sequence? (a) P S R Q (c) S R P Q

Which one of the following is the correct sequence? (a) R S P Q (b) S R P Q (c) R S Q P (d) S R P Q

(b) S P Q R (d) Q S R P

Explanation: The proper way of writing the sentence is “it is well-known that the effect of cinema on children is very bad” This is indicated by the sentence P S R Q and so (a) is the correct answer. 32. A diversified use as a heating or power generation fuel by connverting gas into (P) adding a new dimension to the traditional use of gas (Q) of natural gas is emerging (R) amongst other products,, high quality diesel transportation fuel viirtually free of sulphur (S)

religious and customary practices invoguunein the country (S) Which one of the following is the correct sequence? (a) R Q P S (b) Q R S P (c) R Q S P (d) Q R P S 35. Public interest Litigations as they are subjected to massive misuse (P) buut today they contribute to the backlog (Q ) were instituted as a means to help ordinaryy (R) people sidestep judicial delayed to seccure justice also called postcard petitions (S) Which one of the following is the correct sequence? (a) R S Q P (b) S R P Q (c) R S P Q (d) S R Q P

2.68

▌ English 38. With pressure

36. Among

to submit a report on the retreat of

the soldier's mindset from fighting (P)

glacierrs in Uttaranchal and also its impact (P)

the do octriner's other directives is the need to reorient (Q )

mouunting from every corner (Q )

namely terrorists hiding among civilians the enemy (R)

a committee comprising scientists, geologists and technicall experts (R)

to fighting his own people (S)

the state government has finallly constituted (S)

Which one of the following is the correct sequence? (a) Q R S P (b) S Q P R (c) S Q R P (d) Q S P R

Which one of the following is the correct sequence? (a) Q S P R (b) S Q R P (c) Q S R P (d) S Q P R 39. This could

37. Indeed

while out sea-right form an engine

on how to nurture young talent at the grassrroot level (P)

breakdown to a human problem (P) they shoulder great responsibility

as the power house of women's hockey (Q )

and have to take care of any eventuality (Q)

is an instructive lesson form Indian hockey bosses (R)

because while seafarers do look forward to some fun on the decks (R)

the non-descript town''s emergence (S)

not be further from the truth (S)

Which one of the following is the correct sequence? (a) Q S R P (b) S Q P R (c) S Q R P (d) Q S P R

Which one of the following is the correct sequence? (a) P R Q S (b) S Q R P (c) P Q R S (d) S R Q P

anSWer Key 1. 2. 3. 4. 5.

(a) (a) (d) (a) (d)

6. 7. 8. 9. 10.

(c) (b) (a) (d) (c)

11. 12. 13. 14. 15.

(c) (b) (d) (a) (b)

16. 17. 18. 19. 20.

(a) (d) (c) (a) (c)

21. 22. 23. 24. 25.

(d) (b) (c) (b) (d)

26. 27. 28. 29. 30.

(a) (b) (c) (b) (d)

31. 32. 33. 34. 35.

(a) (a) (b) (a) (d)

36. 37. 38. 39.

(b) (c) (c) (d)

C

H

6

A

P

T

E

R

Jumbled Sentences in a Paragraph Format In the following questions, the first and the last sentences of the passage are in order and numbered 1 and 6. The rest of the passage is split into four parts and named P, Q, R, and S. These four parts are not given in their proper order, as they should appear in the paragraph. Read the passage and find out which of the four combinations (marked a–d) correctly forms the sequence of sentences in the passage: (1) S1: There was once a hardworking and poor, but well contended, farm worker. P: The farm worker buried the silver coins under the ground. Q: He used to work all day in his farm and enjoy the gifts of nature. R: The village Sarpanch gave him a bag containing some silver coins to keep. S: His rich Sarpanch took pity upon the farm-worker. S6 : From that day the poor farm-worker’s happiness vanished. (a) SQPR (b) QSRP (c) RQPS (d) QSRP (2) S1: Anything you do that gives you some form of mental relaxation and peace will benefit your heart and body. P: In fact, never choose an activity just because it is good for you.

anSwer Key 1. (d)

2. (b)

3. (d)

Q: By all means do some relaxing exercises, and even tire yourself to some extent perhaps. R: If you can combine these with enjoyment as a routine then so much the better. S: You will soon get bored of it. S6: But do not indulge in very hard exercises; tune into the warning signals from your body. (a) QSRP (b) PRSQ (c) QRPS (d) PSRQ (3) S1: His son fell into the bad company of village boys. P: The father wanted to wean his son from the bad company. Q: A few days later all the bananas got spoiled. R: His father was much worried about his son’s bad company. S: To give a lesson to the son, his father gave him a few good bananas along with a rotten one. S6: The boy understood the lesson his father wanted to teach him. (a) PSRQ (b) SQPR (c) RPQS (d) RPSQ

2.70

▌ English

PraCtICe eXerCISe Directions: In the following items some parts of the sentence have been jumbled up. You are required to rearrange these parts which are labelled P, Q, R and S to produce the correct sentence. Choose the correct sequence. 1. Even if he consistently underperforms (P)/his place for more than 25 matches(Q)/performances ensure that a player can retain(R)/we have created a system where a few good(S). (a) S P Q R (b) Q R S P (c) S R Q P (d) P S R P 2. For a truly customer-centered(P)/and customer-based strategy,(Q)/analysis of the competitor(R)/has to be made(S) (a) P R S Q (b) P Q R S (c) P S R P (d) Q R S P 3. Ironically, at global forums against the(P)/for a country that has been crying itself hoarse(Q)/India appears to be least prepared for a quota-less world(R)/unfair quota regime imposed on developing countries,(S) (a) Q P S R (b) S R Q P (c) Q R S P (d) S P Q R

9. The bulk of people, for enforcement of law and order(P)/ remain generally passive spectators(Q)/of the war between terrorists and the state organs(R)/through peaceful and wellintentioned(S) (a) S Q R P (b) P R Q S (c) S R Q P (d) P Q R S 10. Granted upon easier terms(P)/com monly spend life in pursuit,(Q)/for excellence is not often(R)/who attain any excellence(S) (a) R P S Q (b) S Q R P (c) R Q S P (d) S P R Q 11. The supreme excellence(P)/of your enemies without even(Q)/ having to fight them(R)/is to subdue the armies(S) (a) P S Q R (b) Q R P S (c) P R Q S (d) Q S P R 12. Plucked up courage to(P)/to reflect global realities(Q)/ government has at last(R)/increase domestic fuel prices(S) (a) R S P Q (b) Q P S R (c) R P S Q (d) Q S P R

4. Bandwidth of handled devices come down(P)/demand will also depend(Q)/to a large extent on(R)/how quickly prices of(S) (a) P R S Q (b) Q S R P (c) P S R Q (d) Q R S P

13. The making to curb the(P)/in various government departments(Q)/practices of nepotism(R)/measures are in(S) (a) R P S Q (b) S Q R P (c) R Q S P (d) S P R Q

5. Personal chemistry(P)/plays a role in America’s relations to a degree(Q)/with other countries(R)/that is far greater than in acknowledged(S) (a) P S Q R (b) P R Q S (c) P Q S R (d) R S Q P

14. With the new bill in the parliament,(P)/exports are expected(Q)/ the passage of(R)/to get a fillip(S) (a) Q S R P (b) R Q S P (c) Q P R S (d) R S Q P

6. All the same, the kind of pressure it was(P)/in the next four years(Q)/never subject to in relations with the US(R)/India is certain to come under(S) (a) Q S P R (b) R P S Q (c) Q P S R (d) R S P Q 7. These developments of which devotional music was the next to flourish(P)/paved the way for the emergence(Q)/ marketed cassette-based music(R)/of other varieties of commercially(S) (a) R S Q P (b) P Q S R (c) R P Q S (d) Q S R P 8. Biotechnology describes basic level so that they develop some specific properties(P)/by means of which life-forms can be manipulated and altered at the most(Q)/a number of powerful techniques, (R)/many of them only a few years old(S) (a) R Q S P (b) P S Q R (c) R S Q P (d) P Q S R

15. It is a proves of global production in so far as it(P)/wellaccepted fact that(Q)/ brings together the various factors of production(R)/international transportation is crucial to the whole(S) (a) Q S P R (b) P R Q S (c) Q R P S (d) P S Q R 16. Polluted on aqua life while(P)/Kerala rivers threatens the indigenous varieties(Q)/rivers are wreaking havoc(R)/the introduction of exotic fish in the(S) (a) R P S Q (b) Q S P R (c) R S P Q (d) Q P S R 17. Typically despite the overwhelming evidence of the negative role(P)/or for the destruction of natural habitats,(Q)/played by commercial logging and mining interests(R)/it is local communities who are blamed for deforestation(S) (a) P R S Q (b) S Q P R (c) P Q S R (d) S R P Q

Jumbled Sentences in a Paragraph

▌ 2.71

18. If the idyllic and rehabilitation work being carried out across several nations(P)/only a cursory mention in the massive restoration(Q)/island of Car Nicobar has found(R)/part of the reason could be inaccessibility of these islands(S) (a) R Q P S (b) S P Q R (c) R P Q S (d) S Q P R

27. That so suggests that there are certain cultural resources,(P)/ many Indians excel as expatriates(Q)/that enable them to succeed while living away from home(R)/perhaps unique to Indians,(S). (a) R Q P S (b) Q R P S (c) Q P R S (d) R P S Q

19. Humayun’s son Akbar,(P)/the real founder and (Q)/was Babur’s grandson(R)/the greatest ruler of the Mughal Empire(S) (a) S P Q R (b) Q R S P (c) S R Q P (d) P Q S R

28. We entertained and informed ourselves (P)/the advent of satellite television was a silent(Q)/and completely changed the way(R)/revolution which created a whole new industry(S) (a) Q R S P (b) P S R Q (c) Q S R P (d) Q P S R

20. In the neighbourhood of the equator, there is little need of clothes or fire(P)/stretched out on the bare ground beneath the shade of a tree(Q)/and no danger to health, to pass the lifelong day(R)/and it is possible with perfect comfort(S) (a) P R S Q (b) Q S R P (c) P S R Q (d) Q R S P 21. Hospitality to commendation on this account(P)/is a virtue for which the natives of the east(Q)/and the people of Egypt are well entitled(R)/in general are highly and deservedly admired(S) (a) P Q S R (b) P R S Q (c) Q R S P (d) P S R Q 22. It is possible between Finland and Sweden, which was an international accord(P)/to find a solution to the Kashmir issue(Q)/setting a territorial dispute on the basis of status quo(R)/on the lines of the agreement on the Aland Islands(S) (a) Q R P S (b) P S Q R (c) Q S P R (d) P R Q S 23. Shanghai has metamorphosed into(P)/in China’s economic development strategy(Q)/a major international city which is a key element(R)/once an obscure fishing town,(S) (a) R P S Q (b) S Q R P (c) R Q S P (d) S P R Q 24. When factors to the growth of biotechnol ogy come together(P)/it heralds the arrival of a new venture(Q)/and a government that is dedicated(R)/like rich biodiversity, a conducive climate, a network of established research institutes, educated and skilled manpower(S). (a) Q P R S (b) S R P Q (c) Q R P S (d) S P R Q 25. During the cricket match(P)/excitement was visualized(Q)/ between India and Pakistan a wave of(R)/even before the first ball was bowled(S). (a) P R Q S (b) Q S P R (c) P S Q R (d) Q R P S 26. NASA fact-finding mission on microgravity in 1997.(P)/by becoming the fist Indian women to rocket into space on a(Q)/to reach out for the stars, created history(R)/Kalpana Chawla, a Karnal girl who wanted (S). (a) S R Q P (b) P Q R S (c) S Q R P (d) P R Q S

29. The aspirations of the Scheduled Castes and the Scheduled Tribes(P)/with political parties, industry and other bodies(Q)/the Government would initiate a dialogue(R)/on how best the private sector could fulfill. (a) S P R Q (b) R Q S P (c) S Q R P (d) R P S Q Directions: In the following items, some parts of the sentence have been jumbled up. You are required to rearrange these parts which are labelled P, Q, R and S to produce the correct sentence. Choose the correct sequence. 30. The United States, to sell Uranium, nuclear turbines, fighter P planes to India is now offering

which denied even

Q helicopter nuts and screws and even Patriot missiles R

S

(a) R Q S P (c) Q R S P

(b) Q R P S (d) R Q P S

31. For the what it was when they were just a cluster of districts P inhabitants of these fledgling the quality of life remains pretty Q

R

much but resource-rich States of India (a) Q S P R (c) Q S R P

S

(b) S Q R P (d) S Q P R

32. Some beautiful woman commands places attract courageous P

Q

men from faraway lands in much the same manner R an immediate obeisance S (a) Q R P S (c) Q R S P

(b) R Q S P (d) R Q P S

2.72

▌ English

33. Today cinema do not demand that the logic that the themes of P

(a) Q S R P (c) P S R Q

Q

(b) P R S Q (d) Q R S P

37. Reduction in economic growth of be the result of the concerned

contemporary the lack of grandeur in the music with

(P)

R

state alone poverty may or may not

music directors often explain S (a) S Q R P (c) S R Q P

(R) (a) S P Q R (c) S Q P R

(b) P Q R S (d) P R Q S

34. Apparently, in response to stress, heart rate goes up, in response P

(S)

(b) R Q P S (d) R P Q S

38. Even young in feel Delhi continues to remain though it is P

Q

Q

among the oldest capitals in the world and modern in thought

to things and hormones and humans show a range of

R (a) P Q R S (c) P S R Q

R psychological signs. S (a) S Q P R (c) S Q R P

P

Q

agreements, especially the nuclear pact meant to transform

35. Drug abuse dies down soon after that follows every expose P

S (b) R S P Q (d) R Q P S

39. The recent relations between the two countries Indo-US

(b) P Q R S (d) Q S P R

R

Q

have been the subject of much controversy

but the hue and cry is an increasing vice among R

(a) R Q S P (c) R Q P S

party-hopping elites S (a) Q P S R (c) Q R S P

(Q)

S

(b) Q R P S (d) Q R S P

40. What religion for money or marry meaning does social freedom

(b) S R Q P (d) S P Q R

P

Q

their girl child off to the highest bidder hold when people

36. The rebellion standing up for their rights may be slow and

R are tempted to change

sporadic youngsters become the unlikely heroes but it is

S (a) R S P Q (c) R P S Q

happening as

(b) Q P S R (d) Q S P R

anSwer Key 1. 2. 3. 4. 5.

(c) (b) (a) (d) (b)

6. 7. 8. 9. 10.

(a) (b) (c) (a) (b)

11. 12. 13. 14. 15.

(a) (c) (d) (b) (b)

16. 17. 18. 19. 20.

(a) (b) (a) (d) (c)

21. 22. 23. 24. 25.

(a) (c) (d) (b) (a)

26. 27. 28. 29. 30.

(a) (d) (c) (b) (d)

31. 32. 33. 34. 35.

(c) (a) (c) (b) (b)

36. 37. 38. 39. 40.

(a) (c) (d) (b) (d)

Jumbled Sentences in a Paragraph ▌ 2.73

nDa BoX Directions: In the following items each passage consists of six sentences. The first and the sixth sentence are given in the beginning. The middle four sentences in each have been removed and jumbled up. These are labelled P, Q, R and S. You are required to find out the proper sequence of these four sentences and mark accordingly on the answer sheet.

S: The darker side is that a lot of pornographic material and other unwanted information is being spread all over the world through Internet. The proper sequence should be (a) R S P Q (b) Q S P R (c) R P S Q (d) Q P S R

1. S1: Information technology has made information about any event or phenomenon available to any person anywhere in the world at any time. S6: In education, the contents of various courses can be prepared by a virtual bank of experts. C.173 P: In the case of formal education, quality of teaching material determines the quality of education. Q: Information is the basis of knowledge. R: In a conventional system, the teaching material is designed and developed by the expert resources available in a particular university. S: The knowledge acquisition process has, therefore, tremendously benefited with the advent of information technology. The proper sequence should be (a) Q P S R (b) R S P Q (c) Q S P R (d) R P S Q

4. S1: By a Royal Proclamation on October 16, 1905 Bengal Province was reduced in size and population and a new province called East Bengal was created. S6: The courage with which Bengal faced its trial stirred the whole of India. P: In these meetings Hindus and Muslims joined in the protests with equal zeal and earnestness. Q: The Indian National Congress unanimously condemned the partition. R: The scheme was condemned not only in India but also by the British press. S: In the two Bengals, over 2,000 public meetings were held. The proper sequence should be (a) S P R Q (b) R Q S P (c) S Q R P (d) R P S Q

2. S1: Colombia, the only nation named after Colum bus, situated in the north-west of South America, extends up to the Isthmus of Panama. S6: Bolivar united New Granada, as it was called, with Venezuela and Ecuador in the Greater Colombia Confederation in 1819. P: In 1819, Simon Bolivar, the first President broke the hold of Spain. Q: Colombia was discovered by Ojeda in 1499. R: It is the only South American country that borders on both the Atlantic and Pacific Oceans. S: It was once a part of the South American Spanish Empire. The proper sequence should be (a) R Q S P (b) S P R Q (c) R P S Q (d) S Q R P 3. S1: What do people use Internet for? S6: The conservatives feel that the impact of Internet on moral and cultural aspects of the society is devastating. P: It also provides entertainment in the form of music and videos. Q: E-mail, internet browsing, chatting with known and unknown people, collection of information and knowledge are the major uses. R: Parents are worried about the exposure their children get to such material.

Directions (For the following 6 items): In the following items each passage consists of six sentences. The first sentence (S1) and the sixth sentence (S6 ) are given in the beginning. The middle four sentences in each have been removed and jumbled up. These are labelled P, Q, R and S. You are required to find out the proper sequence of the four sentences and mark accordingly on the Answer Sheet. 5. S1: Nick sat in my classroom after school, fiveyears ago, wanting help on a research paper that stood between him and passing. S6: I could easily tell the difference between a parent’s writing style on homework and the student’s style on in-class essays. P: Not wanting the child to fail, and often desperately wanting him to succeed, some parents step in and take over. Q: An average student, he felt overmatched by the assignment. R: Many parents find themselves in the same position: A child feels lost with homework and turns to you for help. S: His green eyes pleaded with me, his English teacher, his hope. The correct sequence should be: (a) P R S Q (b) Q S R P (c) P S R Q (d) Q R S P 6. S1: Superstitions are blind and baseless beliefs. S6: Whenever these forces were angry, he tried to pacify them. P: In old times man had no knowledge of certain happenings such as earthquakes, floods, storms and cyclones.

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▌ English Q: They are born out of ignorance and fear of forces of nature. R: He regarded them as the doings of certain’hidden, mysterious and divine forces. S: Such happenings came to have a mysterious significance for him. The correct sequence should be: (a) Q P S R (b) S R Q P (c) Q R S P (d) S P Q R

7. S1: To drive under the influence of alcohol has been an offence in our country since 1926. S6: Every year nearly 10,000 fatal accidents occur because of drunken driving. P: It is a well-established fact that alcohol in blood stream affects the reflexes and impairs our judgement. Q: Why do we need such a law at all? R: As a result, a drunk driver is a danger at the wheel. S: In India, at least, drinking by drivers plays important part in accidents. The correct sequence should be: (a) Q S R P (b) R P Q S (c) Q P R S (d) R S Q P 8. S1: I was born in a Hindu family and I am a Hindu S6: Religions are like rivers, all of them following’ towards the ocean of God. S: No religion teaches us to follow evil ways. Q: Fortunately, my parents are not very orthodox and they encourage me to think for myself about the teachings of the Hindu religion. R: All religions have the same central purpose they teach man to obey God and to follow paths which will please Him. S: They have taught me to believe that, though we ourselves are Hindus, no religion in the world is to be thought of as being bad. The correct sequence should be: (a) P R S Q (b) Q S R P (c) P S R Q (d) Q R S P 9. S1: In other parts of the world, the best intellect of the country and the leading men of business are called to devise means for increasing the wealth of the country. S6: Can we not take to heart the lesson thus taught? P: But they have universal education and up-todate universities. Q: They are not rich in natural wealth. R: There is little distress or poverty in such small countries as Norway and Sweden. S: So the miracle has been accomplished by them through science by fully utilizing all the available resources of the country. The correct sequence should be: (a) P Q R S (b) R S P Q (c) P S R Q (d) R Q P S

10. S1 : Happiness is an inner state of mind. S6 : But my point is that it is not wealth but coordination of one’s thoughts and actions which removes inner conflicts. P : Some of the most miserable persons I have come across in my life are rich people. Q : Happiness has very little to do, for instance, with whether you are rich or not. R : It is little dependent on outside environment. S : It is true that poverty makes one miserable in a very acute way. The correct sequence should be: (a) S Q P R (b) R P Q S (c) R Q P S (d) S P Q R Directions (For the following 7 items): In the following items, each passage consists of six sentences. The first sentence (S1) and the sixth sentence (S6) are given in the beginning. The middle four sentences in each have been removed and jumbled up. These are labelled P, Q, R and S. You are required to find out the correct sequence of the four sentences and mark accordingly on the Answer Sheet. 11. S1: Now that I am getting old and stiff in the joints, I like to meditate, while grazing in the pasture, on my foal days. S6: When I was old enough, the trainer came and, to my great indignation, fastened a long rope to my head, and then began driving me round and round in circles with his long whip. P: I had no work to do, and could run about after my mother, who was a fi ne white Arab mare, without any restraint. Q: I think that was the happiest part of my life. R: But that could not last forever. S: Most of my time was spent in the fields, where I nibbled the tender grass and capered about, while my mother was steadily grazing. Which one of the following is the correct sequence? (a) S-R-Q-P (b) Q-P-S-R (c) Q-R-S-P (d) S-P-Q-R 12. S1: A stamp is, to many people, just a slip of paper that takes a letter from one town or country to another. S6: An album, a packet of hinges, a new supply of stamps, and the time passes swiftly and pleasantly. P: But they do not realise that there are many who do buy, many who find the effort worthwhile and many who, if they did no spend their time collecting stamps, would’ spend it less profitably. Q: They are unable to understand why we stampcollectors find so much pleasure in collecting them. R: To them it seems a waste of time, a waste of effort and a waste of money. S: We all seek something to do in our leisure hours and what better occupation is there to keep us out of mischief than that of collecting stamps?

Jumbled Sentences in a Paragraph ▌ 2.75 Which one of the fallowing is the correct sequence? (a) S-P-R-Q (c) S-R-P-Q (b) Q-P-R-S (d) Q-R-P-S 13. S1: The British wanted it to be their answer to the American White House. S6: In the end it earned him a place in the history books. P: Today, the Head of the Indian Republic occupies just a handful of the 340 rooms. Q: No wonder Edwin Lutyens didn’t mind that the 17-year assignment earned him just 5,000. R: But the 354-acre complex is an apt example of the expertise of a man who wanted the Rashtrapati Bhavan to be an object of admiration forever. S: Though the answer was good enough, the British didn’t anticipate that within 17 years of building the Viceroy House, they would have to leave the country and the architectural wonder would be renamed Rashtrapati Bhavan. Which one of the following is the correct sequence? (a) R-Q-S-P (c) S-Q-R-P (b) S-P-R-Q (d) R-P-S-Q 14. S1: And the biggest wonder about the Harappan cities is, simply put, their brick. S6: In other words, the brick was just like the platinum rod kept in Paris to define the world standard of time. P: Some archaeologists have ventured further. Q: If at all the size varied in some sites, it still retained the same length-breadth-height pro-portion everywhere! R: It had a standard size, be it in Mohenjodaro or Harappa, Kalibangan or Lothal. S: They say that even the constructions the houses and the cities were built in the same proportion as the brick. Which one of the following is the correct sequence? (a) R-Q-P-S (c) R-P-Q-S (b) S-P-Q-R (d) S-Q-P-R 15. S1: Freedom and power bring responsibility. S6: That future is not one of ease or resting but of incessant striving so that we may fulfil the pledges we have so often taken and the one we shall take today. P: Some of these pains continue even now. Q: Before the birth of freedom we have endured all the pains of labour and our hearts are heavy with the memory of this sorrow. R: evertheless, the past is over and it is the future that beckons to us now. S: That responsibility rests upon this assembly, a sovereign body representing sovereign people of India. Which one of the following is the correct sequence? (a) P-R-S-Q (c) P-Q-S-R (b) S-R-P-Q (d) S-Q-P-R 16. S1: The other day we heard someone smilingly refer to poets as dreamers. S6: Dreams are the sunrise streamers heralding a new day of scientific progress, another forward surge.

P: We must not be so superficial that we fail to discern the practicableness of dreams. Q: Where they differ from the logician and the scientist is in the temporal sense alone; they are ahead of their time, where logicians and scientists are abreast of their time. R: The truth is that poets are just as practical as people who build bridges or look into microscopes; and just as close to reality and truth. S: Now, it is accurate to refer to poets as dreamers, but it is not discerning to infer, as this person did, that the dreams of poets have no practical value beyond the realm of literary diversion. Which one of the following is the correct sequence? (a) S-P-Q-R (c) S-R-Q-P (b) Q-R-S-P (d) Q-P-S-R 17. S1: What was his great power over the mind and heart of man due to? S6: That truth made the service of the poor and the dispossessed the passion of his life, for where there is inequality and discrimination and suppression there is injustice and evil and untruth. P: That truth led him to proclaim without ceasing that good ends can never be attained by evil methods, that the end itself is distorted if the method pursued is bad. Q: Even we realize that his dominating passion was truth. R: That truth led him to fight evil and untruth wherever he found them, regardless of the consequences. S: That truth led him to confess publicly whenever he thought he had made a mistake-Himalayan errors he called some of his own mistakes. Which one of the following is the correct sequence? (a) S-R-Q- P (b) Q-P-S-R (c) Q-R-S-P (d) S-P-Q-R Directions (For the next 8 items): In the following items, each passage consists of six sentences. The first sentence (S1) and the sixth sentence (S6) are given in the beginning. The middle four sentences in each have been removed and jumbled up. These are labelled P, Q, Rand S. You are required to find out the correct sequence of the four sentences and mark accordingly on the Answer Sheet. Example ‘X’ has been solved for you. X. S1: There was a boy named Jack. S6: At last she turned him out of the house. P: So the mother asked him to find work. Q: They were very poor. R: He lived with his mother. S: But Jack refused to work. Which one of the following is the correct sequence? (a) R-Q-P-S (b) S-P-Q-R (c) R-P-Q-S (d) S-Q-P-R

2.76

▌ English

Explanation: The correct sequence in this example is R-Q-P-S which is marked by (a). Therefore, (a) is the correct answer. 18. S1: Political empowerment apart, the state should recognise the right to life of every citizen. S6: The state should prepare for the seasons ahead, not just winter, and create shelters on a war footing. P: Places of religious worship play an important role in this respect, providing food and shelter. Q: Where lives are at stake, resource crunch cannot work as an excuse R: It can still intervene to save lives, by ensuring that offices, schools and other institutions that function only during the day are used as night shelters. S: However, civil society organizations cannot be expected to compensate for state neglect. Which one of the following is the correct sequence? (a) S-Q-R-P (b) R-P-S-Q (c) S-P-R-Q (d) R-Q-S-P 19. S1: This is the age of knowledge. S6: Not surprisingly then that we Indians are often labeled as being overly superstitious. P: Logic and intuition are no longer enemies. Q: It is also the Age of Aquarius. R: The two live together comfortably in every Indian household. S: So why should superstitions be considered out of place in this age of nanotechnology and computers? Which one of the following is the correct sequence? (a) Q-P-S-R (b) R-S-P-Q (c) Q-S-P-R (d) R-P-S-Q 20. S1: Sachin has scored centuries against all oppositions, in all countries. S6: But largely, he has been the dictator, giving nightmares to quality bowlers of the world. P: There have been those rare occasions when he looked entangled at the crease. Q: In fact, some of his most memorable essays came in adverse conditions. R: Gavaskar called him “the closest thing to batting perfection this game has ever seen”. S: Sachin has established his stamp over all types of attack. Which one of the following is the correct sequence? (a) P-Q-S-R (b) R-Q-S-P (c) P-S-Q-R (d) R-S-Q-P 21. S1: Jagan has been working in our home for over 25 years. S6: Then came the question of raising his children on the paltry sum of money he earned washing cars and sweeping homes. P: And another. Q: Till finally his wife delivered a son and he declared his innings. R: I remember the day his first daughter was born. S: And then came another.

Which one of the following is the correct sequence? (a) R-P-S-Q (b) Q-S-P-R (c) R-S-P-Q (d) Q-P-S-R 22. S1 : Diabetes is a silent killer and because it does not seem as potent as cancer or AIDS is very often overlooked without too much heed by patients. S6: Though diabetes can never be totally cured it can be definitely be controlled from causing serious consequences. P: Worse, as many as a third of them don’t even know it. Q: Untreated diabetes can lead to heart and kidney failure, amputations and even death. R: Millions of people suffer from diabetes. S: The early symptoms of diabetes are often confused with other less grave conditions. Which one of the following is the correct sequence? (a) S-Q-R-P (b) R-P-S-Q (c) S-P-R-Q (d) R-Q-S-P 23. S1: Ships, built in dry docks, are launched amidst chanting of Atharva Veda. S6: The latest INS Mumbai is the tenth reincarnation of its original INS Bombay. P: Even now newer warships are given names of old decommissioned warships. Q: One of the enduring superstitions of the Navy is that old ships don’t die. R: In Europe, they break champagne bottles on the bow; in India we break coconuts. S: It derives from the animistic belief that a ship has a soul that lives on and that it is reborn. Which one of the following is the correct sequence? (a) R-Q-S-P (b) S-P-R-Q (c) R-P-S-Q (d) S-Q-R-P 24. S1: Jaswant Singh of 4 Garhwal Rifles was apparently relaxing at 10,000 feet when he spotted a whole battalion of Chinese troops advancing toward an Indian Army post. S6: Versions of the story vary. P: They surrounded him, captured the three an beheaded them. Q: Finally the enemy sent a scout party to ascertain real strength of the Indian defence, and they fond just a rifleman and two girls. R: After the war, the Chinese, impressed by the rifleman’s valour, gave his head back to the Indians, who set up a temple for him. S: With just one 303 and ammunition supplied by two girls from an abandoned dump, he mowed down about 50 enemy troopers. Which one of the following is the correct sequence? (a) P-Q-S-R (b) S-R-P-Q (c) P-R-S-Q (d) S-Q-P-R

Jumbled Sentences in a Paragraph ▌ 2.77 25. S1: Each creature of the sea has its own significance for seamen. S6: A dead dolphin is a bad omen. P: One reason could be that dolphins and sharks do not cohabit. Q: Sighting a dolphin is supposed to bring luck. R: A dolphin swimming with the ship brings good Luck S: Upon sighting the first dolphin on a voyage even the admiral will come up on deck to watch it. Which one of the following is the correct sequence? (a) S-R-Q-P (b) Q-P-S-R (c) S-P-Q-R (d) Q-R-S-P Directions (For the 8 items which follow): In the following items. each passage consists of six sentences. The first sentence (S1) and the final sentence (S6) are given in the beginning. The middle four sentences in each have been removed and jumped up. These are lebelled P, Q, R and S. You are required to find out the proper sequence of the four sentences and mark accordingly on the Answer Sheet. Example ‘X’ has been solved for you X. S1: There was a boy named Jack. S6: At last she turned him out of the house. P: So the mother asked him to fi nd work. Q: They were very poor. R: He lived with his mother. S: But Jack refused to work. Which one of the following is the correct sequence? (a) R Q P S (b) P Q R S (c) Q P R S (d) R P S Q Explanation: The correct sequence in the example is RQPS which is marked by (a). Therefore, (a) is the correct answer. 26. S1: Several times in history of the world, some countries and cities have attained a high degree of civilization. S6: For the barbarians are always liable to break in on you and with their greater numbers and rude vigour, scatter your civilization to the winds. P: One of the reasons why they did not last was that they were confined to a very few people. Q: Now it is no good being civilized if everybody around you is barbarous. R: Yet none of these civilizations, important as they were, have lasted. S: They were like little oases of civilization in deserts of barbarism. Which one of the following is the correct sequence? (a) P R Q S (b) R P S Q (c) P R S Q (d) R P Q S

27. S2: Savages believe firmly in magic. S6: Accordingly, much magic is practiced in the hope of curing diseases P: Some of these are supposed to make the crops grow or to bring rain. Q: Death is always supposed to be due to some sort of magic or witchcraft. R: A great deal of their lives is concerned with performing magical rites. S: In some Parts of Africa, even today, death is not attributed to any natural causes. Which one of the following is the correct Sequence? (a) S P R Q (b) R Q S P (c) S Q R P (d) R P S Q 28. S1: We drove first to the foot of the mountains. S6: It seemed at times that we were climbing almost perpendicularly. P: A water course rushed down the mountain beside us, and a bright hot sun burst through the clouds. Q: Then with one guide in front and the other behind, we set-off on foot along an uncompromising track. R: But after the first few minutes. I gave up looking at any of these splendid things. S: Instead, I kept my head resolutely downwards, concentrating upon the next step ahead. Which one of the following is the correct Sequence? (a) P Q R S (b) Q P S R (c) P Q R S (d) Q P R S 29. S1: If I could find him I might find a clue which would help me to discover what was at the bottom of the mysteries which surrounded us. S6: It would indeed be triumph for me if I could succeed where my master had failed. P: Holmes, my master, had missed him in London. Q: If this man were inside I should force him to reveal who he was and why he had pursued us so long. R: Then and there, I determined to explore every hut upon the moor until I found the right one. S: And if I found the right hut but without its occupant, I must wait there, howsoever long it might be till he returned. Which one of the following is the correct sequence? (a) S P R Q (b) R Q S P (c) S Q R P (d) R P S Q 30. S1: The man who is described as “the first truly modern scientist” was an Italian astronomer. Galileo Galilei, who was born in Pisa, in the year 1564. S6: Galileo saw, too, that all the plants, including the earth travel round the sun.

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▌ English P: On the sun he saw dark spots that moved slowly across its surface; and this proved that the sun, like the earth, spin round itself. Q: With these he studied the heavens, and there he saw many wonderful things that men had never seen before R: He studied at Pisa University, and became a Professor of Mathematics there, and afterwards at Padua. S: Galileo made many important scientific discoveries, and from the year 1609 onwards, he was the maker of the first good telescopes. Which one of the following is the correct sequence? (a) R S P Q (b) P Q S R (c) R Q S P (d) P S Q R

32. S1: While English must be actively promoted, it would be foolhardy to let regional languages fall off the radar. S6: In the 21st century market place, languages are the new bargaining chips. P: To be able to integrate socially and excel in great Indian marketplace, it makes good sense to be conversant with regional languages other than just one’s mother tongue. Q: Indeed, most urban Indians are bilingual and smattering of at least a third language. R: A willingness to adopt and adapt to another language will enrich the individual. S: Migration to where opportunity beckons is order of the day and Indians are constantly shifting base within the country. Which one of the following is the correct sequence? (a) R P S Q (b) Q S P R (c) R S P Q (d) Q P S R

31. S1: As soon as we loosened the string the lid of the basket flew into the air. S2: The boys stood up on their desks and yelled hysterically. P: The teacher pushed himself back on the chair and fell over. Q: With a loud hiss he attacked the teacher’s face. R: There is black Cobra, whose eye burnt like redhot coal and hood was tight and undamaged. S: He landed on the floor and stared at the Cobra, paralysed with fear. Which one of the following is the correct sequence? (a) R S P Q (b) P Q R S (c) R Q P S (d) P S R Q

33. S1: It takes years to build trust. S6: The company take periodic litigation and nonlitigation actions against errant resellers who indulge in the pirated or unlicensed software. P: After all, your customers come back to you because they trust you to give them the best. Q: And just one wrong to undo it. R: They follow your advice as you have consistently showcased true quality in every aspect of your relationship. S: Our company respects this and is working towards ensuring that only its genuine software is available to customers. Which one of the following is the correct sequence? (a) Q S R P (b) R P Q S (c) Q P R S (d) R S Q P

anSwer Key 1. 2. 3. 4. 5.

(c) (d) (d) (a) (b)

6. 7. 8. 9. 10.

(a) (c) (b) (d) (c)

11. 12. 13. 14. 15.

(b) (d) (b) (a) (d)

16. 17. 18. 19. 20.

(c) (b) (b) (a) (d)

21. 22. 23. 24. 25.

(c) (b) (a) (d) (b)

26. 27. 28. 29. 30.

(b) (d) (c) (b) (c)

31. (c) 32. (a) 33. (a)

C

H

7

A

P

T

E

R

Sentence Improvement Format oF the QueStIonS In certain competitive examinations, there are questions in which the candidate is required to improve the given sentence grammatically, using the responses given under each sentence. In the following questions each sentence is given three possible solutions for the bold part. If one of them (a), (b) or (c)—fits better than the bold part, indicate your response on the answer sheet against the corresponding alphabet (a), (b) or (c). If none of the substitutions improve the sentence, indicate, as your response on the answer sheet (d) that is “no improvement” or “no correction required”. 1. But man very soon found that it was much better to divide the work among themselves so that one had built houses, another made chairs and tables, and yet another grew corn.

(a) (b) (c) (d)

some people built houses some people built a house one built houses no improvement

2. A hundred rupee are all that he wants. (a) were (b) was (c) is (d) no improvement 3. I had hardly reached there when he arrived. (a) Hardly had I (b) Hardly I had (c) I have hardly (d) no improvement

anSwer Key 1. (c)

2. (c)

3. (a)

Sentence Improvement Direction: In each of the statements below, a part of the sentence is highlighted. Out of the four options which follow, choose the one which best replaces the highlighted phrase. 1. Such people never have and never will be trusted. (a) never have and never will be trust (b) never have and will be trusted (c) never have been trusted and never will be trusted (d) never have had anyone trust them and never will have anyone

2. At the end of the play about women’s liberation, the leading lady cautioned the audience not to judge womanhood by the way she dresses. (a) she dresses (b) she dressed (c) it dresses (d) they dressed 3. As no one knows the truth as fully as him, no one but him can provide the testimony. (a) as fully as his, no one but him (b) as fully as he, no one but him

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▌ English (c) as fully as he does, not one but he (d) as fully as he does, no one but be alone

4. The use of radar as well as two-way radio paging make it possible for state police to intercept most speeders. (a) make it possible (b) makes it possible (c) allows the possibility (d) makes possible 5. No sooner had he entered the room when the lights went out and everybody began to scream. (a) when the lights went out (b) than the lights went out (c) and the lights went out (d) then the lights went out 6. The alarming report of the building collapse had everyone spellbound. (a) alarmed report (b) reporting alarm (c) reported alarm (d) no improvement 7. Yogic exercise seem to be help the urban population deal effectively with stress. (a) seems to be helpful (b) seems to be helped (c) seems to help (d) no improvement 8. The pedestrians must have to be very cautious while crossing the road. (a) must had been (b) ought be (c) are required to be (d) no improvement 9. I am sure that he has recovered from his illness and he will accompany us to the picnic spot. (a) and he will company (b) and that he will accompany (c) but he will accompany (d) no improvement 10. They feel very proudly that their team had won the match. (a) feels very proudly (b) felt very pride (c) felt very proud (d) no improvement 11. Your good gesture will highly appreciate. (a) will highly appreciated (b) will be highly appreciative (c) will be highly appreciated (d) no improvement

14. The small child does whatever his father was done. (a) was done (b) did (c) had done (d) no improvement 15. The moment they saw me, they were delight. (a) was delight (b) were delighted (c) are delighted (d) no improvement 16. If he has to spend five hours in the queue, it was really a wastage. (a) was real a wastage (b) is real a wastage (c) is really a wastage (d) no improvement 17. You can always dependent on them. (a) depend on them (b) be dependent with them (c) depend them (d) no improvement 18. Why did you not threw the bag away ? (a) did you threw (b) had you not threw (c) did you not throw (d) no improvement 19. They are not beware of all the facts. (a) are not beware (b) are not aware of (c) are not to be aware (d) no improvement 20. Can you tell me why did you not speak the truth ? (a) why did not you speak (b) that why did you not speak (c) why you did not speak (d) no improvement 21. This is one of the most important inventions of this century. (a) invention of this century (b) invention of these centuries (c) inventions of centuries (d) no improvement 22.

(a) (b) (c) (d)

that why he has not that why had he not why he had not no improvement

23. He hesitated to listen to what his brother was saying. (a) hesists to listened to (b) listened to hesitate (c) hesitates to listening (d) no improvement

12. Speculations and hypothesizing are the most essential and well-known aspects of inventions. (a) Hypothesized (b) hypothesizing needs (c) hypothesis (d) no improvement

24. Though we have kept in mind to try and maintain most facilities, we would like to request you to kindly bear with us incase of any inconvenience (a) had kept in mind to try and maintain (b) must keep in mind to try and maintain (c) would keep in mind to try and maintain (d) no improvement

13. Increased productivity necessary reflects greater efforts made by the employees. (a) Increase productivity necessarily (b) Increased productivity is necessary (c) Increased productivity necessarily (d) no improvement

25. We met him immediately after the session in which he had been given a nice speech. (a) has been given (b) would be giving (c) had given (d) no improvement

Sentence Improvement ▌ 2.81 26. The drama had many scenes which were so humorous that it was hardly possible to keep a straight face. (a) hardly impossible to keep (b) hardly impossible keeping (c) hardly impossible for keeping (d) no improvement 27. Acquisition of certain specific skills can be facilitated from general awareness, education an exposure to novel situation. (a) can be facilitate from (b) may facilitate through (c) can be facilitated by (d) no improvement 28. The research study is an eye-opener and attempts to acquaint us with the problems of the poor nations. (a) attempting to acquaint (b) attempts to acquainting (c) attempted to acquaint (d) no improvement 29. If I would have realised the nature of the job earlier, I would not have accepted it. (a) If I would (b) In case I would have (c) Had I (d) no improvement 30. The man who has committed such a serious crime must get the mostly severe punishment. (a) got the mostly severely (b) get the most severe (c) have got the most severely (d) no improvement 31. He confidently asked the crowd if they thought he was right and the crowd shouted that they did. (a) that he did (b) that they had (c) that he is (d) no improvement 32. The moment the manager came to know of the fraudulent action of his assistant, he order immediately dismissed him. (a) order immediate dismissed him (b) ordered his immediate dismissal (c) immediately order dismissal of his (d) no improvement 33. The tea estate is in such a mess there is no one to set things right. (a) in a mess there (b) in a such mess that there (c) in such a mess that there (d) no correction required 34. Because of his ill health, the doctor has advised him not to refrain from smoking. (a) not to refrained (b) to resort to (c) to refrain (d) no improvement 35. What happens to all those travellers on the ship was not known. (a) What happen to (b) That is what happens to (c) What happened to (d) no improvement 36. Despite of their differences on matters of principles, they all agree on the demand of hike in salary. (a) Despite of there (b) Despite for their (c) Despite their (d) no improvement

37. The orator had been left the auditorium before the audience stood up. (a) had been left (b) was left (c) had left (d) no improvement 38. It is unanimously resolved that the parties should unitedly undertook launching of popular programmes. (a) should be unitedly undertook (b) should be unitedly undertaken (c) should unitedly undertake (d) no correction required 39. One of my drawbacks is that I did not have tolerance of ambiguity. (a) did not had (b) do not have (c) am not (d) no improvement 40. Their earnings are such that they find it difficult to make both ends to meet. (a) to make both end to meet (b) to make both ends for meeting (c) to make both ends meet (d) no correction required 41. If neither of them are to be trusted it is not at all advisable to invest further. (a) were to be trusted (b) has to be trusted (c) is to be trusted (d) no improvement 42. Don’t touch that door as it has just been painted. (a) had just been painted (b) is just painted (c) have been just painted (d) no improvement 43. The District Collector gave up the prizes to the winners at the end of the function. (a) give up (b) gave away (c) gave back (d) no improvement 44. Let’s have a party some time the next week. (a) sometime at next week (b) some time in the next week (c) sometimes next week (d) no improvement 45. I have been telling her that she had better consulted a good doctor. (a) consulting (b) consult (c) been consulting (d) no improvement 46. One day you will repent over what you have done. (a) of (b) for (c) about (d) no improvement 47. He would have lent me a pen, if he was knowing that I didn’t have one. (a) was known (b) has been knowing (c) had known (d) no improvement 48. The train was late for fifty minutes. (a) from (b) by (c) around (d) no improvement

2.82

▌ English

49. Call in this evening, if you can, I have something important to discuss. (a) Called in (b) Call off (c) Call for (d) no improvement 50. These days, it is difficult to make both ends meet. (a) to have both ends meet (b) to make all ends meet (c) to make the ends meet (d) no improvement 51. Before I could stop him, the boy was throwing the box down the stairs. (a) were throwing (b) threw (c) did throw (d) no improvement 52. My father is suffering from diabetes for the past three years. (a) is suffer (b) has been suffering (c) has suffered (d) no improvement 53. However, great you may be, you cannot trifle on a man’s feelings. (a) trifle along (b) trifle upon (c) trifle with (d) no improvement 54. She was more beautiful than either of her three sisters. (a) either of her three sister (b) any of her three sisters (c) all of her three sisters (d) no improvement 55. If I shall pass this examination, my mother will be very happy. (a) shall passed (b) pass (c) would pass (d) no improvement 56. Although India is still by far a poor country, it can become rich if its natural and human resources are fully utilised. (a) by forth (b) by and large (c) by and by (d) no improvement 57. The more they earn, more they spend. (a) more they earned, more they spend (b) the more they earn, the more they spend (c) more they earn, the more they spend (d) no improvement 58. But in all these cases conversion from one scale to another is easy because scales have well formulated. (a) have well-formulate (b) are well formulated (c) well formulated (d) no improvement 59. Five-years ago today, I am sitting in a small Japanese car driving across Poland towards Berlin. (a) am sat (b) sat (c) was sitting (d) no improvement 60. The old man felled some trees in the garden with hardly no effort at all. (a) hardly no efforts (b) hardly any effort (c) a hardly any effort (d) no improvement 61. She says she’s already paid me back, but I can’t remember, so I’ll have to take her word. (a) to takes her word (b) to take her at her word (c) to take her word for it (d) no improvement

62. The workers are hell bent at getting what is due to them. (a) hell bent through getting (b) hell bent for getting (c) hell bent upon getting (d) no improvement 63. You are warned against committing the same mistake again. (a) against not commiting (b) for committing (c) against to commit (d) no improvement 64. While we would like that all Indian children to go to school, we need to ponder why they do not. (a) that every Indian children (b) if all the children of India (c) all Indian children (d) no improvement 65. Due to these reasons we are all in favour of universal compulsory education. (a) Due to the reasons (b) for these reasons (c) by these reasons (d) no improvement 66. When it was feared that the serfs may go too far to gain their freedom from serfdom, the Protestant leaders joined the princes at crushing them. (a) at crush (b) in crushing (c) without crushing (d) no improvement 67. In India today many of our intellectuals still talk in terms of the French revolution and the Rights of Man, not appreciating that much has happened since then. (a) much was happened (b) much had happened (c) much might happen (d) no improvement 68. Taxpayers are to be conscious of their privileges. (a) have to (b) needs (c) ought (d) no improvement 69. I would have waited for you at the station if I knew that you would come. (a) I known (b) was knowing (c) had known (d) no improvement 70. No one could explain how a calm and balanced person like him could penetrate such a mindless act on his friends. (a) penetrated (b) perpetrate (c) precipitate (d) no improvement 71. The teacher told us that the prize would be presented the next day. (a) would be presented the next day (b) would have been presented the next day (c) shall be presented tomorrow (d) should be presented tomorrow

Sentence Improvement ▌ 2.83 72. His speech was optimistic, but at the end of it he stroke a note of caution. (a) strokes a note of caution (b) strut for a note of caution (c) striked a note of caution (d) struck a note of caution

81. Their attempt of rioting was foiled because of the police squad arrived on time. (a) of the police squad being arrived (b) of the timely arrival of the police squad (c) the police squad arrival (d) of the police squad had arrived

73. Men have been known how important the sun is to them. (a) Men have been knowing (b) Men have to know long (c) Men had long known (d) Men have long known

82. How can one mobilize support from colleagues without being cordially to them? (a) being cordially for (b) been cordially to (c) being cordial to (d) cordially being to

74. I did not like his comments on my paper but I had no alternative as I have agreed to keep quiet. (a) I have to agree to keep quiet (b) I had agreed to keep quiet (c) I had agreed for keeping quiet (d) I have to agree for keeping quiet

83. For every citizen it is mandatory to help the civic administration for keep up the city clean. (a) for upkeep clean the city (b) for clean and keep the city (c) for keeping of the city cleanliness (d) to keep the city clean

75. Please give him medicines if his temperature will rise. (a) if his temperature will arise (b) if his temperature would rise (c) if his temperature rises (d) unless his temperature rises

84. Anyone who known to India’s villages knows the meaning of scarcity. (a) knowing India’s villages knows (b) is known to India’s villages knows (c) knows India’s villages knows (d) knew India’s villages knowing

76. Despite being tried his best to persuade people to give up smoking, he could not attain success. (a) Despite his best trying (b) Despite of his best (c) Inspite of being tried his best (d) Despite trying his best

85. In our country women have opportunities to rise to top in every walk of life. (a) have been having opportunities (b) have had opportunities for a raise (c) have opportunities to rise (d) No correction is required

77. His suggestions were so trivial and hence nobody took any cognisance of them. (a) so trivial that and have (b) very trivial and hence so (c) too trivial to and hence (d) very trivial and hence

87. If Indian people are united, this nation can become a source of strength for the entire world. (a) could become a resource of (b) can become resourceful for (c) would be a source in (d) No correction required

78. But for your time of helping, we could not have accomplished our goal in such a small time span. (a) But for your timely help (b) Because of your timely helping (c) Despite your time of helping (d) But your time for helping

88. All their efforts were direct to promote harmony among various groups of people. (a) were directing to promote (b) were direct at promote (c) have directed for promote (d) were directed to promoting

79. He failed in his attempt to disperse the mob before the miscreants sets the fire on the bus. (a) set the bus on fire (b) setting fire on the bus (c) set fire on the bus (d) set the fire on to the bus

89. What did happen there in the first place is not a matter of our concern. (a) What happens (b) What would have happened (c) What happened (d) What should have happened

80. Even at most critical moments, he is calm, but today he appears very much disturbed. (a) appeared very much disturb (b) appears very much to disturb (c) appeared to be very much disturbing (d) No correction required

90. The speaker highlighted the contribution of women for bringing about social changes. (a) for bringing in (b) in bringing of (c) for the brought over (d) in bringing about

2.84 ▌ English 91. Such inequalities can be founded enshrined in the constitutions of other countries also. (a) could have been founded (b) can be found (c) could be founded (d) could have found

96. The new facts he has discovered could not change my opinion. (a) has not changed (b) could not be changed (c) cannot be changed (d) No correction required

92. The local authorities were never bothering to enquire about the unfortunate happenings. (a) were never bothered to (b) never bothered to (c) never were bothering to (d) were never bothering to

97. On firm ground the tent had held in place by pegs driven into the ground. (a) has been held at (b) was held in (c) was being held with (d) should be held at

93. India’s outlook on the word is composing of these various elements. (a) is composed of (b) is composed by (c) is composed with (d) has been composing at

98. In the midst of his laborious work, came a stroke of good fortune. (a) comes a strike of (b) came stroke of (c) a strike comes of (d) came a stroke for

94. How religious intolerance can blight social peace and harmony can be gauged from events take place around the globe. (a) taken place over (b) taking place around (c) took place around (d) taken place in

99. Tourists have still come to Egypt and are probably still inscribing their names on the old stones. (a) are still coming (b) have come still (c) still have come (d) will have still come

95. The fact-finding team which had been at the area found villagers giving information to the police. (a) which had been for (b) that led to (c) which led to (d) that had been to

100. There was a pause when we had finished and then a strangely subdued voice broke the silence. (a) the strange subdueing (b) a strange subdueing (c) the strange subdued (d) No correction required

anSwer Key 1. (c)

2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

(c) (b) (b) (b) (d) (c) (c) (a) (c) (c) (d) (c)

14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.

(c) (b) (c) (a) (c) (b) (c) (d) (c) (d) (d) (c) (d)

27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39.

(a) (d) (c) (b) (d) (b) (c) (c) (c) (c) (c) (c) (b)

40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52.

(c) (c) (d) (b) (b) (d) (d) (c) (b) (d) (d) (b) (b)

53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65.

(c) (b) (b) (b) (c) (b) (b) (b) (c) (c) (d) (c) (b)

66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78.

(b) (d) (a) (c) (b) (a) (d) (d) (b) (c) (d) (d) (a)

79. 80. 81. 82. 83. 84. 85. 86. 87. 88. 89. 90. 91.

(a) (d) (b) (c) (d) (c) (d) (d) (d) (c) (d) (b) (b)

92. 93. 94. 96. 97. 98. 99. 100.

(a) (b) (d) (d) (b) (b) (a) (d)

Sentence Improvement ▌ 2.85

nDa BoX Directions: Look at the underlined part of each sentence.Below each sentence are given three possible substitutions for the underlined part. If one of them (a), (b) or (c) is better than the underlined part, indicate your response on the answer sheet against the corresponding letter (a), (b) or (c). If none of the substitutions improve the sentence, indicate (d) as your response on the answer sheet. Thus, a ‘No improvement’ response will be signified by the letter (d). Errors may be in grammar, appropriate word usage or idioms. There may be a necessary word missing or there may be a word which should be removed. 1. It is a well-acclaimed principle that one should hold fast to his beliefs. (a) hold fast to others’ beliefs (b) hold fast to their beliefs (c) hold fast to one’s beliefs (d) No improvement 2. It was my business to cross the bridge to explore the bridgehead and to find out the extent to which the enemy had advanced. (a) much (b) the extent where (c) the point where (d) No improvement

3. The world is becoming increasingly polarised between the rich and the poor. (a) among rich (b) around the rich (c) in between the rich (d) No improvement 4. I laid all the facts before him so that he could make his own judgement. (a) laid all the facts in front of him (b) lay all the facts before him (c) lay all the facts in front of him (d) No improvement 5. If only she had told me about her problem I would have helped her. (a) would help (b) could help (c) had helped (d) No improvement 6. Apollo was worshipped as long as the Roman Empire continued. (a) was continued (b) ruled (c) lasted (d) No improvement

anSwer Key 1. (c)

2. (b)

3. (d)

4. (a)

5. (a)

6. (a)

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Reading Comprehension FoRmat A passage of about 300–500 words is taken from published material and a series of questions are asked about the content of the passage. Some of the questions test the ability to understand directly stated information, some test the ability to recognize implications and draw inferences, and some test the ability to comprehend the text and recognize the main idea presented by the author. Note that since the questions asked are to test the ability to read a passage and comprehend its meaning, you have to judge the correctness of an answer on the basis of what is stated or implied in the passage. Even if the information in the passage runs counter to what you believe to be correct, your answers should be based only upon what the author says in the passage.Unlike in the vocabulary section, no answer options are given here. In RC questions, all the information needed to answer a question is in the passage itself, either stated clearly or hinted at. Passage Read not to contradict and confute, nor to believe and take for granted, nor to find talk and discourse, but to weigh and consider. Some books are to be tasted, others to be swallowed, and some few to be chewed and digested; that is, some books are to be read only in parts; others to be read, but not curiously; and some few to be read wholly, and with diligence and attention. Some books may also be read by deputy, and extracts made of them by others; but that would be only in the less important arguments and the meaner sort of books; else distilled books are like common distilled waters, flashy things. Reading maketh a full man, conference a ready man,

and writing an exact man. And, therefore, if a man write little, he had need have a good memory; if he confer little, he had need have a present wit; and if he read little, he had need have much cunning to seem to know that he doth not. ‘Histories make men wise, poets witty, the mathematics subtile, natural philosophy deep, moral, grage, logic and rhetoric able to contend.’ (From Of Studies—Francis Bacon) Questions 1. What does the art of writing do to man’s character? (a) It makes him a full man (b) It makes him a ready man (c) It makes him an exact man (d) It make him a good author 2. How should you read those books that are to be ‘read wholly’? (a) By making extracts from them (b) They should be chewed and digested (c) They should be read in parts (d) They should be read with diligence and attention 3. If a man does not write much, what must he do? (a) Should have extracts made of them by others (b) A good memory (c) Must chew and digest the contents (d) A good reading and comprehension speed

answeR Key 1. (c) makes him an exact man (as given at the end of the fourth sentence of the passage). 2. (d) with diligence and attention (as given in the end of the second sentence). 3. (b) a good memory (see the fifth sentence).

2.88

▌ English

Reading CompRehension Passage 1 Not only does the lack of education among women make the dissemination of nutrition education difficult, it appears also to be a major obstacle to campaigns for family planning. It is significant that one of the more successful family planning efforts has been in Korea, where literacy is over 80 per cent. Thailand, Singapore, Hong Kong and Taiwan have also had more satisfactory results than, for example, Afghanistan, Pakistan, Bangladesh, India or Indonesia, where a large proportion of the female population is illiterate. The education level of women is significant, also because it has a direct impact on their chances of employment; and the number of employed women in a country’s total labour force has a direct bearing on both the Gross National Product and the disposable income of the individual family. 1. The passage suggests that Korea has been successful in family planning because Korean women are (a) well employed (b) well educated (c) literate enough to understand its significance (d) exposed to nutrition education 2. Countries where family planning programmes have been satisfactory are (a) Thailand, Singapore, Korea, Hong Kong and India (b) Thailand, Afghanistan, Korea and Pakistan (c) Korea and Indonesia (d) Taiwan, Hong Kong, Korea, Thailand and Singapore 3. According to this passage, a women’s lack of education has a direct bearing on (a) the GNP of the country (b) her chances of getting a job (c) the total labour force of the country (d) her knowledge of the nutritional value of food 4. One of the main disadvantages of lack of education among women is that they (a) cannot be persuaded to plan their families (b) do not know how to cook nutritious food (c) cannot earn more money (d) cannot communicate well Passage 2 Gandhiji recognised that, while all men should have equal opportunity, all did not have the same capacity. Some had the ability to earn more than others. But, he believed that those who had talent would be performing the work of society if they used their talent wisely and well. Gandhiji said that he would allow a man of intellect to earn more and not suppress his talent. But it was his view that the bulk of his larger earnings should go to the common fund. Those with talent and opportunity would find their fulfilment as trustees. Gandhiji extended this concept of trusteeship to cover all fields of life.

5. Gandhiji never believed in (a) social equality (b) equality of opportunities (c) political equality (d) complete equality in all respects 6. Gandhiji’s trusteeship concept is (a) a philosophy (b) applicable to social life only (c) applicable to all fields of life (d) irrelevant today 7. According to Gandhiji, one can serve the society (a) if one is talented (b) if one used his talent wisely (c) if one earned well (d) if one worked honestly 8. The title of the passage should be (a) Gandhiji’s Character (b) Gandhiji’s Views (c) Gandhiji’s Services (d) Gandhiji’s Philosophy 9. The meaning of “trustee” is a (a) person who has to hold his property in trust (b) State official who executes wills and trusts (c) person having confidence (d) number of trusts 10. Gandhiji believed that a man of intellect should (a) earn as much as he needs (b) earn for others, not himself (c) earn more and not suppress his talent (d) live on charity Passage 3 So Tiziano continued to draw. But one thing troubled him greatly—all the pictures he made were black, drawn with his piece of black charcoal. Yet around him glowed a perfect glory of colour—the beautiful blue of the sky; the delicate, changing pink of the great jagged peaks above him; the red, blue and yellow wild flowers; the golden brilliance of sunshine; and the rich, soft, mellowed tints in the old houses of the town. Colour! Tiziano loved it more than anything else in the world. Yet, how was he to reproduce it and get it into his pictures? He had no money to buy paints, and paints were expensive in those days. His father, who was a mountaineer, would never listen to anything so foolish as buying paints for a boy when the family needed food, clothing and fuel to keep them warm. Let Tiziano make shoes! Theat was a trade for a man! All the same, Tiziano continued to dream of painting, and to wonder if there was not some way he could make a picture in colours. The day before the festival of flowers; Tiziano chanced to pass the spot where the garlands had been woven the evening before. Suddenly, he noticed stains on the stones of the walk before the

Reading Comprehension ▌ 2.89 inn. They were every colour that a painter needed! In a moment the feast and the fun went out of Tiziano’s mind. Catarina saw her brother hastening out of the village. She ran to bring him back and found him in a meadow looking like a variegated quilt from the brilliance of the wild flowers. “Tiziano!” she called, “Why are you running away from the feast?” The boy did not answer for a moment. Too often he had been teased by his family and the villagers for the crazy dreams in his head. At last he answered bluntly, “I have found that the stains of flowers make colours and I am going to paint a picture.” 11. Tiziano’s “crazy dreams” that are mentioned in the second paragraph refer to his desire to (a) make the best shoes in the land (b) find a method for recycling the flowers from the feast (c) bring prosperity to his family and his village (d) paint pictures in colour 12. Tiziano’s father wanted his son to be a (a) painter (b) cobbler (c) florist (d) mountaineer 13. In the second paragraph, the word “variegated” means (a) marked with patches of different colours (b) having different designs (c) flowery (d) of different varieties 14. Which of the following would best describe Tiziano’s father ? (a) miserly (b) artistic (c) insensitive (d) generous 15. At the end of the passage, it is clear that Tiziano had discovered (a) that he could fulfil his dream of painting with colours obtained from crushed petals (b) that painting was a better profession than making shoes (c) a way to make his sister happy (d) a way to add to the fun of the Festival of Flowers Passage 4 Nationalism, of course, is a curious phenomenon, which at a certain stage in a country’s history gives life, growth and unity at the same time. It has a tendency to create oneness, because one thinks of one’s country as something different from the rest of the world. One’s perspective changes and is continuously thinking of one’s own struggles and virtues and failing to come to the conclusion of other thoughts. The result is that the same nationalism, which is the symbol of growth for people becomes a symbol of a cessation of that growth in the mind. Nationalism, when it becomes successful, sometimes goes on spreading in an aggressive way and becomes a danger, internationally. Whatever line of thought you follow, you arrive at the conclusion that some kind of balance must be found. Otherwise something that was good can turn into evil. Culture, which is essentially good, becomes not only static but aggressive and something that breeds conflict and hatred when looked at

from a wrong point of view. How are you to find a balance, I don’t know. Apart from the political and economic problems of the age, perhaps, this is the greatest problem today because behind it there is a tremendous search for something that it cannot find. We turn to economic theories because they have an undoubted importance. It is a folly to talk of culture or even of God when human beings starve and die. Before one can talk about anything else one must provide the normal essentials of life to human beings. That is where economics comes in. Human beings today are not in the mood to tolerate this suffering, starvation and inequality when they see that the burden is not equally shared, leaving others to profit while they only bear the burden. 16. “Others” in the last sentence refers to (a) other people (b) other nations (c) other communities (d) other neighbours 17. A suitable title for this passage can be (a) Nationalism breeds unity (b) Nationalism—a road to world unity (c) Nationalism is not enough (d) Nationalism and national problems 18. Aggressive nationalism (a) breeds threat to international relations (b) leads to stunted growth (c) endangers national unity (d) isolates a country 19. The greatest problem in the middle of the passage refers to the question (a) how to mitigate hardship to human beings (b) how to contain the dangers of aggressive nationalism (c) how to share the economic burden equality (d) how to curb international hatred 20. Negative national feeling can make a nation (a) selfish (b) self-centered (c) indifferent (d) dangerous Passage 5 Since the world has become industrialized, there has been an increase in the number of animal species that have either become extinct or are nearing extinction. Bengal tigers, for instance, which once roamed in the jungles in vast numbers, now number only 2,300 and by the year 2025 their population is estimated to go down to zero. What is alarming about the case of Bengal tiger is that this extinction would have been caused almost entirely by poachers who according to some sources, are not interested in material gain but in personal gratification. This is an example of the callousness that is part of what is causing the problem of extinction. Animals like the Bengal tiger, as well as other endangered species, are a valuable part of the world’s ecosystem. International laws protecting these animals must be enacted to ensure their survival, and the survival of our planet. Countries around the world have begun to deal with this problem in various ways. Some countries, in order to circumvent the problem, have allocated large amounts of land to animal reserves. They then charge admission fee to help defray the costs of maintaining the parks and often must

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also depend on world organisations for support. When they get the money, they can invest in equipment and patrols to protect the animals. Another solution that is an attempt to stem the tide of animal extinction is an international boycott of products made from endangered species. This seems fairly effective, but it will not by itself prevent animals from being hunted and killed. 21. What is the author’s main concern in the passage? (a) Problems of industrialisation (b) The Bengal tiger (c) Endangered species (d) Callousness of man 22. According to the passage, poachers kill for (a) material gain (b) personal satisfaction (c) Both (a) and (b) (d) None of these 23. Which of the following words is closest in meaning to the word ‘alarming’ ? (a) serious (b) dangerous (c) distressing (d) frightening 24. Certain species are becoming extinct because of (a) Industrialisation (b) Poaching (c) Love of products made from them (d) All of these 25. The phrase ‘stem the tide’ means (a) save (b) stop (c) touch (d) spare 26. Which of the following best describes the author’s attitude ? (a) concerned (b) vindictive (c) surprised (d) generous Passage 6 The Conservative is not an extreme individualist. He may be willing to concede numerous arguments of the unqualified individualists, for his own respect, because the dignity of the individual is not surpassed by that of any other man. Yet, he cannot agree to the full implications of individualism, which is based, so he thinks, on an incorrect appraisal of man, society, history and government. In his own way, the individualist is as much a perfectionist as the Socialist, and with perfectionism the Conservative can have no truck. In particular, the Conservative refuses to go all the way with economic individualism. His distrust of the unfettered man, his recognition of groups, his sense of the complexity of the social process, his recognition of the real services that the government can perform—all these sentiments make it impossible for him to subscribe to the dogmas and shibboleths of economic individualism—laissez faire, the negative state, enlightened selfinterest, the law of supply and demand, the profit motive—The Conservative may occasionally have kind words for each of these notions, but he is careful to qualify his support by stating other, more important social truths. For example, he does not for a

moment deny the prominence of the profit motive, but he insists that it be recognised for the selfish thing it is and be kept within reasonable socially imposed limits. 27. The Conservative is (a) a perfectionist (b) an economist (c) a socialist (d) none of these 28. The Conservative is against economic individualism for all the following reasons except (a) he does not trust free men (b) he believes in the authority of the government (c) he believes in groups (d) he feels that social processes are important 29. The author mentions all the following catchwords of economic individualism except (a) free trade (b) the profit motive (c) balance of trade (d) the negative state 30. Which of the following words can replace the word ‘truck’? (a) dealing (b) bargain (c) debate (d) transport 31. Which of the following statements is true? (a) The Socialist and the Individualist tend to be broadly similar in their views (b) The Conservative believes that profit motive originates in selfishness (c) The Conservative is also an extreme Individualist (d) None of the above 32. Which of the following could be an appropriate title for the passage ? (a) Anarchy And Freedom (b) Progress And The Conservating (c) A Conservative Apology (d) The Conservative Stand Passage 7 There are many types of problem molecules in the body, and each type is specific to its function. For example, proteins such as keratin and collagen give strength and elasticity to hair, as well as to skin and tendons; haemoglobin and myoglobin are the oxygen-binding proteins of the blood and the muscle, respectively; and ovalbumin, the principal protein of egg white is responsible for the setting and foaming properties of eggs. A particularly important group of proteins known as enzymes, directs all the body cells to produce chemical reactions. These reactions provide the basis of every type of cell activity, including growth, repair, the production of energy and the excretion of waste products.

Reading Comprehension ▌ 2.91 33. From the passage it is clear that there are several types of proteins that (a) can create problems in our body (b) carry out specific tasks (c) are found in all organisms (d) have interchangeable functions 34. Proteins are classified according to (a) the number of molecules (b) their names (c) their functions (d) the part of the body they are found in 35. The function of the enzymes is to (a) direct chemical reactions in body cells (b) bind oxygen in the blood (c) give strength and elasticity to hair (d) set and foam properties of an egg Passage 8 My father and mother complemented each other. My mother was fragile, my father robust. Her face was responsive, my father’s impassive. My mother’s face rippled with emotions as water to the wind. My mother was not an intellectual. Her natural propensity was intuitive. Her face illustrated the saying: “Appearances are deceptive”, for it did not show the immense strength of her moral convictions. No one could have inferred from it that she was capable of such fanaticism as she showed. 36. By saying that his mother complemented his father, the author suggests that she was (a) like his father (b) opposed to his father (c) different from his father (d) indifferent to his father 37. The words “her face was responsive” mean that (a) her face clearly showed her feelings (b) her face was very attractive (c) her face was passive (d) one had to look at her face when she spoke 38. The author’s mother held strong views based on (a) religious scriptures (b) legal provisions (c) social customs (d) ethical judgements Passage 9 While it is true that there is no law that compels us to say please, there is a social practice much older and much more sacred than any law that enjoins us to be civil. The first requirement of civility is that we should acknowledge a service. ‘Please and Thank you’ are the small exchanges with which we pave our way as social beings. They are the little courtesies by which we keep the machine of life oiled and running sweetly. They put out intercourse on the basis of friendly cooperation, and easy give and take, instead of on the basis of superiors dictating inferiors. It would be a very vulgar mind that would wish to command where he can have the

service for the asking and have it with willingness and good feeling instead of resentment. 39. According to the author (a) we should not say “Please” as it is against the law (b) we may or may not say “Please” according to our mood (c) to say “Please” is an outdated custom (d) we must say “Please” as it is a civilised custom 40. The writer thinks that (a) courtesies make life oily and sticky (b) the less the courtesies, the more the frankness in our life (c) courtesies make life pleasant (d) courtesies make life mechanical Passage 10 Among the natural resources that can be called upon in national plans for development, possibly, the most important is human labour. Since the English language suffers from a certain weakness in its ability to describe groups composed of both male and female members, this is usually described as ‘manpower’. Without a productive labour force, including effective leadership and intelligent middle management, no amount of foreign assistance or natural wealth can ensure successful development and modernization. The manpower for development during the next quarter century will come from the world’s present population of infants, children and adolescents. But we are not sure that they will be equal to the task. Will they have the health, the education, the skills, the socio-cultural attitude essential for the responsibilities of development? For far too many of them the answer is no. The reason is basic. A child’s most critical years, with regard to physical, intellectual, social and emotional development, are those before he reaches five years of age. During those critical formative years he is cared for almost exclusively by his mother, and in many parts of the world the mother may not have the capacity to raise a superior child. She is incapable of doing so because of her own poor health, ignorance and lack of status and recognition of social and legal rights, and economic parity of independence. One essential factor has been overlooked or ignored—the role of women. Development will be handicapped as long as women remain second-class citizens, uneducated, without any voice in family or community decisions, without legal or economic status, married when they are still practically children, and thereafter producing one baby after another, often only to see half of them die before they are of school-going age. We can enhance development by improving “woman power”, by giving women the opportunity to develop themselves. Statistics show that the average family size increases in inverse ratio to the mother’s years of education—is lowest among college graduates, highest among those with only primary school training or no education. Malnutrition is most frequent in large families and increases in frequency with each additional sibling. The principle established seems that an educated mother has healthier and more intelligent children, and that this is related to the fact that she has fewer children. The tendency of educated,

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upper-class mothers to have fewer children exists even without access to contraceptive services. The educational level of women is significant also because it has a direct influence upon their chances of employment; and the number of employed women in a country’s total labour force has a direct bearing on both the Gross National Product and the disposable income of the individual family. Disposable income, especially in the hands of women, influences food purchasing and therefore the nutritional status of the family. The fact that this additional income derives from the paid employment of women provides a logical incentive to restrict the size of the family. 41. Among the natural resources that can be called upon in national plans for development (a) the most important is certainly human labour (b) the most important is possibly human labour (c) the least developed is certainly human labour (d) the least developed is undoubtedly human labour 42. Without a productive labour force, including effective leadership and intelligent middle management, (a) no productive work is possible (b) entrepreneurs will incur heavy losses (c) economic development will not keep pace with nationalist movements (d) no amount of foreign assistance or natural wealth can ensure successful development and modernisation 43. The manpower development during the next quartercentury (a) will be adversely affected by the threat of war (b) will come from the world’s present population of infants, children and adolescents (c) will be well taken care of by the current emphasis on free education for women (d) will be adversely affected by the country’s economic losses and political instability 44. The writer made only one of the following statements; indicate which one. (a) the world’s present population of infants, children and adolescents is very healthy (b) the world’s present population of infants, children and adolescents is very sickly (c) the world’s present population of infants, children and adolescents may not be equal to the task (d) the world’s present population of infants, children and adolescents is likely to become morally bankrupt 45. The writer directly or indirectly made three of the following statements; indicate the one he didn’t make. (a) Many of the world’s present population of infants, children and adolescents will not have the health, the education, the skills, the socio-cultural attitudes essential for the responsibilities of development. (b) A child’s most critical years with regard to physical, intellectual, social and emotional development, are those before he reaches five years of age.

(c) Development will be handicapped as long as women remain second-class citizens. (d) In the advanced countries of the world, women are regarded as first-class citizens but in developing countries they are third-class citizens. 46. According to the writer, we can enhance development by (a) giving women the opportunity to develop themselves (b) making greater capital investments in agriculture and industry (c) establishing a Ministry of Economic Planning and Development (d) increasing wages and improving general condition of service for all workers Passage 11 Two principles are involved in the controversy about the presence of foreign controlled media in the country; the free flow of ideas and images across national borders and the need to safeguard the national interest and preserve cultural autonomy. Both are valid but both are at loggerheads because each has been used to promote less lofty goals. The first principle conforms to a moral imperative : freedom of expression cannot rhyme with restrictions imposed by any government. But the free flow rhetoric also clouds the fact that the powerful Western, and especially American media, can and often do present, subtly or brazenly, news in a manner that promotes Western political, ideological and strategic interests. Besides, Western entertainment programmes present lifestyles and values that run counter to the lifestyles and values cherished by traditional societies. All this explains why so many Indian newspapers, magazines and news agencies have sought protection from the courts to prevent foreign publications and news agencies from operating in the country. Their arguments are weak on two counts. As the bitter debate on a new world information and communication order demonstrated in the late seventies and early eighties, many of those who resent Western ‘invasion’ in the fields of information and culture are no great friends of democracy. Secondly, the threat of such an ‘invasion’ has been aired by those media groups in the developing countries that fear that their business interests will be harmed if Western groups, equipped with large financial and technological resources and superior management skills, are allowed to operate in the country without let. The fear is valid but it goes against the grain of the economic reform programme. The presence of foreign newspapers and television channels will increase competition, which, in the course of time, can only lead to the upgradation of dynamic Indian newspapers and television channels, even while they drive the rest out of the market. One way to strike a balance between the two antagonistic principles would be to allow foreign media entry into the country, provided the Indian state treats them at par with the domestic media on all fronts. On the import of technology, for instance, foreign media cannot be allowed duty concessions denied to their Indian counterparts. Foreign media will also have to face legal consequences should they run foul of Indian laws. Why, for example, should the BBC, or Time magazine or The Economist get away by showing a map of Kashmir, which

Reading Comprehension ▌ 2.93 is at variance with the official Indian map? Why should they go scot-free when they allow secessionists and terrorists to air their views without giving the government the right to reply, or when they depict sexually explicit scenes, which would otherwise not be cleared by the Censor Board? Since the government can do precious little in the matter, especially about satellite broadcasts, what if it should consider attaching the properties of the offending parties? Demands of this kind are bound to be voiced unless New Delhi makes it clear to the foreign media that they will have to respect Indian susceptibilities, especially where it concerns the country’s integrity and its culture. It may be able to derive some inspiration from France’s successful attempts in the recent GATT to protect its cinematography industry. 47. Which of the following is one of the points weakening the argument to prevent the entry of foreign media? (a) Such entry would be against traditional culture (b) The threat being voiced by those whose business will be harmed by such an entry (c) The arguments being put forth are at loggerheads (d) The foreign media may not be treated on par with the domestic media 48. What will be the impact of increasing competition? (a) The domestic media will not be able to withstand it (b) The foreign media will not be allowed duty concessions on import of technology (c) It will improve Indian newspapers and television (d) The Indian newspapers and news agencies will seek protection from the court 49. Which of the following has been cited as having succeeded in protecting country? (a) GATT (b) News Agencies (c) Television (d) Cultural traditions 50. Which of the following has been the major recommendation regarding the entry of foreign media? (a) It should not be allowed (b) It should be welcomed without putting any restrictions (c) Allow entry, treating them on par with domestic media (d) Allow entry, provided they do not ask for duty concessions on import of technology 51. In the controversy involving two principles regarding allowing foreign media, which of the following is against its entry ? (a) Free flow of ideas (b) Preserve culture (c) Government restrictions (d) Security across national borders 52. According to the passage, which media in particular promotes Western interests? (a) American (b) Foreign (c) French (d) Western

53. Which of the following is the meaning of the phrase “without let”, as used in the passage? (a) with no difficulty (b) without confinement (c) with strings (d) without restrictions 54. Why would the entry of foreign media harm local interests? (a) They are better equipped managerially and technologically (b) Our cultural heritage will be lost (c) Economic reform programmes will get a setback (d) Different sets of laws and rules were made applicable for foreign media 55. Which of the following is the meaning of the phrase “at variance”, as used in the passage? (a) discrepancy (b) at large (c) in conformity (d) differing 56. Which of the following seems to be the most likely purpose of writing this passage? (a) To criticise foreign media (b) To highlight the exploitation by developed nations (c) To highlight the steps and caution to be taken about the entry of foreign media (d) To make the public aware of the technological and managerial superiority of western media 57. Which of the following is the meaning of the phrase “at loggerheads”, as used in the passage? (a) in league with (b) unimportant (c) out of place (d) opposite to each other Choose the word that is most closely the opposite in meaning to the word given in capitals taken as from the passage. 58. COUNTER (a) similar (c) unresponsive 59. ANTAGONISTIC (a) counteract (c) equal

(b) downwards

(b) coincidental (d) corresponding

Choose the word or group of words that is most similar in meaning to the word given in capitals taken from the passage. 60. SUSCEPTIBILITIES (a) norms (c) influences

(b) weaknesses (d) sensitivities

61. RHYME (a) mingle (c) recur with

(b) confirm (d) go with

62. RHETORIC (a) rhyming words (b) persuasive speaking (c) dull monologue (d) tongue-in-cheek

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Passage 12 Today perhaps your only association with the word ‘polio’ is the Sabin Oral Vaccine that protects children from the disease. Fifty-five years ago, this was not so. The dreaded disease, which mainly affects the brain and the spinal cord, causes stiffening and weakening of the muscles, crippling and paralysis—which is why I am in a wheelchair today. If somebody had predicted, when I was born, that this would happen to me, no one would have believed it. I was the seventh child in a family of four pairs of brothers and sisters, with a huge 23 year gap between the first and the last. I was told that, unlike the others, I was so fair and brown-haired that I looked more like a foreigner than a Dawood Bohri. I was also considered to be the healthiest of the brood. 63. In this passage, the word ‘brood’ refers to (a) polio victims (b) foreign children (c) children in the family (d) Indian children 64. In his childhood, the narrator said “more like a foreigner than a Dawood Bohri”. This as because he was (a) a foreign child (b) a very healthy boy (c) tall and smart (d) fair and brown-haired 65. The narrator was the seventh child in a family of (a) 8 children (b) 16 children (c) 23 children (d) 4 children 66. In this passage, the narrator is a patient of (a) heart disease (b) polio (c) paralysis (d) nervous weakness 67. In his childhood, the narrator was (a) a weakling (b) very healthy (c) tall and slim (d) short and stout Passage 13 India has come a long way since the Bengal Famine of 1943. The food situation in India, once characterised by chronic shortages and the spectre of famines, has changed dramatically over the years. From being the biggest recipient of PL-480 during the 1950s and 1960s, India today is relatively self-sufficient in foodgrain at the given level of incomes and prices; in fact, it has marginal surpluses. The General Agreement on Tariffs and Trade (GATT) has been signed, with India as one of the signatories, under which all countries will have to gradually open up their agricultural sectors. It is, therefore, neither feasible nor desirable to keep India’s foodgrain sector insulated from world markets. In fact, this is an appropriate opportunity for India to integrate its agriculture with global agriculture, and make use of private trade (both domestic and foreign) as an important instrument for efficiently allocating her resources as well as providing food security to her people at the lowest economic cost. The time to change gears in food policy has come. Food security, in a broader context, means that people have physical and economic access to food. Since foodgrains have the largest share in the food basket of the poor in a developing country like India, it is the availability of foodgrains that lies at the heart of the concept of food security. The first step in this direction,

therefore, is to make foodgrains physically available to the people. This can be done by augmenting production, or through imports and transportation of grain to people wherever they are. There are several ways of achieving these targets. One may rely on private entrepreneurship by letting the individual farmers produce, traders trade/import and make it available to consumers far and wide; or the Government may directly intervene in the production and/or the trade process. In the former case, the Government follows policies that provide appropriate market signals while in the latter, it acts as producer, importer and trader itself. Indian policy makers have followed a mix of both these options. For production, they have relied on farmers while the Government has retained control over imports. For distribution, it created public agencies to do the job along with private trade, thus creating a dual market structure. Providing economic access to food is the second part of the concept of food security. This can be best obtained by adopting a cost effective technology in production so that the real price of foodgrains come down and more people have access to it. In case it still fails to reach the larger sections of the population, the Government can directly subsidise food for the poor, launch a drive to augment their incomes, or try a combination of the two strategies. India has followed both these policies. 68. India has come a long way means _____ (a) many years have passed after the Bengal famine (b) the food position is now largely improved (c) India now handles such problems under PL-480 Scheme (d) India has advanced in science and technology 69. The author seems to be advocating which of the following views regarding GATT? (a) India should seize the chance and make efforts to fulfill its objectives (b) India should not have signed it to insulate our foodgrains sector (c) India should hand over issue of food security to only private sector (d) India should have signed GATT excluding the agricultural sector 70. According to the author, why is it necessary to make available food security to people? (a) To sustain economic growth (b) As per PL-480 guideline (c) To enable us to export food grains (d) Not given in the passage 71. Which of the following forms the most essential part of the concept of food security in India? (a) Availability of effective technology of food production to poor farmers (b) Availability of all food items in the market for urban poor (c) Easy access of foodgrains to the weaker sections at affordable price (d) Providing subsidy on all food items for rural poor

Reading Comprehension ▌ 2.95 72. If the private agencies are entrusted the work of making food grains available to people, what facilitative role should the Government undertake? (a) Nationalise all distribution system (b) Take decisions that give appropriate message to the market (c) Undertake responsibilities of production and distribution (d) Make efforts to increase the income of the farmers 73. Which of the following can be inferred about the general view of the author in the context of this passage? (a) Fully advocate liberalisation and privatisation (b) Supported State controlled, socialistic but closed economy (c) A very practical and pragmatic approach to guard our economy in view (d) Theoretical view, not concerned with ground realities 74. According to the author, food could be made available to the poor at affordable prices by which of the following measures? (a) Reducing the cost of production of foodgrains by using appropriate technology (b) Offering foodgrains at lower cost and offering economic support for maintaining lower cost (c) Raising the earning of the poor (d) None of these 75. According to the author, why have India’s policy makers created a mixed market structure? (a) This is a practice followed by many developing countries (b) This is a natural fallo ut of GATT agreement (c) India is now poised for globalisation and privatisation (d) India is now a technologically advanced country Passage 14 The shoemaker had for ages suffered from a heart condition and five years ago, after an attack, it had appeared as though he would have to either sacrifice his business upon the auction block and live on a pittance thereafter; or put himself at the mercy of unscrupulous employees who would in the end probably ruin him. But just at the moment of his darkest despair, a Polish refugee, Sobel, appeared one night from the street and begged for work. He was a stocky man, poorly dressed, with a bald head, severely plain face and soft blue eyes prone to tears over the sad books he read. Though he confessed he knew nothing of shoemaking, he said he was apt and would work for very little if Feld taught him the trade. Feld took him on and within six weeks the refugee rebuilt as good a shoe as he, and not long thereafter expertly ran the business for the shoemaker. Feld could trust him with anything, and did frequently, going home after an hour or two at the store, leaving all the money in the till knowing Sobel would guard every cent of it. The amazing thing was that he demanded so little. His wants were few; in money he was not interested—in nothing but books, it seemed. These he lent one by one to Feld’s daughter Miriam together with

his profuse queer written comments, manufactured during his lonely evenings, which his daughter, from her fourteenth year, read page by page. Feld’s conscience bothered him for not insisting that his assistant accept a better wage than he was getting, though Feld had honestly told him he could earn a handsome salary if he worked elsewhere, or maybe opened a place of his own. But the assistant answered, somewhat ungraciously, that he was not interested in going elsewhere. Feld frequently asked himself what kept him there, why did he stay ? He finally told himself that the man no doubt because of his terrible experiences as a refugee, was afraid of the world. 76. After his heart attack Feld feared that he would have to (a) take in several employees to help him in his work (b) teach his daughter, Miriam, the trade of shoemaking (c) give up the business immediately and rest in a hospital (d) sell his business for very little and live as a poor man 77. Sobel begged for work for a pittance (a) because he confessed that he knew nothing of shoemaking (b) because he admitted that he was a poor man (c) because he clearly said that he belonged to Poland (d) because he declared that he was a man of honesty 78. Feld trusted Sobel and (a) he left the money to the latter’s care (b) he sent him out on business errands (c) he found that Sobel never told a lie (d) he felt that people of Poland were honest 79. Feld was a man of conscience (a) because he had love for the poor (b) because he wanted to sell his shoes at a low price (c) because he felt that Sobel could get a better salary elsewhere (d) because he had given employment to Sobel Passage 15 Many of the underdeveloped countries will promote the growth of their economies in one way or another, whether they receive substantial outside aid in the process or not. The character of that development, however, is likely to be strongly influenced by the types and amounts of aid available. The outcome is much more likely to be favourable, from the standpoint of the objectives set up previously for successful development, if there is substantial international aid than if there is not. By substantial aid I mean not only large amounts of technical assistance but also of capital. Initially, the capacity of an underdeveloped country to use capital productively may be surprisingly small—limited by lack of organisation, trained personnel, and other social obstacles. At this stage, technical assistance is its main need from outside, with comparatively small amounts of capital, much of which may have to be in the form of grants for non self-liquidating projects in education, health, access roads to rural areas, and the like. If at this stage, substantial capital is available from outside to supplement what can be

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formed internally (and to stimulate internal capital formation, for it does that too) the rate of economic growth can be considerably increased, and the strains, frustrations and political risks of the development process are likely to be considerably less. It is possible for underdeveloped economies to modernise themselves with very little capital from outside. Japan’s import of capital was small, though some of it came at crucial times. The contribution of foreign direct investments to the advancement of technical know-how also was greater than would be indicated merely by the size of the investment. The Soviet Union industrialised its economy with practically no aid from foreign investment capital, except for the foreign-owned installations confiscated after the revolution, though it imported machinery in the early days on short-term or intermediate-term credits and hired services of foreign experts. Both Japan and Russia achieved their development in an authoritarian political and social framework. The outcome in both cases, from the standpoint of peace in the world and democratic ideals, was highly unfavourable. In the absence of outside aid, the only way to accumulate capital is to increase production without taking much of the benefit in more consumption, or even while pushing consumption standards down. Where the people are already near subsistence level this may mean extreme hardship. Somehow the people must be motivated to change their accustomed ways quickly, to work hard, and to forego present consumption so that capital investment can be made. 81. The passage says (a) Without foreign aid no underdeveloped country can grow (b) Underdeveloped countries must refrain from seeking foreign aid (c) The economies of underdeveloped countries are more likely to grow faster with substantial foreign aid than without (d) Underdeveloped countries are economically backward because their governments have not got their priorities right

(c) Japan rejected offers of substantial foreign capital investment at the early stages of its economic development (d) In the early days of its economic development, the Soviet Union imported machinery on short-term or intermediate-term credits and hired foreign experts 85. Which of the following points or statements did the writer actually make? (a) Japan and Russia achieved their development in a democratic framework (b) Japan and Russia achieved their development in an authoritarian political and social framework (c) Japan and the Soviet Union would have developed faster had they relied on democratic methods (d) Japan and the Soviet Union are still among the underdeveloped countries of the world 86. In the absence of outside aid, the only way to accumulate capital is to (a) Increase tax and import duties (b) Launch an internal campaign for the conservation of goods and property (c) Increase production without taking much of the benefit in more consumption (d) Make conditions attractive to foreign investors Passage 16 At that time everyone believed that the Earth was at the centre of the universe. But Copernicus realised that this picture did not agree with astronomical observations. He worked out that the Sun was at the centre with all the planets moving around it. He said that: our Earth takes an year to travel around the Sun, and also revolves once every 24 hours. Copernicus believed that the planets moved around the Sun in perfect circles. Fifty years later, Kepler used Tycho Brahe’s extremely accurate measurements to show that they do not.

82. Substantial aid in this context means (a) technical assistance in the form of trained personnel (b) capital in the form of bank loans and overdrafts (c) large amounts of technical assistance and capital (d) a cheap and plentiful supply of labour

87. Copernicus’ belief that the Sun was at the centre of the universe was based on (a) common sense (b) his astronomical observations (c) mathematical calculations (d) observations of other astronomers

83. The availability of substantial capital from outside (a) can help to stimulate internal capital formation (b) does encourage wasteful tendencies (c) seldom helps to accelerate the rate of international growth (d) tends to discourage local capital formation

88. The belief that the earth was at the centre of the universe was held at that time by (a) everybody (b) Copernicus (c) the uneducated persons (d) none of these

84. The passage says that (a) Japan imported substantial capital before it became modernised (b) The Soviet Union industrialised its economy with plenty of foreign investment capital

89. Copernicus’ belief that the planets moved in perfect circles (a) was self-evidently wrong (b) was perfectly correct (c) was based on astronomical observations (d) was proved wrong on further astronomical observations

Reading Comprehension ▌ 2.97 Passage 17 The omnipotent dictator is the least likely to have any authority. Gandhiji had no power to compel, punish or reward. His power was nil, his authority was enormous. It came out of love. Living with him, one could see why he was loved; he loved. Not merely in isolated incidents, but day in and day out, morning, noon and night, for decades, in every act and word he had manifested his love of individuals and of mankind.

After a while she got to her feet again, allowed the kid to drink for a minute, poised carefully on the brink, and jumped down while the kid again ran backwards and forwards above her. Seven times in the course of the next half-hour this procedure was repeated until finally the kid, abandoning itself to its fate, jumped, and landing safely beside its mother was rewarded by being allowed to drink its fill. The lesson for her young, that it was safe to follow where she led, was over for that day.

90. “His power was nil, his authority was enormous.” Select the most likely meaning of this statement. (a) he had no power; therefore, nobody recognised his authority (b) he exercised extraordinary control over others because he had enormous power of control (c) his authority over others was enormous because he could generate fear in others (d) he did not exercise his power as a dictator does, but he could make others obey his wishes

93. Which of the following excerpts from the passage best bears out the description of the thar as a sure-footed mountain goat? (a) “Seven times in the course of the next half-hour this procedure was repeated.” (b) “The lesson for her young _________ was over for that day.” (c) “________ the mother went to what looked like a mere crack in the vertical rock face _________ rejoined her young.” (d) “I was too far away to hear whether the mother was encouraging her young _________ I believe she was doing so.”

91. Gandhiji was loved by all because (a) he had enormous power to punish or reward (b) he was a highly respected person (c) he himself had enormous love for others (d) he had always preached non-violence and love of mankind 92. ‘The omnipotent dictator is least likely to have any authority.’ Select the most likely reason for this. (a) a powerful dictator exercises control over others only temporarily (b) a ruler with only unrestricted authority is not likely to win the love and respect of his people (c) a person with supreme authority generates obedience through fear in the minds of the people (d) a ruler with absolute power is seldom trusted Passage 18 I was lying on a ridge scanning, with field glasses, a rock cliff opposite me for thar, the most sure-footed of all Himalayan goats. On a ledge halfway up the cliff, a thar and her kid were lying asleep. Then, the thar got to her feet, stretched herself, and the kid immediately began to feed. After a minute the mother freed herself, took a few steps along the ledge, poised for a moment, then jumped down on to another but a narrower ledge some twelve to fifteen feet below her. As soon as it was left alone the kid started running backwards and forwards, stopping every now and then to peer down at its mother, but unable to summon the courage to jump down to her, for below the narrow ledge was a sheer drop of a thousand feet. I was too far away to hear whether the mother was encouraging her young, but from the way her head was turned I believe she was doing so. The kid was getting more and more agitated and, possibly fearing that it would do something foolish, the mother went to what looked like a mere crack in the vertical rock face and, climbing it, rejoined her young. Immediately on doing so she lay down, presumably to prevent the kid from feeding.

94. The mother goat feared that the kid “would do something foolish” like (a) drink more than its share of milk (b) throw itself off the cliff (c) follow her down the vertical rock face (d) continue to pace up and down indefinitely 95. Which of the following phrases can replace “abandoning itself to its fate”, used in the second paragraph? (a) Making the most of the situation (b) Taking the bull by the horns (c) Facing the music (d) Accepting consequences 96. One way in which the kid was encouraged to follow its mother was by (a) not being allowed to have its fill of milk until it had jumped (b) running backwards and forwards on the ledge before it jumped (c) getting very agitated before it jumped (d) being rejoined by its mother several times 97. The mother goat taught her kid the lesson through (a) agility and sure-footedness (b) severity and punishment (c) patience and perseverance (d) praise and reward Passage 19 Once, an ant who had come to drink at a stream fell into the water and was carried away by the swift current. He was in great danger of drowning. A dove, perched on a nearby tree, saw the ant’s danger and dropped a leaf into the water. The ant climbed onto this, and was carried to safety. Sometime after this, a hunter, creeping through the bushes, saw the dove asleep, and took a careful aim with his gun.

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He was about to fire when the ant, who was nearby, crawled forward and bit him sharply on the ankle. The hunter missed his aim, and the loud noise of the gun awakened the dove from her sleep. She saw the danger and flew swiftly away to safety. Thus, the ant repaid the dove for having saved its life in the foaming current of the stream.

99. The dove dropped a leaf into the water to (a) drown the ant (b) save the ant (c) help itself (d) perch on it 100. The dove was in danger because (a) there was a bush nearby (b) a hunter was about to shoot it (c) it had fallen off the branch (d) a hunter wanted to care for it

98. The ant came to the stream to (a) fall into it (b) to drink from it (c) look at the swift current (d) carry back some water

answeR Key 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

(c) (d) (d) (a) (d) (c) (b) (b) (a) (b) (d) (b) (a)

14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.

(c) (a) (a) (c) (d) (b) (b) (c) (b) (d) (d) (b) (a)

27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39.

(d) (d) (a) (a) (b) (d) (b) (c) (a) (c) (a) (d) (d)

40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52.

(c) (b) (d) (b) (c) (d) (a) (b) (c) (d) (c) (b) (a)

53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65.

(d) (a) (d) (c) (d) (a) (d) (d) (d) (b) (c) (d) (a)

66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78.

(b) (b) (b) (c) (d) (c) (c) (a) (c) (c) (d) (b) (b)

79. 80. 81. 82. 83. 84. 85. 86. 87. 88. 89. 90. 91.

(a) (c) (c) (c) (a) (d) (b) (c) (b)

92. 93. 94. 95. 96. 97. 98. 99. 100.

(c) (a) (b) (d) (d) (a) (b) (b) (b)

(a) (d) (d) (c)

pRaCtiCe eeRCise Passage 1 It was in Germany and France that the first successful attempts were made to produce an internal-combustion engine driven by petrol. In England, people were strangely timid about horseless vehicles. English inventors were handicapped by a quaint old law that forbade any such vehicle to attain a greater speed than four miles an hour, and compelled each one to be preceded by a man carrying a red flag. This law was not repealed until 1896. The earliest motor cars were looked upon as mere jokes, or as rather dangerous playthings, by everyone except their inventors. Some of them were single-seaters, others would carry two or even three people; but all were noisy, clumsy, queer-looking things. When in 1888, Carl Benz, a German, produced a three-wheeled, internal-combustion car, a great forward stride had been made. Another German, whose name, Daimler, is often seen on motor cars to this day, was experimenting about the same time, and testing a petrol-driven engine. It is easy to understand how the introduction of the petroldriven engine revolutionised road transport throughout the world. Until then the necessary power to push a vehicle along could not

be obtained without the cumbersome tanks, boilers and furnaces of the steam engine. The internal-combustion engine is light in weight and small in size by comparison; the fuel is burned in it, so that there is no waste, like the dusty cinders of a coal fire. 1. How did most people regard early motor cars? (a) Not better than horse-driven vehicles (b) A mere joke, or as rather dangerous playthings (c) A mere scientific experiment (d) A cumbersome vehicle 2. What were all early motor cars? (a) A plaything or toy (b) A vehicle better than horse-driven vehicle (c) Noisy, clumsy, queer-looking things (d) Dangerous and risky 3. What made the English inventors handicapped? (a) The general public did not welcome the invention (b) The quaint old law that forbade any such vehicle to attain a greater speed than four miles/hour

Reading Comprehension ▌ 2.99 (c) Non-availability of adequate fuel to power the engine (d) None of these 4. What does ‘repealed’ mean? (a) repeated (b) abolished (c) contradicted (d) enforced 5. Which among the following words is as closely opposite to ‘clumsy’? (a) unhandy (b) refined (c) unusually large (d) unusual 6. What is incorrect about the internal combustion engine? (a) fuel burnt in it as waste (b) light in weight (c) noisy and clumsy (d) small in size Passage 2 At one time, in the history of India, most women knew very well how to bring up their infants and they lived a perfectly healthy life, free from diseases. The overall standard of women and children in the country was much better than those of other civilisations of that period. But ever since India was exposed to frequent invasions by foreign nations, life became unsafe and property was unprotected, and people were forced to congregate in towns in such a compact way, that it led to awful insanitation and diseases. The traditional knowledge of domestic and personal health and hygiene was ignored. Women were confined indoors for fear of insults and a train of social and unhealthy dangers followed. It is a problem now, how we can restore the original conditions of healthy and happy life in India. This is a socio-economic problem which needs to be given priority to bring back the original culture and restore welfare of women and children in India. 7. What was the main cause of the poor health conditions of women in India? (a) Women were confined indoors (b) Illiteracy among women (c) Frequent foreign invasions (d) Awful sanitation 8. What question has the writer posed before the readers? (a) How to check foreign invasion? (b) Why has the traditional knowledge been ignored? (c) What should be done for infants and women? (d) How can the original conditions of healthy and happy life be restored in India? 9. How did life become unsafe and property unprotected? (a) Because of awful insanitation and diseases (b) Because the women were confined indoors (c) Exposure to frequent foreign invasion (d) People were forced to congregate in towns 10. Why were the women confined indoors? (a) To improve sanitation and healthy atmosphere (b) To keep them away from disease (c) Because of foreign invasions (d) None of these 11. When were women leading a perfectly healthy life? (a) After being confined indoors

(b) Before foreign invasions (c) After restoring original conditions of healthy and happy life (d) Cannot be determined from the passage 12. What does the word ‘congregate’ mean in the passage? (a) forced (b) assemble (c) live (d) settle 13. How are foreign invasions responsible for poor health conditions? (a) Foreigners maltreated the public (b) People were forced to congregate in towns in a compact way, leading to insanitation (c) Foreigners brought in unhealthy ways of life (d) People were made to serve as labour 14. Select from the answer choices the word that is as nearly opposite in meaning to ‘confine’. (a) forced (b) directed (c) to keep out (d) reject Passage 3 An important aspect of the growth of modern towns and cities is the increasing population density. Mr J. P. Orr, the Honorable Chairman of the Mumbai Improvement Trust, pointed out in a recent lecture on the subject, how it affected the health and prosperity of the inhabitants. Life in most thriving towns is intimately connected with the local trades and industries. Unhealthy conditions in factories and workshops, and offensive trades have been prevented in big towns, and this had led to better health. The question of density is more difficult to deal with in India, because, older towns have been built on different principles. People not only want to live healthly but also live in communities that give them greater social convenience, comfort and safety. They care for the health, comfort and beauty of the town, and these conditions of a well-built and well-arranged town are still insisted on by the people. So as long as individuals obey the laws of health, and their habits and customs are free from insanitary effects, mere density of population does not perhaps tend to increase the death rate and harbour diseases. But in the present day, it is apparent that the habits of people have changed greatly and they do not obey the laws of health and cleanliness as well as their forefathers used to do in days when cities expanded and people flourished. It is, therefore, necessary to modify the old plans of city building in the light of modern sanitary laws and requirements. In his lecture, Mr Orr spoke mainly of the density of the city of Mumbai. But his observations are of considerable interest to all those whose population is huddled in narrow quarters, without adequate air and light. 15. The density of population does not increase the death rate and harbour diseases as long as (a) unhealthy conditions in factories and workshops are prevented (b) people obey the laws of health and their habits are free from insanitary effects

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(c) towns are well-built and well arranged (d) the older towns are properly expanded 16. Why is it necessary to modify the old plans of city buildings? (a) Because the population of India is increasing (b) Because the density of population has increased (c) Because the habits of people have greatly changed and they do not obey the health laws. (d) Because people want to live in good and healthy conditions 17. Life in most thriving towns is connected with the (a) density of population (b) size of the town and its planning (c) local trade and industry (d) size of population 18. In addition to health, what do the people want in towns? (a) Lesser density of population (b) Social convenience, comforts and safety (c) Well-planned and well-arranged towns (d) Good factories and good working condition 19. What had led to better health in big towns? (a) Lower density of population (b) revention of unhealthy conditions in factories and workshops (c) Well-built and well planned towns (d) Sanitation and cleanliness 20. What should be kept in mind while modifying the old plans of city buildings? (a) Habits of people living there (b) Convenience and comforts of people living there (c) Modern sanitary laws and requirements (d) Size and density of population 21. The word ‘thriving’ used in the above passage means (a) populated (b) modern (c) growing (d) well planned 22. Select from the answer choices a suitable synonym to ‘huddle’. (a) grow (b) heaped together (c) scatter (d) sporadic 23. Select from the answer choices the word that is as closely opposite in meaning (antonym) to ‘harbour’. (a) shelter (b) cherish (c) banish (d) infectious 24. Which of the following statements is incorrect according to the passage. (a) Life in most thriving towns is intimately connected with the local trades and industries (b) It is necessary to modify the old plans of city building (c) The question of density is more difficult to deal in India because the older towns have been built on different principles (d) The density of population tends to increase death rate and harbour diseases

Passage 4 Read the following passage carefully. Note the words italicised in the passage and try to understand their meanings in the context of the passage and answer the questions given below it. On August 15, 1947, India was granted independence from British colonial dominance. This was an event of international significance. From this day onwards, the Indian Union assumed a role in world politics that was appropriate enough to modify the thinking of nations. Concepts like non-alignment, tolerance, non-violence and peaceful coexistance were introduced by India into the international vocabulary. “Our dreams concern India”, said the first Prime Minister of the country, Pt Jawaharlal Nehru, on the eve of achieving freedom, “but they also concern the whole world, for the world has become one”. Mahatma Gandhi chose Nehru to become the first PM of independent India. With his vision of unity of the nation, he became the architect of modern India. It was to be a federal state based on secularism—the commitment to an Indian identity, which was above all races, castes, and religions. It was a vision of extraordinary dimensions. Both Nehru as well as his daughter and successor, Smt Indira Gandhi, were well aware that only a firm anchorage in the Indian identity could unite the nation and enable it to make an impact on the world. In these days of mutual interdependence, the western powers and America cannot afford to ignore what is happening in this region. It is possible that the fate of the Asian democracies would one day be decided on the banks of the Ganges. If India fails to remain a democracy, this would trigger the fall of the whole of South and South-east Asia. That is why the Indian Prime Minister has to play a role that goes well beyond duties related to internal affairs only. It is of international political significance. 25. What made Pt Jawaharlal Nehru the architect of modern India? (a) He secured independence for India (b) His vision of the unity of the nation (c) Because he was the first Prime Minister of India (d) He wanted India to make an impact on the world 26. Which of the following concepts have been introduced by India into the international vocabulary? (a) Mutual interdependence and democracy (b) Non-alignment and tolerance (c) Non-violence and peaceful co-existence (a) A and C only (b) A and B only (c) B and C only (d) All the above 27. Which word in this passage is used to describe the phrase “to live together with mutual tolerance”? (a) non-alignment (b) coexistence (c) secularism (d) anchorage

Reading Comprehension ▌ 2.101 28. Which among the following words best describes the meaning of the word ‘trigger’ used in the passage? (a) to set in action (b) to become uncontrollable (c) to become inactive (d) aggravate

35. By spending his time in the garden, Tagore developed the habit of (a) meditation (b) observing things carefully (c) taking life easy (d) writing and relaxing

29. Select from the answer choices, the word that is synonymous to ‘Dimensions’. (a) support (b) size (c) importance (d) height

Passage 6 Until very recently, it was universally believed that men are congenitally more intelligent than women; even so enlightened a man as Spinoza decided against voting for women on this ground. Among white men, it is held that white men are by nature superior to black men; in Japan, on the contrary, it is thought that yellow is the best colour. In Haiti, when they make statues of Christ and Satan, they make Christ black and Satan white. Aristotle and Plato considered Greeks so innately superior to barbarians that slavery was justified as long as the master was Greek and the slave a barbarian.

30. Which of the following words can be used to replace the word ‘repercussion’, used in the passage, without altering the meaning of the writer? (a) impression (b) reaction (c) suppression (d) alteration 31. Why does the role of the Indian Prime Minister go beyond the duties related to internal affairs of the country? (a) To create an impact on foreign countries (b) Because happenings in India influence politics and democracies of South and South-east Asia (c) To avoid interference of America and other big powers (d) None of these 32. Select the word that is as nearly opposite in meaning to ‘dominance’ used in this passage. (a) government (b) rule (c) subordination (d) importance Passage 5 Among the chief sources of education available to Tagore was a quiet garden adjoining his family house. Here he used to spend much of his time, absorbing the peace and beauty of nature. It was through this early contact with nature that he acquired the serenity of mood that distinguished him all his life. It was in this garden that he came to understand the principle of harmony that was at work throughout the Universe. At the same time, he formed the habit of observing and reflecting on things. 33. How did Tagore spend much of his time in the garden adjoining his family house? (a) Reading literary books (b) Plucking flowers and enjoying their fragrance (c) Enjoying natural peace and beauty (d) Sleeping in natural air and thinking in poetry 34. How did the garden near Tagore’s house serve him? (a) As a means of peace (b) As a means of education (c) As a source of entertainment (d) A good passing time to relax in natural beauty

36. In Haiti, people believe that (a) Christ was black (c) all black men were evil

(b) black was good (d) all white men were evil

37. Aristotle and Plato supported slavery because they thought (a) Greeks to be superior to barbarians (b) the barbarians were hardworking (c) the barbarians to be superior to Greeks (d) the barbarians had no other means to earn

38. Spinoza decided against voting for women because according to him (a) (b) (c) (d)

they did not deserve to have votes they were generally unintelligent they were naturally less gifted with intelligence they were not trustworthy

Passage 7 Mikhail Gorbachev’s ouster, though dramatic in every respect, is on no account a surprise. Both his foes and his closest friends have been warning of it with a heightened sense of urgency for the past several months. Its consequences, however, are wholly unpredictable. The USSR could well witness protracted violence should the reformists and those republics which have sought varying degrees of sovereignity for themselves choose to defy central authority. It is possible that the country after an initial period of uncertainty, and perhaps even violence, could revert to the pre-perestroika system. Equally uncertain is the course of East-West relations. These are bound to deteriorate though the extent of deterioration must remain a matter of conjecture. Hailed abroad as a leader who had dared to free Soviet citizens from fear, enabled the countries of Eastern Europe to become democracies even as they regained their full sovereign status, paved the way for the reunification of Germany and exposed the moribund and totalitarian character of communism, he, at home appeared to come under fire from all sides. 39. The removal of Mikhail Gorbachev from power is (a) dramatic but expected (b) uncalled for and unexpected

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(c) strange and cruel (d) good for the country 40. During Gorbachev’s reign, the Soviet people were (a) not free to express (b) afraid to speak against the government (c) committed to communism (d) not secure 41. The relations between the Soviet Union and the Western countries (a) are likely to remain unaffected (b) will definitely get worse (c) may improve considerably (d) will improve but slowly 42. The post-Gorbachev era may witness (a) a more open economy (b) reversal of perestroika (c) greater role for economic reform (d) weak and fragile economy 43. As a result of Gorbachev’s policies, the countries of Eastern Europe became (a) democratic and truly independent (b) authoritarian and inhuman (c) united and totalitarian (d) democratic but with a monarchy Passage 8 Sunita was extremely happy when she got a nice little house in an area near her place of work. But her happiness was shortlived. Her maid servant began to bring home rumours of thefts and other sordid happenings in the area. Since Sunita was alone in that house and a stranger to the place, she decided to call her younger brother to come and stay with her for a few days. But he was too tied down with office work to be able to spare a few days of leave. So Sunita became very nervous. She did not know what she would do if anything happened. She took special care to hide her house keys. Every night she hid them at a different place. She also had an alarm fixed so that she could use it in case of an emergency. 44. Why did Sunita’s happiness not last long? (a) Because she was lonely (b) Because she did not find herself safe in the house (c) Because of rumours (d) Because her brother did not come to stay with her 45. Why did Sunita want her brother to come? (a) Because she loved him very much (b) Because she was lonely and her servant had frightened her (c) Because she wanted to help in studies (d) She was advised to call her brother by the maid servant 46. Sunita hid her keys in a different place every night because (a) she did not want to lose them (b) she was advised by her maid servant to do so

(c) she did not want thieves to know where her keys were (d) she had no wardrobe or cupboards in the house Passage 9 Swami Dayanand was the first to use the word ‘Swaraja’, even before Dadabhai Naoroji and Tilak used it. The British regarded Arya Samaj as a seditious organisation and during the martial law days (1919) students of DAV College, Lahore, were ordered to go for roll call by the police thrice a day. It is now being increasingly accepted that during the Sepoy Revolt of 1857 Dayanand played a historic role, not as a solider but by inspiring the freedom fighters. It was (and is even now) taught in school textbooks that the Aryans migrated here from Central Asia. Dayanand was the first to debunk that view and assert that the Aryans were the original residents of Arya Varta (India); now many historians agree with him. Indians told the British rulers: “You quit because you are foreigners.” Their reply was: “You too were migrants from other lands. Only you came much before us. So first you quit and then we will see.” They invented this theory of Aryan migration as a part of their imperial policy of ‘divide and rule’. The aim was to put the Dravidians—according to them original residents of this country—against the Aryan invaders. Swami Dayanand said: “Ours is not any independent religion. I am under the Vedas and my duty is to preach the Truth.” Dayanand in his own humble way said: “I am not a knowall. If you discover any error in me, you may correct it after reason and argument.” “Anything that cannot be justified by reason and commonsense must be rejected as irreligious”—an idea too revolutionary for his age. 47. Swami Dayanand considered it his duty to (a) send back the British from India (b) guide the Sepoy Mutiny of 1857 (c) preach Vedas and the Truth (d) preach the doctrine of reasoning in religion 48. Swami Dayanand wanted to tell the British that (a) their policy of ‘divide and rule’ was not acceptable (b) the Aryans were the original residents of India (c) the Aryans migrated from Central Asia (d) he would soon launch Quit India Movement to remove the British rule 49. The British considered Arya Samaj as a/an (a) seditious organisation (b) foreign organisation (c) organisation to fight the Dravadians (d) anti-Hindu organisation 50. During the Sepoy Revolt of 1857, what was the role of Swami Dayanand? (a) to fight as a brave soldier (b) to inspire the freedom fighters (c) to lead the mutiny (d) to ensure that there were no merciless killings

Reading Comprehension ▌ 2.103

answeR Key 1. 2. 3. 4. 5. 6. 7.

(b) (c) (b) (b) (b) (c) (c)

8. 9. 10. 11. 12. 13. 14.

(d) (c) (c) (b) (b) (b) (c)

15. 16. 17. 18. 19. 20. 21.

(b) (c) (c) (b) (b) (c) (c)

22. 23. 24. 25. 26. 27. 28.

(b) (c) (d) (b) (c) (c) (a)

29. 30. 31. 32. 33. 34. 35.

(c) (b) (b) (c) (c) (b) (b)

36. 37. 38. 39. 40. 41. 42.

(d) (a) (c) (a) (c) (b) (b)

43. 44. 45. 46. 47. 48. 49.

(a)

50. (b)

(b) (b) (c) (c) (b) (a)

nda BoX Directions: In this section you have three short passages. After each passage you will find several questions based onthe passage. First, read Passage 1, and answer the questions based on it. Then go on to the other passages. Passage 1 There is a natural tendency in every animate being to restrictions. If you block the growth of a plant, it will try to pierce through the blockade. If you enclose a worm fish in a glass jar, you will find it restless moving hither and thither to find a way out. Similarly, if you confine a bird or an animal in a cage or a room, you will find it constantly trying to break through the cage or the room in a fit of escape. Only when it gets tired or identifies itself with the environment of the prison, will it sit silent. Thus, the urge to be free is inborn in all animate beings. It gets suppressed only on its identification with the situation. The urge to be free is also present in inanimate objects but it is latent and ultimately gets fulfilled. It is for this reason that matter constantly changes its shape. The change is in way a fulfillment 1. The natural tendency among the living beings is to (a) submit to the circumstances (b) strive for freedom (c) place restrictions in the way of others (d) create blockades 2. A worm in a glass jar first tries to (a) adjust itself (b) break the jar (c) move out of it (d) ignore the change 3. An animal will sit quietly in a cage after it has (a) eaten well (b) struggled enough (c) identified itself with the changed circumstances (d) given up hope of freedom 4. Matter constantly changes its shape because it (a) is inanimate (b) also has an urge to be free (c) is moulded into different shapes by men (d) has no sou.

Passage 2 At what age should school begin? The answer must depend upon the home, its topography rather than upon its oral or psychological character. A child who lives on a farm in the countryside can happily and profitably spend his time wandering about, watching animals, reaping, threshing etc. until the time comes when it is necessary to begin formal instruction. But for the urban child whose parents live in a cramped apartment it is quite otherwise. For him, school is desirable as an escape into freedom—freedom of movement, freedom of noise, and freedom of companion ship. 5. Topography means the science of (a) drawing maps (b) description of features of a place (c) shooting photographs (d) description of topics 6. formal instruction here implies (a) school discipline (b) rigid code (c) regular education at school (d) disciplined conduct of teachers 7. The author believes that (a) A child in a countryside need not have formal instruction (b) an urban child likes freedom of all kinds (c) countryside child is benefited by free environment (d) an urban child must begin formal instruction late 8. The author thinks that the problem as to when the children should go to school depends rather on the (a) psychological character of home (b) moral character of home (c) topographical character of home (d) spiritual character of home Passage 3 If you want your mate to stop guessing about your feeling and motives, you have to be prepared to reveal yourself. And in

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order to reveal yourself, you have to know yourself. You can’t talk openly and honestly with your mate until you have tried being honest with yourself first. Take time off to be alone. Use that time not simply to engage in passive meditation but to carry on an active inner dialogue with yourself. Between the person you think you are and the inner you that operates at gut level. Communicating with yourself. involves revelation, self-analysis and re-evaluation. Change is impossible otherwise. No architect in any field will attempt to make new plans without assessing the presents situation. So, each of us can and must make an assessment of our own assets or liabilities. 9. Your mate shall stop guessing about your feelings when (a) you are honest (b) you know yourself (c) he reveals himself to you (d) you talk openly 10. You can talk openly and honestly to your mate only when you (a) reveal your emotions and feelings (b) are honest with him (c) do not engage yourself in passive meditation (d) are honest with yourself 11. You should take time off to be alone so that (a) you talk to yourself (b) you engage yourself in passive meditation (c) your motives and feelings are clear (d) you do not change yourself 12. No architect plans without assessing the present situation because (a) each of us can and “must make an assessment (b) self-analysis and re-evaluation are necessary for him (c) change is impossible otherwise (d) no assessment of our own assets and liabilities is Possible Directions (For the following 10 items): In this section you will have three short passages. After each passage you will find several questions based on the passage. First, read Passage 1. and answer the questions based on it. Then go on to the other passages. You are required to select your answers based on the contents of the passage and opinion of the author only. Passage 1 As society becomes more and more affluent, people would prefer to work shorter hours at their regular employment so that they may enjoy longer hours of leisure during which they can indulge in more congenial activities. They will not like to waste their leisure time performing routine domestic chores which tend to sap their energy. The idea of a robot slave which would be capable of performing many of these monotonous chores is one that will appeal to many. Already robots are used in mines and factories to perform tasks which are too risky for human beings. It takes a lot of research by dedicated scientists and sufficient financial backing for a model to be developed to perform a new

task. This in turn depends on the number of people who feel the need for a robot to perform these specific tasks. The problem of designing a robot capable of performing various routine tasks around the house is not insurmount able. A domestic robot slave with its own computer and response mechanism and source of power, which can be programmed to perform certain specific tasks is not an impossibility. 13. In an affluent society, people prefer to work shorter hours because (a) they are lazy (b) they do not have the strength to work (c) they want time to carry out other activities (d) they abhor their work 14. Domestic chores (a) are welcome by housewives (b) are too demanding (c) lead to family discord (d) are too time-consuming 15. Robots are already used in mines and factories to perform (a) hazardous tasks (b) tasks with greater efficiency than human beings (c) tasks which lead to economy in the operations (d) tasks which are forbidden by the labour laws 16. Developing a robot capable of performing a new task (a) is difficult for the scientists (b) is impossible for the scientists (c) requires huge capital (d) depends on the number of people who require it Passage 2 The energy crisis has been with us for a long time, and will be with us even longer. Whether Arab oil flows freely or not, it is clear that the world industry cannot be allowed to depend on so fragile a base. The supply of oil can be shut off at whim any time. In any case, the oil wells will run dry in about thirty years. New sources of energy must be found, and this will take time. But it is not likely to restore that sense of copious energy availability we had in the past. 17. In the writer’s opinion, the energy crisis (a) is an outcome of unregulated supply of oil to industries. (b) will subside as soon as the Arab oil begins to flow again. (c) is very likely to lead to the world war. (d) will remain with us for an almost indefinite period of time. 18. The phrase ‘so fragile a base’ refers to the (a) current energy crisis in the world. (b) uncertainty about the flow of Arab oil. (c) delay in finding the new energy sources. (d) drying up of the oil wells in future. 19. The type of writing in the given passage could be called (a) official (b) scientific (c) popular (d) political

Reading Comprehension ▌ 2.105 Passage 3 All who have gone before you have submitted to the stroke of death. All who come after you shall undergo the same fate. The great and the good, the prince and the peasant, the renowned and the obscure travel alike the road which leads to the grave. At the moment when you expire, thousands throughout the world shall, with you, be yielding up their breath. Can that be held to be a great calamity which is common to you with everything that lives on the earth which is an event as much according to the course of the nature, as it is that leaves shall fall in autumn, or that fruit should drop from the tree when it is fully ripe. 20. Death is a natural phenomenon because (a) the moment you die, thousands of people all over the world die with you (b) it is as natural as the fall of leaves or fruits from the trees (c) many people have died in the past (d) the great and the good, the prince and the peasant, the renowned and the obscure, all die 21. The author compares death with the fall of the ripe fruit from a tree to show that death (a) occurs in nature also (b) is a kind of fall from a great height (c) gives freedom from the ordeals of life (d) occurs with the achievement of fullness 22. Which one of the following sentences shows that death is a great leveller? (a) All who come after you shall undergo the same fate (b) The great and the good, the prince and the peasant, the renowned and the obscure travel alike on the road which leads to the grave. (c) At the moment- when you expire, thousands throughout the world shall, with you, be yielding up their breath. (d) All who have gone before you have submitted to the stroke of death. Directions (For the following 7 items): In this section you have two short passages. After each passage, you will find some questions based on the passage. First, read the passage and then answer the questions based on it. You are required to select your answers based on the contents of the passage and opinion of the author only. Passage 1 Yes, there were giants before the Jam Sahib (the great Indian cricketer, Kumar Shree Ranjit singhji, better known to the world of cricket as Ranji). And yet I think it is undeniable that as a batsman, the Indian will live as the supreme exponent of the Englishman’s game. The claim does not rest simply on his achievements although, judged by them, the claim could be sustained. His season’s average of 87 with a total of over 3,000 runs, is easily the high-water mark of English cricket. Thrice he has totalled over 3,000 runs and no one else has equalled that record. And is not his achievement

astonishing—of scoring two double centuries in a single match on a single day—not against a feeble attack, but against Yorkshire, always the most resolute and resourceful of bowling teams? But we do not judge a cricketer so much by the runs he gets as by the way he gets them. “In literature as in finance,” says Washington Irving, “much paper and much poverty may co-exist.” And in cricket, too many runs and much dullness may be associated. If cricket is menaced with creeping paralysis, it is because it is losing the spirit of joyous adventure and becoming a mere instrument for compiling tables of averages. There are dull, mechanical fellows who turn out runs with as little emotion as a machine turns out pins. There is no colour, no enthusiasm, no character in their play. Cricket is not an adventure to them; it is a business. It was so with Shrewsbury. His technical perfection was astonishing; but the soul of the game was wanting in him. There was no sunshine in his play, no swift surprise or splendid unselfishness. And without these things, without gaiety, daring, and the spirit of sacrifice cricket is a dead thing. Now the Jam Sahib has the root of the matter in him. His play is as sunny as his face. He is not a miser hoarding up runs, but a millionaire spending them, with a splendid yet judicious prodigality. It is as though his pockets are bursting with runs that he wants to shower with his blessings upon the expectant multitude. It is not difficult to believe that in his little kingdom Nawangar, where he has power of life and death in his hands, he is extremely popular for it is obvious that his pleasure is in giving pleasure. 23. Consider the following statements: 1. Yorkshire did not have potent bowling attack because the Jam Sahib was only a specialist batsman. 2. Cricket loses its flamboyant spirit when players like Shrewsbury are at the batting crease. Which of the above statement is/are correct? (a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2 24. Consider the following statements: 1. It is statistics which matter most when true worth of a cricketer is estimated. 2. Surfeit of runs scored in a cricket match may not always lend excitement to a cricket match. Which of the above statements given is/are correct? (a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2 25. Consider the following statements: 1. Many cricketers suffer from paralysis in their old age because of excessive physical stress undergone during their cricketing careers. 2. Boards of cricket of the cricket playing nations should consist of past cricketers. Which of the above statements above is/are correct? (a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2 26. Consider the following statements: 1. While batting, the Jam Sahib was very unselfish since he ensured that his batting partner never got run-out.

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2. The Jam Sahib carried his magnanimity in his nature even while dealing with people in his kingdom. Which of the above statements given is/arc correct? (a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2 Passage 2 Happy is the man who acquires the habit of reading when he is young. He has secured a life-long source of pleasure, instruction and inspiration. So long as he has his beloved books, he need never feel lonely. He always has a pleasant occupation of leisure moments, so that he need never feel bored. He is the possessor of wealth more precious than gold. Ruskin calls books, “Kings’ Treasures”—treasuries filled, not with gold and silver and precious stones, but with riches much more valuable than these—knowledge, noble thoughts and high ideals. Poor indeed is the man who does not read, and empty is his life. The blessings which the reading habit confers on its possessor are many, provided we choose the right kind of books. Reading gives the highest kind of pleasure. Some books we read simply for pleasure and amusement—for example, good novels. And novels and books of imagination must have their place in everybody’s reading. When we are tired, or the brain is weary with serious study, it is healthy recreation to lose ourselves in some absorbing story written by a master hand. But to read nothing but books of fiction is like eating nothing but cakes and sweetmeats. As we need plain, wholesome food for the body, so we must have serious reading for the mind. And here we can choose according to our taste. There are many noble books on history, biography, philosophy, religion, travel and science which we ought to read, and which will give us not only pleasure but also education. And we can develop a taste for serious reading, so that in the end it will give us more solid pleasure than even novels and books of fiction. Nor should poetry be neglected, for the best poetry gives us noble thoughts and beautiful imaginings clothed in lovely and musical language. Books are the most faithful of friends. Our friends may change, or die; but our books are always patiently waiting to talk to us. They are never cross, peevish or unwilling to converse: as our friends sometimes are. No wonder a reader becomes a “book-lover”. 27.

Consider the following statements: 1. Books are far more pleasant than reticent friends. 2. To get maximum pleasure, one has to be very judicious in making a correct selection of books. Which of the above statements is/are correct? (a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2

28. Consider the following statements: 1. Serious reading provides education, enhances intellect but hardly provides any pleasure which books of fiction can offer. 2. Book reading habit provides inspiration only during the formative years of the reader. Which of the above statements is/are correct? (a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2

29. Consider the following statements: 1. Book reading helps a person to overcome solitude. 2. Since poetry is based on imagination, it need not find place in everybody’s reading. Which of the above statements is/are correct? (a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2 Directions (For the next 5 items): In this section you have a short passage. After the passage, you will find some questions based on the passage. First, read the passage and then answer the questions based on it. You are required to select your answers based on the contents of the passage and opinion of the author only. ‘Examples ‘I’ and ‘J’ are solved for you. Passage In our approach to life, be it pragmatic or otherwise, a basic fact that confronts us squarely and unmistakably is the desire for peace, security and happiness. Different forms of life at different levels of existence make up the teeming denizens of this earth of ours. And, no matter whether they belong to the higher groups such as human beings or to the lower groups such as animals, all beings primarily seek peace, comfort and security. Life is as dear to a mute creature as it is to a man. Even the lowliest insect strives for protection against dangers that threaten its life. Just as each one of us wants to live and nor to die, so do all other creatures. I. The authors main point is that (a) different forms of life are found on earth (b) different levels of existence are possible in nature (c) peace and security are the chief goals of all living (d) even the weakest creature struggles to preserve its life C.J. Which one of the following assumptions or steps is essential in developing the author’s position? (a) All forms of life have a single over riding goal (b) The will to survive of a creature is identified with a desire for peace (c) All beings are divided into higher and lower groups (d) A parallel is drawn between happiness and life, and pain and death Explanation: I. The idea which represents the author’s main point is “peace and security are the chief goals of all living beings”, which is response (c). So (c) is the correct answer. J. The best assumption underlying the passage is “The will to survive of a creature is identified with a desire for peace”, which is response (b). So (b) is the correct answer. Passage 1 Today, India looks to be on course to join the league of developed nations. It is beginning to establish a reputation not just as the technology nerve-centre and backoffice to the world, but also as its production centre. India’s secularism and democracy serve as a role

Reading Comprehension ▌ 2.107 model to other developing countries. There is great pride in an India that easily integrates with a global economy, yet maintains a unique cultural identity. But what is breathtaking is India’s youth. For despite an ancient civilization that traces itself to the very dawn of human habitation, India is among the youngest countries in the world. More than half the country is under 25 years of age and more than a third is under 15 years of age. Brought up in the shadow of the rise of India’s service industry boom, this group feels it can be at least as good if not better than anyone else in the world. This confidence has them demonstrating a great propensity to consume, throwing away ageing ideas of asceticism and thrift. The economic activity created by this combination of a growing labour pool and rising consumption demand is enough to propel India to double-digit economic growth for decades. This opportunity also represents the greatest threat to India’s future. If the youth of India are not properly educated and if there are not enough jobs created, India will have forever lost its opportunity. India’s Information Technology and Business Process Outsourcing industries are engines of job creation, but they still account for only 0.2 per cent of India’s employment. The country has no choice but to dramatically industrialize and inflate its economy. According to a recent survey, more than half of India’s unemployed within the next decade could be its educated youth. 30. Consider the following statements: 1. India’s rich cultural heritage prevents India from surging ahead to become an active partner in the global economy. 2. By and large, India’s youth still believe in a thrifty lifestyle. Which of the statements given above is/are correct? (a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2 31. What is the approximate number of people in India who are in the age group 15-25 years? (a) 500 million (b) 350 million (c) 210 million (d) 180 million 32. In the recent past, which sector has witnessed a phenomenal growth ? (a) Heavy industry (b) Service industry (c) Petrochemical industry (d) Textile industry 33. Consider the following statements: 1. Rising consumption demand will retard economic growth. 2. India’s youth are its greatest opportunity as well as threat. Which of the statements given above is/are correct? (a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2 34. Consider the following statements: 1. Information Technology sector provides a relatively large proportion of jobs in India. 2. In the coming decade only uneducated youth will remain deprived of employment opportunity. Which of the statements given above is/are correct?

(a) 1 only (c) Both 1 and 2

(b) 2 only (d) Neither 1 nor 2

Directions (For the 10 items which follow): In this Section, you have three short passages. After each passage, you will find several question based on the passage. First, read a Passage, and then answer the questions based on it. You are required to select your answers based on the contents of the passage and opinion of the author only. Examples ‘I’ and ‘J’ are solved for you. Passage In our approach to life, be it pragmatic or otherwise, a basic fact that confronts us squarely and unmistakably is the desire for peace, security and happiness. Different forms of life at different levels of existence make up the teeming denizens of this earth of ours. And, no matter whether they belong to the higher groups such as human beings or to the lower groups such as animals, all beings primarily seek peace, comfort and security. Life is as dear to a mute creature as it is to a man. Even the lowliest insect strives for protection against dangers that threaten its life. Just as each one of us wants to live and nor to die, so do all other creatures. I. The authors main point is that (a) different forms of life are found on earth (b) different levels of existence are possible in nature (c) peace and security are the chief goals of all living (d) even the weakest creature struggles to preserve its life C.J. Which one of the following assumptions or steps is essential in developing the author’s position? (a) All forms of life have a single over riding goal (b) The will to survive of a creature is identified with a desire for peace (c) All beings are divided into higher and lower groups (d) A parallel is drawn between happiness and life, and pain and death Explanation: I. The idea which represents the author’s main point is “peace and security are the chief goals of all living beings”, which is response (c). So (c) is the correct answer. . J. The best assumption underlying the passage is “The will to survive of a creature is identified with a desire for peace”, which is response (b). So (b) is the correct answer. Passage 1 The young are those to whom we look for future strength and for future good; and the longer we live, the more anxious we become that they who to be the fresh recruits should be morally of right stature. Around them are peculiar temptations and trials, witching, cunning, insidious and forceful: and we are obliged to see thousands falling by the way, whose fall seems needless. They, like ourselves, are to have but one chance in life. We who are somewhat advanced in years, seeing how many perils there are round about that one chance, feel an earnest desire that every advantage should be given to those who are coming onto our places. We can live but once; and life is usually moulded and takes its shape very early.

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35. Which one of the following is correct? The author looks upon the young as (a) handsome and healthy (b) an embodiment of possibilities (c) strong and hardworking (d) a group of boys and girls who are obedient and Dutiful 36. What does the phrase “morally of right stature” mean? (a) Being highly educated (b) Having a good personality (c) Having rectitude (d) Felling superior to others 37. Which one of the following is the correct? The failure of many a young men and women is (a) well deserved (b) un warranted (c) fortuitous (d) sad Passage 2 There are a great many people who have all the material conditions of happiness i.e., health, sufficient income and clout, but who nevertheless, are profoundly unhappy. In such cases it would seem as if the fault must lie with wrong theory as to how to live. We imagine ourselves more different from the animals than we are. Animals live on impulse and are happy as long as external conditions are favourable. Your needs are more complex than those of your pets but they still have their basis in instinct. In civilized societies, this is too apt to be forgotten. People propose to themselves some paramount objective and restrains all impulses that do not minister to it. A business man may be so anxious to grow rich that to this end he sacrifices health and private affection. When at last he has become rich, no pleasure remains to him except harrying other people by exhortations to initiate his noble example. 38. Which one of the following is the correct? The “material conditions of happiness” do not include (a) health (b) money (c) power (d) hope 39. Which one of the following is the correct? Modern man is very unhappy because he (a) is always busy making money (b) feels alienated from his fellow beings (c) cannot communicate with animals (d) suppresses his inner urges

(c) we are really not very different from animals (d) we have the unique ability to control our Impulses 41. What does the phrase “do not minister to it” mean? (a) Do not support it (b) Do not oppose it (c) Are ignorant about it (d) Are careless about it Passage 3 The first of the political causes of the war is war itself. Many wars have been fought, among other reasons, for the sake of seizing strategically valuable piece of territory, or in order to secure a ‘natural’ frontier, that is to say, a frontier which is easy to defend and from which it is easy to launch attacks upon one’s neighbours. Purely military advantages are almost as highly praised by the rulers of nations as economic advantages. The possession of an army, navy and air force is itself a reason for going to a war. ‘We must use our forces now’, so runs the militarist’s argument, ‘in order that we may be in a position to use them to better effect next time.’ 42. Why have wars been fought? (a) To use weapons and make room for fresh purchase (b) Because people want to show their neighbours that they are strong (c) To capture some areas of another country which are of strategic importance (d) To teach neighbouring countries a good lesson 43. What does a ‘natural’ frontier mean? (a) An area on the border from where you can keep watch on or attack your enemy (b) Some place on the border of a country having beautiful natural scenery (c) A borderline that has been naturally chosen by two neighbouring countries (d) A sudden gift of land by nature because of sudden change in the course of a river 44. Which one of the following is correct? ‘Military advantage’ and ‘economic advantage’ (a) are the same for a country (b) may or may not be the same but the rulers make them appear to be the same (c) are completely differently for a country (d) go against each other

40. Which one of the following is the correct? The author is of the opinion that (a) we are superior to animals (b) animals are more impulsive than us

answeR Key 1. 2. 3. 4. 5. 6.

(d) (c) (c) (b) (b) (c)

7. 8. 9. 10. 11. 12.

(b) (c) (b) (d) (a) (c)

13. 14. 15. 16. 17. 18.

(c) (d) (a) (d) (c) (d)

19. 20. 21. 22. 23. 24.

(c) (b) (a) (b) (b) (b)

25. 26. 27. 28. 29. 30.

(d) (d) (b) (d) (a) (d)

31. 32. 33. 34. 35. 36.

(d) (b) (b) (d) (c) (c)

37. 38. 39. 40. 41. 42.

(b) (d) (d) (d) (a) (c)

43. (a) 44. (b)

PART B

GENERAL KNOWLEDGE CHAPTER 1. INDIAN HISTORY CHAPTER 2. GEOGRAPHY CHAPTER 3. INDIAN POLITY CHAPTER 4. INDIAN ECONOMY CHAPTER 5. GENERAL SCIENCE CHAPTER 6. BASIC GENERAL KNOWLEDGE

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Indian History ANCIENT INDIA Indus Valley Civilization Period Between 3000 and 1500 bc . Some Indian historians, however, believe that the Indus Valley Civilization dates back to 3250 bc. Cities Mohenjodaro, Harappa and Lothal. In India, important sites connected with the Indus Valley Civilization are: Lothal near Ahmedabad (in Gujarat), Kalibangan in Rajasthan, Banwali in Hissar district of Haryana and Ropar near Chandigarh (in Punjab).

Characteristics Political It seems there was a central government as per the evidence from the existence of assembly hall and citadels. Weapons of war like axes, spears, daggers, bows and arrows were made of stone. Leadership is said to have been vested with merchants and traders. Socio-economic The Indus Valley Civilization people sowed seeds in the flood plains in November when the flood water receded and reaped their harvest of wheat and barley in April, before the advent of the next flood. Wheat, rice, barley, milk, dates, fish, eggs and animal flesh formed their staple food. Cotton was first produced by the Indus valley people. Spun and woven cotton and wool dresses were used by them. Agriculture, hunting, fishing and rearing of animals/birds was their main source of livelihood. Ornaments of silver, gold, ivory, copper and precious stones were used. They carried on considerable amount of trade in stones, metals, shells, etc., within their cultural zone. They were goods traders and might have carried on all exchanges through barter. They knew the use of wheel and solid wooden wheels fixed by

bullock carts and boats were used as means of road and river transportation. Town planning Great buildings, double-storeyed dwellings, and drainage system were in existence. There were planned cities and roads. Mortar and baked bricks were used for construction of dwellings. Religious They worshipped Mother Goddess, Pashupatinath, stones, trees and animals. Script Not yet deciphered. Decay of Indus Valley Civilization Invasion of the Aryans, recurrent floods and other possible natural causes like earthquakes, are said to be responsible for the disappearance of the Indus Valley Civilization.

The Vedic Period Early Vedic Age Period 1500–1000 bc Who were the Aryans? The Aryans were semi-nomadic pastoral people who originally inhabited the area around the Caspian Sea in Central Asia. The Aryans entered India probably through the Khyber Pass (Hindukush Mountains) around 1500 in more than two waves in search of new pastures. Region The Aryans first settled down in Punjab and later moved eastwards and spread all over the Gangetic plain. The area of this civilization extended from Afghanistan to Punjab and western Uttar Pradesh, called Sapta Sindhava, the land of the seven rivers.

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Characteristics Political The family (kula) was the ultimate basis of the Vedic state. They lived in villages (grams). Each village had a chief or a gramani. An elected panchayat managed the affairs of the village. An aggregate of villages made up the vis (district or clan) and a group of vis made a jana (tribe). Every tribe had a hereditary chieftain, whose duty included protection and welfare of the members of his tribe. The chieftain only ruled over his jana and not over any specified territory. The chief king (rajan) had two assemblies—sabha, constituting elders, and samiti, the national assembly. The king was by no means an autocrat. His powers were limited by the ‘will of people’ expressed in the Sabha or Samiti. The king led the tribe in battle and ensured their protection, in return for which the people rendered him obedience and voluntary gifts or bali. Social The Aryans were a semi-nomadic pastoral people who had a joint family system. They lived in groups and spoke Sanskrit. The father was the head of the joint family and was known as grihapati. Women held a respectable social status. Their dress was simple. They were fond of outdoor activity like dancing, wrestling, boxing, etc. Dwellings were generally made of wood. Economic The economy of the Aryans was largely rural based. They had knowledge of agriculture and it was their main occupation. The Rig Veda mentions artisans such as potters, weavers, carpenters, leather workers, chariot makers, etc. They also knew metal-work. Cattle breeding was another source of livelihood. Cattle was the chief measure of wealth and a wealthy man who owned many heads of cattle was known as gomat. In the Rig Veda gavyuti is used as a measure of distance and godhuli as a measure of time. The use of fire and stone tools is also mentioned in the Rig Veda. Religious Being lovers of nature, Aryans worshipped the sun, water, fire, etc. Yagna or havan was an essential part of their religion. They are said to have been the originators of the Hindu civilization. Religious Books of the Aryans The Vedas are the most sacred books of early Aryans. There were four Vedas and the Brahmanas concerned with these Vedas are: 1. Rig Veda (Brahamani Aitaraya and Kaushitika Brahamana) 2. Sama Veda (Jaminya Brahamana and Tandyamaha Brahamana) 3. Yajur Veda (Satpatha Brahamana) 4. Atharva Veda (Gopatha Brahamana)

Later Vedic Period Period 1000–600 bc Region The history of the later Vedic period is based mainly on the Vedic texts which were compiled in the Post Rig Vedic era.

These texts reveal that the Aryans expanded from Punjab over the whole of western Uttar Pradesh covered by the Ganga–Yamuna Doab. On excavation of these regions, archaeologists have found the remains of few cities of this period out of which Hastinapur, Ahichatra and Kausambi form prominent examples.

Characteristics Political More developed than the early Vedic period, the tiny tribal settlements were replaced by strong kingdoms. Powers of the king, who was called samrat increased phenomenally. Importance of assemblies declined. A regular army was maintained for the protection of the kingdom. The formation of wider kingdoms made the king more powerful. In the later Vedic times, as the popular assemblies lost importance, the royal power increased its influence. Social Growth of big cities like Ayodhya, Indraprastha and Mathura were seen. Women enjoyed freedom and respect but their status deteriorated compared to the early Vedic period. Economic In addition to agriculture and cattle-rearing, trade and industry also gradually began. Religious This was also called the Brahmanical age which came very close to modern Hinduism. Caste system The Hindu society was divided into four castes, initially based on occupation, which later became hereditary, viz., 1. 2. 3. 4.

Brahmins (Priestly class) Kshatriyas (Military class) Vaishyas (Business or Trading class) Shudras (Labour class)

The epic age It was the epic age in which the Aryan tribes established themselves in the entire northern India. The land between the Himalayas and Narmada river was divided into sixteen independent states. Information about the socio-religious life of the people is derived from the two great epics of this period, the Mahabharata and the Ramayana. The rise of Brahmanism During the later Vedic period, the observance of religion was made very complicated by the addition of several rituals; consequently only Brahmins could perform religious ceremonies. This led to a religious monopoly by the Brahmins who were considered the sole mediators between God and man. Many older gods were superseded by the new set of gods—the trinity of gods or trimurti denoted by Brahma, the creator, Shiva, the destroyer and Vishnu, the preserver. It was about this time that the popular myths about the creator of the world, the gods and their various incarnations and other legends were collected in the form of eighteen Puranas or old stories. The revolt against Brahmanism As Brahmins monopolized religion and did not allow others to perform any religious ceremony, the people revolted against Brahmanical exploitation.

Indian History

Growth of Buddhism and Jainism Period Sixth century bc, also called the period of religious unrest. As a result of revolt against the supremacy of Brahmanical priests, several schools of philosophy opposing Brahmanism developed, led by the Kshatriyas of the royal families of Magadha who later helped in the propagation of Jainism and Buddhism. Buddhism The fourth greatest religion in the world originated in India. Buddhism received state patronage of kings like Ashoka the Great, and it spread to neighbouring countries like Myanmar, Sri Lanka, Japan, Vietnam and Thailand. Period Founded around 6th century bc. Causes for the rise of Buddhism: 1. Vedic rights had become very complicated and expensive 2. Domination by Brahmins who monopolized the religion 3. Use of difficult and outdated language in religious ceremonies

Muzaffarpur in Bihar. At the age of 30, he became an ascetic and at the age of 42 he attained perfect knowledge—Kaivalya. He conquered misery and happiness and came to be known as Jina (the conqueror). He died at the age of 72 in 468 BC.

Characteristics Political Jainism weakened the prevailing military spirit and created a peaceful atmosphere. Social Further weakening of the caste system, development of new styles in art and architecture in temples, construction of several inns, hospitals, schools and other institutions of public utility. Decline of Jainism Due to very rigid and high ideals, Jainism suffered a downfall. The austere ways and high ideals of Jainism led to its decline but it has remained vibrant in India.

The Magadha Empire

Founder Founded by Gautama Siddhartha who was Kshatriya prince of the Saka clan. He was born in 563 bc (or 576 bc as is believed by some historians) at Lumbini in Nepal and was the son of Suddhodana, Raja of Kapilavastu. He left his family at the age of 29 in search of truth (also called the Great Renunciation) and wandered about for seven years. He received enlightenment at Bodh Gaya under a pipal tree and became the Buddha. He delivered his first sermon at Sarnath in Benaras. He spread his message for about 45 years and died at the age of 80 in 483 bc at Kushinagar in Deoria district of eastern Uttar Pradesh.

Period 6th century–4th century bc

Influence of Buddhism

Alexander’s Invasion

Political Buddhism destroyed the rising militant spirit and fostered a sense of national unity and universal brotherhood. Social Buddhism struck a fatal blow to the caste system and fostered an atmosphere of peace, stressed on purity of life and mental upliftment. During this period, the development of art and architecture also took place. Educational centres were founded at the Buddhist Viharas and Indian culture spread to regions outside India during the reigns of emperor Ashoka and Kanishka. Decline of Buddhism Buddhism declined as Hinduism reformed with the rise of the Rajputs as a military force. Lack of unity among Buddhists and increased use of Sanskrit led to a decline in Buddhism. Muslim invasions in the 11th and 12th centuries led to its further disintegration. Jainism Jainism is also a non-Brahminical religion founded as a result of revolt against the Brahmanism of the 6th century bc. Founder Founded by Rishabha, who was father of King Bharata, the first Chakravarti of India. Jainism became a major religion under Vardhamana Mahavira who was the 24th Tirthankara or prophet of Jainism. Rishabha was succeeded by 23 other Tirthankaras. Vardhamana Mahavira was a great Kshatriya belonging to the royal families of Magadha. He was born at Kundagrama (Vaishali), near

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Extent From a small kingdom it became a major power in north India embracing the districts of Patna and Gaya in Bihar. Its capital was Patliputra. Characteristics Due to favourable geographical location and the rich, fertile soil of the lower Ganga region trade flourished. Iron deposits in the region proved useful for making better agricultural implements which enabled peasants to produce more surplus and consequently provide more weapons for the army.

Period 356–323 bc The son of Phillip of Macedonia (Greece), invaded India in 326 bc. His major battle was with Porus, the king of Punjab on the banks of Jhelum. Alexander emerged victorious.

Causes of Alexander’s success 1. Good generalship provided by Alexander and a welldisciplined and superior Greek army 2. North-western India was divided into numerous small states 3. Outdated and outmoded techniques of war followed by the Indian soldiers

Influence of Alexander’s invasion 1. It opened up free intercourse between India and the west and strengthened commercial ties 2. Cultural exchange took place, especially in the field of art and architecture 3. Techniques of coinage developed 4. Greeks influenced science and astronomy

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The Mauryan Empire

The Gupta Dynasty

Period 321–298 bc

Period ad 320–550

Founder Chandragupta Maurya was the founder of the Mauryan empire, which overthrew the Nandas. He founded his dynasty in Magadha around 320 bc. His son Bindusara (296–273 bc) succeeded him and annexed the south up to Mysore. Extent of empire He expelled Greeks from Punjab and Sind and brought under his rule the whole of north India from Patliputra to Hindukush mountains in the northwest and to Narmada in the south. His empire included Kabul, Herat, Kandhar, Baluchistan, Punjab, UP, Bihar, Bengal, Gujarat and Kathiawar. Importance of Chandragupta’s rule Chandragupta was the first Indian king who can be called a national ruler, who set up an administration with autocratic and centrally based system helped by a council of ministers. He also carved out an effective espionage organization to keep watch on his enemies. Kautilya (Chanakya) A minister of Chandragupta, his famous work was the Arthashastra, a treatise on statecraft detailing how a good government should be organized. Megasthenes A Greek ambassador to Chandragupta’s court who wrote Megasthenes’ account of India called Indica in Greek detailing the Mauryan dynasty, and the capital city of Patliputra.

Extent of empire From the Brahmaputra to the Yamuna and Chambal, from the Himalayas to the Narmada.

Ashoka the Great Period 273–232 bc Ashoka was the grandson of Chandragupta and son of Bindusara (Bindusara was successor of Chandragupta and died in 273 bc) and is regarded as one of the greatest kings of all times. He was the first ruler to maintain direct contact with the people and ruled for 40 years. Ashoka could occupy the throne of the Mauryan empire only after a war of succession (273–268 bc) between him and his brothers. Extent of empire Askoka’s empire covered the whole territory from Hindukush to Bengal and extended over Afghanistan, Baluchistan and the whole of India with exception of a small area in the farthest south. Kashmir and the valleys of Nepal were also included. It was the biggest Indian empire and Ashoka was the first Indian king who ruled over almost the whole of India. The Kalinga war It was an important war during Ashoka’s rule which changed his attitude towards life. In 265 bc Ashoka invad ed Kalinga (Orissa) and occupied it after widespread destruction and bloodshed. This lead to conversion of Ashoka and he became a Buddhist. This weakened his administration which slowly led to the decline of the Mauryan Empire. Aspects of Ashoka’s reign Ashoka’s empire was divided into provinces with a viceroy in each province. He established dharamsalas, hospitals and sarais throughout his kingdom. Dharma Mahapatras were appointed to preach to the people. Buddhism spread during his reign as a state religion and inscription of Buddhist principles were engraved on rocks. He organized a network of missionaries to preach the doctrine both in his kingdom and beyond. Ashoka sent missionaries to Ceylon, Burma and other South-east Asian regions, notably Thailand to spread the doctrine of Buddhism.

Important rulers 1. Chandragupta I (ad 320–335) founded the dynasty around ad 320 in Magadha and ruled till ad 330. 2. Samudragupta (ad 335–375) was the successor of Chandragupta I, known as the Indian Napolean as he never suffered a defeat in the battlefield. 3. Chandragupta II or Chandragupta Vikramaditya (ad 375–415) was the son of Samudragupta and is also referred to as the Vikram of Ujjain. He removed foreign rule completely from India.

Characteristics of the Gupta dynasty The Gupta dynasty is called the golden age or the classical age of ancient India because 1. there were political units; foreign rule was completely removed and peace and prosperity prevailed 2. of the enlightened character of government, i.e., taxes were light, punishment mild, etc. 3. of revival of Hinduism but there was tolerance of all other religions 4. use of Sanskrit developed and art and literature flourished during the period 5. of the great personages who lived during this period including: Kalidasa, poet and dramatist known as Shakespeare of India. Aryabhatta, Varahmihira and Brahmagupta, the great mathematicians and astronomers lived during this period. Kumarila Bhatta and Shankaracharya, the great preachers of Hinduism, and Dhanwantri, the great physician, also lived during this period. Fa-hein, a Chinese pilgrim who visited India (ad 399–411) during Vikramaditya’s reign and gave an excellent account of the Gupta dynasty and prosperity of the country.

Harshavardhana Period ad 606–646 Extent of empire From the Brahmaputra to eastern Punjab, from the Himalayas to the Narmada. Important ruler Harashavardhana was the last Hindu king of northern India. He belonged to the Pushvabhukti family who ruled in Thanesar, north of Delhi. He established a strong empire conquering Bengal, Malwa, eastern Rajasthan and the entire Gangetic plain up to Assam.

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The Rajputs Period ad 650–1200 After Harshavardhana, the Rajputs emerged as a powerful force in western and central India and dominated the Indian political scene for nearly 500 years from the 7th century.

Important rulers Prithviraj Chauhan ruled over Delhi and Agra and fought two important battles, viz.,





• First battle of Tarain was fought in 1191 between the forces of Prithviraj Chauhan and Mohammad Ghori in which the latter was defeated. • Second battle of Tarain was fought in 1192 when Mohammad Ghori again invaded India in which Prithviraj Chauhan was defeated and captured and later on slain. The kingdom of Delhi fell to Mohammad Ghori. • The battles of Tarain had great significance in the Indian political scene as they led to the establishment of Muslim rule over north India and subsequently in the south for several centuries.

Last Ruler of the Chola Dynasty–Rajendra III Period

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1246–79

Political The king was the central head who was helped by a council of ministers, but the administration was democratic. Land revenue and trade tax were the main source of income. Social The society was divided into Brahamins and nonBrahmins. The temple was the cultural and social entire, where art and literature flourished. The Brihadeswara (Siva) temple at Tanjore was built by Rajaraja I. The giant statue of Gomateswara at Shravanabelagola was built during this period. Decline Malik Kafur invaded this Tamil State in 1310 and extinguished the Chola empire.

Jai Chand Rathor was the last Rajput king who was also defeated and killed by Ghori in 1194.

Rashtrakutas (ad 735–973) were descendants of the nobles who governed under the Andhras. They overthrew the Chalukyas and ruled up to ad 973. Krishna I is the Rashtrakuta king credited to have built the Kailasa temple at Ellora. Other kings of this dynasty were Govinda III, Amoghavarsha and Indra III.

Other Dynasties

The Yadavas

The Andhras (ad 250–225) Considered to be one of the earliest rulers of the Deccan. They gained their independence after the death of Ashoka. Their rule ended in ad 225. The Chalukyas (ad 550–642) Also over the Deccan. The famous rulers of this dynasty were: Pulakesin I (ad 543–561) was the first independent ruler of Badami with Vatapi in Bijapur as the capital. Pulakesin II (ad 610–642) the grandson of Pulakesin I was a contemporary of Harshavardhana and the most famous of the Chalukyan kings. He increased his kingdom by annexing the entire Andhra kingdom. He defeated Harshavardhana on the banks of the Narmada. However, later the Pallava king Narsimhavarman attacked Pulakesin II in ad 642 and captured his capital city of Vatapi.

The Chola Dynasty

Period ad 1191–1318 They ruled from Devagiri, and the important rulers among them were: (i) Singhana (ad 1210–1247) (ii) Ramachandra (ad 1271–1309)

Hindu Kingdom of Vijayanagar Period ad 1336–1646 Founders The five sons of a south Indian Hindu Sangama of whom Harihar and Bukka were prominent. Extent of empire The kingdom lay in the Deccan, to the south of the Bahmani kingdom. They protected the Hindu way of life and checked Mohammedan advance.

Extent of empire The Tamil dynasty ruled over Madras and parts of Karnataka with Tanjore as their capital

Important rulers Krishnadevaraya (ad 1505–29) was the most illustrious ruler of the Deccan.

Founder Rajaraja I (ad 985–1014) was the founder of the Chola dynasty. Rajendra Chola son of Rajaraja I was an important ruler of this dynasty who conquered Orissa, Bengal, Burma and the Andaman and Nicobar islands. The Chola dynasty was at its zenith during his reign.

Battle of Talikota (ad 1564–1565) was fought in February 1565. The Muslim rulers of the Deccan viz., Ahmednagar, Bijapur and Golconda fought the battle against the army of Vijayanagar with their combined forces led by Ramaraja. Ramaraja was defeated and killed. The second battle of Talikota resulted in the complete collapse of the Hindu kingdom of Vijayana

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MEDIEVAL INDIA This includes the Muslim Period (ad 1000–1707) which started with the raids by Mahmud of Ghazni and establishment of the Sultanate of Delhi. Mohammad Bin Qasim invaded India in ad 712 and conquered Sindh which became the province of Omayyad Khilafat. Sultan Mahmud of Ghazni (ad 997–1030) Mahmud Ghazni led about seventeen expeditions to India with twin objectives: (i) To spread Islam in India, and (ii) To enrich himself by taking away the wealth from India. In ad 1025 he attacked and raided the most celebrated Hindu temple of Somnath that lies on the coast in the extreme south of Kathiawar. The temple was destroyed in ad 1026. Mohammad Ghori (ad 1175–1206) Mohammad Ghori invaded India and laid the foundation of the Muslim dominion in India. He may be considered the founder of Muslim rule in India.

Reasons for the Success of Muslims in India 1. Rajputs lacked unity and organization; divided by rivalries. 2. There was no central government. 3. Kingdoms were small and scattered. 4. Muslims were better organized and took advantage of the lack of mutual cooperation among the Rajputs. 5. Buddhism made the Indian masses timid and weakened their military attitude.

The Sultanate of Delhi Period ad 1206–1526 Mohammad Ghori’s conquests became the nucleus of a new politi­ cal entity in India—the Sultanate of Delhi—and the beginning of Muslim rule in India. This period can be divided into five distinct periods viz., Table 10.1 Pre-Mughal Dynasties Dynasty

Period

1. The Slave Dynasty

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1206–1290

2. The Khilji Dynasty

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1290–1320

3. The Tughlaq Dynasty

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1320–1414

4. The Sayyid Dynasty

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1414–1451

5. The Lodhi Dynasty

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1451–1526

The Slave Dynasty This is the first Muslim dynasty in India. Period ad 1206–1290 Founder Qutubuddin Aibak

Important rulers of the Slave dynasty 1. Qutubuddin Aibak (ad 1206–1210) A Turkish slave by origin, he was purchased by Mohammad Ghori who later made him his Governor. After the death of Mohammad Ghori, Aibak became the master of Hindustan and founded the Slave dynasty in 1206. He is considered a generous ruler and popularly called ‘lakh bakhsh’ (given by lakhs). He died in 1210. He was succeeded by his son Aram Baksh (1210–11). Architecture of his time The 72.5 m (238 ft) high Qutub Minar in Delhi was built by him to perpetuate the memory of saint Qutubuddin. Quwwat­ul­Islam mosque was also built during the reign of Aibak. 2. Shamasuddin Iltutmish (ad 1211–1236) He was a slave of Qutubuddin Aibak and occupied the throne of Delhi in 1211 after deposing Aram Baksh. He was a very capable ruler and is regarded as the founder of the Delhi Sultanate because during his long reign he consolidated all Turkish conquests and reasserted his supremacy in the entire north and extinguished the rebellions by reorganizing his army. He introduced the silver coin (tanka) and the copper coin (jital). He also completed the construction of Qutub Minar, which was started by Qutubuddin Aibak. 3. Ruknuddin (ad 1236) He was son of Iltutmish and was crowned by her mother, Shah Tukran, after death of Iltutmish. He was deposed by Razia, daughter of Iltutmish, when he was out of capital to curb a rebellion in Avadh against him. 4. Razia Sultana (ad 1236–1239) She was the daughter of Iltutmish. She was the first and only Muslim lady who ever ruled India. She succeeded her father in 1236 and was a very brave and strong ruler. She fell in love with Jamaluddin Yakut, whom she married. She was killed by her own brother Muizuddin Bahram when she tried to recapture the throne of Delhi from him. 5. Bahram Shah (ad 1240–42) After Razia, Iltutmish’s third son Bahram Shah was put on throne by powerful Turkish Council, Chalisa (or family of forty). He was considered only as de jure ruler, while naib-e-mamlakat (head of Chalisa) was de facto ruler. Bahram Shah lost his life after failed attempt to assert his authority once on the throne. 6. Masud Shah (ad 1242–46) He was son of Ruknuddin but was desposed after Balban and Nasiruddin Mohammad’s mother, Malika-e-Jahan, conspired against him and established Nasir-e-Mohammad as the new Sultan. 7. Nasiruddin Mahmud (ad 1246–1266) He was the son of Iltutmish, and was known as the darvesh king as he was very pious and noble. He died in 1266.

Indian History 8. Balban (ad 1266–87) He dissolved Chalisa and established peace in and around Delhi, instead of expanding the Sultanate. He established diwan-i-arz (Royal Military department) to reorganize his army, introduced rigorous court discipline and new customs to prove his superiority to Turkish nobles. However, in his last days he overlooked Sultanate affairs due to death of his eldest and most loving son, Mohammad, and rebellion by his closest and most loved slave, Tughril. Mohammad died fighting Mongolians in 1285 and Tughril was captured and beheaded. 9. Kaiqubad (ad 1287–90) A grandson of Balban was established on the throne by Fakruddin, the kotwal of Delhi who assumed high political authority during last days of Balban. But Kaiqubad was killed by the Khilji family, which saw end of slave or Ilbari dynasty and beginning of Khilji Dynasty at Delhi throne.

The Khilji Dynasty Period ad 1290–1320 Founder Sultan Jalaluddin Khilji (ad 1290–1296) Extent of empire He brought under his sway all the Rajput kingdoms. Alauddin Khilji (ad 1296–1316) He was a nephew of Sultan Jalaluddin. Alauddin Khilji killed him and succeeded the throne in 1296. He invaded Devangiri and conquered Warangal and Mysore, the Hindu Kingdoms of the south. He was an able administrator and known for the rapid expansion of Khilji domination. He was very severe with Hindus. He set up an efficient espionage system and also introduced various price control measures. Though he himself was illiterate, he was a patron of art and learning. Amir Khusrau, the poet-musician was his favourite court poet, Alauddin Khilji built the Siri and Arai Minars near Qutub Minar in Delhi. Rani Padmini was the queen of Rana Bhim Singh of Mewar. Alauddin Khilji was so attracted by her beauty that he invaded Chittor in 1303 to capture the queen. Though he captured Chittor the queen committed Jauhar along with other Rajput women. Kafur (ad 1315) In 1315, after death of Alauddin, Kafur, a regent in the Alauddin’s court seized the throne. Before Kafur died, he nominated Shiabuddin (Alauddin’s six-year old prince) as king but imprisoned eldest prince Mubarak Khan. He was killed by the loyalists of the Royal family of Alauddin. Kafur himself was illiterate but gave encouragement and accolades to great poets of Khilji era, Amir Khusrau and Amir Hassan. Mubarak ( ad 1316–1320) After the death of Kafur (1316), Mubarak Khan was freed from prison and worked as regent for Shiabuddin. He captured the throne at the first opportunity he got, but could rule only for 4 years as he sank into debauchery and could not give up his dissipated lifestyle. He awarded his lover Hassan authority over army and palace guards, who soon

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obtained full control over Sultan’s palace. Hassan was given the title Khusrau Khan by the Sultan and within months Khusrau killed Mubarak Khan and assumed the title of Nasiruddin in mid-1320. Khusrau Khan (ad 1320) Khusrau Khan was killed by Ghazi Khan, governor of Dipalpur, when he tried to oppose rebellion by Ghazi Malik and his son Fakhruddin Jaima. This brought the end of Khilji dynasty in Delhi and rise of Tughlaq dynasty at the throne of Delhi.

The Tughlaq Dynasty Period ad 1320–1414 Founder Ghiyasuddin Tughlaq (ad 1320–1325)

Important rulers 1. Ghiyasuddin Tuglaq (ad 1320–1325) Khusrau Khan, the last king of the Khilji dynasty was killed by Ghazi Khan, and Ghazi Khan ascended the throne assuming title Ghiyasuddin Tuglaq. He died in an accident and his son Ulugh Khan succeeded him under the title Mohammad Bin Tughlaq. 2. Mohammad Bin Tughlaq (ad 1325–51) Prince Jauna, son of Ghiyasuddin Tughlaq ascended the throne in 1325. During his reign, the capital was shifted from Delhi to Devangiri in the south in 1327 and renamed Daulatabad but he brought it back to Delhi again. Mohammad Bin Tughlaq introduced token coins of brass and copper which had the same value as silver coins. 3. Firoz Shah Tughlaq (ad 1351–88) He was a cousin of Mohammad Bin Tughlaq. After his death the nobles and theologians of the court selected Firoz Shaha as the next sultan. His reign was a period of peace and prosperity. He established several new towns such as Firozepur, Firozabad, Hissar-Firoza and Jaunpur. He died in 1388. After Firoz Shah Tughlaq (ad 1388–1414) The Tughlaq dynasty could not survive much after Firoz Shah’s death. The Malwa, Gujarat and Sharqi kingdoms broke away from the Sultanate. During Timur’s invasion Punjab, Jammu, Del hi, Haridwar, Nagarkot fell without much opposition marking end of Tughlaq dynasty. Ibn Battuta was an African traveller, who visited India in 1333 during the reign of the Tughlaqs. He was appointed as Chief Qazi of Delhi by the Sultan and subsequently as his ambassador to China in 1342. Invasion by Timur Timur, a Turk invaded India in 1398 during the reign of Nasiruddin Tughlaq. His army mercilessly sacked and plundered Delhi. Timur returned to central Asia, leaving a nominee to rule to Punjab which ended the Tughlaq dynasty.

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The Sayyid Dynasty Period ad 1414–1450

Important rulers of the Sayyid dynasty 1. Khizr Khan (ad 1414–1421) Timur’s nominee captured Delhi and was proclaimed the new Sultan and the first of the Sayyid dynasty. They ruled over Delhi and surrounding districts for about 37 years. 2. Mubarak Shah (ad 1421–1434) He succeeded Khizr at the throne after his successful expeditions against Mewatis, Katihars and the Gangetic Doab area. He was killed by the nobles in his own court. 3. Mohammad Shah (ad 1434–1443) The nobles put Mohammad Shah on the throne, but he could not survive the in­fighting among the nobles in the court. He was authorized to rule a meager area of around 30 miles and rest of the Sultanate was ruled by nobles who were never at agreement at any political or social issue, thus marking the first signs of fall of Sayyid dynasty. 4. Alam Shah (ad 1443–1451) The last Sayyid king descended in favour of Bahlul Lodhi and himself retired. Thus began the Lodhi dynasty, which confined to Delhi and a few surrounding areas.

The Lodhi Dynasty Period ad 1451–1526 Founder Bahlul Lodhi (ad 1451–1488) Bahlul Lodhi was one of the Afghan Sardars who established himself in Punjab after the invasion of Timur.

Important rulers 1. Sikander Lodhi (ad 1489–1517) He was the son of Bahlul Lodhi who conquered Bihar and western Bengal. He transferred his capital from Delhi to Agra. 2. Ibrahim Lodhi (ad 1517–1526) He was the last king of the Lodhi dynasty and the last sultan of Delhi. He was the son of Sikander Lodhi. First Battle of Panipat The first battle of Panipat was fought in 1526 between Ibrahim Lodhi, the Sultan of Delhi and Babur, the ruler of Kabul. Babur invaded India and established the Mughal dynasty. Decline of Delhi Sultanate The main causes of decline of Delhi Sultanate were: 1. Despotic and military type of governments which did not have the confidence of the people 2. Degeneration of the Delhi sultans

3. The Sultanate became too vast and could not be controlled effectively 4. Financial instability 5. Number of slaves increased to 1,80,000 in Firoz Shah’s time which was a burden on the treasury

Religion in Medieval India Muslim invasions brought Islam to India, which led to conflict between Hinduism and Islam. This resulted in the development of two popular movements during the period: (i) the Sufi movement and (ii) the Bhakti movement. Sufi movement The Sufi doctrine was based on union with God which can be achieved through love of God, prayers, fasts and rituals, without reference to Hindu or Muslim. The Sufis formed orders under a Pir or Shaikh. Sufi Sant Nizamuddin Auliya was respected both by the Hindus and Muslims. Bhakti movement The Bhakti movement was based on the doctrine that the relationship between God and man is through love and worship rather than through performing any ritual or religious ceremonies. Kabir ( ad 1440–1518) Sant Kabir criticized both Hindu and Muslim ideas of God and tried to equate them by stressing that both religions are identical. He expressed his ideas in ‘dohas’ or poetic couplets. Nanak (ad 1469–1539) described God without any reference to Hindus or Muslims. He wanted his followers to eat together in a common kitchen called a ‘langar’, without consideration of any caste or religion. He expressed his ideas in the form of verses in his book Adi granth. The Khalsa emerged as a strong military group under Guru Gobind Singh (the tenth Guru) and distinguished themselves from other people by five Ks, viz., 1. Kesha (hair), 2. Kanga (comb), 3. Kara (iron bracelet), 4. Kirpan (dagger) and 5. Kachchha (undergarment). Guru Gobind Singh named the Granth Sahib as the everlasting Guru of the Sikhs.

The Mughal Dynasty Period ad 1526–1540 and ad 1555–1857 Extent of Empire The empire stretched from Punjab to Bengal including Jaunpur and Bihar in the 16th century. It included Kabul in the north­west, Kashmir in the north, Sindh, Multan, Ajmer and Gujarat in the west, Malwa and Benar in the south and Orissa and Bengal in the east at the time of Akbar’s death. Stretched from Kabul, Kandhar and Peshawar in the north of Kaveri in the south by the end of the 17th century.

Important rulers 1. Babur (ad 1526–1530) Is said to have founded the Mughal empire. He defeated Ibrahim Lodhi in the First Battle of Panipat on 20 April, 1526 and became emperor of Delhi. In 1527, he defeated Rana Sanga at Khanwa

Indian History ▌ 2.119 near Fatehpur Sikri and occupied Agra. In 1527, in the Battle of Gorge, he defeated the Afghans and thus became the master of the entire India. He wrote his autobiography, Tuzuk-i-Babri in which he gives an excellent account of India and his empire. He died in 1530. 2. Humayun (ad 1530–1540 and 1545–1556) He was the son of Babur and ascended the throne in 1530. His succession was challenged by his brothers Kamran, Hindal and Askari along with the Afghans. He fought two battles against Sher Shah at Chausa (1539) and at Kannauj (1560) and was completely defeated by his enemies. He escaped to Persia where he passed 12 years of his life in exile. After Sher Shah’s death, he invaded India in 1555 and defeated his brothers and the Afghans. He once again became the ruler of India. He died in an accident in 1556, just two years after he regained his kingdom. Sher Shah Suri was an Afghan who ruled the country for a brief period from 1540–1545 after defeating Humayun. He, thus, caused a brief disruption in the Mughal rule. His empire extended from the Brahmaputra in the east of the Indus in the west, from the Himalayas in the north to the Narmada in the south. During his reign of five years he introduced a brilliant administration, land revenue policy and several other measures to improve economic conditions of his subjects. He issued the coin called 'Rupia' and fixed standard weights and measures all over the empire. He also improved communications by building several highways. He built the Grand Trunk Road (G.T. Road), that runs from Peshawar to Calcutta. He also introduced military reforms; he recruited and paid the soldiers directly and every soldier had his chehra (face) recorded and his horse branded with the imperial sign. He set up cantonments in various parts of his empire and a strong garrison was posted in each cantonment. 3. Akbar (ad 1556–1605) The eldest son of Humayun he ascended the throne at the young age of 13 on February 14, 1556 and his tutor Bairam Khan was appointed as the regent. Second Battle of Panipat The second battle of Panipat was fought between Hemu, a Hindu leader and Akbar's regent Bairam Khan. Hemu was defeated on November 5, 1556, captured and slain by Bairam Khan. This war ended the Mughal—Afghan contest for the throne of Delhi in favour of the Mughals and enabled Akbar to reoccupy Delhi and Agra. Expansion of Akbar’s empire Akbar ended the regency of Bairam Khan in 1560 and at the age of 18 assumed the reins of the kingdom. In his bid to expand his empire he conquered various towns and forts such as Gwalior, Ajmer and Jaunpur and

also annexed the kingdom of Malwa which brought him into the neighbourhood of the Rajput kingdom. The Rajput kingdom of Mewar put up a fierce defence under Rana Uday Singh and his son Rana Pratap. Akbar conqurered Gujarat (ad 1572–73), Bengal (ad 1574–76) and by 1595 he conquered Kashmir, Sindh, Orissa, central Asia and Kandhar (Afghanistan). Akbar and the Rajputs Akbar tried to win over the Rajputs wherever possible and inducted Rajput kings into Mughal service and treated them at par with Mughal nobility. By marrying Jodha Bai, daughter of Raja Bihari Mal of Jaipur in 1562 Akbar displayed his secular policy with the Hindus. Most of the Rajput kings recognized the supremacy of Akbar, except Rana Pratap Singh and his son Amar Singh of Mewar. Battle of Haldighati Was fought in 1576 near Gogundo (Haldighati) between Rana Pratap Singh of Mewar and the Mughal army led by Man Singh of Amber. Rana Pratap Singh was defeated but he did not submit and continued the struggle. Fatehpur Sikri A place near Agra. It is said that Akbar had no son for a long time. Sheikh Salim Chisty of Fatehpur Sikri, a Sufi Saint blessed Akbar with a son who was named Salim (Jahangir). In honour of Sheikh Salim Chisty, Akbar shifted his court from Agra to Fatehpur Sikri. Din-e-Ilahi As a revolt against the orthodoxy and bigotry of religious priests, Akbar proclaimed a new religion, Din-e-Ilahi, in 1581. The new religion was based on a synthesis of values taken from several religions like Hinduism, Islam, Jainism and Christianity. It did not recognize the Prophet. Birbal was the only Hindu among the 18 persons who followed this new religion. Din-e-Ilahi, however, did not become popular. Architecture, art and literature Akbar built Fatehpur Sikri, forts at Agra, Lahore and Allahabad and Humayun’s Tomb at Delhi. He was patron of the arts and in his court, scholars and statesmen like Abul Fazl, Faizi, and Todar Mal, the witty administrator Birbal and great musicians like Tansen flourished. Tulsidas, the great Hindi poet, who wrote Ramcharitmanas, also lived during Akbar’s period. Sikandra When Akbar died, he was buried at Sikandra near Agra. Significance of Akbar’s reign Akbar is considered the real founder of the Mughal empire in India because Babur and Humayun could not consolidate the empire as Akbar successfully did. He was the first Muslim ruler who divorced religion from politics. His attitude towards Hindus was very conciliatory. 4. Jahangir (ad 1605–1627) Salim, son of Akbar, came to the throne after Akbar’s death in ad 1605. He is known for his strict administration of justice. Nur Jahan In May 1611, Jahangir married Mihr-un-nisa, widow of Sher Afghan, a Persian nobleman of Bengal. Later on she was given the title ‘Nur Jahan’. Nur Jahan took an active interest in the matters of the state and also ruled the empire when Jahangir was ill for a long time. Jahangir issued coins jointly in Nur Jahan’s name and his own.

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Relations with foreigners In 1608, Captain William Hawkins, a representative of the East India Company came to Jahangir’s court. In 1615, Sir Thomas Roe, an ambassador of King James of England, also came to his court. Though initially Jahangir resisted, he later on granted permission to the English to establish a trading post at Surat. Revolts His reign was marked by several revolts. Guru Arjan Dev was martyred during his period, thus alienating the Sikh community. Political success A political triumph during Jahangir’s reign was the submission of Rana Amar Singh of Mewar (1615). Jahangir captured the strong fort of Kangara (1620) after a long siege. A part of Ahmednagar was also annexed. Malik Ambar ceded back to the Mughals the territory of Balaghat. 5. Shahjahan (ad 1628–1658) Son of Jahangir, he ascended the throne in 1628 after his father’s death. He was best known for his Deccan and foreign policies. Death of Mumtaz Mahal Three years after his accession, his beloved wife Mumtaz Mahal died in 1631. To perpetuate her memory he built the Taj Mahal at Agra in 1632–53. Extent of empire In addition to Jahangir’s empire, Nizam Shahi’s dynasty of Ahmednagar was brought under Mughal control (1633) by Shahjanhan. The Deccan Sultanate of Bijapur and Golconda accepted his suzerainty in 1636. He reconquered Kandhar (Afghanistan) in 1638 and routed the Uzbeks at Balkha in 1647. He also annexed the region of Kamarpura in Assam. Significane of Shahjahan’s reign Shahjahan’s reign is said to have marked the pinnacle of the Mughal dynasty and empire. He is known for the promotion of art, culture and architecture during his time. The Red Fort, Jama Masjid and Taj Mahal are some of the magnificent structures built during his reign. Imprisonment Shahjahan’s failing health set off the war of succession among his four sons in 1657. His third son, Aurangzeb emerged the victor who crowned himself emperor in July 1658. Shahjahan was imprisoned by his son Aurangzeb in the Agra Fort where he died in captivity in 1666. 6. Aurangzeb (ad 1658–1707) After imprisoning Shahja­ han, his son Aurangzeb was crowned at Delhi under the title Alamgir. He ruled for 50 years till his death in February 1707 in Ahmednagar. Extent of empire Aurangzeb’s empire extended from Kashmir in the north to Jinji in the south, and the Hindukush in the west to Chittagong in the east. During the first 23 years of the rule (1658–81) Aurangzeb concentrated on north India. During this period the Marathas under Shivaji rose to power and were a force to reckon with. As a result, Aurangzeb left the north and for the next 25 years (1682–1707) made desperate bids to crush the Marathas. The execution of Sambhaji in 1689 saw the collapse of the Maratha empire. Aurangzeb’s empire now extended from Afghanistan to Bengal, and from Kashmir to Karnataka. But in certain parts of south India (Mysore, Maharashtra, etc.) he was

not entirely successful in thwarting his enemies. Aurangzeb never returned to the north and died in Ahmednagar in February 1707. Intolerance of other religions He was Muslim fanatic and thus was an intolerant autocrat. Many Hindu temples were demolished and religious festivals, idol worship and pilgrimages were banned during his reign. Sikh revolt Aurangzeb captured Guru Teg Bahadur, the 9th Guru of Sikhs in 1675 and executed him when he refused to embrace Islam. Guru Gobind Singh, son of Guru Teg Bahadur, organized his followers into a militant force called ‘Khalsa’ to avenge the murder of his father. Guru Gobind Singh was, however, murdered in 1708 by an Afghan in the Deccan. Banda, the militant successor of Guru Gobind Singh continued the war against Mughals but he too was put to death. War with Shivaji Shivaji was the most powerful Maratha king and an arch enemy of Aurangzeb. When Aurangzeb could not eliminate him, he conspired with Jai Singh of Ambar, a Rajput, to eliminate Shivaji. In 1665, on an assurance given by Jai Singh, Shivaji visited Aurangeb’s court. Shivaji was imprisoned by Aurganzeb but he managed to escape and in 1674 proclaimed himself an independent monarch. He died in 1680 and was succeeded by his son Sambhaji, who was executed by Aurangzeb. Sambhaji was succeeded by his brother Rajaram and after his death in 1700, his widow Tarabai carried on the movement. Decline of the Mughal empire After Aurangzeb, the Mughal empire rapidly declined. Important causes for the decline were: 1. Aurangzeb’s Rajput, Deccan and religious policies; 2. Stagnation in agricultural production, trade and manufacture; 3. Rapidly rising demands of the ruling classes, leading to attempts to realize more from Jagirs causing peasant and Zamindari discontent; 4. Jagirdari crisis: Nobels tried to corner the most profitable Jagirs leading to corruption; 5. Factionalism among nobility after Aurangzeb; development of powerful Irani, Turani, Deccani, Hindustani blocs in the court who vied for power in order to destabilize the central administrative machinery; 6. Rise of independent kingdoms; 7. Rise of European power in India; 8. Nadir Shah’s invasion in 1739. Bahadur Shah (1707—1712) Bahadur Shah, son of the last emperor, succeeded to the throne in June 1707 after defeating his two brothers. He assumed the title of Bahadur Shah or Shah Alam I. He tried to reconcile with the Marathas and in his bid released Sahu, the son of Sambhaji. After another civil war Bahadur Shah was crowned emperor but after a short reign, was deposed and succeeded by his nephew Farukhsiyar with the help of Abdullah Khan and Hussian Ali, the two Saiyad brothers of Barah, known in the Indian history as the king-makers. Mohammad Shah (1710–1748) succeeded Farukhsiyar.

Indian History Invasion of Nadir Shah In 1739, during the reign of Mohammad Shah, a Persian king, Nadir Shah invaded India and mercilessly massacred the people in Delhi. Nadir Shah, however, reinstated Mohammad Shah and went back to Persia. Nadir Shah’s invasion broke the Mughal empire. As a result the Marathas became very powerful in the Deccan. Sadat Ali Khan, Governor of Awadh, and Alivardi Khan, Governor of Bengal and the Rohillas in the Ganges valley also became independent. The empire of Mohammad Shah’s successor Ahmad Shah (1748–54) only consisted of the lands adjoining Delhi and some districts in Uttar Pradesh. Imal-ul-mulk became Ahmad Shah’s Wazir after terrorizing him, and ultimately placed his son Mohammad Azim Uddaulah on the throne as Alamgir II. He was later on murdered and another Mughal prince was installed on the throne. The heir-apparent, prince Shah Alam, sought refuge with Nawab Wazir of Awadh. Shah Alam was recognized as emperor by Abdali, the successor of Nadir Shah. He lived under the protection of the English after his defeat at their hands at the Battle of Buxar in 1764. In 1771 he returned to Delhi at the invitation of the Marathas who placed him on the throne. Shah Alam II was succeeded by Akbar Shah II (1806–1837), and the last ruler, famous as Bahadur Shah Zafar, ruled between 1837–1858 as a titular king under Britishers.

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The Mughal emperors ruled only in name until 1857–58 and real political power in the 18th century had shifted to new kingdoms.

The Hindu Dynasty Marathas Period ad 1649–1748 Mission of Marathas To liberate India from Muslim rule. Shiviji was the most powerful among them. He conquered the forts of Torna, Rajgarh, Purandhar, etc., and became an independent ruler. His method of warfare was guerilla warfare. Zenith of Maratha power Marathas became powerful after the departure of Nadir Shah under the leadership of the Peshwas, viz., Balaji Vishwanath, Bajirao I and Balaji. They conquered Deccan kingdoms and aspired to bring under their control Delhi and Punjab. They entered into an alliance with the Delhi court against Najibuddaulah. In 1757 they captured Delhi and in 1758 conquered Punjab by defeating Timar Shah (Abdali’s son). Third Battle of Panipat The third battle of Panipat fought in 1761 between Ahmed Shah Abdali and the Marathas, ended Maratha power.

MODERN INDIA Portuguese traders were the first to discover a sea­route to India free from Turkish threat in 1498. They were followed by the Dutch in 1595 and the English in 1600 and finally the French in 1664 who also came to India for trade. The Portuguese In 1498, it was Vasco-da Gama, a Portuguese sailor who first discovered a sea­route to India via the Cape of Good Hope. He arrived at Calicut on May 27, 1498. The Portuguese soon established political power along the west coast of India. He was succeeded by Captain General Alfonso de Albuquerque who conquered Goa in 1510. The Dutch The first fleet of the Dutch reached India in 1595 and Dutch East India Company was formed in 1602, but their influ­ ence soon vanished. In 1605 they established their first factory in Masulipattam, followed by more factories in Pulicat (1610), Surat (1616), Bimilpatam (1641), Karikal (1645), Chinsura (1653), Kasimbazar (1658), Baranagore (1658), Patna and Balasore (both 1658) and Cochin (1663). Till 1690, Pulicat was their chief trade centre and afterwards it shifted to Negapatnam. The Anglo-Dutch rivalry was at high during late 17th and early 18th century till the Dutch collapsed with their defeat by the English in the battle of Bedera in 1759. The English The English East India Company was formed in 1600 through a Charter signed by Queen Elizabeth I granting permission to trade with India. Captain Hawkins paid a visit to the court of Jahangir in 1608 but failed to secure trading rights. However, in

1613, on Sir Thomas Roe’s visit they were permitted to establish their first factory at Surat. Gradually, the Company established its trading centres at Bombay, Calcutta and Madras. The English established their settlements/factories in Masulipattam (1611), Agra, Ahmedabad, Baroda, Broach (all 1619), Armagaon near Pulicat (1626), Hariharpur and Balasore (1633), Patna, Dacca, Kasimbazar in Bengal and Bihar (1835), Madras (1639) to establish Fort St George, Hugli (1651), a network of settlements in Bihar, Bengal and Orissa (1658), Bombay (1668), Sutanuti (1690), Kalikota and Govindpur (1698). Sutanuti, Kalikota and Govindpur were later joined together to a new city ‘Calcutta’ and the factory at Sutanati was fortified in 1700 and named ‘Fort William’. During 1686, the English declared war against Mugal Emperor Aurangzeb in which the English lost all control of its settlement and factories in India to Mughals in 1688–1689. In 1690, the surrendering British were pardoned by Mughal Emperor. In 1691, the English were granted ‘farman’ by Aurangzeb which exempted the British Company from payment of customs duties in Bengal. Faruk Siyar granted British another ‘farman’ in 1717 thus extended the privilege to British in Gujarat and Deccan. The French The French came to India in 1664 and set up centres near Madras and Chandernagore on the Hooghly to trade with India. They also established naval bases in the islands of Bourbon and Mauritius in the Indian Ocean. They flourished initially till 1706, but afterwards decline occurred till 1720 and it was governors Lenoir and Dumas that after 1720 the French regrouped in

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India. However, during 1742 the French governor Dupelix started repulsing the English power which resulted in ‘Carnatic Wars’ and finally resulting in French defeat. The Danish In 1616, the East India Company, Denmark reached Indian coasts and established settlements in Tranqubar in Tamil Nadu (1620) and Serampore in Bengal (1676). However, due to rising presence of the British they had to sell all their settlements to the British during 1845.

East India Company and British Rule On arriving in India the East India Company had to face Dutch and French opposition as they were the main contestants for political supremacy over India. But the British were successful in destabilizing them and soon the Comapany’s functions expanded into political ambition. Robert Clive He led the English forces to capture Arcot and other regions. He was instrumental in laying the foundation of the British empire in India. In the Carnatic Wars between the French and the English, the latter finally defeated the French in the Battle of Wandiwash to gain control over South India. The English Conquest of Bengal Nawab Alivardi Khan was an independent ruler of Bengal between 1740–1756. He in fact extended protection to the European merchants in carrying on their trade. Alivardi Khan nominated his grandson (daughter’s son) Sirajuddaula as his heir since he had no son. He died in April 1756. In the meantime the Company constructed fortifications at Calcutta and violated the terms under which they were allowed to trade. Sirajuddaula took prompt action and occupied an English factory at Kasimbazar and later captured Calcutta in June 1756. Black-hole Tragedy English prisoners were said to have been crowded into a small chamber which had a single, tiny window on a hot summer night of 20 June 1756. As a result several English prisoners died of suffocation and wounds. In December 1756, Colonel Clive and Admiral Watson reached Bengal from Madras and captured Calcutta. Mir Jafar, brother-in-law of Alivardi Khan, had a secret pact with Clive who promised him the same state of Bengal. Mir Jafar nevertheless also assured his support to Sirajuddaula against the English. Battle of Plassey (1757) Robert Clive led the Company’s forces against Sirajuddaula’s army on 23 June 1757, and defeated them with the help of his conspiracy with Mir Jafar. This proved to be the first step towards territorial supremacy and paved the way for the British conquest of Bengal and eventually the whole country. The Nawab was captured and executed and Mir Jafar was installed as the Nawab of Bengal. He ceded Zamindari rights to Twenty four Parganas and got Rs 16,700,000 as compensation. This was the first British acquisition of Indian territory. Battle of Buxar (1764) At the instigation of Mir Qasim, successor of Mir Zafar, this battle was fought by Nawab Sujauddaula of Awadh and Shah Alam II (Mughal) on one side and the English forces led by Clive on the other side. Clive’s forces were victorious resulting in the capture of Bihar and Bengal.

The Carnatic Wars First Carnatic war (1746–1748) The French and the British companies clashed at Carnatic. Dupleix was then the chief official of the French Company at Pondicherry. The French opened hostilities by sacking Fort St George and expelled all Englishmen. The Nawab of Carnatic sent an army but was defeated. Second Carnatic war (1749—1754) The British were able to consolidate themselves by taking hold of Bengal, Bihar and Orissa. The second battle between the French and the British took place in 1760 in which the French were defeated. It ended with the Treaty of Paris in 1763 which foiled the dreams of the French to have an empire in India. War with Marathas (1775—1782) The First Anglo-Maratha war took place during the governor-generalship of Warren Hastings. The war ended with the Treaty of Salbai, 1782, and status quo restored. The Mysore war Mysore was a powerful state under Haider Ali. In 1769 the first Anglo­Mysore war was fought in which the British forces were defeated. Haider Ali occupied almost the whole of Carnatic. However, in 1781, Haider Ali was defeated at Porto Novo and saved Madras. After Haider Ali, the war was carried on by Tipu Sultan. A peace treaty was then signed. However, in 1789 another war was launched and Tipu Sultan was defeated in 1792.

Governor-Generals of India and Reforms First Governor In 1758, Robert Clive was appointed the first Governor of Bengal by the East India Company. Clive remained in England from 1760–1765 and on his return in 1765, the emperor ceded to the Company the Diwani of Bengal, Bihar and Orissa. Warren Hastings (1772–1785) Warren Hastings succeeded Clive in 1772 and became the first governor-general of India. He introduced several reforms, established civil and criminal courts, and courts of appeal. He passed The Regulating Act 1773, giving a legalized working constitution to the company’s dominion in India. It envisaged a Council of Ministers headed by the governorgeneral. Pitt’s India Act of 1784 was passed by the British Parliament to put the Company’s affairs in permanent centralized control of the British Parliament. Lord Cornwallis (1786–93) Lord Cornwallis succeeded Hastings in 1787. He introduced a new revenue system under the Permanent Settlement of Bengal in 1793 with a view to stabilize land revenue and create a loyal contented class of Zamindars. This abolished periodic auction of Zamindari rights and established permanent zamindari rights to collect land revenue from the tenants and payment of a fixed amount to the Government treasury every year. Lord Wellesley (1798–1805) During the governor-generalship of Lord Wellesley, the Fourth Mysore War (1799) was fought. This was the last Mysore war. Tipu Sultan, after regaining lost strength, set out again on his plan to oust the British from India with the

Indian History ▌ 2.123 help of Napolean and the Persian king. Lord Wellesley visualizing danger, sought an alliance with the Nizam and the Marathas and defeated Tipu Sultan in 1799 who died valiantly fighting the British. Besides war, Wellesley depended on a system of subsidiary alliances to expand British territories whereby the ruler of an aligning state was compelled to accept permanent stationing of a British force within his territory and pay subsidy for its maintenance. Sometimes a territory was added in lieu of payment. A ruler also had to accept a British resident. They were not allowed to employ any European without British approval nor negotiate with any Indian ruler without consulting the governor-general. Thus, subsidiary allied lost sovereignty in external matters, while the British resident interfered in internal administration, thus the rulers lost control over their territories. Lord Hastings (1813–23) Under the governorship of Lord Hastings Nepal was defeated in 1814, resulting in Nepal ceding Garhwal and Kumaon to the British. In 1818, the Marathas made a last attempt to regain their independence. This led to the third Anglo-Maratha war in which the Marathas were completely crushed. During Hastings’ tenure various reforms were initiated such as the Ryotwari settlement according to which direct settlement was made between the government and the Ryots (cultivators). The revenue was fixed for a period not exceeding 30 years on the basis of quality of soil. Half the net value of the crop was to be given to the government. During this period special attention was paid to education, building of roads, bridges and canals. Lord William Bentinck (1828–35) He was famous for the social reforms he introduced, such as abolition of Sati (1829), suppression of Thuggee, suppression of female infanticide and human sacrifices. English was, introduced as a medium of higher edu­ cation on the advice of his council member, Thomas Babington Macaulay. Lord Bentinck also made a pact with Maharaja Ranjit Singh, the ruler of Punjab. By the Charater Act 1833, the Company ceased to be a trading company and became an administrative power. He also adopted some corrective measures in the civil services. However, it was Cornwallis who founded the British Civil Service in India. Raja Rammohun Roy lived during his period. He was a religious and social reformer who helped Bentinck in the abolition of Sati. In 1829 a new society called Brahmo Samaj was started by Rammohun Roy which discarded idol worship, caste system and several complicated rites and rituals. Sir Charles Metcalfe (1836–44) He was notable for removing restrictions on the press and media. Lord Hardinge (1844–48) During his period the First Sikh War (1845) was fought between the Sikhs and the British. The Sikhs were defeated and were brought under British control. Lord Dalhousie (1848–56) Lord Dalhousie succeeded Lord Hardinge in 1848. During his period the Second Sikh War (1849) was fought, in which the Sikhs were defeated again and the Dalhousie was successful in annexing the whole of Punjab to the British administration.

The Doctrine of Lapse was introduced by Lord Dalhousie, whereby in the absence of a natural heir, the sovereignty of Indian states was to lapse to the British and such rulers were not permitted to adopt a son to inherit their kingdoms. Reforms The first railway line between Bombay and Thane was opened in 1853 and in the same year Calcutta and Agra were connected by telegraph. Other reforms include setting up of P.W.D. and passing of the Widow Remarriage Act (1856). Ramakrishna and Vivekananda Ramakrishna Paramahansa (1836–1886) a priest at a temple in Dakshineshwar near Calcutta emphasized that there are many roads to God and salvation and that service to man was service to God. His great disciple, Swami Vivekananda (1863–1902) popularized his religious message and founded Ramakrishna Mission in 1896. Arya Samaj The Arya Samaj was founded in 1875 by Swami Dayanand Saraswati in order to reform Hindu religion in north India. Swami Dayanand believed that there was only one God who was to be worshipped in spirit and not in the form of idols and images. He also wrote Satyarth Prakash.

First War of Independence Period Summer of 1857 during the viceroyalty of Lord Canning. Also called the Revolt of 1857 or Sepoy Mutiny or the First War of Independence. The Revolt at Meerut and the capture of Delhi was the precursor to a widespread mutiny by the sepoys and rebellion almost all over the North India, as well as Central and Western India. South remained quiet and Punjab and Bengal were only marginally affected.

Causes of the revolt The main causes are as listed below: 1. Economic exploitation by the British 2. British land revenue policies and system of law and administration 3. British attitude of supremacy towards Indians 4. British policy of annexation of territories, the annexation of Awadh and the Doctrine of Lapse created panic among provincial rulers 5. Immediate cause of the revolt was dissatisfaction in the army due to poor status of soldiers 6. The introduction of Enfield rifles whose cartridges were said to have a greased cover made of animal fat sparked off the revolt. Both Hindus and Muslims were enraged as their religious feelings were hurt 7. Fear of spread of Christianity The Beginning of the Revolt On 29 March 1857, an Indian sepoy of the 34 regiment, Mangal Pandey, killed two British officers on parade at Barrackpore. The Indian soldiers present, refused to obey orders to arrest Mangal Pandey. However, he was later on arrested, tried and hanged. The news spread like wildfire to all cantonments in the country and very soon a countrywide sepoy revolt broke out in Lucknow, Ambala, Berhampur and Meerut.

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Significance of the revolt The important element in the revolt lay in Hindu–Muslim unity. People exhibited patriotic sentiment without any touch of communal feelings. All rebels, irrespective of their religion, recognized Bahadurshah, as their emperor. It no doubt began as a mutiny of soldiers, but soon turned into a revolt against British rule in general. Important leaders connected with the revolt The heroine of this war of Independence was Rani Lakshmi Bai of Jhansi who died on 17 June 1858, while fighting the British forces. Other notable leaders were: Ahmed Ullah of Awadh, Nana Sahib of Kanpur and his loyal commander Tantia Tope, Rao Singh, Azimullah Khan, Kunwar Singh of Jagdishpur, Firuz Shah, Maulwi Ahmed Shah of Firozabad and the Begum of Awadh (Hazrat Mahal) and Ahmadullah (advisor of ex-nawab of Awadh), Khan Bahadur Khan of Bareilly, Maulavi Ahmadullah of Faizabad. The nominal leader of the revolt, Bahadur Shah II, and General Bakht Khan, who led the revolt of sepoys at Delhi have their own places in the history of 1857 revolt. Failure of the revolt The rebels were dealt a powerful blow when the British captured Delhi on 20 September 1857 and imprisoned Emperor Bahadur Shah. The British military then dealt with the rebels in one centre after another. The Rani of Jhansi died fighting on 17 June 1858. Nana Saheb refused to give in and finally escaped to Nepal in January 1859 hoping to renew the struggle. Kunwar Singh died in May 1858 trying to escape from the British, and Tantia Tope, who successfully carried out guerrilla warfare against the British until April 1859 was betrayed by a fellow rebel and was captured and put to death by the British, thus restablishing British authority over India.

Causes of failure of the revolt 1. Disunity of Indians and poor organization 2. Lack of complete nationalism—Sindhia, Holkars, the Nizam and others actively helped the British 3. Lack of coordination between sepoys, peasants, zamindars and other classes 4. Many had different motives for participating in the revolt.

British Rule After the 1857 Revolt Government of India Act 1858 Queen Victoria issued a proclamation on 1 November 1858, placing India under direct government of the Crown, whereby: 1. A viceroy was appointed in India 2. Princes were given the right to adopt a son (abolition of Doctrine of Lapse) 3. Treaties were honoured 4. Religious freedom was restored and equality treatment promised to Indians The Proclamation was called the ‘Magna Carta of Indian Liberty’. The British rule in India was strongest between 1858 and 1905. The

British also started treating India as its most precious possession and their rule over India seemed set to continue for centuries to come. Because of various subjective and objective factors which came into existence during this era, the feeling of nationalism in Indians started to grow. The Indian National Congress The Indian National Union was formed in 1885 by A. O. Hume, an Englishman and a retired Civil Servant, in association with various national leaders who called for a conference in Pune in December 1858. The conference received the unanimous support of all Indian leaders, but the venue was shifted to Bombay for various reasons. Further, the leaders decided to rename the Indian National Union as Indian National Congress. Among some of its founders were leaders like Dadabhai Naoroji, Badruddin Tyabji, Anand Mohan Bose, R. C. Dutt, Ferozeshah Mehta, Gopal Krishna Gokhale, G. Subramaniam Iyyer, Dinshaw Wacha, Bal Gangadhar Tilak, Mahadev Govind Ranade and Madan Mohan Malaviya. The first session of the Congress was held in Bombay under the presidentship of W. C. Bannerjee, a veteran lawyer of Calcutta. It was attended by about 72 delegates from all over India. From 1885 onwards the Indian National Congress met every year and its cause spread rapidly among middle class Indians. With the foundation of the Indian National Congress in 1885, the struggle for India’s independence was launched in small, hesitant and mild but organized manner. The first two decades of Indian National Congress are described in history as those moderate demands and a sense of confidence in British justice and generosity. Their aim was not to be aggressive for attaining independence lest the British should suppress this. This resulted in Indian Council Act in 1892 which allowed some members to be indirectly elected by Indians but keeping the official majority intact. The moderate period extended from 1885 to 1905. However, certain changes at the turn of the century resulted in changing the mood of Congress. Among them the most important was the change in attitude of the British. Some of the repressive and oppressive measures of the British promoted the growth of extremism within the Congress. Extremist leaders like Bipin Chandra Pal, Bal Gangadhar Tilak and Lala Lajpat Rai called on the people to show courage and self-reliance in the cause of India’s Nationalism. This era of militant nationalism was within and outside the Congress from 1905–1919. Partition of Bengal The partition of Bengal came into effect on 16 October 1905, through a Royal Proclamation, reducing the old province of Bengal in size by creating a new province of East Bengal, which later on became East Pakistan and present day Bangladesh. Swadeshi Movement (1905) The Swadeshi movement has its genesis in the anti-partition movement which was started to oppose the British decision to divide Bengal. With the start of the Swadeshi movement at the turn of the century, the Indian National Movement took a major leap forward. On 7 August 1905, a resolution to boycott British goods was adopted at a meeting of the Indian National Congress held in Calcutta. It had many advantageous consequences: (a) It encouraged Indian industries especially the

Indian History ▌ 2.125 small and medium industries; (b) Many Swadeshi banks and insurance companies were launched; (c) Journalism and national poetry developed which inculcated the feeling of nationalism in the Indian masses.

August Declaration of 1917 The control over the Indian government would be transferred gradually to the Indian people. This was the result of Hindu–Muslim unity exhibited in the Lucknow Pact.

Muslim League (1906) In 1906, All India Muslim League was set up under the leadership of Aga Khan, Nawab Salimullah of Dacca and Nawab Mohsin­ul­Mulk. The League supported the partition of Bengal, opposed the Swadeshi Movement, and demanded special safeguards for its community and separate electorates of Muslims. This led to communal differences between Hindus and Muslims.

The Gandhian Era (1918–1947) Mahatma Gandhi dominated the Indian political scene from 1918–1947. This period of the Indian National Congress is also referred to as the Gandhian Era. It was the most intense and eventful phase of India’s freedom struggle. Mahatma Gandhi provided the leadership of the highest order and his philosophy of non-violent Satyagraha became the most potent weapon to drive out the British from the Indian soil.

Swaraj In December 1906 at Calcutta, the Congress under the leadership of Dadabhai Naoroji adopted Swaraj as the goal of Indian people. Naoroji in his presidential address declared that the goal of the Indian National Congress was “Self-government of Swaraj like that of United Kingdom”. Surat Split The Indian National Congress split into two groups—the extremists and the moderates—at the Surat session in 1907 held on the banks of the river Tapti. The extremists were led by Tilak, Lajpat Rai and Bipin Chandra Pal and the moderates were led by Gopal Krishna Gokhale. At the Surat session, the moderate and extremist delegates of Congress met in an atmosphere surcharged with excitement and anger. The Session started with the objection made by the extremists to the duly elected president for the year, Ras Behari Ghose. As both sides came to the session prepared for the confrontation, the split was inevitable. Morley–Minto Reforms (1909) Morley–Minto Reforms were introduced in 1909 during the period when Lord Minto was the Governor-General of India. The reforms envisaged a separate electorate for Muslims besides other constitutional measures. The government thereby sought to create a rift within the Congress on the one hand by winning the support of the moderates, and on the other, to win favour of Muslims against Hindus. Home Rule Movement (1915–1916) Dr Annie Besant, inspired by the Irish rebellion, started a Home Rule Movement in India in September 1916. The movement spread rapidly and branches of the Home Rule League were established all over India. Bal Gangadhar Tilak wholeheartedly supported this movement. He joined forces with Dr Besant and persuaded the Muslim League to support this programme. Lucknow Pact 1916 An important step forward in achieving Hindu– Muslim unity was the Lucknow Pact 1916. Anti­British feelings were generated among the Muslims following a war between Britain and Turkey which opened way for Congress and Muslim League unity. Both the Congress and the Muslim League held sessions at Lucknow in 1916 and concluded the famous Lucknow Pact. The Congress accepted the separate electorates, and both organizations jointly demanded dominion status for the country. Hindu–Muslim unity weakened the British attitude and forced the government to announce its future policy. In 1916 a British policy was announced whereby association of Indians was increased and there was to be a gradual development of local self-governing institutions.

Rowlatt Act (1919) During the viceroyalty of Lord Chelmsford, a sedition committee was appointed by the government in 1918 with Justice Rowlatt which made certain recommendations to curb seditious activities in India. The Rowlatt Act 1919, gave unbridled powers to the government to arrest and imprison suspects without trial. The act caused a wave of anger among all sections of the people. Even before the act was passed, popular agitations had begun against it. Gandhi decided to fight against this Act and he gave a call for Satyagraha on 6 April, 1919. He was arrested on 8 April, 1919. This led to further intensification of the agitation in Delhi, Ahmedabad and Punjab. Jallianwala Bagh Massacre (13 April 1919) The arrest of Dr Kitchlu and Dr Satyapal on 10 April 1919, under the Rowlatt Act in connection with Satyagraha caused serious unrest in Punjab. The people of Amritsar took out processions to protest against the arrest. A public meeting was held the next day, 13 April 1919 in a park called Jallianwala Bagh in Amritsar where thousands of people including women and children assembled. Before the meeting could start General O’Dwyer ordered indiscriminate heavy firing on the crowd and the people had no way out to escape. As a result hundreds of men, women and children were killed and more than 1200 people wounded. The massacre was a turning point in Indo-British relations and inspired the people to provide a more unrelenting fight for freedom. It gave strength to Gandhi’s mission, which ultimately led to British expulsion from India. Khilafat Movement (1920) The Caliph, Sultan of Turkey, was looked upon by the Muslims as their religious head. During the First World War, when the safety and the welfare of Turkey were threatened by the British thereby weakening the Caliph’s position, Indian Muslims adopted an aggressive anti-British attitude. The two brothers, Mohammad Ali and Shaukat Ali launched an anti-British movement in 1920—the Khilafat Movement—for the restoration of the Khilafat. Maulana Abul Kalam Azad also led the movement. It was supported by Gandhi and the Indian National Congress which paved the way for Hindu–Muslim unity. Non-Cooperation Movement (1920) With the Congress support of the Khilafat movement, Hindu–Muslim unity was achieved which encouraged Gandhi to launch his non-violent, noncooperation movement. At the Calcutta Session in September 1920, the Congress resolved in favour of the non-violent, noncooperation movement and defined Swaraj as its ultimate aim. The

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movement envisaged: (i) surrender of titles and honorary officers; (ii) resignation from nominated offices and posts in the local bodies; (iii) refusal to attend government darbars and official functions and boycott of British courts by the lawyers; (iv) refusal of general public to offer themselves for military and other government jobs, and boycott of foreign goods, etc. The unrest and defiance of authority engendered by the non­cooperation movement contributed to the rise of many local movements in the different part of the country, movements which did not often adhere strictly either to the programme of the non-cooperation movement or even to the policy of non-violence. Chauri-Chaura Incident (1922) The Congress session held at Ahmedabad in December 1921 decided to launch a Civil Disobedience Movement while reiterating its stand on the non-violent, non­cooperation movement of which Gandhiji was appointed the leader. Before Gandhiji could launch the Civil Disobedience Movement, a mob of countrymen at Chauri-Chaura, a place near Gorakhpur in UP, clashed with the police which opened fire. In retaliation the mob burnt the police-station and killed 22 policemen. This compelled Gandhiji to call off the Civil Disobedience Movement on 12 February 1922. Despite this Gandhi was arrested and sentenced to six years imprisonment. The Chauri-Chaura incident convinced Gandhi that the nation was not yet ready for the mass-disobedience and he prevailed upon Congress Working Committee in Bardoli on 12 February 1922 to call off the NonCooperation Movement. Swaraj Party (1922) Gandhi’s decision to call off the agitation caused frustration among masses. His decision came in for severe criticism from his colleagues like Motilal Nehru, C. R. Das and N. C. Kelkar, who organized the Swaraj Party. The foundations of the ‘Swaraj Party’ were laid on 1 January 1923, as the ‘Congress­ Khilafat­Swarajya Party’. It proposed then an alternative programme of diverting the movement from widespread civil diso bedience programme to restrictive one which would encourage its member to enter into legislative councils (established under Montford Reforms of 1919) by contesting elections in order to wreck the legislature from within and to use moral pressure to compel the authority to concede to the popular demand for self-government. In the election held in 1923 the Swaraj Party captured 45 of the 145 seats. However, after the death of C. R. Das in 1925 they drifted towards a policy of cooperation with the government. This led to dissension and the party broke up in 1926. Simon Commission (1927) The activities of the Swaraj Party had induced the British government to review the working of the dyarchy system introduced by the Montague–Chelmsford Reforms and to report as to what extent a representative government could be introduced in India. The British government appointed the Simon Commission in November 1927 for the task. All members of this Commission were Europeans. Indian political leaders felt insulted and decided to boycott the Commission. It was the Indian National Congress that turned the boycott into a movement. The action began as soon as Simon and his colleagues landed in Bombay on February 3, 1928. All the major

cities and towns observed a complete Hartal and people were out on the streets participating in the mass rallies, processions and black flag demonstrations. Wherever the Commission went there were slogans ‘Simon Go back’. While leading the demonstration at Lahore, Lala Lajpat Rai was severely beaten in a police lathi­ charge and succumbed to his injuries. It was his death that Bhagat Singh and his comrades were seeking to avenge when they killed a white police official, Saunders, in December 1928. Lahore Session (1929) In December 1929, under the presidentship of Pt Jawaharlal Nehru, the Indian National Congress at its Lahore Session resolved declaring Poorna Swaraj (complete independence) to be the goal of the national movement. On 31 December 1929, the newly adopted tricolour flag was unfurled and January 26 fixed as the Independence Day which was to be celebrated every year, pleading to the people not to submit to British rule any longer. Dandi March (1930) Also called the ‘Salt Satyagraha’. To achieve the goal of complete independence, Gandhi launched another civil disobedience movement. Along with 79 followers, Gandhi started his famous march from Sabaramati Ashram on 20 March 1930, for the small village Dandi to break the Salt Law. On reaching the seashore on 6 April 1930, he broke the Salt Law by picking up salt from the seashore. By picking a handful of salt, Gandhi inaugurated the Civil Disobedience Movement, a movement that was to remain unsurpassed in the history of the Indian National Movement for the countrywide mass participation it unleashed. The First Round Table Conference (1930) It was held in London on 12 November 1930, to discuss the Simon Commission, but was totally boycotted by the Indian National Congress. In absence of the premier political party the First Round Table Conference had to be adjourned to 2 January 1931.

Gandhi–Irwin Pact (1931) Early in 1931 two moderate statesmen, Sapru and Jayakar, initiated efforts to bring about rapproachment between Gandhi and the government. Six meetings with Viceroy Lord Irwin finally led to the signing of a pact between the two on 5 March 1931, whereby the Congress called off the movement and agreed to join the Sec­ ond Round Table Conference. The Second Round Table Conference (1931) Was held in London during the viceroyalty of Lord Willington during September– December 1931 and Gandhi attended it on behalf of Indian National Congress. The conference closed on 11 December 1931, without any concrete result. The Communal Award (1932) While Gandhi was arrested on his return from London after the Second Round Table Conference, Ramsay Macdonald announced his award on communal representation in August 1931. This was another expression of the age-old British policy of ‘Divide and Rule’. Besides containing provisions for representation of Muslims, Sikhs and Europeans, it envisaged communal representation of depressed classes also. Gandhi was deeply grieved by this and underwent a fast in protest against this award since it aimed to divide India on a communal basis.

Indian History Poona Pact (1932) The communal award created immense dissatisfaction among Hindus. According to the Pact, the idea of separate electorate for the Depressed Classes was abandoned but seats reserved for them in the provincial legislatures were increased from 71 in the Award to 147, and in the Central Legislature to 81 per cent of the total. The Third Round Table Conference (1932) It was held in 1932 but again proved fruitless since the national leaders were in prison.

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Mission would visit India to make recommendations regarding constitutional reforms to be introduced in India. The Mission envisaged the establishment of a Constituent Assembly to frame the Constitution as well as an interim government. The Muslim League accepted the plan on 6 June 1946, while maintaining its rights of striving for a separate Muslim state. The Congress also partially accepted the plan.

The Government of India Act (1935) The Simon Commission report submitted in 1930 formed the basis for the Government of India Act 1935. The new Government of India Act received the royal assent on 4 August 1935. The Act continued and extended all the existing features of the Indian constitution. Popular representation, which went back to 1892, dyarchy and ministerial responsibility, which dated from 1921, provincial autonomy, whose chequered history went back to eighteenth century presidencies, communal representation, which first received overt recognition in 1909, and the safeguards devised in 1919, were all continued and in most cases extended. But in addition there were certain new principles introduced. It provided for a federal type of government. Thus, the act:

Azad Hind Fauj In 1943, Subhash Chandra Bose formed the ‘Azad Hind Fauj’ in Singapore (Indian National Army—(INA)) and gave his famous call Dilli Chalo. He aimed a military campaign for the independence of India. The INA was joined in large numbers by the Indian residents of south-east Asia and by the Indian soldiers and officers captured by the Japanese forces in Malaya, Singapore and Burma. Subhash Chandra Bose set up two INA headquarters, one in Rangoon and the other in Singapore, and began to reorganize INA. Recruits were sought from civilians, funds were gathered, and even a women’s regiment called Rani Jhansi regiment was formed. One INA battalion also accompanied the Japanese Army to the Indo-Burma front to participate in the Imphal campaign. But with the defeat of Japan in 1944–45, the INA also died out. Bose is said to have been killed in an air crash on his way to Tokyo in August 1945.

(i) Introduced provincial autonomy (ii) Abolished dyarchy in provinces (iii) Made ministers responsible to the legislative and federation at the centre

Direct Action Campaign Provoked by the success of the Congress, the Muslim League launched a direct action campaign on 16 August 1946, which resulted in heavy communal riots in the country.

The Act of 1935 was condemned by nearly all sections of Indian public opinion and was unanimously rejected by the Congress. The Congress demanded instead, the convening of a Constituent Assembly elected on the basis of adult franchise to frame a constitution for an independent India. Quit India Movement (1942–1945) On 8 August 1942, the Congress in its meeting at Bombay passed a resolution known as ‘Quit India’ resolution, whereby Gandhiji asked the British to quit India and gave a call for ‘Do or die’ to his countrymen. On 9 August 1942, Gandhiji was arrested but the other leaders continued the revolutionary struggle. Violence spread throughout the country, several government offices were destroyed and damaged, telegraph wires were cut and communication paralysed. The movement was, however, crushed by the government. Gandhi’s Fast Gandhi undertook a 21­day fast in jail. His condi­ tion deteriorated after 13 days and all hopes of his surviving were given up. However, as a result of his moral strength and spiritual stamina, he survived and completed the 21-day fast. This was his answer to the government which had been constantly exhorting him to condemn the violence of the people in the ‘Quit India Movement’. Gandhi not only refused to condemn people resorting to violence but unequivocally held the government responsible for it. Cabinet Mission Plan The struggle for freedom entered a decisive phase in the year 1945–46. The British Prime Minister, Lord Attlee, made a declaration on 15 March 1946, that British Cabinet

Interim Government On 2 September 1946, an interim government was formed. Congress members led by Pandit Jawaharlal Nehru joined it but the Muslim League did not as it withdrew its earlier acceptance of the Cabinet Mission Plan. Format of Constituent Assembly The Constituent Assembly met on 9 December 1946, and Dr Rajendra Prasad was elected its Pres­ ident. The Muslim League did not join the Assembly. Mountbatten Plan In March 1947, Lord Mountbatten replaced Lord Wavell. He announced his plan on 3 June 1947. It offered a key to the political and constitutional deadlock created by the refusal of the Muslim League to join the Constituent Assembly formed to frame the Constitution of India. Mountbatten’s formula was to divide India but retain maximum unity. Its acceptance by the Congress and the Muslim League resulted in the birth of Pakistan. The Indian Independence Act 1947 The Bill containing the provisions of the Mountbatten Plan of 3 June 1947, was introduced in the British Parliament and passed as the Indian Independence Act 1947. The Act laid down detailed measures for the partition of India and speedy transfer of political powers to the new governments of India and Pakistan. Partition of India In accordance with the Indian Independence Act 1947, India was partitioned on 15 August 1947 into India and Pakistan. The Act made India and Pakistan independent dominions. Bloodshed and violence marked the exodus of refugees. The state of Kashmir acceded to the Indian Union, after the raiders

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were helped by Pakistan, in October 1947. Lord Mountbatten was appointed the Governor General of free India and M. A. Jinnah the first Governor General of Pakistan.

India After Independence Lord Mountbatten became the first Governor­General of free India. Sir C. Rajagopalachari became the first and the only Indian governor­general of India in 1948. Pt Jawahar Lal Nehru took over as the first Prime Minister in 1950. Sardar Vallabhbhai Patel single-handedly dealt with the accession of all princely states. All states were merged into neighbouring provinces. The state

of Kashmir, Hyderabad and Mysore merged later on. Mahatma Gandhi undertook a fast for the sake of Muslim rights. On 30 January 1948, he was assassinated by Nathuram Vinayak Godse at the Birla House prayer meeting in Delhi. On 13 September 1948, the Indian Army marched into Hyderabad after the violent actions of the Razakars and the state was acceded to the Indian union. On 26 November 1949, the Constituent Assembly passed the new Constitution of India. On the morning of 26 January 1950, India was proclaimed a republic and Dr Rajendra Prasad took over as the first president, Dr S. Radhakrishnan as the vice­president and Pt Jawaharlal Nehru as the first prime minister of India.

PRACTICE EXERCISE 1. Who initiated the move to form the Indian National Congress? (a) A. O. Hume (b) W. C. Bannerjee (c) Annie Besant (d) Gandhiji 2. Who was the first President of the Indian National Congress? (a) A. O. Hume (b) W. C. Bannerjee (c) Annie Besant (d) Motilal Nehru 3. In which year was the Indian National Congress formed? (a) 1835 (b) 1885 (c) 1901 (d) 1875 4. Who among the following belonged to the moderate group of the Indian National Congress? (a) Lala Lajpat Rai (b) Bipin Chandra Pal (c) Gopal Krishna Gokhale (d) Bal Gangadhar Tilak 5. Jallianwala Bagh massacre took place in the city of (a) Agra (b) Meerut (c) Amritsar (d) Lahore 6. The governor-general of India who initiated the introduction of English in India was (a) Lord Curzon (b) Lord Macaulay (c) Lord Bentinck (d) Lord Dalhousie 7. Gandhi’s Dandi March started from (a) Bardoli (b) Calcutta (c) Surat (d) Ahmedabad 8. Quit India movement was adopted by the Indian National Congress at (a) Bombay (b) Ahmedabad (c) Pune (d) Calcutta 9. King George V visited India during the viceroyalty of (a) Lord Ripon (b) Lord Curzon (c) Lord Harding (d) Lord Reading

10. Who introduced ‘Permanent Settlement’ in Bengal? (a) Lord Bentinck (b) Lord Cornwallis (c) Lord Wellesley (d) Lord Hastings 11. After the Chauri-Chaura incident Gandhi called off the noncooperation movement because (a) the people were fed up with the movement (b) he did not want the movement to turn violent (c) he realized that the government had almost crushed the movement (d) the masses did not display the desired enthusiasm 12. Arrange the following in chronological order: 1. Formation of Interim government 2. Recall of Lord Linlithgow 3. Cabinet Mission 4. Cripps Mission 5. Lord Wavell’s proposals 6. Mountbatten Plan (a) 2, 3, 1, 4, 6, 5 (b) 2, 4, 5, 3, 1, 6 (c) 4, 2, 6, 1, 5, 3 (d) 1, 2, 4, 3, 6, 5 13. Match the following: (a) Simon Commission (b) Quit India Movement (c) Formation of Indian National Congress (d) Minto–Morley Reforms ABCD (a) 1 2 3 4 (c) 3 2 1 4

1. 1885 2. 1942 3. 1927 4. 1909 (b) 4 3 2 1 (d) 1 3 2 4

14. In 1942, the Cripps Mission promised to grant (a) complete dominion status to India (b) self-government in India (c) provincial government in India (d) interim government in India

Indian History ▌ 2.129 15. The Sepoy Mutiny of 1857 failed due to I. poor organization II. lack of unity III. improper leadership (a) II and III only (b) I only (c) I, II and III (d) I and II only

27. The first Civil Disobedience Movement was started by Gandhi after the (a) Rowlatt Act of 1919 (b) Jallianwala Bagh massacre (c) Chauri-Chaura incident of 1922 (d) Arrival of Cripps Mission

16. The Muslim League was founded during the period (a) 1901–10 (b) 1911–20 (c) 1921–30 (d) 1931–40

28. Match the following: (a) Civil Disobedience Movement (b) Indian Home Rule (c) Moderate Leader (d) Brahmo Samaj (a) A-3, B-1, C-4, D-2 (b) A-1, B-2, C-3, D-4 (c) A-4, B-3, C-2, D-1 (d) A-1, B-3, C-4, D-2

17. The Swadeshi Movement was started by (a) stopping migration to other countries (b) boycotting foreign goods (c) speaking Hindi only (d) taking interest in Indian culture 18. Who was the Governor-General during the period 1774–85? (a) Lord Ripon (b) Lord Curzon (c) Lord Hastings (d) Lord Bentinck 19. Raja Rammohun Roy was the founder of (a) Arya Samaj (b) Brahmo Samaj (c) Ramakrishna Mission (d) Prarthana Samaj 20. Which important event immediately preceded the Jallianwilla Bagh massacre? (a) Rowlatt Act enactment (b) Communal Award (c) Simon Commission (d) Quit India Movement 21. The first partition of Bengal under Lord Curzon took place in (a) 1901 (b) 1915 (c) 1908 (d) 1905 22. Who among the following was captured and hanged by the British after the Sepoy Mutiny? (a) Lakshmi Bai (b) Tantia Tope (c) Tipu Sultan (d) Nana Saheb

1. B. G. Tilak 2. Raja Rammohun Roy 3. Mahatma Gandhi 4. G. K. Gokhale

29. Which of the following Acts gave representation to Indians for the first time in the legislature? (a) Indian Council Act 1909 (b) Indian Council Act 1919 (c) Government of India Act 1935 (d) Government of India Act 1945 30. Which of the following pairs are correct? 1. The Indian Councils Act 1919—Provincial Autonomy 2. The Government of India Act 1919— Introduction of Dyarchy 3. The Government of India Act 1935—Proposal for federation of India 4. The India Independence Act 1947—Provincial Legislature (a) 1 and 4 (b) 2 and 3 (c) 1 and 3 (d) 2 and 4

23. The Indian National Congress adopted the resolution of ‘Complete Independence’ in its session held at (a) Lahore (b) Calcutta (now Kolkata) (c) Madras (d) Bombay

31. Partition of Bengal was undertaken with a view to (a) suppressing the revolt (b) satisfying the Hindus and Muslims (c) meeting the demand of Muslims (d) dividing Hindus of West and East Bengal and increasing Hindu–Muslim tension

24. “Swaraj is my birth right and I shall have it.” This was advocated by (a) Lala Lajpat Rai (b) Nana Saheb (c) Mahatma Gandhi (d) Lokmanya Tilak

32. Lord Wellesley greatly extended the British Dominion through (a) Doctrine of lapse (b) Subsidiary alliance (c) Partitioning of India (d) Annexing princely states

25. The slogan ‘Inquilab Zindabad’ was raised by (a) Subhash Chandra Bose (b) Bhagat Singh (c) Mahatma Gandhi (d) Jawaharlal Nehru

33. Gandhi started the Dandi March (a) to demonstrate against the policies of the British (b) to break the Salt Law (c) to boycott foreign goods (d) All of the above

26. Who among the following was the first British Viceroy as well as Governor-General of India? (a) Lord Canning (b) Lord Warren Hastings (c) Lord Dalhousie (d) Lord Bentinck (e) dee 'e'

34. The Sepoy Mutiny failed due to (a) lack of effective organization (b) lack of unity of purpose (c) lack of unified system of leadership (d) All the above

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35. Which of the following was the reason for calling off the noncooperation movement by Gandhiji? (a) High pressure from British government (b) Chauri-Chaura incident (c) Round Table Conference (d) His arrest 36. Who among the following was not a moderate? (a) Bal Gangadhar Tilak (b) Ferozeshah Mehta (c) Gopal Krishan Gokhale (d) S. N. Bannerjee 37. Montague–Chelmsford Reforms intended to (a) bring Indians nearer the government (b) create a self-governing government (c) introduce the system of dyarchy (d) stop Indians from getting higher education 38. The first Satyagraha campaign of Gandhi was started in (a) Champaran (b) Bardoli (c) Dandi (d) Baroda 39. Arrange the following in correct chronological order: (a) Jallianwala Bagh Massacre (b) Second Civil Disobedience Movement (c) Congress split at Surat (d) Boycott of Simon Commission (a) C A D B (b) A B C D (c) D C A B (d) D C B A 40. The first Muslim President of the INC was (a) Ajmal Khan (b) M. A. Jinnah (c) Abul Kalam Azad (d) Rahimulla Sayani 41. ‘Ring fence’ is associated with (a) Henry Lawrence (b) Dalhousie (c) Warren Hastings (d) Lord Hastings 42. ‘Sutudri’ mentioned in the Rig Veda is (a) a term used to denote bride (b) the name of a river (c) a plant used in the preparation of Sura (d) a sacrificial animal 43. A board of copper images generally assigned to the Harappan culture phase, has been found at (a) Daimabad (b) Brahmagiri (c) Inamgaon (d) Nasik 44. About the 5th century BC, the Indian satrapy was paying an annual tribute of 320 tolas of gold to (a) Greece (b) Persia (c) Bactria (d) China 45. Adhvaryu mentioned in the Vedic literature was (a) An associate of the Vedic rajan (b) One of the ratnins (c) A type of chariot (d) A category of sacrificial priest

46. After the Revolt of 1857, the British recruited Indian soldiers mostly from the (a) Brahmins of UP and Bihar (b) Bengalis and Oriyas from the east (c) Gurkhas, Sikhs and Punjabis in the north (d) Madras Presidency in the south and Marathas from the west 47. Akbar’s buildings at Fatehpur Sikri are famous for (a) Synthesis of trabeate and arcuate styles (b) Construction of the first bulbous dome in India (c) The use of marble on an extensive scale (d) Frequent employment of pietra dura 48. Among the following sites, sculptural evidence regarding the first representation of the Buddha in human form has come from (a) Bharhut (b) Mathura (c) Sanchi (d) Amaravati 49. Aranyani, Goddess of the forest, is first mentioned in the (a) Rik Samhita (b) Atharvaveda (c) Aranyaka texts (d) Upanishadic text 50. Asoka’s contemporary Greek King of Syria mentioned in his edicts, was (a) Antiochus II Theos (b) Ptolemy II (c) Antigonas (d) Alexander 51. Before coming to Akbar’s court, Miyan Tansen was in the service of (a) Raja of Jodhpur (b) Raja of Jaipur (c) Raja of Ramchandra (d) Rani of Gondwana 52. Consider the following passage: “His records proclaim him as a feudatory of the emperor Somesvara III. He had his capital at Talakad in the south and Bankapura in the north. It was in ad 1137 that he performed the great ceremony of Tulapurusha.” The above passage refers to (a) Mayurasarman (b) Anantzavarman Chodaganga (c) Yayati Mahasivagupta I (d) Vishnuvardhana 53. During Akbar’s reign, the land which was cultivated every year and was never left fallow was known as (a) Polaj (b) Parauti (c) Chachar (d) Banjar 54. During the 13th and 14th centuries, peasants in India did not cultivate (a) Wheat (b) Barley (c) Gram (d) Maize 55. During the decline of the Mughal empire, the Jats were organized into a political force by (a) Rajaram (b) Churaman (c) Badan Singh (d) Suraj Mal

Indian History 56. During the period of Harshavardhana, the fields were watered by the pots of the ghatiyantra. This is known from the writings of (a) Hiuen Tsang b) Bana (c) Subhandhu (d) Dandin 57. Ghatiyantra was used in early India for (a) Manufacture of vessels of metal (b) Pouring of water in sacrificial rituals (c) Conducting tantric rituals (d) Irrigation from wells 58. Given below are four industries that developed during the British rule in India: 1. Jute industry 2. Iron and steel industry 3. Textile industry 4. Sugar industry Which one of the following shows the correct chronological order in which the industries developed? (a) 1, 2, 3, 4 (b) 3, 1, 2, 4 (c) 1, 3, 2, 4 (d) 4, 1, 2, 3 59. Given below are some famous Buddhist sites. Which one of them shows the correct chronological sequence in which they came into prominence? (a) Nalanda, Nagarjunakonda, Amaravati, Sanchi (b) Sanchi, Amaravati, Nagarjunakonda, Nalanda (c) Sanchi, Nagarjunakonda, Amaravati, Nalanda (d) Nagarjunakonda, Sanchi, Amaravati, Nalanda 60. Hiuen Tsang found Jainsim flourishing in (a) Orissa (b) Kashmir (c) Bengal (d) Bihar 61. Ibrahim Khan Gardi was an important military officer of (a) Haider Ali (b) Ahmad Shah Durrani (c) Peshwa Balaji Daji Rao (d) Raja Suraj Mal 62. In 1751, Alivardi virtually ceded Orissa to (a) Nanaji Holkar (b) Raghuji Bhosle (c) Mahajadi Sindhia (d) Baji Rao 63. In ancient and early medieval India, the uncultivated and untaxed land was called (a) Sita (b) Kharvatka (c) Khiliakshetra (d) Sitadhyaksha 64. In ancient India, the profession of the Karavara involved work (a) as a prison guard (b) as a farmer (c) as a hunter-gatherer (d) in leather 65. In ancient Indian social structure, the term aniravasita was related to (a) brahmanas and kshatriyas (b) vaisyas and sudras (c) sudras only (d) people outside the varna classification

▌ 2.131

66. In March 1784, the Treaty of Mangalore was concluded mainly at the insistence of (a) Warren Hastings (b) Colonel Fullarton (c) Tipu Sultan (d) Lord Macartney 67. In the early decades of Maratha rule, the most popular unit of land revenue measurement was (a) gaj (b) kathi (c) jurib (d) janjit 68. In the late 17th and early 18th centuries, the export of Indian opium to the Indonesian archipelago was dominated by the (a) Gujarati and Marwari mercantile groups (b) Portuguese (c) Dutch East India Company (d) English East India Company 69. In the Mauryan administration, the official designated as Rupadarsaka was (a) examiner of silver and other metals (b) superintendent of courtesans (c) manager of stage (d) examiner of coins 70. In the Vijayanagara kingdom, the provincial governors who did not belong to the royal family but were appointed on merit were generally known as (a) Dalavays (b) Dandanayakas (c) Amaranayakas (d) Upapradhani 71. In Vajrayana Buddhism, the spouse of the Buddha Bodhisatva was known as (a) Matangi (b) Yogini (c) Dakini (d) Tara 72. In which one of the following spheres, Akbar carried forward the policies first initiated by Sher Shah? (a) Measurement of land for purposes of assessing land revenue (b) Recruitment of nobility from various sections of the society (c) Appointment of rulers who accepted his suzerainty to high offices (d) Withdrawal of discriminatory laws against nonMuslims 73. Jivaka, mentioned in the early Buddist literature, was (a) Bodhisatva (b) King (c) Merchant (d) Physician 74. Kumarasambhavam describes the story of the birth of (a) Sanathkumara (b) Kartikeya (c) Pradyumna (d) Abhimanyu 75. Land revenue in the Vijayanagar kingdom ranged from (a) 20% to 25% of the produce (b) 26% to 30% of the produce (c) 31% to 50% of the produce (d) 51% to 60% of the produce

2.132

▌ General Knowledge

76. Match List I (Attributes) with List II (Bodhisatvas), and select the correct answer from the codes given below: List I (a) Pot carrier (b) Thunderbolt carrier (c) Lotus­bearer (d) Sword­holder

List II 1. Vajrapani 2. Avalokiteshwara 3. Maitreya 4. Manjushri

Codes: (a) A-4, B-1, C-2, D-3 (b) A-3, B-1, C-4, D-2 (c) A-3, B-1, C-2, D-4 (d) A-4, B-2, C-3, D-1

List II 1. Deccan 2. Rajasthan 3. Kutch 4. Afghanistan

Codes: (a) A-3, B-4, C-1, D-2 (b) A-1, B-3, C-2, D-4 (c) A-2, B-3, C-1, D-4 (d) A-2, B-3, C-4, D-1 78. Match List I (Mughal Governors) with List II (Territories), and select the correct answer from the codes given below: List I (a) Alivardi (b) Saadat and Safdar Jung (c) Zakir Khan (d) Nizam-ul-mulk

List II 1. Punjab 2. Hyderabad 3. Bengal 4. Oudh 5. Karnataka

Codes: (a) A-4, B-2, C-1, D-3 (b) A-1, B-4, C-3, D-2 (c) A-3, B-2, C-4, D-5 (d) A-3, B-4, C-1, D-2

Codes: (a) A-3, B-1, C-2, D-4 (b) A-3, B-2, C-1, D-4 (c) A-2, B-1, C-4, D-3 (d) A-1, B-4, C-2, D-3

List II 1. Goa 2. Pulicut 3. Bugli 4. Chinsura

81. Match List I with List II, and select the correct answer from the codes given below: List I (a) The First Mysore War (b) The Second Mysore War (c) The Third Mysore War (d) The Fourth Mysore War

List II 1. Cornwallis 2. Lord Hastings 3. Wellesley 4. Warren Hastings 5. Col Smith

Codes: (a) A-5, B-4, C-1, D-3 (b) A-1, B-2, C-3, D-4 (c) A-5, B-4, C-3, D-2 (d) A-4, B-3, C-2, D-1 82. Match List I with List II, and select the correct answer from the codes given below: List I (a) Nyaya (b) Vaiseshika (c) Sankhya (d) Mimamsa

List II 1. Jamini 2. Kapila 3. Kanada 4. Gautama

Codes: (a) A-3, B-4, C-2, D-1 (b) A-2, B-3, C-1, D-4 (c) A-4, B-3, C-2, D-1 (d) A-4, B-1, C-3, D-2

79. Match List I (Tribes who revolted against the British) with List II (Year of revolution), and select the correct answer from the codes given below: List I (a) Khonds (b) Santhals (c) Kols (d) Mudnas

List I (a) Dutch (b) English (c) Portuguese (d) French Codes: (a) A-3, B-4, C-1, D-2 (b) A-1, B-3, C-2, D-4 (c) A-2, B-3, C-1, D-4 (d) A-2, B-1, C-4, D-3

77. Match List I (Items found in Harappan sites) with List II (Possible sources), and select the correct answer from the codes given below: List I (a) Copper (b) Shell (c) Lapis (d) Gold

80. Match List I and List II, and select the correct answer from the codes given below:

List II 1. 1820 2. 1846 3. 1855 4. 1895

83. Match List I with List II, and select the correct answer from the codes given below: List I (a) Koliyas (b) Moriyas (c) Videhas (d) Lichhavis Codes: (a) A-3, B-2, C-4, D-1 (b) A-1, B-4, C-2, D-3 (c) A-3, B-1, C-4, D-2 (d) A-4, B-3, C-2, D-1

List II 1. Pippalivana 2. Vaisali 3. Ramagama 4. Mithila

Indian History ▌ 2.133 84. Match List I with List II, and select the correct answer from the codes given below: List I (a) Chandidasa sect (b) Chaitanya (c) Husain Shah Philosophy (d) Surdas philosophy

List II 1. Radhavallabh 2. Sahajiyas 3. Advaita 4. Satyapir 5. Bhedabheda

Codes: (a) A-2, B-3, C-4, D-5 (b) A-2, B-5, C-4, D-1 (c) A-1, B-5, C-3, D-4 (d) A-3, B-4, C-2, D-1 85. Match List I with List II, and select the correct answer from the codes given below: List I (a) Revolt of the Bhils (b) Revolt of the Gujars (c) Kolhapur Rising (d) Santhal Rebelion

List II 1. 1855–56 2. 1817–18 3. 1839–45 4. 1824

Codes: (a) A-2, B-4, C-1, D-3 (b) A-1, B-3, C-2, D-4 (c) A-2, B-4, C-3, D-1 (d) A-2, B-1, C-3, D-4 86. Match List I with List II, and select the correct answer from the codes given below: List I (a) Double burial (b) Fire altars (c) Workmen’s Quarters (d) Bead making

List II 1. Chanhudaro 2. Kalibangan 3. Lothal 4. Banawali 5. Harappa

Codes: (a) A-3, B-2, C-5, D-1 (b) A-3, B-4, C-1, D-2 (c) A-4, B-2, C-3, D-5 (d) A-2, B-3, C-5, D-1 87. Megasthenes was succeeded as ambassador to the Mauryan court by (a) Hegesander (b) Deimachos (c) Athenaeus (d) Nearchus 88. Monopoly of the East India company to trade with China was abolished by the Charter Act of (a) 1793 (b) 1813 (c) 1833 (d) 1853 89. Nagarasresthi and Sarthavaha are referred to in the context of urban administration in inscriptions of the Gupta period from (a) Madhyadesa (b) Pundravardhana (c) Avanti (d) Saurashtra

90. Qutubuddin Aibak is said to have built, besides the Qutub mosque and the Qutub Minar, (a) The Arhai­din­ka Jhopra mosque at Ajmer (b) The Khirki masjid at Jahan­Panath (c) The mausoleum of Khan-i-Jahan Tilangani (d) The Kila-i-Kuhna mosque at Delhi 91. Raj Ballabh, Ghasiti Begam, Shaukat Jung were the arch foes of (a) Alivardi Khan (b) Husain Shah (c) Mir Qasim (d) Sirajuddaulah 92. Religion-wise the sculptures found at Kankali Tila in Mathura are (a) Buddhist (b) Jaina (c) Shakta (d) Vaishnava 93. Rig Vedic people invoked Indra for (a) Acquisition of knowledge (b) A life after death (c) Release from the cycle of births and deaths (d) Material conforts and victory 94. Since 1858, the supreme court of financial administration in India was vested in the (a) Supreme Government in India (b) British Parliament (c) Board of Control in England (d) Secretary of State in Council 95. Sramana is paired in early sources with (a) Ajivika (b) Saugata (c) Brahmana (d) Nirgrantha 96. Svyamvara was a special form of (a) Gandharva marriage (b) Paisaca marriage (c) Raksasa marriage (d) Brahma marriage 97. Man in some areas of India lived in underground pits is suggested by excavation in (a) Bihar (b) Kashmir (c) Karnataka (d) Rajasthan 98. The Dal Khalsa was founded by (a) Guru Gobind Singh (b) Kapur Singh (c) Ranjit Singh (d) Teg Bahadur 99. The battle of Koppan was fought between (a) The Western Chalukyas and the Pallavas (b) The Western Chalukyas and the Cholas (c) The Cholas and the Hoysalas (d) The Pandyas and the rulers of Ceylon 100. The Besnagar inscription of Heliodorus refers to (a) Samkarshana and Vasudeva (b) Samkarashana, Pradyumma and Vasudeva (c) Vasudeva only (d) All the Panchaviras

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▌ General Knowledge

101. The bill to cancel the exemption given to the European community from the jurisdiction of the magistrates and the criminal courts established by the East India Company was proposed in (a) 1849 (b) 1861 (c) 1879 (d) 1882 102. The Chauluka king Kumarapala is said to have instituted a Jaina reformation under the guidance of (a) Mallinatha (b) Sthulabhadra (c) Nayachandra (d) Hemchandra 103. The conflict between the Mughals and the Marathas began during the reign of (a) Akbar (b) Jahangir (c) Shah Jahan (d) Aurangzeb 104. The crops cultivated by the Harappans included (a) Wheat, barley and mustard (b) Barley, groundnut and rice (c) Wheat, rice and sugarcane (d) Wheat, cotton and sugarcane 105. The doctrine of void (Shunyata) was first expounded by the Buddhist philosopher (a) Nagarjuna (b) Nagasena (c) Ananda (d) Ashvaghosha 106. The earliest Indian coins are not earlier than (a) 7th Century BC (b) 4th Century BC (c) 3rd Century BC (d) 5th Century BC 107. The existence of the store houses as famine relief measure in the Mauryan period is known from (a) Rock edict II from Girnar (b) Pillar edict II (c) Minor edict from Sasaram (d) Sohgaura plate inscription 108. The export of slaves from India was banned by (a) Mohammad Bin Tughlaq (b) Firoz Tughlaq (c) Alauddin Khilji (d) Sikandar Lodi

112. The first Tamil Sangam is said to have been instituted by (a) Tiruvalluvar (b) Parasurama (c) Mamulanar (d) Agastya 113. The gold coins of Muhammad Ghori usually bear (a) Goddess Lakshmi on the observe and his name in Nagari characters on the reverse. (b) Kalima on one side and his name on the other in Arabic characters (c) His name on the observe and name of mint and year of minting on the reverse (d) Zodiac signs on the obverse and his name in Persian characters on the reverse 114. The Government of India Act of 1919 clearly defined the (a) Separation of power between the judiciary and the legislature (b) Jurisdiction of the central and provincial governments (c) Jurisdiction of the Viceroy and the Governors of the provinces (d) Powers of the Secretary of State of India and the Viceroy 115. The greatest painter of birds at Jahangir’s court was (a) Khwaja Abdus Samad (b) Sayyed Ali Tabrizi (c) Basawan (d) Mansur 116. The group which figures prominently in the context of cattle aids in the Rigveda was that of the (a) Miechchhas (b) Dasas and Dasyus (c) Panis (d) Nishadas 117. The Gupta gold and silver issues were initially based on the coins of the (a) Romans and the Saka Kshatrapas (b) Kushanas and Yaudheyas (c) Kushanas and Saka Kshatrapas (d) Romans and Kushanas 118. The head of a guild in ancient and early medieval India was called (a) Adhipati (b) Gahapati (c) Jetthaka (d) Adhikari

109. The fertile agrarian tract in the early Tamil country is denoted by the term (a) Palai (b) Marudam (c) Mullai (d) Neydal

119. The Hindu God who has sometimes been identified by Greek authors with Herakles is (a) Skanda (b) Siva (c) Krishna (d) Parjanya

110. The first Mughal building with complete marble facing was (a) Humayun’s tomb (b) Itmaduddaulah’s tomb (c) Taj Mahal (d) Moti Masjhid at the Red Fort, Delhi

120. The Indian ruler who defeated the English in the early stage of English rule in India was (a) Ranjit Singh (b) Mahadaji Sindhia (c) Shah Jahan (d) Hyder Ali

111. The first surviving true dome in India is (a) The Alai Darwaza (b) Sultan Firuz’s madrasa (c) Balban’s tomb (d) Arhai-din-ka Jhopra

121. The Iqta under the Delhi Sultans was a territorial assignment and its holder was designated Muqti who (a) Had no claims on the peasant other than the due land tax (b) Was the owner of the iqta land (c) Was entitled to claim forced labour from the cultivars (d) Had claim on the person and property of the peasants

Indian History ▌ 2.135 122. The Jaina system of thought closely resembles (a) Vedanta (b) Sankhya (c) Vaisesika (d) Yogachara 123. The Kalyana-mandapa was a notable feature of temple architecture in the kingdom of (a) Pallavas (b) Cholas (c) Kadambas (d) Vijayanagara 124. The Kingdom of Ranjit Singh included (a) Delhi (b) Srinagar (c) Makram (d) Kabul 125. The land revenue in Mughal India was a (a) Tax on land (b) Share in the produce (c) Tax on cultivator (d) Tax on the owner of the land 126. The largest concentration of Harappan sites has been found along the (a) Sutlej (b) Indus (c) Ghaggar-Hakra (d) Ravi 127. The manuscript of Dastaan-i-Amir Hamza was illustrated by (a) Abdus Samad of Shiraz (b) Faruq Qalmaq (c) Mir Sayyid Ali of Tabraz (d) Miskin 128. The Mazhabi Sikhs were originally (a) Landed magnates (b) Peasants (c) Artisans (d) Menials 129. The Mughal emperor who drove away the Portuguese from Hughli, the premier port of Bengal, was (a) Akbar (b) Jahangir (c) Shah Jahan (d) Aurangzeb 130. The Mughal firman of free trade in Bengal was granted to the English in (a) 1770 (b) 1756 (c) 1650 (d) 1717 131. The name of the mother-goddess Uma appears on the coins of (a) Kunindas (b) Indo-Greeks (c) Kushanas (d) Gupta 132. The Nawab of Bengal who transferred the capital from Dhaka to Murshidabad was (a) Mir Qasim (b) Sirajuddaulah (c) Murshid Quli Khan (d) Mir Jafar 133. The nimbus formed a special feature of Mughal portraiture under (a) Akbar (b) Jahangir (c) Shah Jahan (d) Aurangzeb 134. The office of the wakil­us­Mughal administration signified the (a) Land owned by the Emperor himself (b) Entire Imperial Establishment

(c) Land from where revenue was collected for the imperial treasury (d) Religious land grants 135. The original canon of the Jainas is believed to have consisted of (a) 6 angas (b) 8 angas (c) 10 angas (d) 12 angas 136. The Pitt’s India Act of 1784 sought to (a) Introduce some parliamentary control over the English East India Company (b) Reduce the armed forces of the East India Company to curtail the company’s trading rights (c) Curtail the company’s trading rights (d) Convert the company into a business firm without any political authority 137. The planning of the capital in the Arthasastra clearly indicates that the industrial and commercial classes were (a) Not given any consideration (b) Provided specific quarters in the township (c) Allotted space outside the township (d) Allotted space inside and outside the township 138. The post-Gupta social structure in north India was marked by the (a) Growing importance of women (b) Lessening of untouchability (c) Proliferation of castes (d) Revival of slavery 139. The predominant funerary practice at Harappa was (a) Cremation (b) Burying ash kept in an urn (c) Extended burial of the body with pottery (d) Making a memorial to mark the site 140. The rate of land revenue was highest during the reign of (a) Ghiyasuddin Balban (b) Alauddin Khilji (c) Ghiyasuddin Tuglaq (d) Mohammad Bin Tughlaq Directions (Q 141–155) Given below are two statements, one labelled as Assertion (a) and other labelled as Reason (R). In the context of the above two statements which one of the following is correct? (a) Both A and R are true and R is the correct explanation of A (b) Both A and R are true but R is not a correct explanation of A (c) A is true but R is false (d) A is false but R is true 141. Assertion (A): The Sabha and Samiti, the two political bodies of the Rig Vedic Aryans, began to lose their strength during the time of the Mahajanapadas. Reason (R): These tribal assem blies could not fit into organizational framework of the large territorial states.

2.136

▌ General Knowledge

142. Assertion (A): According to the foreign traveller Paes, the Devdasis held a highly respectable position in the Vijaynagar society. Reason (R): The Devdasi system of offering dancing girls to the temples in South India came into existence for the first time under the Vijaynagar rulers. 143. Assertion (A): The partition of India resulted from the failure of the Hindu and Muslim communities to agree on how and to whom power was to be transferred. Reason (R): The political leadership of independence movement of India was divided on sectarian lines and strongly believed in the two-nation theory. 144. Assertion (A): Major Rock Edict No. XIII mentioning Ashoka’s conquest of Kalinga, does not appear at Dhauli. Reason (R): Ashoka did not want the people of Kalinga to be reminded of his conquest. 145. Assertion (A): There are instances of rural resistance in early medieval Deccan and South India. Reason (R): Grants of agrahara and devadana rights to brahmanas and temples respectively undermind rights of villagers. 146. Assertion (A): The Dutch East India Company began to trade with India in the early seventeenth century. Reason (R): The Dutch wanted to barter cotton piece goods for pepper and spices in the Indonesian archipelago. 147. Assertion (A): The month scales were introduced by Shah Jahan to fix the minimum military obligation of a mansabdar according to his salary. Reason (R): The Mughals were always faced with the problem of hasil falling far below the jama. 148. Assertion (A): After their occupation of Bengal, Bihar and Orissa, the English used various means to bolster their trade which resulted in the decline of Indian handicrafts industry. Reason (R): The British monopolised the trade of Bengal through various legislative enactments. 149. Assertion (A): The iqta system went a long way in establishing Islam in North India. Reason (R): The Sultans, through this institution, wanted to control the Turkish nobles. 150. Assertion (A): Gautamiputra Satakarni was opposed to the system of our varnas. Reason (R): Gautamiputra Satakarni extended patronage to Buddhist monks. 151. Assertion (A): After the Kushans, the Turks brought the technique of constructing the true on a massive scale. Reason (R): The Turks introduced into India the technique of preparing line-mortar. 152. Assertion (A): Despite initial victory, the Sepoy Mutiny could not succeed in overthrowing the Raj. Reason (R): The rising middle class sided with the Raj.

153. Assertion (A): In the swadeshi and boycott movement of 1905–1908 in Bengal, not only students and wom en but peasants also actively participated. Reason (R): This movement was a powerful expression of resentment against Bengal Partition. 154. Assertion (A): Mahabat Khan rebelled and made Jahangir and Nurjahan prisoners. Reason (R): Mahabat Khan wanted to install Prince Khurram on the Mughal throne. 155. Assertion (A): There was an upsurge of peasants in Telangana in the erstwhile state of Hyderabad in 1942. Reason (R): It was an extension of the Quit India Movement of 1942. Freedom Struggle/Indian National Movement 156. The culmination of the traditional opposition to British rule came with the Revolt of 1857 which started from: (a) Allahabad (b) Poona (c) Calcutta (d) Meerut 157. The 1857 Revolt which began with the mutiny of the sepoys (Indian soldiers of the East India Company) commenced from: (a) 26 January 1857 (b) 15 August 1857 (c) 10 May 1857 (d) 10 August 1857 158. What was the important element in the strength of the Revolt of 1857? (a) Leadership of Nana Sahib (b) Leadership of Rani of Jhansi (c) Support of Bahadur Shah (d) Hindu–Muslim unity 159. The revolt of 1857 was led by Nana Sahib at (a) Delhi (b) Meerut (c) Kanpur (d) Lahore 160. As a result of 1857 Revolt, which ruler of native states of India was made prisoner? (a) Begum of Avadh (b) Rani of Jhansi (c) Bahadur Shah (d) Raja of Vizianagaram 161. The Revolt of 1857 was the first great struggle of the Indian people for freedom from British imperialism but it failed. The main reason for failure was (a) Paucity of financial resources (b) All the native rulers did not join the revolt (c) There were less number of fighters (d) Imprisonment of Bahadur Shah and death of Rani of Jhansi while fighting with sword in hand (e) All of the above 162. Who was one of the first Indian leaders to start an agitation for political reforms (i.e., freedom of press, trial by jury, etc.)? (a) Tantia Tope (b) Nana Sahib (c) Kunwar Singh of Bihar (d) Raja Rammohun Roy

Indian History 163. The germs of modern political consciousness were sown in the 1820’s and 1830’s by (a) Rani of Jhansi and Nana Sahib (b) Jats of Western UP and Rajputs of Bilaspur (c) Rammohun Roy and Derozians from Bengal (d) Zamindars of Bengal and Maharashtra 164. Which was the first political association to be started in India and where? (a) East India Association in 1866 (b) Landholders’ Society at Calcutta in 1838 (c) Poona Sarvajanik Sabha in 1870 (d) Indian Association in 1876 in Bengal 165. Which of the following statements in respect of the Grand Old Man of India (Dadabhai Naoroji) is correct? (a) He organized the East India Association in England in the year 1866 (b) He showed that the poverty and economic backwardness of India were not inherent in local conditions (c) He was a successful businessman and dedicated his entire life and wealth to the national movement (d) All the above statements are true 166. The doctrine of Swadeshi was first preached _____ and why? (a) In 1866 to save Indian economy (b) In 1866 to generate feeling of patriotism (c) In 1870 to protect Indian industries against onslaught of British manufacturers (d) In 1876 to create feeling of Indianization 167. The Swadeshi movement was specially marked by (a) Indians becoming aware of economic exploitation by Britishers (b) Active participation of women in movement (c) Opening of several cottage industries (d) Creation of hatefulness towards British Industrialists 168. The Indian National Congress was sponsored by (a) W. C. Bannerjee in December 1885 (b) A. O. Hume (an Englishman and a retired civil servant) in 1885 (c) S. N. Bannerjee in 1883 (d) Aurobindo Ghosh 1882 169. Who was the first President of Indian National Congress and when was it first organized? (a) A. O. Hume 1857 (b) M. G. Ranade 1887 (c) Aurobindo Ghosh 1885 (d) W. C. Bannerjee 1885 170. Why the movement of Indian National Congress remained most successful struggle file freedom? (a) Due to leadership of W. C. Bannerjee (b) Due to unity among Hindu–Muslim (c) Because the movement was sponsored by a great Englishman (d) As there was no distrust among the members the British authori ties could not find ways of sup pressing the movement

▌ 2.137

171. Why was B. G. Tilak arrested in 1897 due to which he became popular all­India leader and was given the title of Lokmanya by the people? (a) Due to his support to the struggle for human rights waged in South Africa (b) Because of his association with M. K. Gandhi to support human rights struggles in South Africa (c) On the charge of spreading disaffection against the British Indian Government (d) All of the above 172. Who was the first Indian to use the word Swarajya in 1906? (a) B. G. Tilak at the Calcutta session of the Congress (b) Dadabhai Naoroji at the Calcutta session of the Congress (c) Gopal Krishna Gokhale at Mumbai in 1885 (d) Lala Lajpat Rai in the first session of Indian National Congress 173. Whose name is associated with the Muslim communalism which helped the British Government to follow the policy of ‘Divide and Rule’? (a) Sayyid Ahmed Khan (b) Rahimutullah Mohammad Sayani (c) Badrudin Tyabji (d) B. G. Tilak 174. Who was the Indian leader who opposed organization of Indian Congress in 1885 along with Raja Shiva Prasad of Benares? (a) Badrudin Tyabji (b) Nawab AbduI Latif (c) Sayyed Ahmed Khan (d) Mohammed Ali Jinnah 175. The concrete shape and setting to the communal theory was given when the All India Muslim League was set up in the year: (a) 1905 (b) 1904 (c) 1924 (d) 1906 176. Under whose leadership the All India Muslim League was set up? (a) Mohammad Ali Jinnah (b) Sayyid Ahmed Khan (c) Aga Khan (d) Nawab Salimullah of Dacca 177. The Ahrar Movement was started as a (a) Pro-British movement (b) Pro­Muslim League movement (c) Pro­Muslim League and militant movement (d) Anti­communal and anti­Muslim League movement 178. Who was the most prominent Muslim leader of the Ahrar movement? (a) Hakim Ajmal Khan (b) Maulana Mohammad Ali (c) Maulana Abul Kalam Azad (d) Nawab Mohsin-ul-Mulk

2.138

▌ General Knowledge

179. Who started the Home Rule League movement and when? (a) B. G. Tilak in 1916 upon return from Mandalay (b) Mrs Annie Besant in September 1916 (c) Gandhiji (d) Lala Lajpat Rai 180. What was the main aim of Home Rule Movement? (a) To remove British rule (b) To turn out Britishers (c) To keep unity among Hindu–Muslims and fight for freedom (d) To attain self-government within the British Empire 181. Why was Khilafat Movement started after the First World War by Ali brothers? (a) To turn foreigners out of the country (b) To attain freedom from British Rule (c) To give cooperation to various organizations formed in the country to attain independence (d) To agitate for preservation of office of Khalifa and better treatment for Turkey where British Government imposed harsh terms after the war 182. Khilafat movement, fully supported by both Hindus and Muslims and adopted by the Congress through Gandhiji subsided because of (a) Muslim League’s opposition to Indian National Congress (b) Special concessions given to Muslims by the British government (c) Office of Khalifa was abolished in Turkey itself and better terms given to Turkey (d) Internal fiction between Congress and Muslim League 183. When and where was the Ghaddar Party organized by Indian Revolutionaries? (a) In 1913, by Indian soldiers serving the British empire outside India (b) In 1913, by Indians living in Canada and America (c) In 1912, when the Bomb on the Viceroy Lord and Lady Harding was thrown by revolutionaries (d) In 1916, when Home Rule movement started 184. Which of the following statements is not true about Ghaddar Party? (a) The party decided to send a shipload of arms to Punjab where a revolt was to take place (b) The secret mentioned in point (a) leaked out and the ship and the leaders were arrested (c) The name of the party was taken from the name of a weekly paper published in 1913 in commemoration of the uprising of 1957 (d) It was formed in 1912 by Indian Revolutionaries living abroad 185. Which of the following statements is correct in respect of Lucknow Pact? (a) Through this pact a way was to achieve Hindu–Muslim unity

(b) As per Gandhiji, it was a pact for power between the educated and rich Hindus and Muslims (c) It did not involve the Hindu and Muslim masses (d) All above statements are correct 186. When was the non-violent non-cooperation movement started and by whom? (a) In 1919 by Motilal Nehru (b) In 1919 by Mahatma Gandhi (c) In 1920 by Mahatma Gandhi (d) In 1920 by Lala Lajpat Rai 187. Which of the following statements is correct about Satyagraha Movement? (a) It was started by Mahatma Gandhi (b) Its object was to offer peaceful resistance to cruel and unjust laws (c) Sunday, the April 6, 1919 was observed as Satyagraha day and there was an all-India Hartal (d) All the above statements are correct 188. With what the 13 April 1919 is associated in the history of freedom struggle of India? (a) Start of Civil disobedience movement (b) Countrywide non-cooperation movement (c) Adoption of pledge for complete freedom of the country (d) Massacre of Jallianwala Bagh 189. Firing to Jallianwala Bagh was ordered by (a) Lord Simon (b) Rowlatt (c) O’Dwyer (d) Curzon-Wyllie 190. ‘Inquilab Zindabad’ slogan was given by (a) Iqbal (b) S. C. Bose (c) Bhagat Singh (d) Chandra Shekhar Azad 191. Non-Cooperation Movement was withdrawn mainly because of the (a) Friction between the Moderates and the Extremists (b) Withdrawal of support by Muslim League (c) Chauri-Chaura incident (d) Both (b) and (c) 192. In which year non-cooperation movement was withdrawn? (a) 1917 (b) 1820 (c) 1920 (d) 1923 193. How did ‘National Liberal Federation of India’ come into existence? (a) Differences between Muslim League and Congress. (b) Clash between the Moderates and the Extremists in the Congress (c) Failure of Muslim League and Congress to achieve the goals. (d) ‘Divide and Rule’ policy of the British Government

Indian History ▌ 2.139 194. The Moderates decided to break with the Extremists in the year (a) 1918 (b) 1916 (c) 1919 (d) 1907 195. While Moderates decided to break with the Extremists as early as 1907, final separation took place only in the year (a) 1910 (b) 1912 (c) 1918 (d) 1917 196. In which year National Liberal Federation of India was formed by the Moderates? (a) 1918 (b) 1919 (c) 1920 (d) 1921 197. Who is known as ‘Sher­e­Punjab’ among the freedom fighters? (a) Bhagat Singh (b) Lala Lajpat Rai (c) Chandra Shekhar Azad (d) Lala Hardayal

198. What is the importance of Bombay sessions of All India Congress Committee which held in August 1942? (a) It passed resolution for Purna Swaraj (b) It was attended by all people including the oderates, the Extremists and even some important British officials (c) It passed the famous ‘Quit India’ resolution (d) All of the above 199. Who among the following leaders who struggled for the Freedom of the country was not a moderate? (a) Bipin Chander Pal (b) Gopal Krishna Gokhale (c) W. E. Bannerjee (d) All of these 200. Most of the political leaders who struggled for freedom in earlier years were not very rich and had to maintain themselves on their own earnings, which were often meagre. Which of the following leaders do not belong to this category? (a) Dadabhai Naoroji (b) Gokhale (c) Tilak (d) Surendranath Banerjee

ANSWER KEY 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

(a)

21. 22. 23. 24. 25.

(d)

(b) (b) (d) (c) (c) (d) (a) (c) (b) (b) (b) (c) (a) (c) (a) (b) (c) (b) (b) (b) (a) (d) (b)

26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45.

(a)

46. 47. 48. 49. 50.

(c)

(a) (a) (c) (b) (d) (b) (b) (d) (b) (a) (c) (a) (a) (d) (c) (b) (a) (b) (d) (a) (b) (a) (a)

51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70.

(b)

71. 72. 73. 74. 75.

(d)

(d) (a) (d) (d) (a) (d) (c) (b) (c) (c) (b) (c) (d) (c) (d) (b) (c) (c) (c) (a) (d) (b) (b)

76. 77. 78. 79. 80. 81. 82. 83. 84. 85. 86. 87. 88. 89. 90. 91. 92. 93. 94. 95.

(c)

96. 97. 98. 99. 100.

(a)

(d) (d) (b) (c) (a) (c) (c) (b) (c) (a) (b) (c) (d) (a) (d) (b) (d) (b) (b) (b) (b) (b) (c)

101. 102. 103. 104. 105. 106. 107. 108. 109. 110. 111. 112. 113. 114. 115. 116. 117. 118. 119. 120.

(d)

121. 122. 123. 124. 125.

(b)

(d) (a) (a) (a) (d) (d) (d) (b) (b) (a) (d) (a) (b) (d) (c) (d) (c) (c) (d) (b) (b) (b) (b)

126. 127. 128. 129. 130. 131. 132. 133. 134. 135. 136. 137. 138. 139. 140. 141. 142. 143. 144. 145.

(c)

146. 147. 148. 149. 150.

(a)

(a) (d) (c) (d) (c) (c) (b) (b) (d) (a) (b) (c) (c) (d) (a) (c) (a) (d) (a) (a) (c) (b) (c)

151. 152. 153. 154. 155. 156. 157. 158. 159. 160. 161. 162. 163. 164. 165. 166. 167. 168. 169. 170.

(a)

171. 172. 173. 174. 175.

(c)

(b) (d) (c) (c) (d) (c) (d) (b) (c) (b) (d) (c) (b) (d) (c) (b) (b) (d) (c) (b) (a) (c) (d)

176. 177. 178. 179. 180. 181. 182. 183. 184. 185. 186. 187. 188. 189. 190. 191. 192. 193. 194. 195.

(c)

196. 197. 198. 199. 200.

(b)

(d) (c) (b) (d) (d) (c) (b) (d) (d) (b) (d) (d) (c) (a) (c) (c) (b) (d) (c) (b) (c) (a) (d)

2.140

▌ General Knowledge

NDA BOX 1. Who among the following was the leader of the extremist group in the Indian National Congress? (a) Tilak (b) Gokhale (c) Abul Kalam Azad (d) Motilal Nehru 2. Who was political guru of the Father of Nation? (a) Malviya (b) Gokhale (c) Dadabhai Naoroji (d) Raja Ram Mohan Roy 3. In 1922 a new party was formed known as ‘Congress Khilafat Swaraj Party’. Who of the following were the main leaders of this new party? (a) C. R. Dass (b) Motilal Nehru (c) Vithalbhai Patel (d) All the above 4. Who were the President and Secretary of the Congress Khilafat Swaraj Party formed by Motilal Nehru in 1922? (a) Malaviya and Jayakar, respectively (b) Vithalbhai and Malaviya, respectively (c) Motilal Nehru and Malaviya, respectively (d) C. R. Dass and Motilal Nehru, respectively 5. To which professions earlier leaders who struggled for Freedom of India mainly belonged? (a) Lawyers (b) Teachers (c) Journalists (d) All of the above Directions (Q 6–25) Given below are two statements, one labeled as Assertion (a) and other labeled as Reason (R). In the context of the above two statements which one of the following is correct? (a) Both A and R are true and R is the correct explanation of A (b) Both A and R are true but R is not a correct explanation of A (c) A is true but R is false (d) A is false but R is true 6. Assertion (A): Annie Besant worked together with the Congress and the Muslim League during the Home Rule Movement. Reason (R): She felt that this was necessary to get the support of the masses for the Home Rule Movement. 7. Assertion (A): The moderate dominated Congress cared for the grievances of the peasants and workers. Reason (R): They were the worst affected classes during the British rule. 8. Assertion (A): The Revolt of 1857 cannot be said to be a national war of independence. Reason (R): The concept of a nation state had not emerged then. 9. Assertion (A): The Wahabi movement turned into a militant political movement. Reason (R): They wanted to get rid of the evils which had crept into the Muslim society.

10. Assertion (A): Ali Brothers founded the Swaraj Party. Reason (R): The Swarajists were in favour of council entry and obstruction from within. 11. Assertion (A): There was a split in the Surat session of congress in 1907. Reason (R): Moderates did not favour boycott and Swadeshi movement. 12. Assertion (A): Khilafat Movement was launched in India to drive out the British from India. Reason (R): After the World War I, the British rule in India became more repressive. 13. Assertion (A): The Britishers referred to Bal Gangadhar Tilak as ‘the Father of Indian Unrest’. Reason (R): Bal Gangadhar Tilak was the first to demand freedom as his birth right. 14. Assertion (A): The Revolt of 1857 was a failure. Reason (R): Few Indian Princes supported it. 15. Assertion (A): Ghadar Party grew into a movement to liberate the country from Britishers. Reason (R): La1a Hardaya1 was one of the organizers of the Ghadar Party. 16. Assertion (A): The Revolt of 1857 was a failure. Reason (R): It broke out before pre scheduled time. 17. Assertion (A): The British defeated Tipu Sultan in the fourth Anglo-Mysore war. Reason (R): The British had superior weaponry and superior skill. 18. Assertion (A): General Dyer fired at Jallianwa1a Bagh crowd without any warning and till the ammunition was exhausted. Reason (R): This shooting was not only to boost the morale of the army, but also to strike terror in Punjab. 19. Assertion (A): The Swaraj Party was founded at the Gaya Session of the INC. Reason (R): C. R. Dass and Motilal Nehru advocated the entry of the legislative councils under the 1919 Act while C. Rajagopalachari and his supporters opposed it. 20. Assertion (A): In June 1941, the Communist Party of India declared its support to the war efforts of Britain. Reason (R): Britain supported Russia against the German invasion. 21. Assertion (A): The enactment of the Age of Consent Bill was opposed by Lokmanya Tilak. Reason (R): Lokmanya Tilak believed that society should take steps for eradicating social malpractices. 22. Assertion (A): The British introduced in India the modern concept of the rule of law. Reason (R): The British did not recognise the existing Personal Civil law in India.

Indian History ▌ 2.141 23. Assertion (A): Gandhiji observed fast at Poona in 1932. Reason (R): Gandhiji was opposed to the provision of separate electorate rights for Harijans in the Communal Award 24. Assertion (A): R. C. Dutt authored the book ‘Poverty and UnBritish Rule’. Reason (R): He wanted to expose India’s growing poverty. 25. Assertion (A): Lala Lajpat Rai died of police lathi­charge during the march against Simon Commission. Reason (R): Simon Commission did not consist of even a single Indian as a member. 26. The royal title Kaisara was derived from (a) Persian (b) Scythian (c) Chinese (d) Roman 27. The sardeshmukhi of the Marathas was (a) A tax levied on deshmukhs (b) A capitation levy (c) A share in revenue collected by other authorities (d) Village lands set aside for the deshmukhs 28. The statement “If by worshipping stones one can find God, I shall worship mountains”, was made by (a) Ramanand (b) Namdev (c) Guru Nanak (d) Kabir 29. The story of the migration of Videgha Mathava to the east with his priest is mentioned in the (a) Aitareya Brahmana (b) Satapartha Brahmana (c) Gopatha Brahmana (d) Brithadaranyaka Upanishad 30. The system of competitive examination for civil service was accepted in principle in the year (a) 1833 (b) 1853 (c) 1858 (d) 1882 31. The term brahmadeya occurs for the first time in (a) Early Vedic texts (b) Early Buddhist texts (c) Pre-Gupta inscriptions (d) Post-Gupta inscriptions 32. The term devamtrika refers to (a) the divine mothers (b) one’s homeland (c) lay with divine qualities (d) country with rainfall

36. The Uprising of 1857 was described as the first Indian war of independence by (a) S. N. Sen (b) R. C. Mazumdar (c) B. G. Tilak (d) V. D. Savarkar 37. The use of spinning wheel (Charkha) became common during the (a) 9th Century ad (b) 10th Century ad (c) 12th Century ad (d) 14th Century ad 38. The Vijayanagara king who employed skilled archers of the Turkish clan and raised the fighting capacityofhis bowmen was (a) Bukka I (b) Devaraya I (c) Krishnadevaraya (d) Ramaraya 39. The Yaudheyas, a kshatriya tribe, worshipped exclusively the God (a) Indra (b) Vasudeva (c) Pasupati (d) Karthikeya 40. The zamindari in Mughal India was not (a) A saleable right (b) Hereditary (c) Ownership of land (d) Mortgageable 41. Though Ashoka had many sons, the inscriptions mentioned only one who is not mentioned in any other source. He is (a) Kunala (b) Tivara (c) Mahendra (d) Jalauka 42. The treaty of Mangalore was signed between (a) The English East India Company and Haidar Ali (b) The English East India Company and Tipu Sultan (c) Haidar Ali and the Zamorin of Calicut (d) The French East India Company and Tipu Sultan 43. What is the chronological sequence of the following three wazirs of the Mughal court in the mid-eighteenth century 1. Safdarjung 2. Imad-ul-mulk 3. Javid Khan

33. The term Nirgrantha is associated with (a) Ajivikas (b) Charvakas (c) Jainas (d) Pasupatas

Choose the correct answer from the codes given below: (a) 2, 1, 3 (b) 1, 3, 2 (c) 2, 3, 1 (d) 3, 1, 2

34. The theory of Sullh-i-kul was propounded by (a) Nizamuddin Auliya (b) Akbar (c) Zainul Abidin (d) Shaikh Nasiruddin Chiragh

44. What is the correct chronological order of the following Mughal emperors? 1. Ahmad Shah 2. Shah Alam II 3. Alamgir II 4. Akbar II

35. The ultimate ownership of land during the post-Gupta period lay with (a) The cultivator (b) The village community (c) The king (d) The joint family

Choose the answer from the codes given below: (a) 2, 1, 3, 4 (b) 1, 2, 4, 3 (c) 1, 3, 2, 4 (d) 4, 3, 1, 2

2.142

▌ General Knowledge

45. What is the correct chronological sequence of the following events relating to Shivaji? 1. Visit of Agra 2. Attack of Poona 3. Coronation 4. Signing of the Treaty of Purandar Choose the answer from the codes given below: (a) 1, 4 ,3, 2 (b) 2, 4, 1, 3 (c) 2, 1, 4, 3 (d) 3, 1, 2, 4 46. What is the significance of Kayavarohana in the history of Saivism? (a) It is the same as Pasupara-Vi mokshana, one of the principal characteristics of the Pasupata sect (b) It was the birth place of Lakulisa (c) It was one of the Vratas of the Kapalikas (d) It was a ritual prescribed for those adhering to the Kalamukha 47. What was considered as Varna Samkara? (a) Illegitimate birth (b) Revolt against the varna system (c) Transgression of Varna boundaries for marriage (d) Championing the Varnasrama dharma 48. What was the main cause of the growth of money-lending class during the British rule? (a) Failure of crops, caused by the natural calamities (b) Extravagant habits of the people (c) Demand of high revenue from the peasants (d) Need of capital for the cultivation 49. Which of the following are charac teristics ascribed to Buddism? 1. Rejection of the authority of the Vedas 2. Emphasis on the role of the individual 3. Belief in the categories of jiva and ajiva 4. Theory of Prakriti and Purusha Choose the correct answer from the codes given below: (a) 1 and 2 (b) 1, 3 and 4 (c) 2 and 3 (d) 3 and 4 50. Which of the following are mentioned in Asokan inscriptions? (a) Cholas, Cheras, Pandyas and Pallavas (b) Cholas, Satyaputras, Cheras and Pandyas (c) Pandyas, Satyaputras, Andhras and Cheras (d) Cholas, Pandyas, Bhojas and Andhras 51. ‘No Taxation without Representation’ was the popular slogan of (a) Indian freedom struggle (b) American war of independence (c) Russian Revolution (d) French revolution 52. Aruna Asaf Ali was associated with which on of the following? (a) Bardoli Satyagraha (b) Quit India Movement

(c) civil Disobedience Movement (d) Khilafat Movenent 53. Between whom was the battle of Chausa fought? (a) Bahadur Shah of Gujarat and Humayun (b) Humayun and Sher Khan (c) Akbar and Rana Pratap (d) Jahangir and Rana Amar Singh 54. Between whom was the Battle of Khanwa (1527) fought? (a) Ibrahim Lodi and Rana Sanga (b) Babur and Rana Sanga (c) Humayun and Sher Khan (d) Humayun and Nusrat Shah 55. By virtue of which Act, diarchy was introduced in India? (a) Government of India Act, 1909 (b) Government of India Act, 1919 (c) Government of India Act, 1935 (d) Government of India Act, 1947 56. Consider the following declaration and identify the person who made it using the codes given below The times has come when badges of honour make our shame glaring in their incongruous context of humiliation and I, for my part, wish to stand shorn of all special distinction, by the side of my countrymen who , for their so­called insignificance are liable to suffer degradation not fit for human beings. (a) Mahatma Gandhi (b) Jawaharlal Nehru (c) Dadabhai Naoroji (d) Rabindranath Tagore 57. Consider the following statement: “We believe that it is inalienable right of the Indian people as of any other people, to have freedom and to enjoy the fruits of their toil and have the necessities of life so that they may have full opportunities of growth.” Identify the correct context of this statement from below: (a) Pledge of independence to be publicly taken all over Indian on 26 January, 1930 (b) Preamble of the Constitution of India adopted in 1950 (c) Congress Working Committee Resolution adopted at the Special Calcutta Congress Session in 1920 (d) Part of Speech delivered by Subhash Chandra Bose at the launch of ‘Azd Hind Fauj’ 58. Consider the following statement: 1. panchsheel was signed between people’s Republic of china and India after the Sino-Indian War in 1962. 2. India and China supporting each other in international for a is one of the principles of panchsheel. Which of the statements given above is/are correct? (a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2 59. Consider the following statement and identify with the help of the code given below the viceroy who made the statement and when In my belief, Congress is tottering to its fall and one of my great ambitions while in India is to assist it to a peaceful demise.

Indian History ▌ 2.143 (a) Lord Curzon, in a letter to the Secretary of States in 1900 (b) Lord Curzon, while announcing the partition of Bengal (c) Lord Dufferin, during the farewell speech at Calcutta (d) Lord Minto, while addressing the Muslim delegation Which met him at shimla in 1906 60. Consider the following statements: 1. The Buddhists in Deccan excavated rock-cut Chaityas and the Vaishnavas, Shaivas and Jainas imitated these in late centuries. 2. The Vaishnavas, Shaivas and Jainas excavated temples at sites for distant from rock-cut Chaityas. Which of the statements given above is/are correct? (a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2 61. Consider the following statements: 1. The champaran Satyagraha marked Gandhiji’s second appearance in Indian politics as a leader of the masses. 2. The Champaran Satyagraha was launched to address the problems faced by Indigo plantation workers. Which of the statements given above is/are correct? (a) 1 Only (b) 2 Only (c) Both 1 and 2 (d) Neither 1 nor 2 62. Consider the following statements: 1. Charles Wood’s Dispatch of 1854 laid exclusive emphasis on the development of higher education in India and neglected Primary and secondary education. 2. The Carlyle Circular issued by RW Carlyle sought to check the spread of revolutionary activates in edudcational institutions. Select the correct answer using the codes given below: (a) 1 Only (b) 2 Only (c) Both 1 and 2 (d) Neither 1 nor 2 63. Consider the following statements about Gandhiji’s thinking on environment. 1. His environmental thinking is rooted in his larger philosophical and moral thinking. 2. He preferred sustainable environmental practices to nourish the soil and the natural world. 3. He laid emphasis on the rigorous ethic of non­injury in our treatement of animals. Which of the statements given above is/are correct? (a) 1 only (b) 1 and 2 only (c) 1, 2 and 3 (d) 2 and 3 only 64. Consider the following statements relating to Gandhian strategy of Satyagraha 1. Under the Gandhian strategy, which may be described as Struggle-Truce-Struggle (S-T-S) , phase of a vigorous extra-legal mass movement and confrontation with colonial authority alternate with phases, during which direct confrontation is withdrawn 2. The entire political process of S-T-S was an upward spiraling one, which also assumed that the freedom struggle

would pass through several stages ending with the transfer of power by the colonial regime itself Which of the statements given above is/are correct? (a) I Only b) II Only (c) Both I and II (d) Neither I nor II 65. Consider the following statements relating to Jawaharlal Nehru’s attitude towards science and technology. 1. Nehru was convinced that science and technology were Crucial to the solution of India’s problems relating to underdevelopment and poverty 2. To emphasize the importance of science and scientific research, Nehru himself became the chairman of the Council of Scientific and Industrial Research Which of the statements given above is/are correct? (a) I only (b) II only (c) Both I and II (d) Neither I nor II 66. Consider the following statements with regard to Champaran Satyagraha (1917). 1. The Champaran Satyagraha marked Mahatma Gandhi’s first appearance in Indian politics as a leader of the masses. 2. Young nationalists like Rajendra Prasad and J.B Kriplani and had worked with Mhatma Gandhi at Champaran. 3. It was during this movement that Mhatma Gandhi formulated his philosophy of Satyagraha. Which of the statements given above is/are correct? (a) 3 Only (b) 1 and 2 (c) 2 and 3 (d) 1 and 3 67. Direction: The following questions consist of two statements, statements I and Statement II. You are to examine these two Statements carefully and select the answers to these items using the code given below: Code: (a) Both the statements are individual are individually true and Statement II is the correct explanation of Statement I. (b) Both the Statements are individually true but statement II is not the correct explanation of Statement I. (c) Statement I is true but statement II is false. (d) Statement I is false but Statement II is true. Statement I Mahatma Gandhi launched the Rowlatt Satygraha as an all-India mass Protest. Statement II Rowlatt Act was passed by the British Government in 1919 to authorize the government to imprison any person without trial. 68. Direction: The following questions consist of two statements, statements I and Statement II. You are to examine these two statements carefully and select the answers to these items using the code given below. Code: (a) Both the statements are individual are individually true and Statement II is the correct explanation of Statement I.

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▌ General Knowledge

(b) Both the Statements are individually true but statement II is not the correct explanation of Statement I. (c) Statement I is true but statement II is false. (d) Statement I is false but Statement II is true. Statement I the Private trade of the East India Company’s officials in the 18th century had flowered with the indirect patronage of the authorities of the Company. Statement II The extra­legal power enjoyed by the foreign merchants and the duty–tree nature of their private trade virtually edged the indigenous merchants out of competition. 69. Directions: The following questions consist of two statements, one labelled as ‘Assertion (A)’ and the other as ‘Reason (R)’. You are to examine these tow statements carefully and select the answers to these itmes using the code given below Code: (a) Both A and R are individually true and R is the correct explanation of A (b) Both A and R are individually true but R is not the correct explanation of A (c) A is true but R is False (d) A is false but R is true Assertion (A) There was a countrywide agitation against the Simon Commission. Reason (R) The members of the Simon Commission discontinued their work in face of the country-wide agitation. 70. Due to the leadership and success in which one of the following did Vallabhbhai Patel get the title of ‘Sardar? (a) Kheda Satyagraha (b) Non-Cooperation Movement (c) Bardoli Satyagraha (d) Civil Disobedience Movement 71. During Aurangzeb’s reign, whose duty was it to see that people live their lives in accordance with the ‘Sharia’? (a) Ahadis (b) Mansabdars (c) Muhtasibs (d) Walashuhis 72. During the regin of Alauddin Khilji, who were amils? (a) Local landlords at village level (b) Government agents to collect land revenue (c) Khurasani and Multani trades of foodgrains (d) Banjaras who carried the food­grains from villages to towns 73. Gautam Buddha delivered his first Sermon at which one of the following places? (a) Kushinagar (b) Sarnath (c) Bodh Gaya (d) Lumbini 74. Identify the correct sequence of the following events of Indian history. 1. Foundation of the Indian Muslim League 2. Surat Split 3. Partition of Bengal 4. Transfer of capital from Calcutta to Delhi

Select the correct answer using the code given below: (a) 1-2-3-4 (b) 2-3-4-1 (c) 3-1-2-4 (d) 4-3-2-1 75. Identify, using the codes given below, the nationalist leader who was sentenced in the words outlined below: “ The fact that in the eyes of millions of your countrymen you are a great patriot and great leader but as a man subject to the law, who has, by his own admission broken the law you are subject to six years imprisonment.” (a) CR Das (b) Mahatma Gandhi (c) Subhas Chandra Bose (d) Jawaharlal Nehru 76. In Hind Swaraj, Mahatma Gandhi was critical of railways, because they are 1. Carriers of plague germs 2. Instruments for frequency of famines 3. Responsible for creating class division in the society 4. Accident-prone Select the correct answer using the code given below: (a) 1, 2 and 3 (b) 1 and 2 (c) 2 and 3 (d) 1 and 4 77. Mahatma Gandhi and Rabindranath Tagore did not define India as a ‘nation’, because they were (a) Appreciative of cultural divisiveness (b) Opposed to the idea of homogeneity (c) supportive of ‘oneness’ (d) critical of hegemonic culture 78. Mahatma Gandhi undertook ‘fast unto death’ against the Communal Award. He however, discontinued the fast, because (a) The congress leaders successfully prevailed upon Mahatma Gandhi to accept the Award (b) The British Government with drew the Communal Award (c) Of the conclusion of the Poona pact (d) The weaker classes openly opposed the Award 79. Match List I with List II and select the correct answer using the codes given below the lists: List I (Battle)

List II (Year)

A.

1 Battle of Panipat

1.

1565

B.

Battle of Plassey

2.

1526

C.

Battle of Haldighti

3.

1757

D.

Battle of Talikota

4.

1576

st

Codes: A B C D (a) 2

3

4

1

(b) 2

4

3

1

(c) 1

4

3

2

(d) 1

3

4

2

Indian History ▌ 2.145 80. Match List I with List II and select the correct answer using the codes given below the lists. List I A. Ryotwari Settlement B. Formation of Azad Hind Fauj C. All-party conference resolution to boycott School, college and law courts in 1920 D. Referendum held for Provinces to join Pakistan

List II 1. Singapore 2. Allahabad 3. North-western frontier province 4. Madras and Bombay presidency

Codes A B C D (a) 4

1

2

3

(b) 1

2

3

4

(c) 4

2

1

3

(d) 1

3

2

4

81. Neel Darpan is a play based on the story of Indigo Rebellion of Bengal of 1860–61. Who authored it? (a) Dinabandhu Mutra (b) Bankim Chandra Chattopadyaya (c) Sisir kumar Gohs and Motilal Ghosh Jointly (d) Mdhusudan Data 82. Other than Annie Besant, who among the following also launched a Home Rule Movement in India? (a) Aurobindo Ghosh (b) Bal Gangadhar Tilak (c) Gopal Krishna Gokhle (d) Moti Lal Nehru 83. Rajmohan Gandhi’s book ‘A Tale of Two Revolts’ (2009) deals with the (a) Taiping and Indigo revolts (b) Revolt of 1857 and Deccan Uprising (c) Revolt of 1857 and the Deccan Uprising (d) American Civil War and Santhal Uprisng 84. Sati was declared illegal and punishable by the Regulation XVII during the Governor Generalship of (a) Lord William Bentinck (b) Lord Canning (c) Lord Ripon (d) Lord Dalhousie 85. The agenda for the Round Table Conference held in London was to (a) Discuss the report of the Simon Commission (b) From a constitution the India (c) Find a solution of the communal problem in India (d) Find a solution of the communal problem in India find a solution for the problems of the ‘depressed classes’ of India 86. The first Anglo­Burmese war was concluded by a treaty. Which one among the following is that? (a) Treaty of Burma (b) Treaty of Tamu (c) Treaty of Bhamo (d) Treaty of Yandaboo

87. The Government of India Act, 1935 was based on (a) The principle of federation and parliamentary system (b) The principle of secession of the British Indian provinces (c) Acceptance of independence of India (d) Acceptance of the idea of a constituent assembly to draft a constitution 88. The Haripura Congress (1983) remains a milestone in Indian freedom struggle, because (a) It declared war on the British Empire (b) It anointed Jawahar Lal Nehru as the future prime Minister of India (c) Of the introduction of the idea of a Planning commission (d) Of the acceptance of the Government of India Act, 1935 by the Congress 89. The international body which was formed in Europe (in 1889) to coordinate the efforts of socialists all over Europe was called (a) Social Democratic Party (b) Commune (c) Second International (d) Labour Party 90. The notion of saptanga that was introduced in Arthashastra includes (a) kings, territory, administration and treasury (b) Music, dance ragas and wrestling (c) Ministers, civil servants, subalterns and those involved in espionage (d) aristocrats, acharyas, tradesrs and monks 91. The Passive Resistance Association was first formed by Mahatama Gandhi in South Africa in 1906 to organize a campaign against the introduction of (a) Poll Tax (b) Certificate of Registration (c) Legislations Against Indian Businessmen (d) Legislations to disenfranchise the Indians 92. The Rowlatt Act was passed to (a) bring about aeration reforms (b) curtail the nationalist and revolutionary activates (c) have a favourable balance of trade (d) put Second World War criminals on trial 93. The themes of the murals of the Ajanta caves are 1. Decorative designs which include a variety of animals, trees and flowers. 2. Portraits of various Buddhas and Bodhisattvas. 3. Narrative scenes portraying Jataka stories. Select the correct answer using the code given below: (a) 1, 2 and 3 (b) 2 and 3 only (c) 1 and 3 only (d) 1 and 2 only 94. The was the effect of the Government of India Act of 1935 on the national Movement? (a) It strengthened the movement (b) It weakened the movement by satisfying the people

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▌ General Knowledge

(c) It weakened the movement, because separate electorate for the Muslims and Hindus created dissensions (d) It weakened the movement, because the Act introduced federal system 95. Tipu Sultan’s major interest in building ships was to (a) Fight the English at sea (b) Promote trade and commerce (c) Maintain Diplomatic relation with Muscat (Oman) (d) Establish an overseas settlement at port Basra (Iraq), Which would be under his control, on lines of the European companies? 96. Vande mataram, the nationalist song, was a part of (a) Durgesh Nandini (b) Saraswati Chandra (c) Grihadaha (d) Anand Math 97. What is the correct sequence of the following? 1. Wavell Plan 2. Radcliff Award 3. Cripps Mission 4. Cabinet Mission Select the correct answer using the codes given below: (a) 1-2-3-4 (b) 3-1-4-2 (c) 2-1-3-4 (d) 3-4-1-2 98. Where did Buddha attain Parinivana? (a) Bodh Gaya (b) Kushinagara (c) Rajgriha (d) Vaisali 99. Which among the following statement are correct with regard to the Portuguese in India? I. They had the monopoly over the Eastern trade in the 16th Century II. They possessed Mumbai in the beginning III. They had trading settlements at Cochin, Diu and Daman IV. The Mughals denied them any trading concessions Select the correct answer using the code given below: (a) I, II and III (b) II, III and IV (c) I, II and IV (d) I and III only 100. Which among the following statements with regard to Raja Rammohan Roy are correct? I. He started the Atmiya Sabha II. He wrote the Gift of Monotheist III. He published the precepts of jesus IV. He founded the Brahmo Sabha Select the correct answer using the code given below: (a) II, III and IV only (b) I, II and III only (c) I, III and IV only (d) I, II and IV 101. Which of the following statements regarding consolidation of British power in 18th century Bengal is/are correct? 1. The Nawab granted the Company, the Nizamat powers in 1765. 2. The duties of collecting revenues and administration of justice were entrusted with the European officials of the? East Indian Company.

Select the correct answer using the coed given below: (a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2 102. Which of the following statements regarding Mughal paintings is/are correct? 1. The paintings of Akbar’s time were mostly confined to book illustrations and portraits. 2. Mir Sayyid Ali of Heart and Khwaja Abdus Samad formed the nucleus of the Mughal School of Select the correct answer using the code given below: (a) 1 Only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2 103. Which of the following statements relating to the Government of India Act of 1935 are correct? 1. It introduced provincial autonomy. 2. It proposed a federation of India. 3. It proposed for the establishment of Federal Court. Select the correct answer using the code given below (a) 1, 2 and 3 (b) 1 and 2 only (c) 2 and 3 only (d) 1 and 3 only 104. Which of the following statements with regard to freedom struggle are correct? 1. The British rule could prevail in India on the basis of the consent or acquiescence of many sections of Indian people. 2. The social basis of the colonial regime was among the Zamindars and upper classes. 3. The Indian National Army forced the British to withdraw from India. 4. The Hindu Mahasabha supported the partition of India. Select the correct answer using the code given below: (a) 1 and 2 (b) 2 and 3 (c) 3 and 4 (d) 1, 2, 3, and 4 105. Which one among the following is common to the Treaty of Yandaboo (1826) the Treaty of Salbai (1782) and the Treaty of Gandamak (1879)? (a) With these treaties, various Indian powers formed alliances to defeat the British (b) These treaties enabled the British to control the South Asian powers (c) These treaties expedited the spread of Indian Culture abroad (d) These treaties gave an essential boost to enhanced trade in South Asia 106. Which one among the following is not the characteristic feature of the Harappan settlement? (a) Doorways and Widows generally faced the side lanes and rarely opened onto the main streets (b) Houses generally has separate bathing areas and toilets (c) The citadel was walled but the lower town was and toilets (d) Drains and water chutes form the second storey were often built inside the wall

Indian History ▌ 2.147 107. Which one among the following is not true with regard to Rig Veda Samhita? (a) There are about 300 non-Indo-European words in Rig Veda (b) There is a reference to dasarajna (battle of ten kings) in the Rig Veda (c) It is mentioned in the Rig Veda that the Bharata Chief Sudas Fought against a confederacy of ten tribes (d) Purus sided with Bharata in the battle of en kings 108. Which one among the following newspapers was published first? (a) The Madras Mail (b) The Indian Social Reformer (c) The Bengal Gazette (d) The Times of India

(c) Civil Disobedience Movement (d) Quit India Movement 115. Which one of the following slogans is attributed of Subash Chandara Bose? (a) Jain Jawan Jai Kisan (b) Vande Matram (c) Jai Hind (d) Inqilab Zindabad 116. Which one of the following statements about Chandragupta II is not correct? (a) Kalidas and Amarsigh were famous scholars in his court (b) He expanded his empire both through matrimonial alliances and conquests (c) He married a Naga princess named Kubera Naga (d) He married his daughter Prabhawati to an Ahom prince

109. Which one among the following statements is not correct? (a) Gandhara School of Arts owed its origin to the Indo-Greek rulers but the real patrons of the school were the Kushans, especially Kanishka (b) Rich carving elaborate ornamentations and complex symbolism were not the main features of the Gandhara sculpture (c) The Greeco­Roman architectural impact modified the structure of the Buddhist stupas? (d) The artists of the Amravati School of arts mainly used white marble

117. Which one of the following statements with regard to ‘Direct Action Day’ is correct? (a) Hasan Suhrwadrdy presided over the ‘Direct Action Day’ (b) ‘Direct Action’ took place in Delhi (c) ‘Direct Action Day’ led to the Bihar riot (d) ‘Direct Action’ was endorsed by the Congress Party

110. Which one among the following was fought earliest? (a) Battle of Ambur (b) Battle of Buxar (c) Battle of Plassey (d) Battle of Wandiwash

119. Which one of the following with regard to the Poona Pact, 1932 is not correct? (a) Adequate representation of depressed sections in government jobs (b) Reservation of seats for the depressed classes in the provincial legislature (c) Acceptance of joint electorate system (d) Reservation of seats for the depressed classes in the central legislature

111. Which one among the following was not (1885–1905)? (a) Indianozation of the superior grades of the administrative Services (b) Purna Swaraj/complete independence (c) Abolition of the Arms Act (d) Autonomy within the empire 112. Which one among the following was the major demand of the Bardauli Satyagraha (1928) organized under the leadership of Sardar Vallabhbhai patel? (a) Land to the Tiller (b) Increase in the rates of labour wage (c) Rollback of newly enhanced revenue rate (d) Supply of agricultural inputs to the farmers at subsidized rate 113. Which one of the following is the correct chronological order of the given rules of ancient India? (a) Ashoka-Kanishka-Milinda (b) Milinda-Ashoka-Kanishka (c) Ashoka-Milinda-Kanishka (d) Milinda-Kanishka-Ashoka 114. Which one of the following movements started from Dandi? (a) Swadeshi Movement (b) Non-cooperation Movement

118. Which one of the following was the official language of Gupta period? (a) Pali (b) Magadhi (c) Prakrit (d) Sanskrit

120. Which one the following dynasties built the Khajuraho temples? (a) Chanellas (b) Chauhans (c) Paramaras (d) Tomars 121. Which Sultan declared himself as Siknadar-i Sani, the second Alexander? (a) Balban (b) Kaiqubad (c) Iltumish (d) Alauddin Khalji 122. Who among the following was the first muslim Military Commander who annexed Bihar and Bengal by overthrowing the Sena Dynasty? (a) Mahmud Gawan (b) Baktiya Khilji (c) Adil Shahos (d) Shams-ud-din

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▌ General Knowledge

123. Who among the following annulled the Partition of Bengal? (a) Lord Chelmsford (b) Lord Curzon (c) Lord Minto (d) Lord Hardinge 124. Who among the following established Fergusson college at pune in the year 1885? (a) Decan Education Society (b) Bhartiya Sewak Samaj (c) Samaj Sewa Sangh (d) Theosophical Society 125. Who among the following first used the word ‘Swarajya’? (a) Raja Rammohan Roy (b) Bal Gangadhar Tilak (c) Mahtma Gandhi (d) Swami Vivekanand 126. Who among the following had constructed the Red Fort in Delhi? (a) Akbar (b) Jahangir (c) Shah Jahan (d) Aurangzeb 127. Who among the following had repealed the Vernacular Press Act? (a) Lord Lytton (b) Lord Ripon (c) Lord Dufferin (d) Lord Landowne 128. Who among the following has authored the book Hind Swaraj? (a) Bal Gangdhar Tilak (b) Mahtma Gandhi (c) Gopal Krishna Gokhale (d) MG Ranade 129. Who among the following national leaders did not opt to defend the case dealing with the trial of the soliders of the Indian National Army in 1945? (a) Sir Taj Bahadur Sapru (b) Bhulabhai Desai (c) C Rajagroplachari (d) Jwahrlal Nehru 130. Who among the following was a contemporary of Gautam Buddha? (a) Nagarjuna (b) Kanishka (c) Kautilya (d) Mahavir 131. Who among the following was a founder of Swaraj Party? (a) Vallabhai Patel (b) Rajendra Prasad (c) CR Das (d) Narendara Deb

132. Who among the following was not associated with the Lahore conspiracy case? (a) Bhagat Singh (b) Surya Sen (c) Rajguru (d) Sukhdev 133. Who among the following was the author of Tarikh-I-Alai, which contains the details of the first few years of Sultan Alauddin Khilji? (a) Ziauddin Barani (b) Shams Siraj Afif (c) Amir Khusrau (d) Yahiya bin Ahmad 134. Who of the following is a contemporary of Gautama Buddha? (a) Ashvaghosa (b) Nagarjuna (c) Parsvanath (d) Vardhmana Mahavira 135. Who of the following started Swaraj Party? (a) Bipin Chandara Pal and Rajendra Prasad (b) Bipin Chandara Pal and Subhash Chandra Bose (c) CR Das and Motilal Nehru (d) CR Das and Madan Mohan Malaviya 136. Who of the following was a contemporary of Alexander the Great? (a) Bimbisara (b) Chandragupta Maurya (c) Ashoka (d) Pushyamitra Sunga 137. Who of the following was associated with the August Offer? (a) Lord Wavell (b) Lord Irwin (c) Lord Linlithgow (d) Lord Mountbatten 138. Who prescribed the separate electorates for India on the basis of the Communal Award in August, 1932 (a) Lord Irwin (b) Ramsay MacDonald (c) Lord Linlithgo (d) Winston Churchill 139. Who was the first Indian native ruler to accept the system of Subsidiary Alliance? (a) Scindia of Gwalior (b) Nizam of Hyderabad (c) Dalip Singh of Punjab (d) Gaikwad of Baroda 140. Who was the Governor General of India during the launch of civil Disobedience Movement? (a) Lord Chelmsford (b) Lord Reading (c) Lord Irwin (d) Lord Wavell

Indian History ▌ 2.149 141. With reference to the Indian Freedom struggle, which one of the following is the correct chronological order of the given events? (a) Partition of Bengal­Lucknow Pact­Surat Split of Congress (b) Partition of Bengal­Surat Split of Congress­Lucknow Pact (c) Surat split of Congress­Partition of Bengal­Lucknow pact (d) Surat Split of Congress­Lucknow pact­Partition of Bengal Direction: The next three (03) items are based on the passage given below. The myth is that the Indian National Congress was started by A.O Hume and others under the official direction, guidance and advice of no less a person than Lord Dufferin, the Viceroy, the provide a safe, mild peaceful and constitutional outlet of safety-valve for the rising discontent among the masses, which was inevitably leading towards a popular and violent revolution. Consequently, the revolutionary potential was nipped in the bud. Most writes accept the core of the myth, that a violent revolution was on the cards at the time and was avoided only by the foundations of the congress …. All of them agree that the manner of its birth affected the basic character and future work of the congress in a crucial manner. 142. The colonial rulers, according to the safety valve thesis, were trying to use Congress as

(a) A constitutional outlet for the mass discontent against the authorities (b) An instrument for the spread of British culture in India (c) An instrument to expand the Indian market for the products of British industries (d) An agency for social change in India 143. The author describes the safety-valve thesis as a myth because (a) Evidence are insufficient to establish the validity of the thesis (b) The author has concluded that in the light of available materials the thesis does not hold ground (c) The congress was a nationalist organization right from the start (d) The violent anti-colonial movements did not come to an end with the foundation of the congress 144. The core of the safety-valve thesis rests on the assumption that (a) a violnet anti-colonial revolution was bound to occur unless a constitutional platform like the congress was founded (b) The congress would assist the British in administering the colony (c) The congress leaders would constitute the colonial elite (d) the congress as a party would assist the colonial elite

Direction: The next four (04) items are based on the table given below: Match List I With List II, List III and List IV and select the correct answer using the code given below: List I (Ruler)

List II (Important Battle)

List III (Dynasty)

List IV (Significant work)

A) Ashoka

1. Against several rulers of Northern and Southern India

I. Kushana

i. Writer and poet, patron of learning besides and empire builder

B) Samudragupta

2. Kalinga war

II. Maurya

ii. Political conquest, patron of learining

C) Kanishka

3. Battle of Pundravanrdhana

III. Pushpavati

iii. Holding the 4th Buddhist Council and patron of learning

D) Harshvardhana

4. Against the Chinese and the parthians IV. Gupta

145.

List I

List II

List III List IV

(a)

A

1

II

(iv)

(b)

A

2

II

(iv)

(c)

B

1

IV

(d)

B

2

IV

List I

List II

List III List IV

(a)

C

4

I

(iii)

(b)

D

3

III

(i)

(c)

C

4

I

(i)

(d)

D

3

III

(ii)

147.

iv. Spread of Dharma, ahimsa and welfare activities List I

List II

List III List IV

(a)

B

1

III

(i)

(b)

C

4

IV

(ii)

(iii)

(c)

C

1

IV

(ii)

(iii)

(d)

C

4

III

(i)

List I

List II

List III List IV

(a)

A

1

II

(iii)

(b)

D

3

III

(ii)

(c)

A

1

II

(iii)

(d)

D

3

III

(i)

146.

148.

2.150

▌ General Knowledge

ANSWER KEY 1. (a) 2. (b) 3. (d) 4. (d) 5. (d) 6. (a) 7. (a) 8. (a) 9. (a) 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

(d) (a) (d) (b) (a) (a) (a) (d) (a) (a)

20. 21. 22. 23. 24. 25. 26. 27. 28.

(a)

29. 30. 31. 32. 33. 34. 35. 36. 37. 38.

(b)

(a) (c) (a) (d) (b) (d) (a) (d) (b) (c) (d) (c) (b) (c) (d) (d) (b)

39. 40. 41. 42. 43. 44. 45. 46. 47.

(d)

48. 49. 50. 51. 52. 53. 54. 55. 56. 57.

(c)

(a) (b) (b) (d) (c) (b) (d) (a) (a) (b) (b) (b) (b) (b) (b) (d) (a)

58. 59. 60. 61. 62. 63. 64. 65. 66.

(c)

67. 68. 69. 70. 71. 72. 73. 74. 75. 76.

(a)

(a) (a) (b) (a) (c) (c) (a) (b) (a) (c) (c) (c) (b) (b) (c) (b) (b)

77. 78. 79. 80. 81. 82. 83. 84. 85.

(b)

86. 87. 88. 89. 90. 91. 92. 93. 94. 95.

(d)

(c) (a) (a) (a) (b) (b) (a) (a) (a) (c) (c) (a) (b) (b) (a) (d) (a)

96. 97. 98. 99. 100. 101. 102. 103. 104.

(d)

105. 106. 107. 108. 109. 110. 111. 112. 113. 114.

(b)

(b) (b) (a) (d) (a) (c) (a) (a) (d) (d) (c) (b) (a) (b) (c) (c) (c)

115. 116. 117. 118. 119. 120. 121. 122. 123.

(c)

124. 125. 126. 127. 128. 129. 130. 131. 132. 133.

(a)

(d) (a) (d) (a) (a) (d) (b) (c) (b) (c) (b) (b) (c) (d) (c) (b) (c)

134. 135. 136. 137. 138. 139. 140. 141. 142.

(d)

143. 144. 145. 146. 147. 148.

(d)

(c) (b) (c) (b) (b) (c) (b) (a) (a) (b) (a) (d) (d)

C

H

2

A

P

T

E

R

Geography PHySICAL AND WORLD GEOGRAPHy SOLAR SySTEM In the vastness of the Universe, the Earth, the Sun and planets are tiny dots. Thousands of millions of other galaxies, each with its own star system, are moving apart from each other at great speeds. Our galaxy, the Milky Way, of which sun and its solar system form part, belong to the cluster of three dozen galaxies called the Local Group. The solar system revolves around the centre of the Milky Way at a speed of 285 km/s and completes one revolution in 224 3 105 years. Our solar system which has a disc-like shape, consists of a star (sun), eight planets and countless fragments of rock and metal left overs called asteroids. The diameter of the solar system is about 1,170 crore km. Origin Even before the astronomers knew the true extent and the age of the solar system, they began to speculate on its origin. Perhaps the first truly scientific system was put forward by Marquis de Laplace (France) in 1796 who propounded a theory: The sun spinning rapidly as it contracted after its birth, threw off from its centre, rings of gaseous material, each of which condensed into a separate planet. The most likely explanation of the origin of the planets is that they are formed from a disc of gas and dust spinning around a newly formed star, the sun. Thus, a whirling cloud of gas and dust is believed to have been the origin of our solar system. Around the young sun, the remains of the gas cloud were gradually smeared out into a disc, or solar nebula. Before the sun began to glow brightly, the disc of gas resembled a freezing fog, with the lighter material on the outer edges. Matter within the disc collided, and so coalesced into larger lumps. The lumps eventually swept each other up, forming large bodies, called proto-planets, in orbit around the sun. Sometimes, several proto-planets merged; and sometimes one body was captured in orbit around the other, like the moon and earth—thus giving final shape to our solar system.

Age Scientists have been able to determine the probable age of the solar system and to calculate an age for the Earth by assuming that the Earth and the rest of the solid bodies in our solar system formed at the same time and are, therefore, of the same age. Collision Theory At the beginning of 20th century, Sir James Jeans (England) suggested that gravitational pull of a passing star, or perhaps a grazing collision, when a passing star could have physically touched or collided with the sun had distorted the shape of the sun, drawing out filaments of gas that later broke into great droplets to form planets. A major objection to this ‘Collision Theory’ is that the composition of planets such as earth is very different from that of sun. Also, the orbits of the planets are remarkably rounded, whereas a chance of collision would be expected to throw the material into wildly elongated orbits. Another scientist, Otto Schmidt, in 1944 suggested that the sun might have passed through one of the many clouds of dirt and gas in our galaxy, and collected a disc of matter that later evolved into planets. All these earlier theories and many others about origin of planets in the solar system were ruled out due to one reason or the other.

Thousands of meteorites, which are fragments of asteroids that fall to Earth, have been recovered. These rocks from space provided the first extra-terrestrial sample long before men brought back specimens from the moon. The rate at which radioactive atoms decay is known, and by measuring the amount of such decay in meteorites scientists learned that the sun and the planets (hence, the solar system) formed together about 4,600 million years ago, a figure confirmed by the study of moon rocks.

Characteristics The planets of the solar system are divisible into two groups: 1. Terrestrial planets—the planets of the inner circle, and 2. The giant planets—the planets on the outer circle.

▌ General Knowledge

2.152

The inner circle comprises four planets (Mercury, Venus, Earth and Mars), whereas the outer circle also contains four planets (Jupiter, Saturn, Uranus, and Neptune). The planets of the solar system, in order of their mean distance from sun are—Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, and Neptune. Both Uranus and Neptune are visible through good binoculars. Table: Planets Inner planets

Outer planets

1. Mercury

5. Jupiter

2. Venus

6. Saturn

3. Earth

7. Uranus

4. Mars

8. Neptune

Jupiter, the biggest planet, is at the centre of the planets in our solar system and the size of the planets becomes smaller and smaller as we move on either side of Jupiter (except Mars). All planets and planetoids (asteroids) revolve around the sun in the same direction. Also, the planets have rotatory motion in the same direction of their revolution around the sun, the exceptions being Venus and Uranus. All planets, except these two, rotate in the anti-clockwise direction (reference being north pole to south pole of the earth). The planets appear to move rapidly among the stars because of being closer. The stars are also in motion, some of them at tremendous Possibility of Another Planet in Our Solar System In July 2005, a possible new planet temporarily called 2003 UB 3/3, was announced by a team from the California Institute of Technology. This most distance object in our solar system, thought to be 1.5 times the size of Pluto, takes 560 years to make a very skewed orbit of the Sun. Most planets circle the sun very near the same plane on Earth's orbit, but the new objects' orbit is off this elliptic plane by about 44 degrees. However, it is considered that just like Pluto, this new body is nothing more than Kupier Belt object.

speeds, but they are all so far away that their motion does not change their apparent positions sufficiently to be perceived. The nearest star is about 9000 times farther away than most distant planet in our solar system. A few of the other larger objects found beyond Pluto are Varuna (2000), Ixion (2001), Quaoar (2002), and Sedna (2004).

The Sun The sun is made up of extremely hot gases. Its glowing surface is called the photosphere. Above the photosphere is the chromosphere (being reddish in colour), and beyond this layer is the corona which is visible during eclipses. The visible white light of the sun is made up of seven colours—violet, indigo, blue, green, yellow, orange and red. Superimposed on these are hundreds of dark lines called the Fraunhofer lines. Like all other stars, the sun is composed mainly of hydrogen. The sun and the surrounding stars generally move in almost circular orbits around the galactic centre at an average speed of

about 285 km/s. The sun at this speed takes 224 million years to complete one revolution around the galactic centre. This period is called a cosmic year. At the core of the sun lies a giant nuclear furnace, converting hydrogen into helium at the rate of 600 million tonnes per second. Here the temperature is approximately 1,50,00,000°C. The sun, the controlling body of earth’s solar system, is often described as an average star. Yet, the sun’s mass and luminosity are greater than 80 per cent of the stars that can be seen in earth’s galaxy. On the other hand, most of the stars that can be easily seen on any clear night are bigger and brighter than the sun. It is the sun’s proximity to earth that makes it appear tremendously large and bright. The sun is 400,000 times as bright as the full moon and gives earth six million times as much light as do all other stars put together. A series of thermonuclear reactions involving the atoms of the elements of which sun is composed produce heat and light which make life possible on earth. The Structure of Sun The globe of gas that forms the sun can be visualized as a series of concentric shells of varying thickness. Within each shell, different processes take place. Chemical Composition of the Sun When sunlight is analysed with a spectroscope, it is found to consist of a continuous spectrum composed of all the colours of a rainbow in order, crossed by many dark lines. The ‘absorption lines’ are produced by gaseous material in the atmosphere of sun. The chromosphere is visible to the naked eye only at times of total solar eclipses, appearing as a pinkish violet layer with occasional prominence projecting above its general level. With proper instruments, the chromosphere can be seen or photographed whenever sun is visible without waiting for the total eclipse. Above the chromosphere is the ‘corona’ also visible to the naked eye only at times of total eclipse. Instruments also permit the brighter portion of corona to be studied whenever conditions are favourable. Iron, nickel and calcium are believed to be the principal contributors to the composition of the corona, all in state of extreme attenuation and high ionization that indicates temperature in the order of a million degrees Fahrenheit. Sunspots The sun’s photosphere is sometimes marked by sunspots—patches of gas cooler than the gas around them. At 4000°C they are still hotter than the surface of many stars. Astronomers believe that sunspots occur when lines of magnetic forces break through the sun’s surface, thus preventing some of the heat from reaching that part of the sun’s surface. When the magnetic field around a sunspot collapses, an eruption of energy occurs, which is known as solar flare. Spheres and Corona The radiating surface of the sun is called photosphere, and just above it is the chromosphere. There is an intimate connection between sunspots and the corona. At times of low sunset activity, the fine streamers of the corona are longer above the sun’s equator, than over the polar regions of the sun. During periods of high sunspot activity, the corona extends outwards fairly evenly from all regions of the sun, but to a much greater distance in space. Sunspots are dark, irregularly shaped

Geography regions whose diameters may reach lengths of tens of thousands of miles. The average life of a sunspot group is from two to three weeks, but some sunspot groups have lasted for more than a year by being carried repeatedly around the sun and rotated upon its axis. The record duration of a sunspot is eight months. Death of the Sun Like all stars, the sun spends most of its life in a state of stable size and light output, fuelled by nuclear reactions at its core that turn hydrogen into helium in a process similar to that which occurs in a hydrogen bomb. In the last 10 per cent of its life, however, the sun will go through a series of death throes that will radically change it. In the first stage, it will run out of hydrogen at its core, and will start to bum its helium. It will then swell into a red giant, roasting and engulfing the planets as far out as Mars. Finally, it will become so swollen and distended that it will lose more than half of its mass in space, a process that will last for about 50,000 years. What is left will be a white dwarf, a tiny star that slowly releases energy from its outer layer and cools. It then becomes an invisible black dwarf. Any planet that remains in orbit around the dead sun will freeze in the intense coldness of space.

The Planets The bodies revolving around the sun (at the same time rotating on their imaginary axis) are called planets. They have no light of their own but shine by radiating the light they receive from the sun. They all revolve around the sun in elliptical orbits. Until about 200 years ago only six planets were known. Three more planets were discovered later, the latest being Pluto (discovered in 1930) which is no longer considered a planet. Eight planets can now be identified. Mercury Mercury is the planet nearest to the sun. It rotates on its own axis in 58.65 earth days. It takes 88 days to complete one revolution round the sun. Thus it is the fastest planet in our solar system. Mercury has no satellite. Venus Also known as the evening star and morning star, is the brightest object in the sky after the sun and the moon. It is slightly smaller (500 km less in diameter) than the earth and is the planet closest to the earth. It is also the hottest planet in our solar system and has a weak magnetic belt. It rotates backwards unlike other planets and takes 224.7 days to complete one orbit round the sun. Venus has no satellites like Jupiter, no rings like Saturn and no ice caps like Mars. It is named after the Roman goddess of beauty. Earth Details about the earth are given under a separate section, ‘Basic Knowledge about the Earth’. Mars Mars is the fourth planet from the sun and is the next planet after the earth. Being favourably situated, it is brighter than most of the stars and, is therefore, known as the Red Planet. It has two small satellites called Phobos (Fear) and Deimos (Terror). It has polar caps similar to those of the earth and due to a tilt in its axis, the polar regions are exposed to sunlight alternately giving each hemisphere summer and winter.

▌ 2.153

Jupiter Jupiter is the largest planet in our solar system. It is about eleven times larger than the earth. Its volume is one and half times the volume of all the planets combined together. The most conspicuous aspect about Jupiter is its Great Red Spot. It is also known as the giant planet because of its huge size. It has 63 satellites. Prominent among them are Europa, Ganymede, and Callisto. Ganymede is the largest satellite in our solar system. Saturn Saturn is an outer planet visible to the naked eye. Second in size to Jupiter, it is the least dense of all the planets. The most spectacular feature of Saturn is its system of rings. The ring system is made up of a variety of separate particles which move independently in circular orbits. It has 47 satellites. Titan is its biggest satellite, having its own atmosphere. Uranus Uranus is the seventh planet from the sun and is not visible to the naked eye. The first planet to be discovered telescopically, it was identified as a planet in 1781 by William Herchel. It has completed only two revolutions round the sun since its discovery, and takes about 84 terrestrial years to circle round the sun. In 1977, astronomers discovered that Uranus was surrounded by a system of five faint rings. It has 27 satellites, and most prominent among them are Miranda, Ariel, Umbriel and Titania. Neptune Neptune is not visible to the naked eye but can be seen through a small telescope as a greenish star. It is eighth in position from the sun. This planet was discovered by I. G. Galle of Berlin in 1846. Till 1930, it was believed to be the farthest planet from the sun and the outermost in our solar system. It has 13 satellites, and Triton and Nereid are the most conspicuous of them. Pluto From its discovery in 1930 until 2006, it was considered the ninth and smallest planet. However, on 24 August 2006, the International Astronomical Union (IAU) has reclassified Pluto as a dwarf planet. Therefore, we now have eight planets in our solar system.

Uranus

Mars Venus Sun Mercury

Saturn Earth Jupiter

Neptune

Figure: Planets and Their Orbits

Satellites Satellite are bodies which revolve around the planets. All planets have one or more satellites, except Mercury and Venus. The moon is the earth’s natural satellite. The total count for identified satellites in the solar system stood at 156 till mid-2006, but search for more satellites is continued. Mercury and Venus do not have any satellites or moon. Earth has one and Mars has two moons.

2.154

▌ General Knowledge

Jupiter leads the group with maximum number of 63 moons while Saturn is second with 47 moons. Uranus and Neptune have 27 and 13 moons, respectively. The Moon The moon is the earth’s natural satellite and is its nearest neighbour in space. It revolves around the earth while rotating on its own axis. Only 59% of its surface is directly visible from the earth. Of all satellites in the solar system, the moon is the largest in proportion to its primary body, that is, the earth. All other satellites have sizes below 1/8 the size of the mother planet. The moon is about 1/4 the size of its mother planet, the earth. It takes about 1.3 seconds for moonlight to reach the earth, whereas sunlight takes about 8 minutes and 16.6 seconds to reach the earth. The moon takes 27 days 7 hours 43 minutes and 11.47 seconds to complete one revolution of the earth. It rotates on its axis in exactly the same time. Hence we see only one side of the moon. The moon is the first member of our solar system to have been visited by man. Neil Armstrong and Edwin Aldrin created history on 21 July 1969 by becoming the first two men to set foot on the moon. The spacecraft was Apollo XI. A total of 12 Americans walked on the surface of the moon during the Apollo programme and brought back about 380 kg of rocks and soil samples. These samples have shown that the moon is about the same age as the earth and was formed some 4600 million years ago.

Mars Jupiter

Asteroid belt

Venus Mercury

Sun

Earth

Meteors Meteors are small bodies often seen in the sky, shooting with great speed producing a trail of light. This results from the entry of a particle of matter called meteoroid into the earth’s atmosphere at a speed between 11 and 72 km/s. The mass of individual particles extends from micron-size dust specks to complete asteroids weighing thousands of tonnes. It may burn up or fall on earth (known as falling stars) in the form of dust or fragments called meteorites creating craters on the earth’s surface. The largest known meteorite, weighing about 70 tonnes and consisting of mainly iron and nickel, was found at Hoba in Namibia where it landed in prehistoric times.

Comets

Asteroids It is believed that Asteroids are debris left over from the formation of the inner planets, and that they are prevented from coalescing into one large body by the strong gravitational pull of nearby Jupiter. They are too small to retain any atmosphere of their own. There are called Trojan Asteroids. They circle around the sun between the orbits of Mars and Jupiter. The duration of their revolution round the sun varies from three to ten years. Asteroids are also called ‘planetoids’ or small planets. The first ten minor planets are: Ceres, Pallas, June, Vesta, Astraea, Heba. There are about 50,000 known Asteroids in our solar system. The largest is Ceres with a diameter of 1,025 km but most of them are much smaller. The smallest asteroid has a diameter of about 4–5 km. Asteroids are too faint to be seen by the naked eye, with the exception of Vesta. Their orbits are of low eccentricity, that is, more circular than elliptical. Some asteroids, however, (e.g., Eros and Icarus) have highly eccentric orbits. Such asteroids have their orbits either a cross or come close to earths’ orbit are labelled Near Earth Asteroids or NEAs.

Comets derive their name from the Greek word Kometes, meaning ‘hair-like’. A comet’s head consists of a coma made up of dust and gas. From the head flows a tail consisting of streams of dust and gas. There are about 1,00,000 comets in our solar system. Hailey’s Comet Named after Edmund Hailey, Hailey’s Comet last appeared in 1986 and it may reappear again after 76 years. In 1705 he stated that the comets seen in 1531, 1607 and 1682 were in fact the same body which orbits the sun every 76 years. His statement that it would reappear in 1758-59 proved correct.

The Auroras There are the luminous occurrences in high altitudes of both hemispheres; called ‘Aurora Borealis’ or ‘northern lights’ in the Northern Hemisphere, and ‘Aurora Australis’ in the Southern Hemisphere. The mechanism that produce auroral display are not completely understood but they are seen in many forms, including luminous curtains, arcs, bands and patches.

Table: Important Data Pertaining to Planets in Our Solar System

Planet

Mean distance from the sun (million km)

Equatorial diameter (km)

Period of revolution round the sun or sideral rotation

Priod of rotation on own axis or axis rotation

No. of satellites

Mass: times of earth

Mean temp. of surface

Other details

58 days, 15 hours and 30 minutes

0

0.055

1 350°C

Nearest to the sun and fastest planet

Mercury

57.9

4,880

88 days

Venus

108.2

12,104

224.7 days

243 days and 14 hours

0

0.815

1 475°C

Brightest, hottest and closest to earth

Earth

149.6

12,756

3651/4 days

23 hours, 56 minutes and 40 seconds

1

—

1 22°C

Most dense

Geography

Planet

Mean distance from the sun (million km)

Equatorial diameter (km)

Period of revolution round the sun or sideral rotation

Priod of rotation on own axis or axis rotation

No. of satellites

24 hours, 37 minutes and 22 seconds 9 hours, 50 minutes and 30 seconds

Mass: times of earth

2

0.108

63

317.9

10 hours and 14 minutes

47

95.2

84 years

16 hours and 1 minute

27

14.6

165 years

18 hours

13

17.2

Mars

227.9

6,787

687 days

Jupiter

778.3

1,42,800

12 years

Saturn

1,427

1,20,000

30 years

Uranus

2,869.6

51,800

Neptune

4,496.6

49,500

Mean temp. of surface

▌ 2.155

Other details

It has largest mountains 2 23°C and deepest valleys. Largest planet: largest 2 123°C moon in solar system. Least dense: smallest 2 180°C moon in solar system. Planet most tilted on its 2 218°C axis (axis tilt of 98° from its orbital plane). 2 228°C

Its moon period has the most eccentric orbit.

BASIC KNOWLEDGE ABOUT THE EARTH The earth is one of the planets of the solar system. It is in the third position from the sun and is the fifth largest planet in the solar system.

Shape The earth is an oblate spheroid (ball flattened at the poles), i.e., almost spherical, flattened a little at the poles with a slight bulge at the centre (equator). Hence, it is flat at the poles.

Structure The earth is made up of a number of concentric layers of material as in the bulb of an onion. The main layers are as follows: 1. Crust: 30–35 km in thickness (average); 2. Mantle: 2, 900 km in thickness (average); 3. Outer core: 2,100 km of molten material (average); 4. Inner core: 1,370 km of solid mass (average). Each layer has its own characteristics and chemical composition. Some 4,500 million years ago, it was made up of molten rock material which cooled down and hardened. As a result, the heavier material sank towards the centre, while the less dense, lighter material remained near the surface. The average thickness of the solid crust of the earth varies between 30 and 35 km. Below the crust is the mantle composed of several layers which is about 2,900 km in thickness. It is a shell of red-hot rocks composed of silicate materials, rich in magnesium and iron. The mantle is partly fluid. The earth’s inner core is believed to be solid and is of a depth of about 1,370 km. It is made up of iron and nickel which accounts for the earth’s magnetism

The upper part of the mantle is called the asthenosphere which is about 250 km thick. The rocks contained in the asthenosphere are partially molten. And it is on this part that the top crust of the earth carrying the continents and oceans rests. The surface of the earth has four distinct spheres, viz., 1. Lithosphere is the top crust which includes the land surface and the ocean floors. 2. Hydrosphere is the water surface which includes oceans, seas, rivers and lakes. 3. Atmosphere is the cover of air that envelopes the earth’s surface. 4. Biosphere is the sphere where life exists, and is spread over all the above three spheres. Atmosphere Biospher Hydrosphere Lithospher (crust) Mantle

Core

Inner core Outer core

Earth’s structure

Mantle Crust Figure: Cross-Sectional View of Atmosphere, Biosphere and Earth

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▌ General Knowledge

Table: Facts about the Earth Diameter

Circumference

At the equator

12,755 km

At the poles

12,712 km

Mean diameter

12,734 km

At the equator

40,075 km

At the poles (Meridianal)

40,024 km

Equatorial radius

6,377 km

Total surface area

510,100,500 km2

Total land area (29.08%)

148,950,800 km2

Total water area (70.92%)

361,149,700 km2

Total mass of the earth

6.0 31024 kg

Total volume of the earth

1,083,208,840,000 km3

Mean depth of hydrosphere

3,554 m

Mean distance from the sun

149,407,000 km

Approximate age of the earth

4,500 million years

Time of rotation on its own axis

23 h 56 min and 4.09 s

Period of revolution round the sun

365 days 6 h 9 min and 9.54 s

Highest point

Mt Everest (8,848 m above mean sea level)

Lowest point

On the earth’s surface

Shores of Dead Sea (Israel, Jordan) 396 m below sea level

Deepest part in the Ocean

Challenger Deep (Pacific Ocean) in the Mariana’s Trench east of the Philippines, 10,900 m below sea level

Density of core

13.09 g/cm3

Temperature at core

4500°C

Composition

North pole

The earth is made up of more than 100 different elements. The following eight elements constitute the important ones: Oxygen Silicon Aluminium Iron Calcium Sodium Potassium Magnesium Other elements

46.5% 27.72% 8.13% 5.01% 3.63% 2.85% 2.62% 2.09% 1.41%

Earth’s Movements The earth has two types of movements, viz., rotation or daily motion and revolution or annual motion.

Eq ua

tor

South pole Figure: Earth's Rotation

Effects of Rotation (i) Occurrence of day and night (ii) The position of a place on earth can be fixed (iii) Change in the direction of wind and ocean currents Longest Days and Nights

Rotation The earth spins on its own imaginary axis from west to east once in 24 h (in precisely 23 h 56 min and 40.91 s). It is also called diurnal or daily motion. The axis is an imaginary line which runs from north to south and passes through the centre of the earth. It always remains inclined at an angle of 66½° to the plane of the earth’s orbit.

Northern Hemisphere

Southern Hemisphere

Longest day (shortest night)

June 22

December 22

Shortest day (longest night)

December 22

June 21

▌ 2.157

Geography

Earth’s orbit

Equal Days and Nights Days and nights are equal at the equator throughout the year, because the circle of illumination always divides the equator into two equal parts.

Days and Nights in India

A

B 2

A

A

C

Longer days and shorter nights: In June Shorter days and longer nights: In December

S 1

3 Perpetual day Direction of Revolution of the Earth Long day and short night

Equal day and night

Perpetual night

Short day and long night

Figure: Northern Summer

Revolution While rotating on its axis, the earth also revolves around the sun, along an almost circular path (elliptical) called its orbit. This journey of the earth around the sun is called a revolution. It is also called annual motion or yearly motion because it takes one solar year to complete the journey, that is, 365¼ days or 365 days 5 h 48 min and 45.51 s.

How Seasons Change? The earth’s axis is inclined at an angle of 66½° to the plane of its orbit. As a result of this, the earth is in different positions while revolving round the sun. During the first half of the year the northern hemisphere tilts towards the sun resulting in longer days and the summer season in the region. During this period the southern hemisphere experiences winter. During the second-half of the year the southern hemisphere tilts towards the sun, and thus experiences summer and the northern hemisphere experiences winter during this period. The Four Seasons The annual motion of the earth together with its inclined axis, therefore, causes the four seasons of spring, summer, autumn and winter. 2 Seasons 4 Seasons

s

ason

Se One

Effects of Revolution

er

Summ

The earth’s revolution causes a change in seasons.

Seasons

ons

4 Seas

Causes of Change in Seasons

2 Seasons

1. Revolution of the earth around the sun 2. A fixed inclination (tilt) of the earth’s axis at 66½° to the plane of its orbit

Figure: Seasons through the world

NP

Path of orbit

Earth Sun Figure: Earth's Orbit

NP

S

SP

SP

(a)

(b)

Figure: (a) Northern Summer (b) Southern Summer

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▌ General Knowledge

Spring On 21 March, the sun is directly overhead the equator. This is the season of spring in the north temperate zone. Summer On 21 June, the sun is directly overhead the Tropic of Cancer. Thus the north temperate zone experiences summer. Autumn On 23 September, the sun returns to the equator, and the north temperate zone, experiences the season of autumn. Winter On 21 December, the sun is at the Tropic of Capricorn, and the north temperate zone experiences winter. In the south temperate zone, the seasons are the reverse of those mentioned above, at the same time of the year.

Equinoxes and Solstices Equinoxes Equinoxes are dates when the nights and days are equal. During these days the sun shines directly over the equator. 21 March is called vernal equinox and 23 September is called autumnal equinox. Solstice The time of the year when the difference between the length of days and the length of nights is the largest is referred to as solstice. During these days the sun shines vertically over a tropic.

Have Day  Nig udes atit ht L l l  A 1

2

When the light of the sun or the moon is obscured by another body the sun or moon is said to be in eclipse. Lunar Eclipse The moon is said to be in eclipse when the earth comes between the moon and the sun, and this is called lunar eclipse. The shadow cast by the earth on the moon is called an eclipse. Lunar eclipse occurs only on a full moon day. However, it does not occur on every full moon day because the moon is not in the same position in relation to the earth and the sun on every full moon day. Solar Eclipse The sun is said to be in eclipse when the moon comes between the sun and the earth. This is called solar eclipse. There is either a partial or total obstruction of the sun’s light when viewed from the earth. A solar eclipse occurs on a new moon day when the moon is in line with the sun. However, due to the inclination of the moon’s orbit, a solar eclipse does not occur on every new moon day.

Figure Eclipses Umbra

Penumbra

Moon Partial Solar Eclipse

S

urs Ho

Eclipses

Day Night Sun

Arctic Circle Day  Night

Earth

Tropic of Cancer Day  Night

Sun Overhead

Moon

Partial Lunar Eclipse

Penumbra

Equator Day  Night

Sun

Tropic of Capricor n Antarctic Circle Day  Night

Earth Umbra

Penumbra

N

Figure: Earth at the Equinoxes

On or around 21 June, the North Pole tilts towards the sun and the sun shines directly over the Tropic of Cancer. This is called summer solstice. On or around 22 December the earth is at the opposite end of its orbit. The South Pole tilts towards the sun and the North Pole away from it. This is called winter solstice. In the northern hemisphere, days are longest and nights are shortest during the summer solstice and alternately the days and nights are shortest and longest respectively during the winter solstice. The reverse is true in the case of the southern hemisphere.

Total Solar Eclipse

Moon

Sun Earth

Total Lunar Eclipse

Umbra

Penumbra

Sun

ATMOSPHERE The atmosphere is a gaseous envelope that surrounds a celestial body. The terrestrial atmosphere, by nature of its composition, control of temperature and shielding effect against solar radiation,

makes life possible on earth. It covers both the land and the water surface. It is bound to the earth by the gravitational pull of the earth.

Geography

Composition The composition of the atmosphere changes as we go higher from the earth’s surface. Up to about a height of 50 km from the earth, the atmosphere is composed of elements as shown in table. Table: The Composition of Dry Air at Sea Level Gas

Volume (%)

Nitrogen (N2)

78.09

Oxygen (O2)

20.95

Argon (Ar)

0.93

Carbon dioxide (CO2)

0.031

Neon (Ne)

0.0018

Helium (He)

0.00052

Krypton (Kr)

0.00011

Xenon (Xe)

0.0000087

Hydrogen (H2)

0.00005

Methane (CH4)

0.0002

Nitric oxide (NO)

0.00005

Ozone (O3)

0.000002 (winter) 0.000007 (summer)

After a height of 50 km above the earth’s surface, the atmosphere is made up of atomic oxygen (O 2), ozone (O 3), helium and hydrogen.

Atmospheric Pressure Atmospheric pressure refers to the weight of the entire air column over a given point. A litre of air weighs about 1.3 g. At the sea level the air pressure is around 1033.6 g cm2. This is referred to as one atmosphere.

Atmospheric Layers These are the layers of air that lie above the earth’s surface. The atmosphere of the earth is arranged into layers as accrued below, viz., Troposphere The troposphere is the layer nearest to the earth’s surface and extends from sea-level to a height of about 15 km. This region is the densest of all the atmospheric layers and contains water vapour, moisture and dust. In this region the temperature decreases as the height increases from the earth. Tropopause Tropopause is the layer which separates the troposphere (lowest layer) from the stratosphere (upper layer). Stratosphere This is the region of uniform temperature extending from an altitude of about 15 km above the earth to a height of about 50 km. It is free from water vapour, clouds and dust. Mesosphere This is a very cold region and lies above the ozonerich layer of the stratosphere. It extends from 50 or 80 km above the earth’s surface.

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Mesopause The mesopause separates the mesosphere from the next layer called the ionosphere. Ionosphere The ionosphere lies immediately above the mesosphere and extends from 60 to 400 km above the earth’s surface. This layer contains ionized (or electrically charged) air which protects the earth from the falling meteorites (shooting stars) as most of them burn out in this region. It also protects the earth from the harmful radiations of the sun. The ionosphere consists of ‘D’, ‘E’ and ‘F’ layers and includes the thermosphere and exosphere. Thermosphere This is the middle layer of the ionosphere. It is the region of the atmosphere where the temperature is above 100°C. Exosphere The exosphere is the uppermost region of the ionosphere and makes up the outer limits of the atmosphere. Here the gravity of the earth is exceedingly weak. The magnetic belt of the earth which is known as magnetosphere, extends to about 64,000 km above the earth’s surface. The exosphere is now considered as part of the magnetosphere. The outer boundary of the magnetosphere or the final boundary between the earth and outer space is known its magnetopause.

Winds Wind is air in motion and plays an important role in climatic variations on the earth. Why do Winds Blow? Wind movement is caused by the difference in air pressure on the earth’s surface. The air near the earth’s surface gets heated and being light in weight, rises to the higher regions, while cold air from the surrounding regions moves into this low pressure area to equalize the pressure of atmosphere. This movement of air from areas of high pressure to areas of low pressure is known as wind. Winds are referred to by the direction from which they blow. More importantly the deflection of wind is caused due to the rotation of the earth. Winds are deflected to the right in the northern hemisphere, and to the left in the southern hemisphere. Types of Winds Winds are divided into three broad groups: (i) Regular winds, e.g., trade winds and westerlies (ii) Periodical winds which blow seasonally, e.g., monsoons (iii) Variable winds, e.g., cyclones and other local winds. Trade winds Trade winds are steady currents of air blowing towards the equator from the north-east and the south-east. They are called trade winds, because prior to the steam age, sailing ships carrying goods of trade and cargo, depended on these winds. The trade winds blow towards the equator between 5´ and 30´ north and south latitude. Westerlies The westerlies are regular winds which blow outside the tropics in the temperate zone. The south-westerly winds blow in the northern hemisphere and the north-westerly winds in the southern hemisphere. South of 40 degrees latitude the absence of any land mass enables these winds to gather force and thus they are known as the roaring forties.

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Monsoons The monsoons are seasonal winds which blow briefly over India, Sri Lanka, China and north-west Australia. In summer, chiefly due to the high temperature over the earth’s surface near the equator the air pressure is low, and thus, moist rain bearing winds blow in from the sea towards the land. A monsoon climate is marked by summer rains and winter droughts. Westerlies

Other Types of Winds Chinook is a hot dry wind blowing on the east or leeward side of the Rocky Mountains (North America). As these winds descend from the mountains they are pressed down and get heated. Similar winds on the north of the Alps (Europe) are called Fohen. Scirocco is a hot, moist wind which blows from the Sahara desert and moves eastwards across the Mediterranean sea. Solano is a similar type of wind blowing from the Sahara towards the Iberian Peninsula.

Anti-trade

Harmattan is a hot, dry wind blowing outwards from the interior of West Africa Trade winds

Figure: Circulation of Air Producing the Trade Winds and Anti-Trades

Mistral is a very cold wind, which blows down from the plateau of Central France. Punas are cold, dry winds blowing down towards the western side of the Andes. Levanter is a easterly wind through strait of Gibraltar.

Westerlies

Vendavel is a westerly wind through strait of Gibraltar.

Sub-tropics N.E. Trades

Marin is a south easterly from Mediterranean to France. Gregale is a north easterly wind blowing in Greece.

Calms Calms Summer

Bora is a cold, dry wind blowing outwards from Hungary to the north of Italy.

Willy-willy is a tropical cyclone in Australia. The more common usage for dust-devils.

Winter

Figure: Movement of the Trade Winds Produces Mediterranean Climate

LITHOSPHERE The lithosphere is the topmost crust of the earth. The thickness of the lithosphere varies, between 35 and 50 km in continents and between 6 and 12 km at the ocean beds. The lithosphere around mountains is the thickest reaching down to 60 km in the high mountain regions. The lithosphere constitutes about one per cent of the total earth’s volume and 0.4% of its mass. The topmost portion of the earth’s crust comprises soil, sand and rocks.

Rocks The earth’s crust is formed of mineral material called rocks. The rocks which form the substructure of our lithosphere are grouped into three broad categories: Igneous Rocks Igneous rocks are formed by the solidification of molten magma from the interior of the earth. About 95% of the

earth’s crust is made of this type of rock. In fact, all other types of rocks originate from these rocks and, therefore, they are also called primary rocks. Igneous rocks are of three types: (a) Granite These rocks are the major continental rocks. (b) Basalt These rocks are found on ocean beds. (c) Volcanic These rocks are formed by the solidification of molten lava ejected by the volcanoes. Sedimentary Rocks These rocks are formed from the sediment deposits on the ocean beds. They comprise only about 5% of the earth’s crust but cover about 75% of the total land surface. These rocks are made up of the weathered remains of igneous rocks but also contain organic matter from the remains of marine organisms. Sedimentary rocks are formed in horizontal layers, called strata and take millions of years to harden into rocks. These rocks are

Geography also known as stratified rocks because of these layers. Examples of sedimentary rocks are: 1. Gypsum, chalk and limestone which are formed by the deposition created by chemical action or chemical sedimentation. 2. Peat, lignite, bituminous coal, anthracite which are formed by deposition of organic matter or marine remains. 3. Conglomerates like gravel, pebbles and shingle, sandstone and shale like layered clay or claystone formed by the deposition of sediments in water. Metamorphic Rocks These rocks were originally either igneous or sedimentary in nature. They metamorphize or change due to pressure, intense temperature or the action of water and chemical activity. Examples of metamorphic rocks are: 1. Slate is formed by compression of sedimentary rocks like shale and mudstone. 2. Quartzite is formed from sedimentary rocks like sandstone. 3. Gneiss is formed by the metamorphosis of igneous rocks like granite. 4. Marble sedimentary rocks like limestone turn into marble under intense heat.

The Continents The land surface of the earth is made up of immense land masses divided into seven continents and a great number of islands. Together, they cover about one quarter of the earth’s surface. Origin of Continents It is believed that originally there was only one land mass called Pangaea. This large land mass split into a northern mass Laurasia and a southern one called Gondwana land. From these two land masses, the continents gradually drifted to where they are now located and the process is still continuing. Important Data Largest

Asia

continent

Area

44,030,000 km2

Length

North to south 8,700 km

Width

West to east 9,700 km

Smallest

Australia

continent

Area

Island continents

Antarctica and Australia

Ice-covered continent

Antarctica in the southern hemisphere

7,687,120 km2

Mountains Mountains are elevated land masses with peaks (summit) smaller than their base and of a height above 300 m or 1000 ft, Similar elevations below 300 m or 1000 ft are called hills.

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Important Data Largest and highest mountain range

Himalaya–Karakoram in Asia, with the highest peak Mt Everest (8,848 m)

Other important mountain ranges

Andes, Alps, Rockies

Young or new mountains have come into being after the continental drift started, with the break up of the large land mass of Pangaea. The Himalayas, the Andes, the Rockies and Alps are examples of new or young mountains. Old mountains are those which were formed in the pre-drift era long before the continental masses came together to form Pangaea. The Pennines (Europe), Appalachians (America) and the Aravallis (India) are examples of old mountains. Types of Mountains There are broadly four types of mountains which are differentiated according to the mode of their origin or formation, viz., Fold mountains The mountains formed by the crumbling or folding of rocks, resulting in huge wrinkles or folds upon the earth’s surface as well as the ocean floors are called fold mountains. These are formed as a result of a series of earthquakes. The following diagrams illustrate the formation of these mountains:

Stage 1 (A)

Stage 2 (B)

Stage 3 (C) Fold mountain

One part is ridged up to form mountains often of great height, while the other part is covered by the waters of the ocean. All big mountain systems have been formed in this way, for example, the Himalayas, Alps, Andes, Rockies. Block mountains (or horst) A mountain mass formed by the lifting up of land between faults (cracks in the rock strata) or by the sinking of land outside the faults, are called block mountains. They are formed when a mass of elevated land under strain, cracks thus, leaving a higher elevation standing between two areas of lower elevation. It may happen that the outer part sinks leaving an elevated central part, a crust block or block mountain. These are usually steep-sided. The Vosges in France and the Black Forest mountains in western Germany come under this class of mountains.

▌ General Knowledge

2.162

Table: Some Facts About the Seven Continents Area in '000 km2

Approx. % of the world’s land

Asia

44,030

29.5

Mt Everest (Nepal–Tibet) 8,848 m

Dead Sea (Israel–Jordan) 400 m

Africa

29,785

20.0

Mt Kilimanjaro (Tanzania) 5,895 m

Lake Assal (Djibouti) 156 m

North America

24,255

16.3

Mt Mckinley (Alaska) 6,960 m

Death Valley (California) 86 m

South America

17,798

11.8

Aconcagua (Argentina) 6,962 m

Peninsular Valdez (Argentina)40 m

13,338.5

9.6

Vinson Massif 5,140 m

Bentley subglacial trench 2538 m

10,498

6.2

Mt Elbrus, (Russia) 5,642 m

Caspian Sea (Russia) 29 m

7,687.12

5.2

Mt Kosciukso (NSW 2,228 m)

Lake Eyre (South Australia)15 m

Name

Antarctica Europe Australia

Highest point

Lowest point (Below sea level)

Cotopaxi in Andes (South America) are examples of volcanic mountains. They are also called mountains of accumulation.

Block Mountain

Crater Fault

Fault Ash Lava

Flows

Volcanic mountains A volcano is a mountain formed of material that has erupted from inside the earth through an opening in the earth’s crust. Super-heated molten rock matter called lava is ejected, forming a hill, conical in shape with a funnel-shaped hollow at its top called a crater. It is estimated that there are about 850 active volcanoes, 80 of which are on the ocean floor. Mt Fujiyama in Japan, Mt Vesuvius in Italy and Chimborazo and

Figure Volcano Residual mountains Mountains that are deeply dissected and reduced by weathering and river action, are called residual mountains. The Catskill mountains in New York are residual mountains. They are also called mountains of denudation.

Table Important Volcanic Mountains Name (Country)

Height (m)

Location

Last notified eruption

Ojos del Salado (Argentina, Chile)

7,084

Andes

1981

Guallatiri (Chile )

6,060

Andes

1960

Cotopaxi (Ecuador)

5,897

Andes

1975

Lascar (Chile)

5,641

Andes

2007

Tupungatito (Chile)

5,640

Andes

1964

Popocatepeti (Mexico)

5,451

Altiplano de Mexico

1996

Nevado del Ruiz (Colombia)

5,400

Andes

1991

Sangay (Ecuador)

5,230

Andes

2006

Geography

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Table: Principal Mountain Peaks Name

Continent

Country

Range

Height (m)

Mt Everest

Asia

Nepal–Tibet

Himalayas

8,848

K2 (Godwin Austen)

Asia

India

Karakoram

8,611

Kanchenjunga

Asia

Nepal–India

Himalayas

8,598

Lhotse

Asia

Nepal–China

Himalayas

8,501

Makalu

Asia

Tibet–Nepal

Himalayas

8,481

Dhaulagiri

Asia

Nepal

Himalayas

8,172

Nanga Parvat

Asia

India

Himalayas

8,126

Nanda Devi

Asia

India

Himalayas

7,817

Illampu

South America

Bolivia

Andes

7,014

McKinley

North America

Alaska

Rockies

6,194

Mt Logan

North America

Alaska

Rockies

6,050

Mt Elias

North America

Alaska

Rockies

5,944

Ararat

Asia

Turkey

Caucasus

5,156

Mont Blanc

Europe

France

Alps

4,810

Monte Rosa

Europe

France

Alps

4,565

Note: Nanga Parvat is India's highest peak in the undisputed Indian territory

Table: Important Islands Name

Location

Area in km2

Kalaalit Nunaat (Greenland)

North Atlantic Ocean

21,75,597

New Guinea

Southwest Pacific Ocean

8,20,033

Borneo

West-Central Pacific Ocean

7,43,197

Malaqasv (Madagascar)

Indian Ocean

5,87,042

Baffin

North Atlantic Ocean (Canadian)

4,76,068

Sumatra

Northeast Indian Ocean (Indonesia)

4,73,605

New Zealand

South Pacific Ocean

2,70,000

Honshu

Western Pacific Ocean (Sea of Japan)

2,30,316

Great Britain

Atlantic Ocean

2,29,993

HyDROSPHERE The hydrosphere is the mantle of water covering a greater part of the earth’s surface. The oceans and seas comprise the hydrosphere. It is estimated that about 361,740,000 km2 or 70.8% of the earth’s surface is covered by water.

Largest Part of the Hydrosphere The Pacific ocean comprises the largest part of our hydrosphere and occupies and area of about 166,240,000 km2 or 35.25% of the earth’s surface, which is more than the total combined area of all the continents.

Important Data

Depth of the Hydrosphere The average depth of the hydrosphere is about 3.5 km and the deepest point is the Challenger deep, a part of the Mariana Trench in the Pacific Ocean, which is about 10,900 m deep. The depth of ocean is measured in fathom, a unit of length which is equal to about 4 ft.

Mean depth of the hydrosphere

3,554 km

Total weight of the water

1.3 3 1018 tonnes or 0.222% of the Earth’s weight

Total water in the oceans

1300 million m3 of 97% the total of the world’s water

Volume of the oceans

1,285,600,000 km3 in area

Why is the Ocean Water Salty? Various salts carried away from the earth’s surface by rivers and other water bodies are deposited in the ocean which make the ocean water salty. Salinity of the water

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▌ General Knowledge

in oceans is affected by the extent of evaporation of its surface and the volume of fresh water added by rainfall, rivers and icebergs. Enclosed seas in tropical areas, like the Dead Sea and Sambar lake of Rajasthan, are more saline because of rapid evaporation and less addition of fresh water. The ocean water contains, about 3.5% of dissolved salts which make the water saline. The important salts dissolved in ocean water are—chlorine, sodium, magnesium, sulphur, calcium, potassium, bromine, strontium and boron and traces of other minor salts.

Tides The waters of the oceans are held on the earth by the force of gravity. But there are other bodies besides the earth, which are trying to draw all things towards themselves. The most important of them are the moon and the sun. Tides or the rise and fall in the level of the oceans and seas is due to the gravitational pull of the moon and the sun. A rise in the water level is called high tide and a fall is called low or ebb tide. The tide produced as a consequence of the moon and the sun pulling the earth in the same direction, is called spring tide. It is the tide of maximum rise, occurring at the new or full moon of each month. The tide produced, when the moon and the sun are at right angles to the earth, thus pulling in opposite directions is called neap tide. It is the tide of minimum rise, occurring during the first and third quarter of the moon. Its amplitude is less than that of the spring tide. S

b

E

M

NP

M

a

Figure Tide Formation Other Factors That Cause Tides In addition to the gravitational pull of the sun and the moon, the rotations of the earth and the moon, also cause tides. When a body rotates on its axis, loose objects tend to break away from the surface. In other words, a centrifugal force is setup by the rotation of the earth. But the moon is, at the same time revolving round the earth, and so the real cause of the tides may be said to be the centrifugal force set up by the rotation of the earth and the moon round their common centres. Moon

Sun

Earth

Sun

Sun Spring Tides Moon

Earth Sun

Earth Neap Tides

Spring Moon

Figure Detailed View of Tide Formation

Neap Moon

Earth

Ocean Currents Ocean currents are streams of water crossing the ocean in the direction of prevailing winds. Ocean currents are of two kinds, viz., (i) Warm currents which flow from a warm to a cold region, and (ii) Cold currents which flow from a cold to a warm region. Important Currents Beginning in the Atlantic Ocean, the westerly winds drive the Cold Antarctica current eastwards. This turns northwards when it reaches the coast of Africa and is known as the Benguela current. The southeast trade winds take this current westwards as the South Equatorial current. Off Cape St Rogue at the corner of Brazil, this current is divided into two. One branch flows southwest into the Gulf of Mexico as the warm South Equatorial current; which leaves the Gulf of Mexico and flows northeast as the Warm Gulf stream. It widens over the Atlantic, a part flows as the North Atlantic Drift and warms the shores of the British Isles and northern Europe. The other part of this current turns southwards and under the influence of the north-east trade winds joins the North Equatorial current. From the Arctic Ocean flows the cold Arctic current flowing along the shores of Greenland, where it is called the Cold Labrador current. In the Pacific Ocean, the currents are very similar to those in the Atlantic ocean but their names are different. The Antarctic current joins the Peruvian current which continues as the South Equatorial current. Off the east coast of Australia, this current divides into the New South Wales current or Kurusiwo current. This is a warm ocean current which flows up the east coast of Asia and under the influence of the westerly winds flows towards North America. It is the counterpart of the Gulf Stream and also called the Black current or Japani current. The ocean currents of the north Indian Ocean flow under the influence of the monsoons. Those of the South Indian Ocean follow the pattern of the Atlantic and Pacific currents. The most important current in this region is the Agulhas current. Influence of Ocean Currents Ocean currents influence the climate of the regions they pass through. Places under the influence of warm currents have a much warmer climate than lands in the same latitude, under the influence of a cold current. The Pacific Ocean is the biggest and deepest ocean in the world Table Oceans Name

Area (km2)

Sea Area (%)

Average depth (m)

Greatest depth (m)

Pacific

155,557,000

45.7

4,280

10,900

Atlantic

76,763,000

22.8

3,926

9,219

Indian

68,556,000

20.3

3,963

8,047

Southern Arctic

20,327

–

–

–

14,056,007

3.9

1,205

5,441

(Southern ocean was approved by International Hydro-graphic Organization in 2000)

Geography

Major Rivers of the World Largest River The Amazon in South America, flowing into the South Atlantic ocean is the largest river in the world. Though the Amazon river is 222 km shorter in length than the Nile, it is still considered the largest river because: • It is navigable by deep sea steamers for over 3000 km • It has the highest river basin in the world (7 million km2)

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• It has the greatest flow of water with an average of 119,000 cubic m per second, rising up to 200,000 cubic m per second in floods. • It has some 15,000 tributaries, with the longest one of a length of 3,200 km. Longest River The Nile in Egypt (6,670 km), flowing from the Mediterranean sea is the longest river in the world.

Table: Important Rivers Length in m

River

Location/ Country

Source

Outflow into

Nile

Africa/Egypt

6,670

Lake Victoria

Mediterranean

Missouri–Mississippi

North America/US

6,300

Red Rock, Montana

Gulf of Mexico

Amazon

South America

6,448

Andes in Peru

Atlantic Ocean Brazil

Yangtze-Kiang

Asia/China

5,490

Tibetan Plateau

East China Sea

Hwang-Ho

Asia/China

4,840

Tibet

Pacific Ocean

Yenisei

Asia/Siberia

5,300

MI Tannuola

Arctic Ocean

Lena

Europe/Russia

4,800

Lake Baikal

Laptev Sea of Arctic Ocean

Niger

Africa/Nigeria

4,800

Sierra Leone

Gulf of Guinea (Atlantic)

Congo

Africa/Zaire

4,800

Confluence of Lualuba and Luapula

Atlantic

Ob

Asia/Siberia

4,150

Altai Mt

Gulf of Ob (Pacific)

Indus

Asia/India and Pakistan

3,180

Mt Kailas

Arabian sea

Brahmaputra

Asia/India

2,960

Tibet

Bay of Bengal

Danube

Europe/Austria, Hungary, Czech, Slovakia and Yugoslavia

2,820

Near Baden in Germany

Black Sea

Murray Darling

Australia

3,720

Alps

Indian Ocean

Volga

Asia/Russia

3,700

Valdai Plateau (Moscow)

Caspia n Sea

Amur

Asia/Siberia

4,510

Pamirs

Pacific Ocean at the sea of Okhots k

INDIAN GEOGRAPHy LOCATION, DIMENSIONS AND FRONTIERS Location India is the seventh largest country and second most populous country of the world accounting for about 2.4% of the total world area. It lies in the northern hemisphere between parallels of latitude 8°4´ N and 37°6´ N and between meridians of longitude 68°7´ E and 97°25´ E and is a part of the Asian continent. If we consider arable/cultivable land, India is second only to USA. India is located mid-way between west Asia and east Asia. Many transoceanic routes pass through India which connect African continent, industrially developed European continent, petroleum rich west Asia and south-eastern countries, namely, China, Japan, Australia and western coast of USA.

Dimensions Important dimenions Distance from north to south Distance from east to west Length of coastline (main land) Length of coastline (incl Lakshadweep and Andaman and Nicobar Islands) Length of land frontier Total geographic land area Percentage of earth surface covered by India

3,214 km 2,933 km 6,100 km 7,516.6 km 15,200 km 32,87,263 km2 2.4%

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▌ General Knowledge

The peninsular table land bulges into the Indian ocean for a distance of 1600 km and divides it into two (i) the Arabian sea (ii) Bay of Bengal.

Boundaries North The Himalayan range and Nepal separate India from Ti-bet. The protected state of Bhutan also lies in the north-east. The boundary line between India and China is called the McMohan Line. India’s border with Nepal spans for (1752 km); with Bhutan (587 km); and with China (3917 km). East Myanmar (Burma) and Bangladesh form an enclave within India and the Bay of Bengal. Its international boundary with

Myanmar is 1458 km long and with Bangladesh is 4096 km long. West In the north-west India shares a boundary mainly with Pakistan and the Arabian Sea on the west. north-West Afghanistan and Pakistan are situated on the north- west of India (Pak–Afghan border line, that is, Durand Line). India– Afghanistan border is only 80 km long whereas Indo–Pakistan border is 3310 km long. (Indo–Pak border line , that is, Radcliff Line) South To the south of India are the Indian Ocean and Sri Lanka. The Gulf of Mannar and Palk Strait separate India from Sri Lanka on the eastern side.

PHySICAL FEATURES India comprises four well-marked physical divisions, viz., 1. The Himalayan Range 2. The Northern Plains 3. The Deccan Plateau 4. The Coastal Plains The Himalayan range The Himalayas, one of the youngest fold mountains in the world surround India on the north, north-west and north-east, extending about 2400 km between the rivers Indus and Brahmaputra, with their width varying between 160–450 km. They are mainly made up of sedimentary rocks. Longitudinally the Himalayas are divided into three parallel ranges or zones, viz., Greater Himalayas also called ‘Himadri’ lies in the north, are of an average height of 6000 m and are always snow-capped. The highest peak on India’s border is Kanchenjunga (8,598 m) which partly lies in Nepal. Most of the rocks found here are metamorphic rocks. It is an asymmetrical mountain with very few southward spurs. Its northern slope is gentle which meets gradually with the river valley which runs parallel to the mountain for long distances. Lesser or middle Himalayas are of an average height of 3700– 4500 m. This area is also called Himachal. The Kashmir valley, Kulu and Kangra lie in this belt. Steep, bare southern slopes, and gentle forest, covers northern slopes of the Himachal present a typical hogback topography. It is composed of highly compressed and metamorphosed rocks. These linear longitudinal ranges have steeper southern and gentler northern slopes, which gives it a Hog-back type look. The outer Himalayas The Siwalik with a similar hogback look forms foothills of the Himalayas. The southern-most or foot-hill zone called Siwaliks are of an average height of 600 m. These are characterized by fault scraps, anti-clinal crests and synclinal hills. Deep gorges are formed in them by the Himalayan rivers.

The Northern Plains Formed by the valleys of the rivers Ganga and Brahmaputra and their tributaries, with a length of about 2500 km and width ranging between 400–600 km. These plains occupy one-third of India’s land surface and form the most fertile region. According to terrain characteristics, this plain consists of two parts, viz. (a) The upland plain, which lies above flood level and is made up of alluvium soil. It is one of the most fertile areas in the world. This plain is called Bangar land. (b) The lowland plain, which is liable to inundation during floods, is called Khadar land. The plains are cut off from the south by the Vindhya mountain.

The Deccan Plateau Lying south of the northern plains, it has an elevation varying between 500 and 1300 m above sea level, and slopes from west to east. The plateau is flanked by mountain ranges called Eastern and Western Ghats on its eastern and western sides. It is geographically the oldest part of India with rocks which are 3000 to 5000 million years old. The higher Deccan peaks reach over 2500 m which include the Nilgiri hills.

Coastal Plains The northern portion of the western coastal plain is called the Konkan and the southern portion is called the Malabar coast. The eastern coastal strip is known as the Coromandal coast.

The Deccan Mountains The mountains of peninsular India include:

Geography

Hi

lay

as

Vindhya

Hi ll

No rth E

ast

Ar ev

all

i

ma

▌ 2.167

ats

ste

rn

Gh

Ea

Western Ghats

Satpura

Figure India : Mountain Ranges The Aravallis The oldest mountain range in India, reaching an elevation of 5650 ft. The Vindhyas Separate the southern part of India from its northern part, with an average height of 2500–4000 ft.

The Satpuras Are situated between the rivers Narmada and Tapti. Western Ghats The western flank of the Deccan tableland is guarded by Western Ghats, portion of which is called the Sahyadari hill. Eastern Ghats Flank the eastern part of peninsular India.

ISLANDS The following are two groups of islands:

Lakshadweep

Andaman and Nicobar

Are a group of 27 coral islands scattered in the Arabian sea, 300 km to the west of Kerala coast; only 10 of which are inhabited. Together they form the Union Territory of Lakshadweep with Kavaratti island as its capital.

The northern cluster of 204 small islands are the Andamans and the southern cluster of 19 small islands are the Nicobar islands. Together they form the Union Territory of Andaman and Nicobar with Port Blair as its capital.

SOILS The Indian Council of Agricultural Research has classified Indian soil into the following categories:

Alluvial Soil Areas in the Indo-Gangetic plain formed by the periodic deposition of silt brought down by the rivers. Almost a quarter of Indian soil is alluvial soil which is very fertile.

Found in: Plains of Punjab, Haryana, Uttar Pradesh, Bihar and West Bengal. Also found in the plains of Assam, and coastal strips of South India along river beds.

Black Soil Formed by the deposition of lava ejected by volcanoes several million years ago. It is rich in minerals but poor in organic matter

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▌ General Knowledge

and retains moisture for longer period than other soils. Black soil is suitable for cotton growing. Found in: Maharashtra, Gujarat, west Madhya Pradesh, Andhra Pradesh and Tamil Nadu.

Red Soil Formed through the weathering of ancient metamorphic and crystalline rocks. It is red in colour due to high iron content. Good for the cultivation of pulses and coarse grains. Found in: Madhya Pradesh, Orissa, Andhra Pradesh, Karnataka and north-eastern hill states.

Other types of soil

Found in

Saline and Alkanline soil

Dry tracts of the north, especially in Bihar, Uttar Pradesh, Punjab and Rajasthan. Harmful for crops.

Peat and Marshy soil

Kerala, coastal tracts of Orissa, Sunderbans and some parts of Bengal, north Bishar and south-east Tamil Nadu.

Desert soil

In arid regions that receive poor water supply such as western Rajasthan.

Forest soil

Hilly areas of Assam, Uttar Pradesh, subHimalayan tract and Coorg in Karnataka. It is formed by the deposition of organic matter derived from forests.

Laterite Soil Formed by the weathering of laterite rock. Found in: areas occupied by the Deccan, Andhra Pradesh, Eastern Ghats, Mysore, Tamil Nadu, coast of Orissa, and Meghalaya.

THE RIVERSySTEMS OF INDIA The rivers of India are broadly classified into:

The Himalayan System

world’s largest delta, Sunderbans, is formed by the Ganga and the Brahmpautra in West Bengal and Bangladesh in the Bay of Bengal covering an approximate area of 7500 km2.

ak

lum Jhe

s du In

Br

da

Narma

Gulf of Cambay

Ma

ha

Tapti

Goda

vari

a

shn

Kri

Arabian Sea

adra

h gab Tun

y er

General There are 14 major river basins in the country, each with drainage area of above 20,000 km2; 44 medium basins of 20,000 to 2,000 km2. The total water in all river systems of India has been estimated roughly at 16,45,000 million cubic metres. The Ganga and the Brahmaputra carry 61 % of total water of the country’s rivers. The

tra

apu

ahm

Gulf of Kutch

uv Ca

There is also a third river system called ‘Rivers of Inland Drainage Basins’. These are small rivers in sandy areas of Rajasthan. Most of them are of an ephemeral character and drain towards individual basins or salt lakes like the Sambhar or are lost in the sands, with no outlet in sea, except Luni, which drains into the Rann of Kutch.

Tsangpo

Kosi

Third River System

Setluj Gand

These are seasonal rivers as their flow mainly depends on rainfall. They carry about 30% of total discharge of Indian rivers. The Deccan system comprises rivers of peninsular India. Godavari is the largest river system in peninsular India. The Cauvery system is the southernmost in the country.

Beas

s nge Ga unana Yam

The Deccan System

C he n

vi Ra

ab

These are perennial snow-fed rivers. During rainy season they are generally flooded. The Himalayan rivers are younger than the Deccan system and carry 70% of the total discharge of river water into the sea.

Indian Ocean

Figure Important Indian Rivers

na

di

Bay of Bengal

Geography

▌ 2.169

Table: Himalayan River System Name

Source

Enters

Indus

Mount Kailas in Tibet

Arabian 2,900 sea km

General direction of flow:west and north-west (Punjab-India), south-west (Punjab, Sind-Pakistan); has five major tributaries (i) Satluj (ii) Beas (iii) Ravi (iv) Chenab (v) Jhelum

Ganga

Gangotri Glacier

Bay of Bengal

2,640 km

General direction of flow: West, south-west, east and south-east, flows through Uttar Pradesh, Bihar and West Bengal. Tributaries: (1) Yamuna (2) Gomti (3) Ghagra (4) Gandak (5) Ram Ganga (all these except Gomti emerge from the Himalayas) (6) Sone (7) Chambal (8) Betwa and (9) Ken join the Yamuna from the south after flowing through the southern hills, eventually all rivers flow into the Ganga. Receives water from an area of about one-quarter of the total area of India. It drains about 25 % of the total area of the country.

Bay of Bengal

2,900 km

Flows through the Himalayas, turns south–west and then to south, joining the easternmost branch of the of the Ganga and the Padma, both emptying together into the Bay of Bengal.

Brahmaputra Mansarovar Lake in Western Tibet

Length

Features

Note: The sacred Ganga is the longest river in the country. The 2900 km Brahmaputra is longer than the Ganga but only a third of the river passes through India

Table: Deccan River System Name

Source

Enters

Length

Features

Godavari, Krishna, Cauvery,

Western Ghats

Bay of Bengal

1,450 km 1,290 km 760 km

General direction of flow: East, south-east; Godavari forms the largest river basin in Deccan and second largest in India; also called Virdha Ganga or Dakshina Ganga it covers an area of about 10% of the total area of India. Krishna basin is the third largest in India.

Mahanadi, Damodar

North west of the Deccan Plateau

Bay of Bengal

890 km

Flow eastwards. Mahanadi forms the fourth largest river basin in India.

Narmada, Tapti

Northernmost extremity of the Deccan Plateau

Gulf of 1,290 km Cambay in the Arabian Sea

Flow westwards. Narmada has an extensive river basin.

Sharasvati, Netravati, Bharlapuzha, Periyar and Pamba

Western Ghats

Arabian Sea

Flow westwards, cross west coast and fall into Arabian sea. These are small rivers of the south with minor basins.

CLIMATE The Indian Meteorological Department recognizes four seasons, viz., (i) Winter (December–March) (ii) Summer (April–May) (iii) Monsoon (June–September) (iv) Season of retreating south-west monsoon (October–November)

(59 inches) a year. The temperature in the south seldom drops below 20°C. The south-east gets most of its rain between October and December and by March, temperature rises throughout the country. In June, 38°C–43°C is common in the north. The lowest average temperature is l3°C in the Himalayan hill stations of Darjeeling and Shimla. Mumbai on the west coast has an annual mean temperature of 28°C and Chennai a mean temperature of 29°C.

India is subject to a wide range of climatic conditions. In Assam the rainfall is as high as 1080 cm (421 inches) a year, whereas in the Rajasthan desert in the west rainfall is a merely 10 cm (4 inches). The Indo-Gangetic plain averages about 150 cm

Coldest Parts The Trans-Himalayan and Greater-Himalayan regions, Dras and Kargil of Ladakh region are the coldest regions in the country

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▌ General Knowledge

where temperature falls below –40°C. This is due to the distance from the equator as well as altitude.

Regions of Heavy Rainfall (above 200 cm)

Hottest Parts In west Rajasthan at Barmer, temperature shoots up to 50°C (122°F) during daytime. This is due to its distance from the sea, absence of rainfall and dry and sandy soil which gets heated up. Rajkot in Gujarat averages a temperature of 33°C.

Wettest Parts Cherrapunji in Meghalaya is the wettest place in India. In fact, Mawsynram, 10 km from Cherrapunji, receives maximum amount of rainfall in the world—as high as 1080 cm. It is due to its funnel-shaped location surrounded by the Khasi, Garo and Jaintia hills.

Driest Part Jaisalmer in western Rajasthan is the driest place which receives the lowest rainfall (less than 10 cm), It is due to its distance from the monsoon winds, of the Bay of Bengal and location in the subtropical high pressure belt.

The Monsoons The climate of India is the monsoonal type fed by two rain bearing winds viz.,

The South-West Monsoon (June–September) The south-west monsoon contributes 86% of the total rain in India. It opens on the west coast around beginning of June and continues till September. The summer rain has a declining trend as it moves inland. The first rain decreases westwards in the northern plains giving an average rainfall of: Kolkata Patna Allahabad Delhi

from the seas which have a comparatively high pressure, and being moisture-laden bring rainfall to the hilly areas of the country.

119 cm 105 cm 76 cm 56 cm

The Western Ghats, the north-eastern hills and the Himalayas cause regional variations in distribution of rainfall.

(i) Assam, Arunachal, Meghalaya, Sikkim and Northern part of West Bengal—being hilly and in the direct path of the Monsoon winds blowing in from the Bay of Bengal. (ii) The Western Ghats—Rain-bearing clouds blow in from the Arabian Sea towards the Western Ghats. (iii) The Himalayan slopes.

Regions of Scanty Rainfall (below 50 cm) (i) Kutch and Western Rajasthan (ii) Southern Haryana, south-east Punjab and Ladakh Table: Seasonwise Distribution of Rainfall Rainfall Season

Duration

Approx.% of Annual Rainfall

South-west Monsoon

June/September

73.7

Post Monsoon

October/December

13.3

Winter or north-west monsoon

January/February

2.6

Pre-monsoon

March/May

10.4

Total

100.00

Forests in India The total forest and tree cover of the country accounts for 778229 km2 in constituting 23.68% of the country’s geographic area. The forest cover of India is 678333 km2 or 20.64% of India’s geographic area and the total tree cover of country (computed area with 70% canopy density) has been estimated as 99896 km2 or about 3.04%. The National Forest Policy has laid down a target of raising the area covered by forests to nearly 33.3%. The SFR-2004 (state of Forest Report) shows an increase of 21219 km2, which is 0.65% of geographic area as compared to 2001 assessment.

The North-East Monsoon (November/December)

Types of Forests in India The five types of forests found in India are:

The north-east monsoon is a comparatively minor monsoon confined to a smaller area of the country, particularly the cast coast. This is also called Winter Monsoon and it is Tamil Nadu’s major monsoon giving about 45 cm of rainfall in November/December as compared to 38 cm in the four months of the summer.

Evergreen forests (Tropical) Found in areas where rainfall ranges between 200–300 cm e.g., Western Ghats and sub-Himalayan regions. They provide hardwood like teak, rosewood, ebony and bamboo, etc. These are coniferous forests having needleshaped leaves.

Cause of Rainfall in India

Deciduous forests (Monsoon forests) Found in areas having rainfall between 150–200 cm e.g., in large parts of the Deccan Plateau stretching over Maharashtra, Madhya Pradesh and Karnataka. They provide fine timber such as teak, sal, sandalwood, etc.

During the hot summer the land mass absorbs a great amount of heat causing the temperature to rise, thereby creating excessive low pressure. It causes winds (monsoon) to blow towards India

Geography ▌ 2.171 Dry forests Found in desert regions of Rajasthan and south of Punjab where rainfall is below 75–100 cm. Mostly consists of shrubs, thickets and stunted trees like kikar, jal, etc. Hill forests Found in southern India and in the Himalayan regions and provide timber like oak, deodar, pines, chir, chestnuts, walnuts, chinar, Maple, etc.

Tidal forests (Mangrove) Found in coastal plains which are generally submerged particularly on deltas of rivers on the east coast (Ganges, Mahanadi, Godavari). The forests on the Ganges delta in Bengal are called Sunderbans after the Sundari trees found in these forests.

AGRICULTURE For the last five decades agriculture has occupied the most prominent place in the Indian economy. Agriculture dominates the economy to an extent that a very high proportion of working population in India is engaged in agriculture. Agriculture and allied sectors contribute 25% of GDP of India. According to recent census figures, approximately 65% to 67% of India’s working population is engaged in agriculture. Agriculture contributes a major share of national income, around 32% in 1996-97, although it has been decreasing continuously, and the share of the manufacturing and service sectors is increasing.

Land Utilization Land utilization statistics are available for 93.1 % of the total geographical area of 3287.3 lakh hectares (ha). Area under forest had increased from 404.8 lakh hectares in 1950-51 to 690.2 lakh hectares, while net sown area increased from 1187.5 1akh hectares to 1412.3 lakh hectares during the same period. Broad cropping pattern indicates that though food grains have a preponderance in gross cropped area as compared to non-food grains, their relative share has come down from 76.7% during 1950-51 to 65.8% during 1999-2000. The area operated by large holdings (10 hectares and above) has been decreasing and was at 14.8% during 199596, while the medium holdings (area 1 hectares and lesser) has been increasing and was about 17% during the same period. This indicates that the farm land is being fragmented.

Agricultural Scenario in India Although India has long since achieved the ability to feed its population, the agricultural sector remains a weak spot in economy for two reasons. First, this owes to the impact of volatile weather patterns, and secondly, it is remarkable for its low productivity patterns. However, overall India has made lot of progress in agriculture since independence in terms of growth in output, yields and area under many crops. It has gone through a green revolution, a white revolution, a yellow revolution and a blue revolution. Today India is the largest producer of milk, fruits, cashews, coconuts and tea in the world, the second largest producer of wheat, vegetables, sugar and fish and the third largest producer of tobacco and rice. The major causes for the reversal of the ongoing stagnations was due to: (i) Steady rise in average yield per hectare (i.e., rise in agricultural productivity); (ii) Steady increase in area under cultivation. But, still even after his marked rise in agriculture productivity the average yield per hectare for India remained the lowest

compared to the world standards. many reasons for this can be summed up as follows: (i) Poor agricultural techniques; (ii) Inadequate irrigation facilities; (iii) Discouraging rural atmosphere; (iv) Overcrowding in agriculture—too many people depending upon agriculture; (v) Small sizes of land holding; (vi) Absence of proper initiatives; (vii) Inadequate non-farm facilities (finance, security, marketing, etc.) Because of these and some more causes, the green revolution fell short of giving the results which it promised.

Advances in Indian Agriculture Scene Policy Framework for Agricultural Extension (PFAE) Public research and extension played a major role in bringing about the Green Revolution. The nature and scope of agriculture extension has undergone fundamental changes since then. Need for extensive reform in extension services has been felt at different fora. These have been duly recognized in the ‘Policy Framework for Agricultural Extension (PFAE)’ finalized by the government after extensive consultations. The PFAE provides policy directions for implementation of agricultural extension reforms. The five major guiding elements of the policy framework are: Reforming Public Sector Extension; Promoting Private Sector to effectively compliment, supplement and wherever possible to substitute public extension; Augmenting Media and Information Technology Support for Extension; Mainstreaming Gender Concerns in Extension; and Capacity Building/Skill Upgradation of Farmers and Extension Functionaries. National Agricultural Technology Project The decentralized and demand-driven technology dissemination system has been successfully pilot tested under Innovation and Technology Dissemination (ITD) component of National Agricultural Technology Project (NATP) with financial support of the World Bank. This experience is proposed to be upscaled in all the States through a new scheme titled ‘Support to State Extension Programme for Extension Reforms’. Kisan Call Centres Impressive growth in telecommunication infrastructure is being utilized to provide information to farming community through mass media. The programmes to be broadcast under this initiative cover both local as well as issues of national importance. Information support to farmers and extension functionaries is also being provided through agriculture exhibitions/ fairs and print media. Agripreneurship Agri-Clinics and Agri-Business centres scheme aims at providing self-employment opportunities to unemployed

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agricultural graduates. They, in turn, supplement the efforts of the government/public sector agencies by providing fee-based extension and other services to the farmers.

Public Private Partnership (PPP) Public Private Partnership (PPP)

has emerged as one of the crucial areas in agricultural extension. DAC has accordingly interacted with the related private sector representatives at various levels. A pilot action plan is being developed to operationalize the PPP issues at cutting edge level. This plan will be shared with the states and private sector representatives shortly for replication with suitable modifications.

Farm Information Dissemination Agriculture information dis-

semination is strengthened by organizing regional and state level Krishi Melas and by participating in national and international exhibitions. Funds are provided to State Agriculture Universities for promoting agriculture in print media in local vernacular, apart from bringing agriculture extension journals in Hindi and English. It has also been decided to honour selected progressive farmers with Krishi Ratna Awards.

Training A network of training infrastructure has been created/

strengthened to provide training support in agriculture extension. It includes National Institute of Agriculture Extension management at the national level, four extension education institutes at the regional level and State Agriculture Management and Extension Training Institutes (SAMETIs) in seven NATP States.

Subsistence Crops These are crops for individual consumption

like food grains. Also called food crops. They cover about 75% of total cultivated area in the country.

Commercial Crops These crops are mainly for ready cash

earn ings and not solely for individual consumption, e.g., sugarcane, oil seeds, cotton, tobacco, groundnuts, etc. Also called cash crops or non-food crops. They also include plantation crops (tea, coffee, rubber) and horticulture crops (fruit, vegetables, coconut, cashewnut, etc). Cover about 25% of total sown area.

cotton and specially potatoes reported a remarkable increase in the average. Crop Patterns Crop pattern of any region depends upon physical characteristics such as soil, climate, weather, rainfall, etc. It also depends upon the nature and availability of irrigation facilities and other non-farm facilities. Then there are factors such as farm size, price and income maximization motives of the farmers, etc., which influence the crop pattern of any economy. The nature of crop distribution in India can be gauged from the shift noted first from increase in food grain production to that of non-food grain production, from 1950s to 1980s which was again reversed from 1980s to 1990s. The same continued till year 2005-06. The above shifts can be thought of as the results brought through appropriate changes in the economic motives. For instance, during eighties, the prices of food grain crops were rising quite fast and the farmers started growing food grain for the markets. Then, during the nineties, the reversal occurred as the non-food grain became more remunerative under the impact of the new technology. Green Revolution The Green Revolution was launched in 1967-68 to improve agricultural productivity whereby: • cultivation area was extended • new and hybrid varieties of seeds were introduced • use of fertilizers was increased • soil conservation schemes and new agronomic practices were encouraged along with these • irrigation potential was increased • research in agricultural science was carried out Table: Main Crops and Their Major Producers Type

Name of Crop

Major Producing States

Cereals

Wheat

Uttar Pradesh, Punjab, Haryana

Rice

West Bengal, Uttar Pradesh, Punjab

Gram

Madhya Pradesh, Tamil Nadu

Barley

Maharashtra, Uttar Pradesh, Rajasthan.

Bajra

Maharashtra, Gujarat, Rajasthan

Sugarcane

Uttar Pradesh, Maharashtra, Tamil Nadu

Poppy Oil Seeds

Uttar Pradesh, Himachal Pradesh

Coconut

Kerala, Tamil Nadu

Linseed

Madhya Pradesh, Uttar Pradesh

Groundnut

Gujarat, Tamil Nadu, Andhra Pradesh

Crop Seasons There are two major crop seasons in India, viz.,

(i) Kharif crops—grown during the rainy season and under wet agricultural practices; (ii) Rabi crops—grown during dry winter season and developed irrigation system is needed to raise it. Table: Types of Crops Crop

Sown in

Havested Examples

Kharif June/July Sep/Oct

Rabi

Oct/Dec

Rice, Jowar, Bajra, Sugarcane, Soyabean, Maize, Cotton, Jute and Groundnuts

Apr/May Wheat, Barley, Linseed, Rapeseed, Mustard, Grams

*Rice, Maize and Groundnut are grown in summer season also. Among food grains, wheat registered an increase of more than 150%, whereas rice production could climb up to only 35–40%. Whereas the commercial crops like oil seeds, sugarcane,

Cash Crops

Rapeseed Rajasthan, Uttar Pradesh and Mustard

Geography

Sesame

Uttar Pradesh, Rajasthan

Sunflower

Karnatka, Andhra Pradesh, Maharashtra

Soyabean

Madhya Pradesh, Maharashtra, Rajasthan

Fibre Crops Cotton

Plantations

Spices

Maharashtra, Gujarat

▌ 2.173

aimed at capturing a commanding share of the liquid milk market in a few metropolitan cities by linking some 27 milk sheds in the country. Operation Flood II (1981–85) After successful implementation of Operation Flood I, Operation Flood II was launched in 1981 which extended to almost all states. More than 34,500 dairy cooperative societies had been organized in 136 milk sheds covering about 36 lakh farmers.

Jute

West Bengal, Bihar

Silk

Karnataka, Kerala

Hemp

Madhya Pradesh, Uttar Pradesh

Coffee

Karnataka, Kerala

Rubber

Kerala, Karnataka

Tea

Assam, Kerala

Fertilizers

Tobacco

Gujarat, Maharashtra, Madhya Pradesh

Pepper

Kerala, Karnataka, Tamil Nadu

A fertilizer is a plant food other than farm manure. Soil needs three major nutrients, viz.,

Cashewnuts

Kerala, Tamil Nadu, Andhra Pradesh

Cardamom

Karnataka, Kerala, Sikkim, Tamil Nadu

Chillies

Maharashtra, Andhra Pradesh

Castor seed

Gujarat, Andhra Pradesh

Cloves

Kerala

Turmeric

Andhra Pradesh, Orissa

Saffron

Jammu and Kashmir

Ragi

Karnataka, Tamil Nadu, Jammu and Kashmir

Operation Flood III (1985–90) Operation Flood III was launched in 1985 under the Seventh Five Year Plan for which the World Bank approved a 300 million dollar loan. It involved a total investment of Rs 681.29 crore. The programmes were implemented under the aegis of the National Dairy Development Board (NDDB) and the Indian Dairy Corporation (IDC).

1. Nitrogen (N)—helps in greening of plants 2. Phosphorus (P)—helps in flowering of plants 3. Potassium (K)—helps in the fruit-forming stage in plants. All crops, therefore, require three doses of fertilizers. Based on these requirements the chemical fertilizers produced are:

Nitrogenous Fertilizers Urea—contains about 46% Nitrogen Ammonium Sulphate—contains about 21% Nitrogen Calcium or Ammonium Nitrate—contains about 20% Nitrogen. Plants need nitrogen for rapid growth and to increase their protein content.

Phosphatic Fertilizers

First green revolution Was launched in 1967-68 and was mainly confined to the progressive, wheat producing states of Punjab, Haryana and western Uttar Pradesh by introduction of high-yielding varieties of Mexican wheat and dwarf rice varieties.

Single Superphosphate (SSP) Triple Superphosphate (TSP) P2O5

Second green revolution In 1983-84 the Green Revolution was extended to eastern and central states including West Bengal, Bihar, Orissa, Madhya Pradesh and Uttar Pradesh—through expansion in supplies of inputs and services to farmers and better farm management techniques. As a result of the Green Revolution, wheat production more than doubled and rice production increased by 53%.

Single Superphosphate (SSP) Triple Superphosphate (TSP) P2O5

Granary of India Punjab is known as the Granary of India. However, wheat production is highest in Uttar Pradesh and Punjab comes next. In Punjab the output of wheat per acre of land is higher than that of Uttar Pradesh (3,593 kg/ha against national output 2,117 kg/ha). Dairy farming Operation Flood I (1970–81) In collaboration with the World Bank, Operation Flood I was launched in 1970 which

Phosphatic Fertilizers

Potassic Fertilizers—K2O There are also complex fertilizers of several grades of NPK (Nitrogen, Phosphorus and Potassium) which improve fertility of soil. Production • • •

India ranks fourth in the world after the US, Russia and China in production of nitrogenous fertilizers. India ranks 8th in phosphatic fertilizer production in the world. In Asia, India ranks second after China in the production of fertilizers. Potassic fertilizers are not produced in India and are therefore imported

▌ General Knowledge

N

BHUTAN

NE

PA L

Above 30

BAY OF BENGAL

EE DW HA A) KS NDI (I

11-20 2-10 Below 1 NIL

46-75

Area Under Wheat Cultivation

K

PA

N

Area Under Rice Cultivation

CHINA

TA

IS

16-45 1-15 NEGLIGIBLE

SRI

SRI

P

LAN

Percentage of total cropped area Above 75

EEP DW SHA LAK INDIA) (

LA

21-30

KA

BANGLA DESH

ARABIAN SEA

Percentage of total cropped area

NE PA L

K

PA

BHUTAN

N

CHINA

TA

IS

NE

BHUTAN

PA L

BAY OF BENGAL

BANGLADESH

BAY OF BENGAL

ARABIAN SEA

KA LAN

SUGARCANE

SRI

A K LAN SRI

EP

Cultivation Pattern: Cotton and Jute

E DW ) HA KS NDIA (I

LA

EP

E DW ) HA KS NDIA (I

LA COTTON JUTE

Sugarcane Cultivation

MYANMAR

MYANMAR BANGLADESH

ARABIAN SEA

MYANMAR

ARABIAN SEA

BHUTAN

NE PA L

MYANMAR

BAY OF BENGAL

MYANMAR

BANGLA DESH

K

PA

CHINA

N

TA

IS

KA

PA

CHINA

TA

IS

K

LAN

2.174

Geography

Irrigation Irrigation Potential and Utilization Out of a geographic area of about 329 million hectares, India had a cultivable area of only 133 million hectares. In this India has an ultimate irrigation potential of about 114 million hectares, but only 70% of the cultivable land (i.e., about 80 million hectares) is being irrigated till today. With a view of narrowing the gap between the available irrigation potential and its utilization, Government of India has launched Command Area Development (CAD) which aims at maximization of productivity in the irrigation command areas through an integrated approach, i.e., by covering farm development works including construction of field channels and field drains to ensure adequate and equitable water supply to the individual farm holdings. Irrigation Projects in India A large number of irrigation projects have been launched since the beginning of the era of planning in India. However, many projects remained incomplete owing to financial constraints of the states. An Accelerated Irrigation Benefits Programme (AIBP) was launched during 1996-97 to give loan assistance to the states to help them complete some of these incomplete projects. Loan assistance amounting to Rs 500 crore and Rs 952 crore was released to 18 states during 1996-97 and 1997-98, respectively, under this programme. Expansion of irrigation facilities along with consolidation of the existing system have been the main strategy for increasing production of food grains. Irrigation support is provided through major, medium and minor irrigation projects and command area development. With sustained and systematic development of irrigation, its potential has increased from 22.6 mha in 1951, when the process of planning began in India, to about 89.4 mha (provisional) at the end of the Eighth Plant (1992–97). Role of Ministry of Water Resources The National Water Policy adopted in September 1987, stresses that “Water is a prime natural resource, a basic human need and a precious national asset. Planning and development of water resources need to be governed by national perspectives.” The Ministry of Water Resources lays down policies and programmes for development and regulation of the country’s water resources. It covers sectoral planning, coordination, policy guidelines, technical examination and techno-economic appraisal of project, providing central assistance to specific projects facilitation of external assistance and assistance in resolution of inter-state water disputes, policy formulation planning and adequate guidance in respect of minor irrigation, command area development and development of ground water resources, etc. Command Area Development Programme A centrally-sponsored Command Area Development Programme was launched in 1974-75 with the main objective of improving utilization of irrigation potential and optimizing agricultural productivity and production from the irrigated areas by integrating all functions related with irrigated agriculture. The Programme involves execution of on-farm development works like construction of field channels and field drains, land

▌ 2.175

levelling and shaping and conjunctive use of surface and groundwater. Adaptive trials, demonstrations and training of farmers are encouraged to disseminate technical know-how among the farmers for establishing suitable cropping patterns and improved farming practices, and for maintaining soil health. Under the Programme, the ministry of water resources is also introducing and promoting participatory Irrigation Management in the CAD Projects by creating awareness and providing financial assistance to farmer’s associations. Reclamation of waterlogged areas in irrigated commands is also an important component of the programme.

Development of Irrigation Infrastructure Irrigation as the main catalyst of agricultural development in India, accounts for the largest share in total investment in the sector. In the five year plan, investment in irrigation has accounted for 8 to 10% of total public investment. Among the three types of irrigation projects, namely minor, medium and major, the last two are highly capital intensive. The participation of both central and state government NGOs, farmers and public is the need of the hour to strengthen and develop irrigation infrastructure which is most essential to promote the economic development of the country.

Methods of Irrigation Wells Used for smaller areas mainly in Uttar Pradesh, Punjab, Tamil Nadu and Maharashtra. Irrigation by wells accounts for about 48% (27% by tubewells and 21 % by ordinary wells) of the total irrigated area in the country. Tanks Used in central and southern India, especially in Andhra Pradesh. About 10% of total irrigated area is irrigated by tanks. As the rivers in these areas are not perennial unlike the Himalayan rivers, irrigation is done by tanks. Canals Major source of irrigation in Punjab, Uttar Pradesh and Haryana, and through storage canals in the Deccan and Madhya Pradesh. About 40% of total irrigated areas are irrigated by canals. Area Irrigated The irrigation potential in the country estimated at 113.5 m hectares. Accelerated Irrigation Benefits Programme (AIBP) launched in 1996-97 to encourage the States for completion of ongoing irrigation projects through Central Loan Assistance (CLA), is now helping accelerate benefits from locked-up investments.

Multipurpose Projects The objectives of multipurpose river projects are: 1. Extension of irrigation facilities by constructing dams to store surplus water during rainy season to be released in summer. 2. Power generation by constructing hydel power stations. 3. Flood control and making rivers navigable. The execution of some of these large works during the five-year plan periods has brought out a number of problems and

2.176

▌ General Knowledge

weaknesses related to these projects: (a) Irrigation facilities offered by large projects have been poorly utilized. (b) The financial returns of these projects have been poor and should be improved through fuller utilization and collection of betterment levy, flood cess, higher water rates, etc. (c) Large works require continuous maintenance. While, it is necessary to expand irrigation facilities by starting new works, it is equally important to look after the existing works in order to maximize benefits. Damodar Valley Project This project has been designed to bring under control the water of river Damodar which flows through West Bengal and Jharkhand. This river is notorious for its propensity to cause floods. There are four storage dams located at Tilaiya, Konar, Maithon and Panchet Hills. Attached to all these dams, except at Konar, are hydel-power houses. There are three thermal power stations at Durgapur, Bokaro and Chandrapura. The Durgapur barrage is 700 metres long from where start canals and distributaries which irrigate 30 hectares of land. This canal joins the river Hooghly and can be used as waterway between Calcutta and the coal fields of Raniganj in Bihar. The Panchet Hills Dam was constructed for flood control. Rihand Valley Project This is the most important multi-purpose scheme in UP. The dam is across the Rihand river at Pipri in UP. A large network of canals will provide irrigation to the region between the Ganga and the Sone rivers. The projects is also meant to control floods in the Rihand and the Sone rivers. Thein Dam Project Thein Dam Project across the river Ravi in Gurdaspur district of Punjab has a power plant on its left bank. An area on 3.48 lakh hectares is getting irrigation benefits. Besides power and irrigation, its other benefits are flood control, and development of industries and fisheries. Sutlej–yamuna Link Canal This 110 km long canal is intended to draw water from Beas for irrigation in Haryana. The Haryana sector of the canal has been constructed. In 1978, Punjab government (Akali Dal) had agreed to construct the Punjab sector of the canal and also accepted Rs 100 crore from Haryana as its share of building the canal. The construction of the Punjab sector is to be taken up in pursuance of the Punjab Accord and the Award of the Eradi Tribunal. The Bodhghat Project It is aimed mainly at harnessing hydel power for the industrial needs of Bastar in Madhya Pradesh. According to present estimates, the Bodhghat dam would involve

submerging of some 5,740 hectares of land, covered with prime forest—mostly sal. Beas Project The Beas Project, another joint venture of Punjab, Haryana and Rajasthan, consists of: (i) The Beas–Satluj Link, (ii) Beas Dam at Pong, and (iii) Beas Transmission System. The Beas–Satluj Link project is principally a power project. The civil works and conductor system (including Pandoh diversion dam) have been completed. All the four units have now been commissioned and work on the extension units has been taken on hand. The Beas Dam at Pong is principally an irrigation project. Its water is for regulated flow for irrigation in Rajasthan and some areas in Punjab. Rajasthan Canal Project The Rajasthan Canal project (now called Indira Gandhi Canal) will provide irrigation facilities to the northwestern region of Rajasthan which is part of the Thar desert. The project will use waters from Pong Dam. The 445 km long Rajasthan canal is entirely in Rajasthan. The project will ultimately irrigate nearly 12.54 1akh hectares. Due to ‘hard pan’ before the surface of canal, problem of water logging has arisen affecting the land along some parts of canal. Dudh Sagar Waterfalls The waterfall is situated near Kastle Rock between Londha and Marmugao Port. The area is rich in flora and fauna. The source of the water is in Karnataka. A controversy has arisen over the move of Karnataka to divert the water for irrigation purposes. Any such diversion will adversely affect the waterfall and in particular the Mahaveer wildlife Sanctuary. Tehri Dam Several eminent geo-scientists in the country see the Tehri dam in UP as a sure invitation to disaster. The rim slopes of the Tehri reservoir are weak, fractured and are capable of absorbing water. This difficulty is likely to increase the danger to settlements above the dam. The Tehri dam is in a highly seismic zone and the dam would induce rock failures and the probability of the rupture zone of a major earthquake traversing the dam site in the area. Even if it were perfectly safe it would be seen by many as one more way of converting poverty ridden Garhwal into a hinterland to be exploited by the people in the plains. The electricity from the dam is to be diverted largely to Western UP Environmentally bengin micro-hydel projects which could have met local needs have either not been started, or in rare cases where they have been they are not in working order. Russia had extended its hand in the construction of this dam. It is said that Russian experts want some modifications in the design of the dam.

MINERAL RESOURCES OF INDIA India’s mineral resources are mainly concentrated in three zones, viz., 1. North-Eastern Zone Highest degree of concentration of mineral deposits found in this zone. Coal, iron, manganese, copper, bauxite (aluminium), chromium, beryllium, uranium and non-metallic minerals like mica and diamond are found in this area.

2. North-Western Zone Most of the ancient mines of open lead and zinc are found here. Sources of crude petroleum and natural gas are also present here. 3. Southern Zone Rich in iron ore, especially in magnetite, and the only zone in the country that produces gold. Jharkhand is the state with highest mineral output.

Geography

▌ 2.177

India is rich in iron, mica, manganese, bauxite; self-sufficient in antimony, building materials, cement materials, clay, chromite, lime, dolomite and gold, but deficient in copper, lead, mercury, zinc, tin, nickel, petroleum products, rock phosphate, sulphur and tungsten. Mineral resources like potassium are totally absent and have to be imported. Minerals like crude petroleum (which accounts for about 80% of the total value of Indian imports) diamonds (uncut), sulphur and rock phosphorus are imported.

Rajasthan. Quality is very poor with metallic content of only 1.6 crore tonnes of zinc and 0.5 crore tonnes of lead.

Other Minerals

Limestone Produced in Madhya Pradesh, Chhattisgarh, Tamil Nadu, Bihar and Karnataka. Approximate reserves: 7,320 crore tonnes.

Gypsum (Plaster of Paris) India has approximately 124.86 crore tonnes of Gypsum deposits, found in Rajasthan (107.08 crore tonnes), Jammu and Kashmir (14.93 crore tonnes) and Tamil Nadu. Lead–Zinc India has approximately 35.85 crore tonnes deposits of Lead–Zinc. It is mainly found in Andhra Pradesh, Gujarat, Maharashtra, Meghalaya, Tamil Nadu, Orissa, Sikkim and

Chromite Main deposits are fond in Orissa, Bihar, Karnataka, Tamil Nadu, Maharashtra, Andhra Pradesh and Manipur. Approximate reserves: 1.53 crore tonnes. Dolomite Mostly found in Madhya Pradesh, Orissa, Gujarat, Maharashtra, UP, West Bengal and Arunachal Pradesh. Estimated reserves: 395 crore tonnes.

Diamond Mainly found in the Panna diamond belt of Madhya Pradesh. The estimated reserves of this belt are around 10 lakh carats. Traces are also found in Bihar, Orissa, Maharashtra and Andhra Pradesh. Salt Maximum salt in India is produced in Gujarat, which is about 60 per cent of country’s total salt production.

Mineral

Approx. reserve

Regions found in

Features

Iron ore

Haematite: 10,052 million tonnes Magnetite: 3,408 million tonnes

Orissa (Sonai, Mayubhanj,Keonjhar) Jharkhand and Bihar (Singhbhum, Hazaribagh, Palamau, Shahbad), Madhya Pradesh, and Chhattisgarh (Raipur, Durg, Bastar, Raigarh, Bilaspur, Jabalpur, Balaghat), Andhra Pradesh (Krishna, Kurnool, Chittor, Cuddapha, Warangal, Guntur) Tamil Nadu (Salem, Tiruchirapalli) Karnataka (Bellary, Chitaldurg, Chikmagalur) Maharashtra (Ratnagiri, Chanda) Goa

• India has the world’s largest reserves approximately one-fourth of world’s known reserves. India ranks fourth in iron ore production. • Jharkhand and Orissa together account for about 75 % of iron output in India

Coal

210,000 million tonnes (total geological reserves)

Bihar–Jharkhand–Bengal Belt (Raniganj, Jharia, Giridih • About one-fourth of India’s coal reserves Bokaro, Karanpur) Madhya Pradesh and Chhattisgarh belt lie in the Damodar Valley across Bihar, (Singrauli, Korba, Raigarh, Sonhat, Sohagpur, Umaria) Orissa Jharkhand and West Bengal. (Desgarh, Talcher) Maharashtra (Chanda) Andhra Pradesh • India is the third largest coal producing (Singreni) Assam, (Makum, Lakhimpur). In small quantities in country in the world Arunachal Pradesh, Meghalaya, J & K and Nagaland

Manganese

167 million tonnes

Orissa (Keonjhar, Kalahandi, Mayurbhanj, Talcher) Madhya Pradesh (Balaghat, Seoni, Chhindwara, Jabalpur) Maharashtra (Nagpur, Bhandara, Ratnagiri) Gujarat (Panchmahal) Karnataka (Chitaldurg, Tumkur, Shimoga, Chikmagalur, Belgaum, North Canara, Dharwar) Jharkhand (Singhbhum) Andhra Pradesh (Vizag) Rajasthan (Udaipur, Bansawara)

• India ranks sixth in the world in manganese production.

59,890 tonnes

Bihar mica-belt (Gaya, Monghyr) Jharkhand (Hazaribagh) Rajasthan mica-belt (Ajmer, Shahpur, Tonk, Bhilwara, Jaipur) Andhra Pradesh (Nellore)

• India has the largest deposits of mica in the world. • India alone contributes about two-thirds of the world’s production.

Bauxite 2,462 million (Aluminium tonnes ore)

Jharkhand (Palamau) Gujarat (Kaira) Madhya Pradesh (Katni, Jabalpur,Balaghat, Bilaspur, Bastar, Tamil Nadu (Salem) Karnataka, J&K (Kotli) (Chitaldurg and Belgaum) Maharashtra (Kolhapur)

• India ranks sixth in Bauxite production and tenth in Aluminium production.

Copper

461 million tonnes

Jharkhand (Singhbhum, Hazaribagh) Rajasthan (Khetri, • Very meagre reserves. Alwar, Bhilwara, Jhunjhunu, Sirohi) Andhra Pradesh • Almost all copper comes from (Guntur, Khamman, Agnigundala) Karnataka (Chitaldurg, Singhbhum and Hazaribagh in Hassan, Chikmagalur, Raichur) Madhya Pradesh (Balaghat) Jharkhand and Khetri in Rajasthan. Gujarat (Banaskantha) Some quantities also found in Sikkim, Punjab, U.P. and Tamil Nadu

Crude oil

32.2 million tonnes

Assam, Tripura, Manipur, West Bengal, Ganga Valley, Himachal Pradesh, Kutch, Orissa, Andhra Pradesh, Maharashtra and Gujarat

Mica

• Orissa is the leading producer of manganese in the country.

▌ General Knowledge

2.178

Mineral

Approx. reserve

Regions found in

Features

Lignite

36,000 million tonnes

• About 87% of lignite reserves are found

Gold

68 million tonnes 245 million tonnes

Tamil Nadu (Neyveli fields) Some deposits also found in Gujarat, Puducherry, Rajasthan (Palana fields), J&K (Riasi fields) Karnataka (Kolar Gold fields, Hutti mines). In small quantities in Andhra Pradesh (Ramgiri Gold Fields), and Anantpur Tamil Nadu (Salem) Uttarakhand (Almora, Chamoli, Pithoragarh) Karnataka (Mysore, Hassan)

CHINA

KI

PA

Jarji

Udaipur Panch Mahal Marukot

BANGLADESH

Talcher

BAY OF BENGAL

Kantapalli ARABIAN SEA

Parrangiri Umralong

Himgir

Kamptee

BHUTAN

PA L

Raniganj

Bokaro Jharia

Singareni

CHINA

ARABIAN SEA

Goa Uttar Kannad

A

SRI L

ANK SRI L

R

PA L

BHUTAN

Lakhimpur Dibrugarh Digbol Naharkatia

Dholka

Mumbai

SedimentaryZone ContinentalShelf Oilfield

A ANK

A ANK SRIL

BAY OF BENGAL

High

EP WE AD KSH A) LA (INDI

R OBA NIC N & DS MA DA ISLAN ) ia (ind

Major Manganese Deposits of India

AN

Salem Tiruchirappalli

ARABIAN SEA

BAY OF BENGAL

BANGLADESH

Mineral Oil Occurrences in India

MYANMAR

MYANMAR

r ha

arh

darg

EP WE AD KSH A) LA (INDI

Ratnagiri Guntur ARABIAN Dhar Kurnool war SEA Goa Bellary Uttar Kannad Kundremukh Chitradurga

Mehesana Mala Ahmedabad Ankleshwar

on

Ke

Sun

Palamau Hazar BANGLARanchi DESH Singhbhum Raipur Mayurbhanj Chandrapur Dur g

SRI L

Jaipur

Kozhikode

NE

BHUTAN

NE PA L

dragarh

Mohen

OBA

Garhwal

IC &N S AN AM LAND IS dia) (in

N TA

CHINA

AN ST

KI PA

CHINA

IS

BAY OF BENGAL

Major Iron-ore Deposits of India

Figure: Location of Coalfields in India

K PA

Vishakhapatnam

Dharwad Bellary Sandur Chitradurga Shimoga

AND

EEP DW SHA A) LAK (INDI

EEP DW SHA A) LAK (INDI

Coal Neyveli

Jhabua Singhbhum Chhindwara Gangapur Chhoat Balaghat Udepur Keonjhar Nagpur Sundergarh Bonai Bhandara BANGLADESH UttarKoraput

MYANMAR

Singarauli PenchValley

N TA

IS

K PA

NE

MYANMAR

Giridih

NamchikNamphuk Makum Najira

A

AN ST

ANK

Magnesite

in Tamil Nadu.

Geography

▌ 2.179

POPULATION India’s population constitutes nearly 16.7% of the total world population in 2.4% of the world’s geographical area. It is the second country in the world, next to China, in terms of population. With this, India became only second country in the world, after China, to cross the one billion mark. India has, as per the 2001 census, 1027 million persons.

First Census The first all-India census was conducted in 1872. However, further censuses were not held on a regular basis. Regular Census The first regular all-India census was conducted in 1881 and thereafter every ten years. The last census was conducted in 2011, the second census of 21st Century and the fourth millennium.

Table: Demographic Features of India (Census 2011*) Total population

1210 million (approx.)

Males

623.7 million

Females

586.4 million

Rural population

833 million (68.84 %)

Urban population

377 million (31.16 %)

Decennial population growth (2001–2011) (a) Absolute

181,455,986

(b) Percentage

17.64 %

Average density of population

382 person/km2

Sex ratio

940 females/1000 males

Birth rate

To be released

Death rate

To be released

Literacy rate Total

778,454,120

(74.04 % of total population)

Males

444,203,762

(82.14 % of total male population)

Females

334,250,358

(65.46 % of total female population)

Decadal Growth Total

(+) 17.64 %

Males

(+) 17.19 %

Females

(+) 18.12 %

Population (0–6) years Persons

158,789,287

(13.12 % of total population)

Males

82,952,135

(13.13 % of total population)

Females

75,837,152

(12.93 % of total population)

Most populous and Least Populous Regions Most populous state

Uttar Pradesh

Most populous Union Territory

Delhi

Least populous state

Sikkim

Least populous Union Territory

Lakshadweep

* As per provisional population totals: Census 2011

199,581,477

(16.49 % of total)

16,753,235

(1.38% of total)

6,07,688

(0.05 % of total)

64,429

(0.01 % of total)

2.180

▌ General Knowledge

Table: Distribution of Population, Sex Ratio, Density and Decadal Growth Rate of Population: 2011 Females

Sex ratio (Females per 1,000 males)

Area ('000 km2)

Density (per km2)

Decadal growth rate

Rank

5

6

7

8

9

10

623,724,248

586,469,174

940

382

17.64

84,665,533

42,509,881

42,155,652

992

2751

308

11.10

5th

1,382,611

720,232

662,370

920

83.7

17

25.92

26th

3. Assam

31,169,272

15,954,927

15,214,345

954

78.4

397

16.93

14th

4. Bihar

103,804,637

54,185,347

49,619,290

916

94.1

1.102

25.07

3rd

25,540,196

12,827,915

12,712,281

991

135.2

189

22.59

16th

State/ UT code

India/State/ Union territory*

1

2

INDIA

Persons

Total population Males

3

4

1,210,193,422

Indian States 1. Andhra Pradesh 2. Arunahal Pradesh

5. Chhattisgarh 6. Goa

1,457,723

740,711

717,012

968

3.7

394

8.17

25th

7. Gujarat

60,383,628

31,482,282

28,901,346

918

196.0

308

19.17

10th

8. Haryana

25,353,081

13,505,130

11,847,951

877

44.2

573

19.90

17th

9. Himachal Pradesh

6,856,509

3,473,892

3,382,617

974

55.7

123

12.81

20th

10. Jammu and Kashmir

12,548,926

6,665,561

5,883,365

883

222.2

124

23.71

18th

11. Jharkhand

32,966,238

16,931,688

16,034,550

947

79.8

414

22.34

13th

12. Karnataka

61,130,704

31,057,742

30,072,962

968

191.8

319

15.67

9th

13. Kerala

33,387,619

16,021,290

17,366,387

1,084

38.9

859

4.86

12th

14. Madhya Pradesh

72,597,565

37,612,920

34,984,645

930

308.3

236

20.30

6th

112,372,972

58,361,397

54,011,575

925

307.7

365

15.99

2nd

16. Manipur

2,721,756

1,369,764

1,351,992

987

22.3

122

18.65

23rd

17. Meghalaya

2,964,007

1,492,668

1,471,339

986

22.4

132

27.82

22nd

18. Mizoram

1,091,014

552,339

538,675

975

21.1

52

22.78

27th

15. Maharashtra

19. Nagaland

1,980,602

1,025,707

954,895

9.31

16.6

119

-0.47

24th

20. Orissa

41,947,358

21,201,678

20,745,680

978

155.7

269

13.73

11th

21. Punjab

27,704,236

14,634,819

13,069417

893

50.4

550

13.73

15th

22. Rajasthan

68,621,012

35,620,086

33,000,926

926

342.2

201

21.44

8th 28th

23. Sikkim

607,688

321,661

286,027

889

71

86

12.36

72,138,958

36,158,871

35,980,087

995

131.1

555

15.60

7th

3,671,032

1,871,867

1,799,165

961

10.5

350

14.75

21st

26. Uttar Pradesh

199,581,477

104,596,415

94,985,062

908

240.8

828

20.09

1st

27. Uttarakhand

10,116,752

5,154,178

4,962,574

963

53.6

189

19.17

19th

28. West Bengal

91,347,736

46,927,389

44,420,347

947

88.8

1.029

13.93

4th

379,944

202,330

177,614

878

9.3

46

6.68

7th

24. Tamil Nadu 25. Tripura

Indian Union Territories 1. Andaman and Nicobar Islands 2. Chandigarh

1,054,686

580,282

474,404

818

0.1

9.252

17.10

1st

3. Dadra and Nagar Haveli

342,853

193,178

149,675

775

0.5

698

55.50

4th

4. Daman and Diu

242,911

150,100

92,811

618

0.1

2.169

53.54

3rd

5. Lakshadweep

64,429

33,106

31,323

946

0.03

2.013

6.23

5th

6. NCT of Delhi

16,753,235

8,976,410

7,776,825

866

1.5

11.297

20.96

2nd

1,244,464

610,485

633,979

1,038

0.5

2.598

27.72

6th

7. Puducherry

*As per provisional population totals: Census 2011

Geography ▌ 2.181

Figure: Density of Population (2011) (States UTs) per km2.

2.182

▌ General Knowledge

PRACTICE EXERCISE 1. Egypt is situated on the river (a) Moskvo (b) Shatt-al-Arab (c) Ural (d) Nile

13. The Prime Meridian passes thro-ugh (a) USA (b) Great Britain (c) Switzerland (d) Finland

2. Which of the following towns are situated on the river Nile? (a) Alexandria and Baghdad (b) Cairo and Baghdad (c) Alexandria and Cairo (d) Baghdad and Budapest

14. In the Northern Hemisphere the longest day is on (a) 21st June (b) 21st March (c) 22nd December (d) 21st September

3. The Temperate Grasslands of South America are called (a) Pampas (b) Steppes (c) Savanna (d) Evergreen grasslands

15. Diego Garcia is an (a) important site on the moon (b) island in Indian Ocean with a large military base (c) island in the Caribbean Sea (d) island in the Arabian Sea

4. Indicate the wrong combination. (a) Angola : Kwanza (b) Cuba : Peso (c) Laos : Kip (d) Iran : Peso

16. Grenada is situated in (a) Pacific Ocean (c) Caribbean Sea

5. Mark the correct combination. (a) Japan : Peso (b) Bulgaria : Kip (c) Guinea : Leu (d) China : Yuan

17. Which types of rocks are mainly found in the Himalayan ranges? (a) Sedimentary (b) Metamorphic (c) Igneous (d) Granite

6. The Netherlands is the new name for (a) Abyssinia (b) Christina (c) Holland (d) New Guinea 7. Which of the following statements is true in respect to the Savanna Regions? (a) Tropical grasslands in both hemispheres (b) Characterized by evergreen forests and winter rams (c) Characterized by dense forests with tall trees existing in the southern hemisphere (d) Characterized by low temperature and heavy rains in winter 8. Temperate grasslands of Asia and Europe are called. (a) Pampas (b) Savanna (c) Evergreen grasslands (d) Steppes 9. Which of the following atmospheric layers helps in radio communication? (a) Exosphere (b) Ionosphere (c) Troposphere (d) Stratosphere 10. Which of the following phenomena is responsible for the formation of the highest mountains? (a) Denudation (b) Fault formation (c) Volcanic activity (d) Folding 11. Which of the following is a periodic wind? (a) Trade Winds (b) Anti-trade winds (c) Monsoons (d) Westerlies 12. Sedimentary rocks are (a) porous (c) rough

(b) hard (d) brittle

(b) Indian Ocean (d) Mediterranean

18. In which of the following rocks are slate, diamond and marble found? (a) Igneous (b) Sedimentary (c) Metamorphic (d) None of these 19. Lines on a map connecting regions of equal geochemical characteristics are called (a) isobars (b) isohytes (c) isocons (d) latitudes 20. Oasis is associated with (a) glaciers (c) islands

(b) desert (d) heiff

21. Which of these crops require hot climate and rainfall above 200 cm? (a) Wheat (b) Maize (c) Cotton (d) Rubber 22. Riyal is the currency of (a) Romania (c) Japan

(b) Saudi Arabia (d) Libya

23. Which of the following is the currency of Romania? (a) Lira (b) Riyal (c) Peso (d) Leu 24. Loti is the currency of (a) Burundi (c) Sudan

(b) Libya (d) Lesotho

25. The term ‘Third World’ refers to (a) developed countries (b) underdeveloped and developing countries (c) oil rich countries (d) Asian countries

Geography ▌ 2.183 26. ‘Middle East’ group of countries includes (a) south Asian countries (b) west Asian countries (c) south-east Asian countries (d) eastern part of Asia 27. Which of the following countries does not fall under the Far East group of countries? (a) China (b) Taiwan (c) South Korea (d) Myanmar 28. Choose the correct combination. (a) Latin America: Panama (b) Latin America: Argentina (c) Middle East: China (d) Far East: Thailand 29. The Third World comprises approximately _____ % of world’s human race? (a) 50 (b) 70 (c) 45 (d) 30 30. Which of the following statements is not correct with respect to the Third World countries? (a) Comprise approximately 60% of earth’s surface (b) Include nearly 100 countries of the world (c) Lead in industrial production and exports (d) Agriculture is the main occupation 31. Which of the following countries is not grouped with the Caribbeans? (a) Cuba (b) Bermuda (c) Jordan (d) Anguilla 32. Which of the following soils is best suited for wheat cultivation? (a) Black soil (b) Red soil (c) Clayey (d) Deep rich 33. Which of the following is not a fibre crop? (a) Rubber (b) Cotton (c) Jute (d) Flax 34. Choose the wrong combination. (a) Food crops: Rice, wheat (b) Cash Crops: Sugarcane, oilseeds (c) Fibre crops: Cotton, maize (d) Beverage crops: Tea, cocoa 35. The four countries with the largest population in order of size are (a) China. India, Indonesia and Japan (b) China, India, Russia and USA (c) China, India, Russia and Japan (d) China, Japan, Indonesia, and USA 36. Who was the first to classify the Seven Wonders of the Ancient World? (a) Aristotle (b) Plato (c) Antipater (d) Socrates

37. Why was number seven chosen for a classification of the world’s great wonders? (a) Because it was easy to remember (b) Because there were only seven wonders (c) Because seven had a metaphysical attraction for the ancient Greeks (d) Seven was considered a lucky number by Greeks 38. Which among the Seven Wonders listed by Antipater has survived? (a) The Colossus of Rhodes (b) The Pyramids of Egypt (c) The Pharos of Alexandria (d) The Mausoleum at Halicar-nassus 39. The Pyramids of Egypt, one of the Seven Wonders of the ancient world is said to have been built more _____ than ago. (a) 2000 years (b) 3000 years (c) 4000 years (d) 5000 years 40. Which among the following is/are not listed among the Seven Wonders of the World by Antipater? (a) The Mausoleum at Halicar-nassus (b) The Pharos of Alexandria (c) The Great Wall of China (d) The Colossus of Rhodes 41. What is the approximate length of the Great Wall of China? (a) 2000 km (b) 2500 km (c) 3000 km (d) 3460 km 42. The Eiffel Tower is located in (a) Germany (b) France (c) Italy (d) Paris 43. The Leaning Tower is located in (a) Germany (b) Italy (c) France (d) Paris 44. Which is the world’s highest dam? (a) New Cornelia Tailings (b) Rohunsky (c) Aswan Dam (d) Grande Dixence 45. The world’s tallest free-standing structure is (a) Eiffel Tower (b) Leaning Tower of Pisa (c) CN Tower (d) Burj Khalifa 46. Which among the following is the tallest building in the world? (a) Imperial Palace (b) Sears Tower (c) Eiffel Tower (d) Burj Khalifa 47. Largest hotel in the world is (a) Hotel Broadway (b) The Las Vegas Hilton (c) Hotel Peachtree Centre Plaza (d) Central Plaza 48. Where is the largest railway platform in the world? (a) Moscow (b) New York (c) India (d) Canada

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49. Where is the world’s largest railway station? (a) Japan (b) England (c) USA (d) Russia

61. Kangaroo is the national emblem of (a) Canada (b) Australia (c) Italy (d) Ireland

50. Which among the following is the largest mosque? (a) Jama Masjid (b) Merdeka Mosque (c) At-Malawiya Mosque (d) Taj-ul-Masjid

62. India’s national emblem is (a) Lotus flower (c) Lion (e) None of these

51. The largest church in the world is the (a) Diocese of New York (b) Basilica of St Peter (c) Bethlehem (d) Aachen Cathedral 52. Where is the world’s largest temple? (a) India (b) Nepal (c) Sri Lanka (d) Cambodia 53. The largest temple in the world is (a) Meenakshi Temple (b) Angkor Vat (c) Temple of Varanasi (d) Temple in Lumbini 54. The world’s largest airport is (a) King Abdul Aziz International airport at Jeddah (b) International Airport at New York (c) International Airport at United States (d) Capetown Airport 55. Which among the following is the world’s longest road? (a) G. T. Road (b) Broadway Street (c) Panam Highway (d) Monumental Axis 56. Which one among the following pairs of states is the leading producer of tobacco in India? (a) Andhra Pradesh and Gujarat (b) Karnataka and Madhya Pradesh (c) Maharashtra and Chhattisgarh (d) Uttar Pradesh and Rajasthan 57. Kurnool-Cuddapah canal is taken off from which one of the following rivers? (a) Cauvery (b) Penneru (c) Palar (d) Tungabhadra 58. Niagara Falls is located between which of the following pairs of lakes? (a) Lake Superior and Lake Huron (b) Lake Huron and Lake Erie (c) Lake Erie and Lake Ontario (d) Lake Ontario and Lake Huron 59. San Andreas fault passes through which of the following? (a) Beaufort Sea and Rocky Mountains (b) Pacific Ocean north of San Francisco and Gulf of California (c) Bering Strait and Alaska Range (d) Mackenzie Mountains and Western Cordillera 60. The New York version of Statue of Liberty was engineered by (a) A. G. Eiffel (b) American Sculptor Henry Smith (c) Sir Edwin Lutyens (d) Le Corbusier

(b) Lioned Capitol (d) Cow

63. ‘Hammer and Sickle’ was the national emblem of (a) United States (b) France (c) Former USSR (d) Japan 64. Water Lily is the national emblem of (a) Denmark (b) Bangladesh (c) Italy (d) Luxembourg 65. ‘Shora’ is the name of the Parliament of (a) Algeria (b) Iran (c) Russia (d) Afghanistan 66. ‘Riksdag’ is the name of the Parliament of (a) Iran (b) Iraq (c) Sweden (d) United States 67. Parliament of Japan is called (a) Dail (b) Sejim (c) Cortes (d) Diet 68. Match the following: Column I (1) Shora (2) Yuan (3) Majlis (4) Sejm (5) Diet

Column II A. Japan B. Poland C. Maldives D. China National E. Afghanistan

Codes: (a) 1-A, 2-B, 3-D, 4-C, 5-E (b) 1-D, 2-B, 3-C, 4-A, 5-E (c) 1-B, 2-C, 3-D, 4-E, 5-A (d) 1-C, 2-D, 3-B, 4-A, 5-E 69. Which country’s Parliament has the largest membership? (a) India (b) China (c) Japan (d) United States 70. Which one of the following has the volcanic Mount St Helens? (a) Aleutian Range (b) Alaska Range (c) Cascade Range (d) Mohave Desert 71. Which medieval geographer made extensive corrections on Ptolemy’s works? (a) da Vinci (b) Al-Masudi (c) Al-Idrisi (d) Al-Biruni 72. Who among the following geographers adopted the ‘deductive method’ of study in geography? (a) Varenius (b) Cluverius (c) Kant (d) Humboldt

Geography ▌ 2.185 73. Where is the Mt Waialeale, one of the heaviest rainfed places in the world, located (11990 mm in a year)? (a) Brazil (b) Ireland (c) Philippines (d) Hawaii 74. Which one of the following countries has Red River delta? (a) Japan (b) Indonesia (c) Malaysia (d) Vietnam 75. The Hkakabo Razi (5885m), the highest point in mainland south-east Asia, is at the border between which of the following? (a) China and Myanmar (b) China and Vietnam (c) Cambodia and Laos (d) Cambodia and Thailand 76. The Minamata incident, an example for environmental degradation, is due to which one of the following? (a) Air pollution (b) Nuclear disaster (c) Water pollution (d) Thermal pollution 77. Among the following continents, which one has the lowest growth rate of population? (a) North America (b) South America (c) Australia (d) Europe 78. Match the following: Column I (A) Lufthansa (B) JAL (C) Sabena (D) SAS

Column II 1. Scandinavia 2. Belgium 3. Japan 4. Germany

Codes: (a) A-1, B-3, C-2, D-4 (b) A-4, B-3, C-2, D-1 (c) A-3, B-2, C-1, D-4 (d) A-1, B-4, C-2, D-3 79. In which geological period did the shallow sea reptile Ichthyosaurus first appear? (a) Silurian (b) Devonian (c) Triassic (d) Ordovician 80. Jute requires hot, moist climate with adequate amount of rain and rich soil. Which country in the world is leading in jute production? (a) Brazil (b) India (c) Thailand (d) Bangladesh 81. Cocoa is a tropical plant and requires heat and moisture. Which of the following countries is leading in cocoa production in the world? (a) Brazil (b) Nigeria (c) Ghana (d) Ivory Coast 82. Coffee requires hot and moist climate and grows best on slopes. Which of the following countries is leading in coffee production? (a) Indonesia (b) India (c) Brazil (d) Columbia

83. More or less like coffee, tea requires warm, moist, tropical climate and grows well on hill slopes. Which of the following countries is leading in tea production? (a) India (b) China (c) Sri Lanka (d) Indonesia 84. For which crop, black volcanic soil and tropical climate is most suitable without excess rainfall? (a) Bajra (b) Tobacco (c) Cotton (d) Jute 85. Higher summer temperature, stiff soil capable of supporting heavy weight of plants are essential for (a) rubber (b) jute (c) rice (d) maize 86. Which country is leading in the production of rubber? (a) India (b) Indonesia (c) Thailand (d) Malaysia 87. While Cuba is the ‘Sugar Bowl’, which country is leading in the production of sugarcane? (a) India (b) Pakistan (c) Indonesia (d) Mauritius 88. Which is the largest cotton producing country in the world? (a) India (b) Egypt (c) USA (d) Indonesia 89. Wheat being a plant of temperate zone, grows well on plains away from sea and having moderately dry climate. World’s maximum production of wheat is obtained from (a) Australia (b) USA (c) Canada (d) Russia 90. Which country leads in production of barley among the following? (a) China (b) India (c) USA (d) Canada 91. Groundnut crop is the monopoly of (a) Indonesia (b) Bangladesh (c) Pakistan (d) India 92. Which among the following countries tops in the production of wool? (a) New Zealand (b) Australia (c) Argentina (d) Canada 93. Coal is an essential commodity for industrial world. Which country tops in coal output? (a) Canada (b) United Kingdom (c) India (d) United States of America 94. Which of the following countries is known as Island of Cloves? (a) Zanzibar (Africa) (b) Kerala (India) (c) Kenya (Africa) (d) Nassau (Bahamas)

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▌ General Knowledge

95. Which country has extensive mechanized agriculture and which is the main crop produced in that country? (a) Australia : Wheat (b) Soviet Union : Maize (c) USA : Wheat (d) North America : Wheat 96. Tropical deserts, which are hot deserts of the world are situated near the tropics in the west of the landmasses.Approximately how much surface area of the Earth is covered by these deserts? (a) Less than one-fourth (b) Nearly one-fourth (c) 10% (d) 15% 97. Which continent leads in contribution to world’s rice production to the maximum and how much? (a) Australia (50%) (b) Africa (42%) (c) Asia (90%) (d) Asia (72%)

107. To which country, India is second in manufacturing carpets? (a) Bangladesh (b) Afghanistan (c) Malaya (d) Iran 108. While India tops in sugarcane production, which country ranks first in sugar export? (a) USA (b) China (c) Cuba (d) France 109. Which country is the largest producer of copper? (a) China (b) USSR (c) UK (d) USA 110. Which is the most important crop of the Nile delta? (a) Sugarcane (b) Maize (c) Cotton (d) Millets

98. Oil seeds are mostly tropical and sub-tropical products. Which country produces major oilseeds? (a) West African countries (b) Brazil (c) China (d) India

111. Which of the following countries is the largest supplier of meat? (a) USA (b) Pakistan (c) Iran (d) Argentina

99. Tobacco, a product of high commercial value these days and which grows under a wide range of climates, is a major plantation of (a) India (b) China (c) Brazil (d) USA

112. Which is the main industrial activity of Tanzania? (a) Cloves export (b) Mining (c) Oil based (d) Sugar/coconut based

100. Which two of the following products have great commercial value in Tropical Monsoon Region? (a) Wheat and Maize (b) Wheat and Rice (c) Rice and Tea (d) Tea and Wheat 101. For which agricultural produce, Sudan is famous and which comes under Tropical Grassland (Savannas) Region? (a) Coffee (b) Millets (c) Barley (d) Cotton 102. Which type of industry is mostly found in Bangladesh? (a) Fishing (b) Tea (c) Jute based (d) Silk 103. Israel/Jerusalem, whose main mineral resource is oil, has which type of main industrial activity? (a) Oil refining (b) Oil digging (c) Diamond cutting (d) Gas energy preservation 104. Myanmar, whose chief mineral resources comprise silver, lead, and tin, is engaged in chief industrial activity of (a) tinning black iron for food containers (b) tinning export (c) rice export (d) wheat crop 105. Besides tobacco, tea and cotton, which is the other agricultural produce of China? (a) Maize (b) Rice (c) Soyabean (d) Millets 106. South Africa is the largest producer of (a) silver (b) teak wood (c) gold (d) lead

113. World’s biggest producer of mica is (a) Zaire (b) India (c) Sweden (d) Indonesia 114. Which country is the biggest producer of mercury in the world? (a) Mexico (b) Canada (c) Italy (d) Germany 115. Uranium is mostly produced in (a) USA (b) Russia (c) Zaire (d) France 116. Name the world famous and finest harbour. (a) New York (b) Sydney (c) Mumbai (d) Kolkata 117. Baku is famous as (a) capital of Georgia (b) oil centre (c) pearl market of the world (d) mining industry 118. Which of the following statements is incorrect? (a) Amsterdam is legislative capital of Netherlands. (b) Amsterdam is famous for diamond-cutting. (c) Amsterdam is the skyscraper of Netherlands. (d) World’s chief diamond market. 119. Which of the following statements is incorrect of Bahrain? (a) An independent country in Persian Gulf. (b) It is rich in oil. (c) Pearl industry is an important commercial activity. (d) A port of Iraq.

Geography 120. Which of the following statements is correct concerning Bridgetown? (a) Chief port of Guinea–Bissau (b) Capital of Barbados, an island country in Atlantic Ocean (c) A famous seaport of Barbados (d) Largest port in Atlantic Ocean 121. African name of Rhodesia which is dominated by white minority is (a) Zomba (b) Zimbabwe (c) Zaire (d) Zanzibar City 122. Which country/currency/capital combination is wrong? (a) Iran/Rial/Teheran (b) Bulgaria/Lev/Sofia (c) Korea/Won/Seoul (d) Mongolia/Kip/Ulan Bator 123. Which one of the following country/capital/currency combinations is correct? (a) Botswana/Usumbura/Franc (b) Bangladesh/Dhaka/Taka (c) Jordan/Jerusalem/Dinar (d) Guinea/Conakry/Drachma 124. Which of the following is an agricul tural produce of Taiwan? (a) Coffee (b) Maize (c) Rice (d) Millets 125. Coal is an essential energy resource for industries. However, some countries have been able to achieve industrialization without depending on coal as a source of energy. For example (a) Poland and Switzerland (b) Switzerland and Holland (c) Sweden and Italy (d) Holland and Poland 126. Switzerland is famous for Swiss watches. What is the Capital of this country? (a) Geneva (b) Vevey (c) Lausanne (d) Berne 127. With which of the following items is Kimberley associated? (a) Diamonds (b) Silver (c) Perfumes (d) Cereals 128. Which of the following countries/capital combinations is not correct? (a) Libya/Tripoli (b) Malawi/Zomba (c) Egypt/Cairo (d) Chad/Bangui 129. Which of the following is the capital and seaport of Philippines? (a) Manila (b) Davao (c) Zamboanga (d) Quezon City 130. Match the following: (A) Myanmar (B) Iraq

1. Franc 2. Dinar

(C) Japan (D) Switzerland Codes: A (a) 3 (b) 1 (c) 4 (d) 4

B 1 2 2 1

C 2 3 3 3

▌ 2.187

3. Yen 4. Kyat D 4 4 1 2

131. Physical geographic features include (a) climate (b) land (c) water (d) All of these 132. Political geography (a) is an application of political science (b) deals with human social activities (c) deals with human social activities that are related to the locations and boundaries of cities, nations and groups of nations. (d) All of these 133. Red Indians are (a) the original inhabitants of North America (b) natives of Nilgiri Hills (c) hill tribe of Assam (d) martial race of South Africa 134. Regional geography concerns with (a) the differences and similarities among the various regions of the earth (b) the differences and similarities among the various layers of the earth (c) various geographical differences on the surface of earth (d) All of these 135. Seafloor spreading is the process (a) by which new oceanic crust is created (b) by which ocean basins are enlarged by the expulsion of magmatic material through cracks in ocean ridge system (c) both (a) and (b) (d) of spreading of oceanic plates 136. Sedimentary rocks are formed by (a) burial, compression and chemi cal modification of deposited weathered rock debris (b) solidification of molten magma from the mantle (c) chemical or physical modification of existing rock (d) Both (a) and (b) 137. Solifluction is (a) the slow movement of soil caused by freeze-thaw action (b) widespread in polar and sub-polar regions (c) occurs when seasonal or daily fluctuations of temperature are above freezing (d) All are correct

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▌ General Knowledge

138. Systematic geography is concerned with (a) physical elements of earth (b) cultural geography (c) physical geography and the cultural elements of earth (d) physical and cultural features that characterize each region 139. The abandoned channels created when meanders are cut off from the rest of the channel because of lateral stream erosion is called (a) ox-bow lake (b) meandering lake (c) braided river (d) riffles 140. The analysis of geographic information includes (a) quantitative analysis such as mathematical or statistical method (b) objective analysis of a large amount of data and variables (c) collection of data and to form a theory to explain the observation (d) All of these 141. Who among the ancient Greek scholars coined the word “Apeiron” to symoblise the prime substance which could not be experienced through senses? (a) Thales (b) Anaximander (c) Hecateus (d) Eratosthenes 142. An eastward flowing sub-surface counter current of the Pacific Ocean is known as (a) Aborhas Current (b) Cromwell Current (c) Leeuwin Current (d) Equatorial Current 143. Who asserted the following. “Space is not a thing or event. It is a kind of framework of things and events; something like a system of pigeon holes, or a filling system, for observation.” (a) Immanuel Kant (b) Richard Hartshorne (c) B. Varenius (d) A. Hettner 144. Who coined the word ‘Kultur landshaft’, landscape created by human culture? (a) Carl O. Sauer (b) Oto Shulutter (c) Immanuel Kant (d) Carl Ritter 145. Any type of rainfall has the following aspect/s (a) ascent and descent of air (b) horizontal movement of air (c) ascent of air leading to cooling and condensation (d) Only (a) and (b) 146. Arctic circle is located at (a) 23° 27′ north of the equator (b) 23° 27′ south of the equator (c) 66° 30′ north of the equator (d) 66° 30′ south of the equator 147. Antarctic circle is the parallel of (a) Longitude (b) Latitude (c) Tropic of Cancer (d) Both (a) and (b)

148. Arrange the following planets according to their increasing distance from the sun. (i) Venus (ii) Mars (iii) Earth (iv) Mercury (a) iii, ii, iv, i (b) i, iii, iv, ii (c) iv, i, iii, ii (d) iv, i, ii, iii 149. Atmosphere is broadly divided into homosphere and heterosphere. The homosphere extends up to a height of (a) 18 km (b) 90 km (c) 8 km (d) 300 km 150. Atmosphere is (a) a gaseous envelope that surrounds a celestial body (b) air content of environment (c) air, gases, suspended particles and water vapours (d) None of these 151. Atmosphere pressure refers to (a) the amount of force exerted on a unit surface area (b) the weight of the entire air column over a given point (c) it is measured by using ebullioscope (d) Only (a) and (b) 152. Carbon dioxide makes up only a _____ ppm (parts per million) of the atmosphere, by volume. (a) few 1,000 (b) few 100 (c) few 10,000 (d) few 10 153. Choose the correct capital and currency of Libya. (a) Manila : Pesco (b) Lagos : Nyere (c) Tropoli : Dinar (d) Tripoli : Baht 154. Choose the correct descending order of cotton producing countries in the world. (a) USA, China, Pakistan, India (b) China, USA, Pakistan, India (c) India, USA, China, Pakistan (d) China, USA, India, Pakistan 155. Choose the correct pair. (a) Selvas: grasslands of Orinoco basin (b) Aganthas: Indian Ocean current (c) Sirocco: cold winds of the Alps (d) Oyashio: hot current of Green-land 156. Citrus fruits are normally grown in (a) Equatorial region (b) Cool temperature climate (c) Mediterranean region (d) Aloine region 157. Consider the following sentences: (1) Sub-polar Low Pressure Belt is dynamically produced by the rotation of the earth. (2) Sub-polar Low Pressure Belt is represented by individual oceanic cells. Icelandic low and Alleutian low in the Northern Hemisphere is January. (3) It is located in the mean between 40 and 50° latitude.

Geography ▌ 2.189 Select the correct answer from the codes given below: Codes: (a) only (1) is true (b) both (1) and (2) are true (c) only (2) is true (d) only (3) is true 158. Consider the following statements: (1) Mid-oceanic ridges are characterized by shallow-faci earthquakes. (2) The age of the rocks of the ocean floor are nowhere found to be older than Cretacious period. (3) Mid ocean ridges of the world are not connected Select the correct answer from the codes given below: Codes: (a) neither of the statements is true (b) only (1) is true (c) only (2) is true (d) both (a) and (b) are true 159. Consider the following statements: (1) In measuring the angular spacing of the objects, sextant does not require any stand or support. (2) Sextant can be used either in vertical or horizontal or oblique planes. Select the correct answer from the codes given below: Codes: (a) only (1) is true (b) only (2) is true (c) neither (1) nor (2) is true (d) both (1) and (2) are true 160. Consider the following statements: (1) Gnomovic Polar Zenithal Projection is mostly used in charts for navigation. (2) It is impossible to draw the map of one hemisphere on this projection. (3) It is conformal and presecuas areas. Select the correct answer from the codes given below: Codes: (a) only (1) is true (b) only (2) is true (c) neither of the two is true (d) both (1) and (2) are true 161. Consider the following statements regarding the International Date Line and select the correct ones. (1) It is along the longitude of 180° (2) It is the Greenwich Meridian, helping in fixing date and time (3) It solves the problem of time and date amongst the world’s nations (4) It is determined by 180° longitude, but not a straight line (a) 1 and 2 (b) 2 and 3 (c) 3 and 4 (d) 2 and 4

162. Decaying vegetation produces organic acids, which cause a breakdown of minerals. This is (a) physical weathering (b) chemical weathering (c) both (a) and (b) (d) chemical erosion 163. Declination is (a) the equivalent of longitude on the celestial sphere (b) the equivalent of latitude on the celestial sphere (c) the equivalent of prime meridian (d) a line of longitude 164. Down warping of the Rajmahal-Garo gap took place in (a) Paleocene (b) Eocene (c) Oligocene (d) Pleistocene 165. During a solar eclipse, which of the following represents the relative positions of the sun, the moon and the earth correctly? (a) the earth between the sun and the moon (b) the moon in between the sun and the earth (c) the sun, the moon and the earth are not in a straight line (d) the sun in between the earth and the moon 166. Each time zone is defined by its distance (a) east of Greenwich, England (b) west of Greenwich, England (c) east or west of Greenwich, England (d) south of Greenwich, England 167. Ecliptic circle is so named because (a) it is the apparent great-circle annual path of the sun (b) eclipses occur only when the moon is on or near this path (c) both of these are correct (d) the circle lies in the plane of the earth’s orbit 168. Equator is (a) an imaginary line of latitude that circles the globe in an east-west direction (b) lies exactly midway between the North and South poles (c) forms a plane that runs perpendicular to the earth’s axis (d) All of these 169. Eucla Basin in Australia is well-known as (a) Murray-Darling Basin (b) Salinaland (c) Barkley Tableland (d) Nullabor Plain 170. Eurasia includes (a) Cyprus, Turkey and Russia and independent states (b) Syria, Cyprus and Russia and independent states (c) Lebanon, Syria, Russia and independent states (d) Lebanon, Cyprus, Russia and independent states 171. Formation of a rift valley is due to (a) the formation of fold mountains (b) the forces of tension on the earth’s crust (c) the subsidence of the floor of a river valley (d) the over-deepening of a valley by ice action

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▌ General Knowledge

172. Greenwich Mean Time (GMT) is (a) the UK standard time (b) based on the local time of the meridian passing through Greenwich, near London (c) the US standard time (d) only (a) and (b) 173. Harvest moon is (a) the full moon at harvest time in the north temperate zone (b) the full moon occurring closest to the autumnal equinox, at about September 23 (c) the time when moon rises at a point opposite to the sun (d) All are correct 174. Which line marks the boundary of the zone where the sun never sets during the June solstice and never rises during December solstice? (a) Antarctic circle (b) Tropic of Cancer (c) Arctic circle (d) Tropic of Capricorn 175. In which of the following forests do the trees shed their leaves during spring and early summer? (a) monsoon deciduous forest (b) dry forests (c) tropical savannah forests (d) Himalayan forests 176. Koepeen type of climate prevails over (a) the interior peninsula of India (b) the Coromandal coast of India (c) Western coast of India, south of Goa (d) Eastern coast of India, south of Kolkata 177. Largest extent of equatorial forests is found in (a) Zaire Basin (b) Nile Basin (c) Irrawadi Basin (d) Amazon Basin 178. Latitude measures the angular distance, expressed in degrees of a point on the Earth’s surface: (a) north of the equator (b) south of the equator (c) north or south of the equator (d) east or west of the prime meridian 179. Life is possible on earth because of (a) terrestrial atmosphere (b) nature of composition of atmosphere (c) temperature control and shielding effect against solar radiation. (d) All are correct 180. Longitude measures the angular distance, expressed in degrees of a point on the earth’s surface (a) east or west of prime meridian (b) north or south of the equator (c) only east of prime meridian (d) only west of prime meridian 181. Lunar eclipse occurs only on a (a) first quarter day (b) full moon day (c) new moon day (d) last quarter day

182. Lunar eclipse occurs when (a) the earth comes between the moon and the sun (b) the moon comes between the earth and the sun (c) any of these (d) earth, moon and sun not in a straight line 183. Major natural regions of the world are delineated primarily on the basis of (a) latitude (b) temperature, rainfall and vegetation (c) altitude (d) distance from the equator 184. Modern industrial development of India is largely due to (a) rationalization of raw materials (b) availability of skilled manpower (c) liberalization of government policies (d) increase in market demand 185. Among the following, which country is reliant on a single export of oil/petroleum? (a) South Africa (b) Zimbabwe (c) Namibia (d) Nigeria 186. Papaghani series of rocks belong to (a) Paleocene (b) Eocene (c) Oligocene (d) Pleistocene 187. Paragraph series of rocks belong to (a) upper Vindhyan (b) lower Vindhyan (c) upper Cuddapah (d) lower Cuddapah 188. Population of India has grown rapidly after 1921, mainly because (a) death rate declined steeply (b) birth rate has increased rapidly (c) Indians have returned from Burma, Sir Lanka and other countries (d) the number of persons migrating from India has declined 189. Port Stanley is the capital of the (a) West Indies (b) Falkland Island (c) Hawaii Island (d) Madagascar (Nayalgasey) 190. Power cuts have become frequent in various parts of India especially during summer, mainly because (a) the demand for power is greater during summer (b) the demand for power has increased owing to spread of electricity to villages (c) starting of new industries has increased the demand (d) hydel reservoirs have low storage and power generation is low 191. Which one of the following countries produces the largest amount of crude steel of the world? (a) Japan (b) India (c) South Korea (d) USA

Geography 197. Solar eclipse occurs on (a) first quarter day (c) new moon day

192. Pressure is measured by (a) galvanometer (b) barometer (c) anemometer (d) ammeter 193. Prime meridian is (a) at 0° longitude (b) an arbitrarily selected line from which all of the other meridians are measured (c) imaginary line of longitude that run from pole to pole in a north-south direction around the globe (d) All of these are correct 194. Regions lying on the western margins of the continents between 30° and 40° north have (a) dry summers and wet winters (b) rainfall all the year (c) dry climate all the year (d) wet summers and warm winters 195. Right ascension is (a) the equivalent of longitude on the celestial sphere (b) the equivalent of latitude on the celestial sphere (c) the equivalent of prime meridian (d) the equivalent of equator 196. Solar eclipse occurs when (a) the Earth comes between the moon and the sun. (b) the moon comes between the Earth and the sun (c) any of these (d) the sun comes between the earth and the moon

▌ 2.191

(b) full moon day (d) last quarter day

198. Spring tide occurs at (a) new moon and full moon when the Earth, the moon and the sun are in a line (b) first and last quarter of the moon when the Earth and the sun are in line (c) new moon and full moon when the Earth and the sun are at right angles (d) first and last quarter of the moon when the Earth and the sun are at right angles 199. The ‘zodiac’ signs or ‘houses of heaven’ are named after (a) star signs (b) ecliptic system (c) the constellations through which the ecliptic passes (d) All of these 200. The absence of which among the following causes the temperature of the atmosphere to decrease with ascent, along with decreasing density of air? I. Vegetation II. Water III. Carbon dioxide IV. Oxygen V. Nitrogen (a) I and II (c) II and III

(b) III and IV (d) II and V

ANSWER KEy 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

(d) (c) (a) (d) (d) (c) (c) (d) (b) (c) (c) (a) (b) (a) (b) (c) (a)

18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34.

(d) (c) (b) (d) (b) (d) (d) (b) (b) (d) (b) (b) (d) (c) (c) (a) (c)

35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51.

(b) (c) (c) (b) (c) (c) (d) (d) (b) (b) (d) (d) (b) (c) (c) (b) (b)

52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68.

(d) (b) (a) (c) (a) (d) (c) (b) (a) (b) (b) (c) (b) (d) (c) (d) (d)

69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81. 82. 83. 84. 85.

(b) (c) (b) (a) (d) (d) (a) (c) (d) (b) (c) (d) (c) (c) (a) (c) (c)

86. 87. 88. 89. 90. 91. 92. 93. 94. 95. 96. 97. 98. 99. 100. 101. 102.

(d) (a) (c) (b) (d) (d) (b) (d) (a) (c) (b) (c) (d) (d) (c) (d) (c)

103. 104. 105. 106. 107. 108. 109. 110. 111. 112. 113. 114. 115. 116. 117. 118. 119.

(c) (c) (b) (c) (d) (b) (d) (c) (d) (b) (b) (c) (c) (b) (b) (c) (a)

120. 121. 122. 123. 124. 125. 126. 127. 128. 129. 130. 131. 132. 133. 134. 135. 136.

(b) (b) (d) (c) (c) (c) (d) (a) (d) (a) (c) (d) (d) (a) (a) (c) (a)

2.192 137. 138. 139. 140. 141. 142. 143. 144.

▌ General Knowledge

(d) (c) (a) (d) (b) (b) (a) (b)

145. 146. 147. 148. 149. 150. 151. 152.

(c) (c) (b) (c) (b) (a) (d) (b)

153. 154. 155. 156. 157. 158. 159. 160.

(c) (d) (b) (c) (b) (d) (d) (d)

161. 162. 163. 164. 165. 166. 167. 168.

(c) (b) (b) (b) (b) (c) (b) (d)

169. 170. 171. 172. 173. 174. 175. 176.

(d) (a) (b) (d) (d) (c) (a) (c)

177. 178. 179. 180. 181. 182. 183. 184.

(d) (c) (d) (a) (b) (a) (b) (c)

185. 186. 187. 188. 189. 190. 191. 192.

(d) (d) (d) (c) (b) (d) (d) (b)

193. 194. 195. 196. 197. 198. 199. 200.

(d) (c) (c) (a) (b) (c) (a) (c)

NDA BOX 1. The Altazimuth system is (a) used to give the position of a celestial object (b) used by astronomers (c) not used by astronomers (d) both (a) and (b)

10. The different phases of moon are caused by (a) changing angle of earth with respect to moon (b) changing angle of sunlight hitting the moon (c) changing angle of sun (d) None of these

2. The altitude of troposphere at equator is (a) 0–20 km (b) 0–10 km (c) 0–18 km (d) 0–16 km

11. The Earth is divided into time zones for the purpose of (a) maintaining a standard time system (b) maintaining same time (c) ease of calculation (d) None of the above

3. The boundary between the troposphere and stratosphere is called (a) ozone layer (b) tropopause (c) mesosphere (d) ionosphere 4. The Colby hypothesis offers a dynamic explanation of (a) urban development (b) land use development (c) sustainable development (d) rural development 5. The densest of all the atmosphere layers is (a) troposphere (b) stratosphere (c) mesosphere (d) ionosphere 6. The density of air normally at higher attitude (a) increases (b) decreases (c) remains constant (d) varies from place to place 7. The density of dry air at sea level is about the density of water. (a) 1/80 (b) 1/8 (c) 1/800 (d) 1/8,000 8. Winds generally blow from areas of (a) low pressure to those of high pressure (b) high pressure to those of low pressure (c) high pressure to those of moderate pressure (d) low pressure to those of moderate pressure 9. The difference in the time per degree longitude between any two places on the globe is (a) 4 minutes (b) 5 minutes (c) 15 minutes (d) 30 minutes

12. The Earth rotates through (a) 2° in 4 minutes (c) 1° in 6 minutes

(b) l° in 4 minutes (d) 2° in 10 minutes

13. The extreme of temperature between summer and winter is quite low in southern part of peninsular India mainly because (a) the sun’s rays are almost vertical throughout the year (b) adjoining oceans moderate the temperature (c) sky is generally cloudy (d) strong winds blow throughout the year 14. Which one of the following mountain chains has two dissimilar types of vegetation on its two slopes? (a) Aravallis (b) Vindhyas (c) Eastern Ghats (d) Western Ghats 15. The four countries with the largest population, in order of size, are (a) China, India, Indonesia and Japan (b) China, India, Russia and USA (c) China, India, Russia and Japan (d) China, Japan, Indonesia and USA 16. The gravitational pull at the lunar surface is that of Earth. (a) 1/3rd (b) 1/4th (c) 1/5th (d) 1/6th 18. The Gutenberg discontinuity is between (a) the crust and the mantle (b) the upper and lower mantles (c) the outer core and the lower mantle (d) the inner and outer cores (e) None of these

Geography

▌ 2.193

19. The heaviest rainfall is recorded in (a) Mediterranean regions (b) temperate regions (c) equatorial regions (d) tropical mountain areas

29. The Mediterranean type of climate is characterized by (a) dry summer and mild, wet winters (b) moist summers and dry winters (c) hot summers and moist winters (d) clear sky and sunshine

20. The hunting and gathering economy can support only (a) 1 person per km2 (b) 3 persons per km2 2 (c) 8 persons per km (d) 5 persons per km2

30. The moon (a) is the natural satellite of Earth (b) has no liquid water (c) has no atmosphere (d) only (b) and (c) are correct

21. The immediate cause for the S.E. trades to cross the equator and blow as S.W. monsoon winds over India is the (a) intense low pressure over Tibet (b) heated Ganga plain (c) intense low pressure area of Thar (Rajasthan) (d) high temperature over Chhota-nagpur plateau 22. The Indus and Brahmaputra rivers are examples of (a) subsequent drainage (b) super-imposed drainage (c) consequent drainage (d) antecedent drainage 23. The infrared radiation by sun are strongly absorbed by (a) carbon dioxide (b) water vapours (c) carbon dioxide and water vapours (d) ozone 24. The International Date Line roughly corresponds to the (a) 180° east-west meridian of longitude (b) 180° east-west latitude (c) 0° east-west latitude (d) 180° meridian 25. The intersecting lines drawn on maps and globes are (a) latitudes (b) longitudes (c) geographic grids (d) latitude grids 26. The irregularity in the amount of rain in different parts of the north Indian plains, during different years, is mainly due to the (a) irregular intensity of low pressure in the north-western part of India (b) difference in frequency of cyclones (c) variations in the location of the axis of the low pressure trough (d) the amount of moisture carried by the winds not being the same every year 27. The local time varies from Greenwich time (London) at the rate of (a) 4 min/degree of latitude (b) 4 min/degree of longitude (c) 14 min/degree of latitude (d) 14 min/degree of longitude 28. The mass of Earth is times greater than the mass of the moon. (a) 40 (b) 85 (c) 91 (d) 81

31. The moon’s volume is about ______ that of Earth. (a) 1/40th (b) 1/50th (c) 1/60th (d) 1/120th 32. The most densely populated country of Africa is (a) Mauritius (b) Rwanda (c) Burundi (d) Sierra Leone 33. The penumbra is (a) the outer portion of the moon’s shadow. (b) formed by the tangent that intersects between the sun and the moon (c) darkest part of any shadow (d) All are correct 34. The main crops of Sikkim are (a) rice, maize, bajra, wheat, barley, cardamom (b) sugarcane, tobacco, wheat, rice (c) chilies, oilseeds, groundnut, pepper (d) wheat, bajra, pepper 35. What is the literacy rate of Tamil Nadu? (a) 69.68% (b) 73.47% (c) 78.66% (d) 57.36% 36. How many districts are there in Tamil Nadu? (a) 26 (b) 27 (c) 28 (d) 30 37. Which city in Tamil Nadu has airport? (a) Chennai (b) Tiruchirapalli (c) Madurai (d) only (a) and (b) 38. Guwahati High court is the judicature of: (a) Tripura (b) Arunachal Pradesh (c) Assam (d) All 39. Tripura is surrounded by _______ in the East. (a) Assam (b) Mizoram (c) Both (a) and (b) (d) Bangladesh 40. What is the population density of Uttar Pradesh? (a) 689/km2 (b) 589/km2 2 (c) 489/km (d) 389/km2 41. How many districts are there is Uttar Pradesh? (a) 70 (b) 80 (c) 50 (d) 60

2.194

▌ General Knowledge

42. Kathak, Nautanki, Jhora and Kajri are the important dances of (a) Uttarakhand (b) Uttar Pradesh (c) Jharkhand (d) Chhattisgarh 43. Uttar Pradesh is surrounded by _______ in North. (a) Nepal (b) Uttarakhand (c) Haryana and Delhi (d) Rajasthan 44. The major minerals found in Uttar Pradesh are (a) Limestone and dolomite (b) Rock phosphate and dolomite (c) Copper and graphite (d) Dolomite and copper 45. Uttarakhand has International boundary with ______ in North, North-East and Eastern side. (a) China (b) Nepal (c) Bhutan (d) Both (a) and (b) 46. The population density of Uttarakhand is (a) 159/km2 (b) 149/km2 2 (c) 139/km (d) 129/km2 47. Which type of industries are found in Uttarakhand? (a) Forest based small-scale industries (b) Engineering goods, automobiles (c) Tea, Jute, Paper (d) All of these 48. What is the literacy rate of West Bengal? (a) 69.22% (b) 72.29% (c) 51.33% (d) 70.13% 49. Baul is the popular dance of (a) Orissa (c) West Bengal

(b) Uttarakhand (d) Jharkhand

50. On which river is the Bhakra Nangal project built? (a) Kosi (b) Jhelum (c) Beas (d) Satluj 51. A biogeography region with significant reservoir of biodi versity that is under threat from humans is called as (a) Bioendangered region (b) Biodiversity hotspots (c) Biodiversity reservoir (d) environmentally endangered region 52. Among the following sedimentary rocks, which one is of organic origin? (a) Gypsum (b) Limestone (c) Nitre (d) Rock Salt 53. Among the following, which one has recorded the highest population Growth rate during 1991-2001? (a) Arunachal Pradesh (b) Manipur (c) Nagaland (d) Sikkim 54. Among the following, which state has highest level of literacy? (a) Mizoram (b) west Bengal (c) Gujarat (d) Punjab

55. Amravati, Bhavani Hemavati and Kabini are tributaries of which one of the following rivers? (a) Mahandadi (b) Godavari (c) Kaveri (d) Krshna 56. Arrange the following states on the basis of ascending dates of the onset of monsoon. 1. Uttar Pradesh 2. West Bengal 3. Kerala 4. Rajasthan Select the correct answer using the codes given below: Code: (a) 2, 3, 1, 4 (b) 3, 2, 1, 4 (c) 3, 1, 2, 4 (d) 1, 2, 3, 4 57. As we proceed from equator to poles, the daily range of temperature tends to (a) Decrease (b) Increase (c) Be constant (d) Fluctuate Directions use the following codes to answer Q 58 to 61 Codes: (a) Both A and R are individually true and R is the correct explanation of A (b) Both A and R are individually true but R is not the correct explanation A (c) A is true but R is false (d) A is False but R is true 58. Assertion (A) Narmada and Tapti rivers do not form deltas. Reason (R) These rivers form estuaries. 59. Assertion (a) The cotton industry in India suffered a major recession soon after the independence. Reason (a) Most of the cotton mills had gone to West Pakistan, India retaining the cotton growing. 60. Assertion (A) The Equator is great Circle. Reason (R) The Equator encircles the whole Earth and its plane divides the Earth into two equal halves. 61. Assertion (A) The tidal effect of Moon on the Surface of Earth is less than that of Sun. Reason (R) Moon’s gravitational pull at Earth’s surface is less compared to Sun’s gravitational pull. 62. Biodiversity is highest in (a) Tundra Zone (c) Torrid Zone

(b) Prairie Zone (d) Tropic Zone

63. Chambal river is part of (a) Sabarmati basin (c) Narmada basin

(b) Ganga basin (d) Godavari basin

64. Cloudy nights are warmer than clear nights because of (a) Greenhouse effect (b) Depletioon of ozone layer (c) Insolation (d) Terrestrial Radiation

Geography 65. Consider the following statements: 1. All cyclones develop an eye at the center. 2. The temperature inside the eye is nearly 10ºC lesser than that of the surroundings. Which of the statements given above is/are correct? (a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2 66. Consider the following rivers: 1. Betwa 2. Kosi 3. Gandak Which of the statements given above is/are correct? (a) 1 only (b) 1 and 2 (c) 1 and 3 (d) 2 and 3 67. Consider the following statements 1. The Himalayan vegetation varies according to both altitude and climatic conditions. 2. There are mainly two types of tropical forests that are found in the Himalayas—the tropical rainforests and the tropical deciduous forests. Which of the statements given above is/are correct? (a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2 68. Consider the following statements: I. In a cyclone, the direction of wind flow is counter clockwise in the Northern hemisphere II. The tropical cyclone fades away when it reaches land because there is no large supply of warm moist air Which of the statements given above is/are correct? (a) I only (b) II only (c) Both I and II (d) Neither I nor II 69. Consider the following statements: I. In India, the largest concentration of roads is found in the Northern plains II. The ration of surfaced road to the total road length is lower in the Northern plains Which of the statements given above is/are correct (a) I only (b) II only (c) Both I and II (d) Neither I nor II 70. Consider the following statements: 1. The vernal equinox falls on March 21. 2. On equinox, the sun is directly overhead at the equator. 3. The changes in the day length on equinox result from the changes in the tilt of the earth with respect to the sun. Which of the statements given above is/are correct? (a) 1 and 2 (b) 1 only (c) 1, 2, 3 (d) 2 and 3

▌ 2.195

71. Consider the following statements: 1. The earth’s rotation axis is not at 90º to its orbital plane. 2. The earth’s rotation axis is inclined at 23.5º. Which of the statements given above is/are correct? (a) 1 Only (b) 2 Only (c) Both 1 and 2 (d) Neither 1 nor 2 72. Consider the following statements: 1. The difference between the moisture-holding capacity of air and its actual humidity is called saturation deficit. 2. The temperature to which air has to be cooled in order to reach saturation is called dew point. Which of the statements given above is/are correct? (a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2 73. Consider the following statements: 1. International Data Line is drown zigzag to avoid landmass. 2. International Data line is 180º W as well as 180º E of Greenwich. 3. A ship sailing westward from Greenwich when crossing International Date Line would put back the date by a day. Which of the statements given above is/are correct? (a) 1 and 2 (b) 1 and 3 (c) 1, 2 and 3 (d) 3 only 74. Consider the following statements: 1. Different Plant species grow together. 2. Light cannot penetrate into the lower strata full of undergrowth. Which one of the following types of vegetation is characterized by the above? (a) Equatiorial moist evergreen (b) Tropical deciduous (c) Mediterranean (d) Warm temperate broad leaved deciduous 75. Consider the following statements: 1. Lamba in Gujarat is famous for wind power Plant. 2. Manikaran in Himachal Pradesh is famous for geothermal energy plant. Which of the statements given above is/are correct? (a) 1Only (b) 2 Only (c) Both 1 and 2 (d) Neither 1 nor 2 76. Consider the following statements: 1. Kandla Port is situated at the head of Gulf of Khambhat. 2. Paradeep Port is situated in the Mahanadi Delta.

2.196

▌ General Knowledge

Which of the statements given above is/are correct? (a) 1only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2 77. Consider the following statements: 1. Coal is a sedimentary rock. 2. Basalt is an igneous rock. Which of the statements given above is/are correct? (a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2 78. Consider the following statements: 1. In tropical grassland regions, rainfall mainly occurs in the short summer season with a long dry season.l 2. In Mediterranean region, the winter rainfall is caused by the passage of cyclones in the Westerly wind belt which lies over this area. Which of the statements given above is/are correct? (a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2 79. Consider the following statements: 1. The approach of cyclone is characterized by a rise in barometric reading. 2. In the cyclones of the Northern hemisphere, the winds circulate in anticlockwise direction. Which of the statements given above is/are correct? (a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2 80. Consider the following statements: 1. Ranjit Sagar Dam is on the river Beas. 2. Hirakud Dam is on the river Damodar. 3. The reservoir Gandhi Sagar is on the river Chambal. Which of the statements given is/are correct? (a) 1 and 2 (b) 3 only (c) 2 and 3 (d) 1 and 3 81. Consider the following statements: 1. International Date Line on the Greenwich Meridian. 2. The date of Alaska is ahead of the date of Siberia. Which of the statements given is/are correct? (a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2 82. Consider the following statements about comets: 1. Most comets have elongated elliptical orbits that take them close to the sun for a part of their orbit and then out into the further reaches of the Solar System of the remainder. 2. If a comet is travelling fast enough, it may leave the Solar System. Which of the statements given above is/are correct? (a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2

83. Consider the following statements about rocks: I. Shale becomes slate through metamorphosis II. Shale is converted to slate when it is subjected to tremendous pressure and high temperature Which of the statements given above is /are correct? (a) I only (b) II only (c) Both I and II (d) Neither I nor II 84. Consider the following statements regarding Andaman and Nicobar Islands. I. It enjoys equatorial Climate II. This is the only place in India Where a volcano is located III. This is the only place in India where coral bed is found Which of the statements given above is/are correct? (a) I, II and III (b) I and II (c) II and III (d) I only 85. Consider the following statements related to stratification of atmospheric layers. 1. All storms and Cloudiness are restricted to stratosphere. 2. Cirrus clouds are formed on the top layers of troposphers. 3. Stratosphere is also an ‘isoclinal layer’. Which of the statements given above are correct? (a) 1 and 2 (b) 1 and 3 (c) 2 and 3 (d) 1, 2 and 3 86. Consider the following statements. I. The tropical year is shorter than the didereal year II. The solar day is longer than the sidereal day Which of the statements given above is/are correct? (a) I only (b) II only (c) both I and II (d) Neither I nor II 87. Consider the following statements: 1. Alamatti dam is on the Cauvery river. 2. Mettur dam in on the Krishna river. 3. Gandhi Sagar Reservoir is on the Chambal river. Which of the statements given above is/are correct? (a) 1 and 2 only (b) 2 only (c) 1 and 3 only (d) 3 only 88. Detroit is famous for which one of the following industries? (a) Textile (b) Food Processing (c) Automobiles (d) Locomotives Doldrums are characterized by (a) Uniform low pressure (b) Uniform high pressure (c) high wind velocity (d) low humidity 89. Doldrums is a (a) Tropical wind belt (b) Tropical Wind deflection belt (c) sub-tropical wind belt (d) Tropical no-wind belt

Geography ▌ 2.197 90. Dry Point settlements are common in (a) Deserts (b) Plateaus (c) Mountain regions (d) Deltas 91. El Nino Current appears along the coast of which one of the following? (a) Brazil (b) Peru (c) Alaska (d) Western Australia 92. Why the summer monsoon winds blow from south-western direction in the Northern hemisphere? (a) The general direction of wind from the Indian Ocean is south-western (b) The presence of the doldrums around the equator (c) The low pressure conditions in North-West India (d) Due to the effect of Coriolis force Xerophytes thrive in (a) Hot and arid condition (b) Cool and wet condition (c) Hot and wet condition (d) Cool and arid condition 93. For which one of the following is Lamba in Gujarat, famous? (a) Cultivation of oil yielding Plants (b) Wind power plant (c) Oil refinery (d) Uranium enrichment plant 94. Foucault experiment is proof of which one of the following? (a) Revolution of Earth (b) Rotation of Earth (c) Rotation of Moon (d) Revolution of Moon 95. From West to East, which one of the following is the correct sequence of Hills? (a) Mahadeo Hills – Maikala Hills – Garhjat Hills (b) Mahadeo Hills – Garhjat Hills – Maikala Hills (c) Maikala Hills – Mahadeo Hills – Garhjat Hills (d) Maikala Hills – Garhjat Hills – Mahdeo Hills 96. From which one of the following is the percentage of reflected amount of radiation highest? (a) Wet ground (b) Thick cloud (c) Forest (d) Snow cover 97. Glaciated regions are associated with (a) V-shaped vally (b) U-shaped valley (c) Sand dunes (d) stalactites 98. Granite and Basalt are the examples of which of the following? (a) Sedimentary rocks (b) Metamorphic rocks (c) Igneous rocks (d) Calcareous rocks

99. Haridwar is well-known for which one of the following industries? (a) Fertilizers (b) Cement (c) Heavy Electricals (d) Silk Textiles 100. Hot deserts like Sahara, Arabia etc, receive very negligible amount of rainfall. This is because they (a) Do not receive moisture-bearing wind from the oceans (b) Are the most rocky and barren areas of the Earth (c) Are located on the tropical high pressure belt of the atmosphere (d) Are not on the path of the monsoons 101. How much of a time period 15º of longitudes account for? (a) 4 min (b) 15 min (c) 30 min (d) 60 min 102. Identify from the following states of India through which the Tropic of Cancer passes and arranged them from East to West. 1. Gujarat 2. West Bengal 3. Uttar Pradesh 4. Jharkhand 5. Madhya Pradesh 6. Bihar 7. Chhattisgarh Select the correct answer using the codes Given below (a) 2-5-7-4-1 (b) 2-4-7-5-1 (c) 3-2-6-7-5 (d) 3-7-4-6-2 103. If it is 12 noon in a city located on 90º W longitude, what would be the time in a city located on 105º W longitude? (a) 13:00 h (b) 12 : 30 h (c) 11 : 30 h (d) 11 : 00 h 104. If the Earth’s axis were perpendicular to the plane of its orbit, which one among the following would not have happened? (a) The North Pole will always lie in dark (b) Days and nights would be equal throughout the year (c) No change of seasons will take place (d) The sun will be perpendicular to the equator 105. If there are four places on the same meridian 500 km apart and the local time at one place is 12.00 noon, what will be the time at the three other places? (a) 12.00 Noon (b) 1.00 pm (c) 2.00 pm (d) Different time at different Places 106. In which country is the active volcano Mount Merapi located? (a) Philippines (b) Japan (c) Indonesia (d) Italy 107. In which of the following Hills is tea cultivated? (a) Makala Hills (b) Nallamalai Hills (c) Nilgiri Hills (d) Shivalik Hills

2.198

▌ General Knowledge

108. In which one among the following states is the percentage of Scheduled Tribes population to its total population highest? (a) Cahhattisgarh (b) Madhya Pradesh (c) Mizoram (d) Rajasthan 109. In which one of the following areas is monsoon climate found? (a) Pacific Coast of Columbia (b) South-Eastern United States (c) Southern Part of South Africa (d) Central California 110. In which one of the following is a great dark plain called ‘maria’ found? (a) Earth (b) Mars (c) Jupiter (d) Moon 111. In which one of the following regions are the Andes mountains located? (a) East Europe (b) West Europe (c) South Africa (d) South America 112. In which one of the following regions does the Indus river originate? (a) Ladakh (b) Lahaul (c) Nepal (d) Tibet 113. India is comparatively not rich in which one of the following Minerals compared to the other three? (a) Bauxite (b) Copper (c) Iron (d) Manganese 114. Indian Standard Time refers to the local time of which one of the following places in India? (a) Allahabad (b) Bhopal (c) Delhi (d) Lucknow 115. Injurious ultraviolet radiations are mostly prevented from reaching the land surface as these are absorbed mostly by which one of the following? (a) Stratosphere (b) Troposphere (c) Ionosphere (d) Mesosphere 116. Iron ore from Kudremukh is most likely to be exported through (a) Goa (b) Kochi (c) Mangalore (d) Ennore 117. Jet streams are usually found in the (a) Ozonosphere (b) Mesosphere (c) tropopause (d) ionosphere 118. Kiel Canal connects (a) Baltic Sea and North Sea (b) Read Sea and Mediterranean Sea (c) Caribbean Sea and Pacific Sea (d) Atlantic Ocean and Pacific Ocean 119. Mahatma Gandhi Hydroelectric Project is on which river?

(a) Godavari (c) Cauvery

(b) Sharavati (d) Krishna

120. Match List I with List II and select the correct answer using the codes given below the lists: List-I (Grassland)

List-II (Country)

A. Pampas

1.

Venezuela

B. Veld

2.

Australia

C. Downs

3.

South Africa

D. Lianos

4.

Argentina

Codes: A

B

C

D

(a)

4

3

2

1

(b)

4

2

3

1

(c)

1

3

2

4

(d)

1

2

3

4

121. Match List I with ListII and select the correct answer using the codes given below the lists: List I (Himalayan Peaks)

List II (States)

A.

Nanga Parbat

1.

Jammu & kasmir

B.

Nanda Devi

2.

Sikkim

C.

Kanchanjunga

3.

Uttarakhand

Codes: A B

C

(a)

1

2

3

(b)

1

3

2

(c)

2

3

1

(d)

3

1

2

122. Match List I with List II and select the correct answer using the codes given below the lists. List – I (National Highway No.)

List – II (cities connected)

A.

NH5

1.

Bhopal – jaipur

B.

NH7

2.

Bhubneshwar – Chennai

C.

NH9

3.

Nagpur – Varanasi

D.

NH12

4.

Pune – Hyderabad

Codes: A B

C D

(a)

2

1

4

3

(b)

4

3

2

1

(c)

2

3

4

1

(d)

4

1

2

3

Geography 123. Match List I with List II and select the correct answer using the codes given below the lists. A.

List I (Current) Kuroshio Current

1.

B.

Peru current

2.

C. D.

Labrador Current

3.

Florida current

4.

List II (Feature) Warm current in the Atlantic Ocean Cold current in the Atlantic Ocean Warm current in the Pacific Ocean Cold Current in the Pacific Ocean

126. Match list I with list II and select the correct answer using the codes given below the lists. List I (Agents of erosion)

A 3 3 1 1

B 4 2 4 2

C 2 4 2 4

D 1 1 3 3

124. Match List I with List II and select the correct answer using the codes given below the lists. List I (Mineral)

1.

Bauxite

1.

Andhra Pradesh

B.

Mica

2.

Odisha

C.

Copper

3.

Madhya Pradesh

D.

Zinc

4.

Rajastan

B.

Glacier

2.

Barchan

C

Wind

3.

Rift Valley

D. Underground water 4.

Doline

5.

Gorge

Codes: (a) (b)

A 5 5

B 1 2

C 2 1

D 4 3

(c) (d)

4 3

2 4

1 12

5

127. Match List I with List II and select the correct answer using the codes given below the lists.

Codes: A

B

C

D

(a)

4

1

3

2

(b) (c) (d)

2 4 2

1 3 3

3 1 1

4 2 4

List –I (places indicated

125. Match List I with List II and select the correct answer using the codes given below the lists. List I (Desert)

List II (Country)

1

1.

Zone V

B.

2

2

Zone IV

C.

3

3

Zone III

D.

4

4

Zone II

Codes: (a)

A

B

C

D

2

1

3

4

Kalahari

1.

Angola

(b)

2

3

1

4

B.

Namib

2.

Sudan

(c)

4

3

1

2

C.

Nubian

3.

Botswana

(d)

4

1

3

2

D.

Atacama

4.

Chile

Codes: C 1 1 2 2

D 3 4 3 4

List –II (Seismic Zone) in the map)

A.

A.

B 2 2 1 1

Cirque

List II (State)

A.

A (a) 4 (b) 3 (c) 4 (d) 3

List II (Topographical Feature)

A. Running water

Codes: (a) (b) (c) (d)

▌ 2.199

128. Match List I with List II and select the correct answer using the codes given below the lists. List –I (Peak)

List –II (State)

A.

Dodabetta

1.

Maharashtra

B.

Guru Shikhar

2.

Madhya Pradesh

C.

Kalsubai

3.

Tamil Nadu

D.

Dhupgarh

4.

Rajasthan

2.200

▌ General Knowledge Codes:

Codes: A

B

C

D (a)

A

B

C

D

2

4

1

3

(a)

3

1

4

2

(b)

2

1

4

2

(b)

3

1

4

2

3

4

1

2

2

1

4

3

(c)

3

4

1

2

(c)

(d)

2

4

1

3

(d)

129. Match List I with List II and select the correct answer using the code givne below the lists. List I (Fishing Bank)

132. Match List I with List II and select the correct answer using the codes given below the lists. List-I (Industry)

List II (Coast)

A.

Dogger Bank

1.

Newfoundland Coast

B.

Grand Bank

2.

Australia Coast

C.

Great Barrier Reef

3.

North Sea

D.

Tonga Trench

4.

East of Fiji Island

Codes:

A.

Aluminium

1.

Combatore

B.

Heavy Electriclas

2.

Renukoot

C.

Petrochemicals

3.

Vadodara

D.

Cotton Textiles

4.

Jagdishpur

Codes:

A

B

C D

A B

C D

(a)

3

1

2

4

(a)

2

4

3

1

(b)

4

2

1

3

(b)

1

3

4

2

(c)

3

2

1

4

(c)

1

4

3

2

(d)

4

1

2

3

(d)

2

3

4

1

130. Match List I with List II and select the correct answer using the codes given below the lists. List I (Volcano)

133. Match List –I with List-II and select the correct answer using the code given below the lists. List (Landform)

List II (Country)

A.

Mount Etna

1.

India

B.

Kilimanjaro

2.

USA

C.

Katmai

3.

Tanzania

D.

Barren Island

4.

Italy

A.

Inselbergs

1.

River

B.

Stalagmite

2.

Glacier

C.

Delta

3.

Underground Water

D.

Moraines

4.

Wind

A B

C D

A

B

C

D

1

3

2

4

(a) 4

2

1

3

1

2

4

(b)

4

2

3

1

(b) 3

(c)

1

3

3

4

(c) 1

2

3

4

1

(d) 4

3

1

2

(d)

4

List –II (Agent of erosion/ deposition )

Codes:

Codes: (a)

List-II (Industrial Centre)

2

2

131. Match List I with List II and select the correct answer using the code given below the lists. List-I (Mineral)

List-II (Mine)

134. Match List –I with List-II and select the correct answer using the code given below the lists. List-I

List-II (State)

A.

Bauxite

1.

Balaghat

A. Mangrove

1.

Madhya Pradesh

B.

Copper

2.

Korba

B. Scrub

2.

Karnataka

C.

Iron ore

3.

Singareni

C. Teak

3.

Rajasthan

D.

Coal

4.

Keonjhar

D. Coniferous

4.

Arunachal Pradesh

Geography

139. On planet Earth, there is no centrifugal force at the (a) Equator (b) Tropic of Cancer (c) Tropic of Capricorn (d) Poles

Codes: A B

C D

(a)

4

1

3

2

(b)

2

1

3

4

(c)

4

3

1

2

(d)

2

3

1

4

140. Port Blair the capital of Andaman and Nicobar Islands, is located in which one of the following islands? (a) North Andaman (b) Little Andaman (c) Middle Andaman (d) South Andaman

135. Match List-I with List-II and select the correct answer using codes given below the lists. List-I (Coal fields of India)

List-II (State)

A.

Jharia

1.

Tamil Nadu

B.

Ranigani

2.

Odisha

C.

Talcher

3.

Jharkhand

D.

Korba

4.

West Bengal

5.

Chhattisgarh

Codes: A B

C D

(a)

3

4

2

5

(b)

2

5

1

4

(c)

3

5

2

4

(d)

2

4

1

5

141. Rain shadow effect is associated with (a) Cyclonic rainfall (b) Orographic rainfall (c) Convectional rainfall (d) Frontal rainfall 142. Savanna natural region is characterized by which one of the following? (a) A distinct wet and dry season with annual range of temperature between 3º C – 8º C (b) Broad-leaf evergreen forest and grasses (c) Uniformly high temperature throughout the year (d) No spatial variation in mean rainfall 143. Shimsha, Hemavati, Arkavati are the tributaries of which one of the following rivers? (a) Tapti (b) Tungabhadra (c) Cauvery (d) Krishna 144. Sirocco is a name used to means (a) a local wind (b) a volcano (c) an island (d) an ocean current

136. Match List-I with List-II select the correct answer using the code given below the lists. List –I (Industry)

List-II (Location )

A.

Petrochemicals

1.

Ranchi

B.

Aluminum

2.

Hyderabad

C.

Electronics

3.

Vadodara

D.

Heavy Engineering

4.

Koraput

Codes: (a)

▌ 2.201

A

B

C

D

3

4

2

1

(b) 1

2

4

3

(c)

3

2

4

1

(d) 1

4

2

3

137. Mount Merapi is located on which one of the islands? (a) Java (b) Sumatra (c) Borneo (d) Celebes 138. No trees are found in Tundra biome near polar region of northen hemisphere. This is due to (a) Snowfall inhibits plant respiration (b) Frozen ice beneath the surface soil (permafrost) restricts root growth (c) less wind movement and inadequate sunlight (d) low temperature which restricts development of reproductive organs

145. Starting from the Best quality of iron ore, which one of the following is the correct sequence of the iron ore based on the contents of pure iron? (a) Magnetite – Hematite – Limonite – Siderite (b) Hematite – Magnetite –Limonite –Siderite (c) Magnetite – Hematite –Siderite –Limonite (d) Hematite – Magnetite – Siderite – Limonite Directions use the following codes to answer Q 146 to 153 Codes: (a) Both A and R are individually true and R is the correct explanation of A (b) Both A and R are individually true but R is not the correct explanation A (c) A is true but R is false (d) A is False but R is true 146. Statement I: Decay and disintegration of rock in situ is called weathering. Statement II: Mechanical weathering is mainly caused by temperature variation. 147. Statement I: During the day, winds blow from sea to land. Statement II: The land gets more heated than the surrounding sea, hence lower pressure develop over land as compared to sea. 148. Statement I: In the Northen hemisphere, the Ocean currents flowing from equator towards the North pole and from pole towards the equator are deflected to their right.

2.202

▌ General Knowledge

Statement II: This happens due to rotation of the Earth on the axis from West to East. 149. Statement I: Pressure gradients determine the velocity of Winds. Statement II: when isobars (lines of equal atmospheric pressure) are closely spaced, the wind velocity would be gentle. 150. Statement I: Temperatures of countries like united Kingdom, Norway, the Netherlands and Denmark are higher as compared to places located on similar latitudes during the winter. Statement II: United Kingdom, Norway, the Netherlands and Denmark are located on the coast. 151. Statement I: The Mediterranean climate is highly suitable for fruit production. Statement II: cool and moist winters in Mediterranean regions enable ample production of fruits. 152. Statement I: The semi-arid tracts of India stretching from Eastern Rajasthan in the North to South central Tamil Nadu are agriculturally less productive. Statement II: The semi arid tracts are homeland to a large number of central Indian Scheduled Tribes Population. 153. Statement I: Winds are deflected to their right in the Northern hemisphere and to their left in the southern hemisphere. Statement II: The Earth’s axis is inclined. 154. Sunda Trench lies in (a) Atlantic Ocean (c) Indian Ocean

(b) Pacific Ocean (d) Antarctic Ocean

155. The tropic of cancer passes through which one of the following? (a) Asom (b) Manipur (c) Mizoram (d) Nagaland 156. The below diagram is of a mushroom rock. In which of the points in the diagram is the intensity of wind strongest?

(a) A (c) C

(b) B (d) D

157. The current produced by upwelling of cold water off the coast of Chile and Peru known as (a) El Nino (b) Humboldt Current (c) Agulhas current (d) Canary Current 158. The earth’s crust is the thinnest (a) Under the mountain ranges (b) Under continental masses (c) At ocean bottoms (d) At mid-oceanic ridges

159. The exceptionally high and tides that occur at the time of the new moon or the full moon when the Sun, the Moon and the Earth are approximately aligned is called (a) Spring (b) Fall (c) Neap (d) Diurnal 160. The Gulf of Mannar is situated along the coast of (a) Tamil Nadu (b) Kerala (c) Karnataka (d) Andhra Pradesh 161. The intensity of insolation depends on (a) altitude (b) nature of terrain (c) wind (d) latitude 162. The interval between two high tides is approximately (a) 4 h (b) 6 h (c) 12 h (d) 24 h 163. The latitude is the angular distance of a point of the Earth’s surface, North or South of the equator as measured form the (a) Centre of the Earth (b) Equator (c) Tropic of Cancer or the Capricorn (d) Poles 164. The mass of water vapour per unit volume of air is known as (a) Relative humidity (b) Specific humidity (c) absolute humidity (d) Variable humidity 165. The Narmada river in the Peninsular plateau flows westward with a remarkably straight channel. It is because the (a) Slope gradient in this part controls the river channel pattern (b) River carries a huge amount of water which has created a straight channel course (c) River forms the boundary between the Central Highlands and the Deccan Plateau (d) river flows through the trough of a rift valley inclined westward 166. The National Highway No. 7 connects which of the following? (a) Mumbai with Varanasi (b) Mumbai with Bhubaneshwar (c) Delhi with Kanyakumari (d) Varanasi with Kanyakumari 167. The Nuclear Power Station Rawatbhata is in which State? (a) Maharashtra (b) Uttar Pradesh (c) Rajasthan (d) Tamil Nadu 168. The phenomenon of ‘Trade winds’ takes place due to (a) conduction of heat (b) Convection of heat (c) radiation (d) None of these 169. The river with highest tidal bore in India is (a) Cauvery (b) Mahanadi (c) Hoogli (d) Krishna 170. The surface temperature of the Sun is nearly (a) 2000 k (b) 4000 k (c) 6000 k (d) 8000 k

Geography ▌ 2.203 171. The time difference between the two cities, city A(30’ N 60’ E ) and City B (30’ N 80’B) would be (a) 80 min (b) 0 min (c) 20 min (d) 34 min 172. The Tropic of Cancer does not pass through (a) Odisha (b) Tripura (c) Chahattisgarh (d) Rajasthan 173. The Victoria Falls in Africa is located on which river? (a) Zaire (b) Orange (c) Zambezi (d) Niger 174. The western coasts of India receive very high rainfall in summer mainly due to (a) Tropical location (b) Nearness to sea (c) western Ghats (d) Himalayas 175. To a perpendicular to the plane of ecliptic, the Earth’s axis of 1 degrees. Had this angle been, 2 which one among the following would result? (a) There would have been no seasons (b) The lengths of day and night would have been the same throughout the year (c) The length of day and night would have been the same all over the Earth (d) All of the above

rotation makes an angle of 23

176. Tropical cyclone of Philippines is termed as (a) Typhoon (b) Willy-willy (c) Hurricane (d) Bahuio 177. Unlike other Meridians, International Date Line is drawn zigzag in order to (a) Permit certain land areas and groups of islands to have the same calendar day (b) Facilitate the sailors to adjust time in their watch (c) Adjust the day in calendar while sailing from East to west and vice versa (d) Make 180º E and 180º W coterminous 178. What are the Westerlies? (a) Permanent winds (c) Local Winds

(b) Seasonal Winds (d) Variable Winds

179. What does the imaginary line passing through Lake Ontario, Lake Erie, Lake Huron and Lake Superior represent? (a) The southern boundary of Canada (b) The Northern limit of iron and steel industry of USA (c) The internal waterway rout to USA (d) The internal waterway route to Canada 180. What is the general direction of cyclones formed in the Bay of Bengal? (a) East to west (b) West to East (c) West to South (d) North to south 181. What is the name given to an almost circular coral reef inside which there is a langoon?

(a) Fringing reef (c) Atoll

(b) Barrier reef (d) Isthmus

182. What is the process that tends to build up the land surface by deposition of solid material in its lower areas, known as? (a) Abrasion (b) Agglomeration (c) Aggradation (d) Attrition 183. What is the reason for India being a major producer of different varieties of fruits and vegetables? (a) Its large population size (b) its large land area (c) Variation in its agro-climatic Conditions (d) Diversity in socio-cultural phenomenon 184. What is the reason for the major hot deserts of the world lying in the western part of the tropical latitude? (a) They are influenced by trade winds (b) They lie in the rain- shadow area of the mountains (c) They are influenced by monsoon winds (d) They are influenced by dry winds 185. What is the sequential order of vegetation types observed while moving from Asom Valley to Rajasthan Plains? (a) Tropical Wet Evergreen Tropical Moist Deciduous Tropical Dry Deciduous Tropical Thorn Forest (b) Tropical Thorn Forest Tropical Dry Deciduous Tropical Moist Deciduous Tropical Wet Evergreen (c) Tropical Moist Deciduous Tropical Wet Evergreen Tropical Dry Deciduous Tropical Thorn Forest (d) Tropical Dry Deciduous Tropical Thorn Forest Tropical Moist Deciduous Tropical Wet Evergreen 186. What is the term used to denote the critical temperature at which the air becomes saturated with vapor and below which the condensation is likely to begin? (a) Condensation Point (b) Evaporation Point (c) Dew Point (d) Point of critical temperature 187. What is the term used to denote the temperature at which the water vapour present in the atmosphere is sufficient to saturate? (a) Condensation point (b) Dew point (c) Sublimation (d) Saturation point

2.204

▌ General Knowledge

188. When it is 12 Noon at Allahabad, the time at Welington (New Zealnad) would b close to which one of the following? (a) 4 : 30 Pm of the same day (b) 4 : 30 pm of the previous day (c) 6 : 30 pm of the same day (d) 6 : 30 Pm of the previous day 189. When one of the following cities is located on the bank of the Irrawady river? (a) Bangkok (b) Hanoi (c) Manila (d) Yangon 190. Which of the following statements regarding the Deccan Traps is/are correct? 1. Intense volcanic activity in the form of fissure eruption took place towards the end of Cretaceous period. 2.The volcanic lava spread out in horizontal sheets. 3. The regur soil found here is rich in nitrogen. Select the correct answer using the codes given below (a) 1 and 2 (b) 1, 2 and 3 (c) 3 Only (d) 1 Only 191. Which among the following is the correct sequence of rivers starting from North to South? (a) Bhima-Godavari-Penganga-Tungabhadra (b) Godavari-Penganga-Tungabhandra-Bhima (c) Penganga-Gadavari-Bhima-Tungabhadra (d) Penganga-Bhima-Godavari-Tungabhadra 192. Which among the following is the major item of export from Paradeep Port? (a) Rice (b) Tea (c) Fish (d) Iron ore 193. Which among the following planets is smaller in size than the earth? (a) Neptune (b) Venus (c) Saturn (d) Uranus 194. Which among the following statements about the North Atlantic drift is/are correct? I. It keeps the west coast of Northern Europe ice free. II. It is responsible of the warm air mass which interacts with the cold air mass from the Polar region and causes rainfall in Western Europe. III. It meets the Labrador current near Vancouver Island and causes dense fog. Select the correct answer using the code given below: (a) I, II and III (c) II only

(b) I and II (d) I and III

195. Which among the following statements characterized EI Nino? I. It occurs at irregular intervals II. It carriers warmer water III. It carries less saline water IV. Its atmospheric equivalent is Southern oscillation

Select the correct answer using the code given below: (a) I and II only (b) II and III only (c) III and IV only (d) I, II, III and IV 196. Which among the following statements provides the best evidence that a river is flowing through a rift valley? (a) the Chambal valley is marked by bad land topography (b) River Tapi does not have Delta but Estuary only (c) River Mahanadi flows through a gorge at Satkosia (d) River Colorado has the Grand Canyon along its valley 197. Which forests lie in the Amazon Basin in South America? (a) Montane forests (b) Tropical rain forests (c) Wet deciduous forests (d) Subtropical mixed forests 198. Which is the rarest naturally occurring element on earth? (a) Zaire (b) Orange (c) Zambezi (d) Niger 199. Which National Highway connects Delhi and Mumbai? (a) NH 6 (b) NH 8 (c) NH 10 (d) NH 12 200. Which of the following are west flowing rivers? 1. Krishna 2. Narmada 3. Mahanadi 4. Sabarmati Select the correct answer using the codes given below Codes: (a) 2 and 4 (b) 1 and 3 (c) 1 and 4 (d) 2 and 3 201. Which of the following crops would be preferred for sowing in order to enrich the soil with nitrogen? (a) Wheat (b) Mustard (c) Sunflower (d) Gram 202. Which of the following features is the product of vulcanicity? (a) Fold Mountain (b) Escarpment (c) Geosynclines (d) Caldera 203. Which of the following has/have been declared as national waterways in India? I. The Allahabad-Haldia Stretch of river Ganga II. The Sadiya-Dhubri stretch of river Brahmaputra III. The Cherla-Rajamundry stretch of river Godavari Select the correct answer using the code give below (a) I and II (b) II and III (c) I Only (d) I, II and III 204. Which of the following instruments is used for measuring the humidity of the air (a) Wind vane (b) Aneroid barometer (c) Wet and dry bulb thermometer (d) Maximum and minimum thermometer

Geography ▌ 2.205 205. Which of the following is not correctly matched? (a) Prime Meridian : 0º (b) International Date Line : 180º (c) Artic Circle : 66

1 N 2

(d) Tropic of cancer : 23

1 S 2

206. Which of the following is/are the chief characteristics of commercial grain farming of the middle latitude grasslands? 1. The size of farms are generally large. 2. Cultivation is highly mechanized. 3. it is type of extensive farming. Select the correct answer using the code given below: (a) 1 and 2 (b) 2 only (c) 1, 2 and 3 (d) 1 and 3 207. Which of the following statements are correct? 1. Ina cyclone, the areas of low pressure is at the centre surrounded by the areas of high pressure 2. In a cyclone, the areas of low pressure surround the area of high pressure 3. In an anti-cyclone, the area of high pressure is surrounded by the areas of low pressure 4. In an anti-cyclone, the area of low pressure is surrounded by the areas of high pressure Select the correct answer using the codes given below: (a) 1 and 2 (b) 1 and 3 (c) 1 and 4 (d) 2 and 4 208. Which of the following statements characterize conversational rainfall? 1. It occurs daily in the afternoon in the Equatorial regions. 2. It is of very short duration. 3. It occurs through Cumulonimbus clouds. Select the correct answer using the codes given below: (a) 1, 2 and 3 (b) 1 and 3 (c) 2 and 3 (d) 1 only 209. Which of the following statements characterize the anticyclones? 1. Anticyclones are low pressure systems. 2. They are characterized by divergent wind circulation. 3. They are indicative of dry weather conditions. Select the correct answer using the codes given below: (a) 2 and 3 (b) 1 and 3 (c) 1, 2 and 3 (d) 1 only 210. Which of the following statements is/are correct with regard to lines of latitude? 1. They are concentric circles numbered from 0º to 90º. 2. They are circles on a globe which are parallel to the Equator and which are to the North and South of the Equator. Select the correct answer using the code given below (a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2

211. Which of the following statements is/are correct with regard to Milky way? 1. it is a spiral galaxy. 2. The solar system resides in one of its spiral arms. Select the correct answer using the codes given below: (a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2 212. Which of the following statements is/are correct? 1. Cyclone is a low pressure system. 2. The wind movement is clockwise in the cyclone of northern hemsisphere. Select the correct answer using the codes given below: (a) 1 Only (b) 2 Only (c) Both 1 and 2 (d) Neither 1 nor 2 213. Which of the following statements is/are correct? 1. The major constituent mineral of granite rock is quartz. 2. The major constituent ‘mineral of sandstone rock is feldspar. 3. The major constituent mineral of limestone rock is dolomite. Select the correct answer using the codes given below: (a) 1, 2 and 3 (b) 3 only (c) 1 and 2 only (d) 2 and 3 only 214. Which of the following statements is/are correct? 1. In comparison to the jupiter, planet Earth display eclipse more frequently. 2. On Mars, only partial solar eclipses are possible. Select the correct answer using the codes given below: (a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2 215. Which of the following statements is/are correct? I. Mumbai receives more rainfall than pune because it is located at the windward side of Western Ghats. II. Vidarbha region experiences semi-arid climate as it is located in a rain shadow region. III. In India, monsoon reaches Kashmir Valley at the last. Select the correct answer using the codes given below: (a) I, II and III (b) I and II (c) II and III (d) I only 216. Which of the following statements is/are correct? 1. Lunar eclipse takes place when the Earth comes directly between the Sun and the Moon 2. Solar eclipse happens when the Moon Comes directly between the Sun and the Earth 3. Lunar eclipse takes place when the Sun comes directly between the Earth and the Moon 4. Solar eclipse happens when the Earth comes directly between the sun and the Moon Select the correct answer using the codes given below: Codes (a) 1, 2 and 3 (b) 3 and 4 (c) 1 and 2 (d) 2 only

2.206

▌ General Knowledge

217. Which of the following statements regarding south-west monsoon in India is/are correct? I. Monsoon reaches the Malabar coast first II. Rajasthan does not get rainfall from south west monsoon III. South-west monsoon retreats when the permanent wind belts start shifting to the south Select the correct answers using the codes given below: (a) I, II and III (b) I and II (c) III only (d) I and III 218. Which of the following two countries are separated by the 49th parallel? (a) North Korea and South Korea (b) South Africa and Zimbabwe (c) USA and Canada (d) France and Germany 219. Which one among the following explains the Earthquakes of the Eastern margins of Asia? (a) Subduction of Pacific plate under Asiatic Plate (b) Subduction of African Plate below European Plate (c) Subduction of Indian Plate under Asiatic Plate (d) Subduction of American Plate under the Pacific Plate 220 Which one among the following is a correct sequence of production of coal in the Indian States in descending order? (a) Jharkhad-madhya Pradesh-West Bengla-Meghalaya (b) West Bengal-Madhya Pradesh-Jharkhand -Meghalaya (c) Jharkhand-West Bengal-Meghalaya-Madhya Pradesh (d) Madhya Pradesh-Jharkhand-west Bengal-Meghalaya 221. Which one among the following is a correct sequence of the Indian ports from North to South? (a) Haldia-Kandla-Paradeep-Kochi (b) Kandla-Haldia-Paradeep-Kochi (c) Kandla-Haldia-Kochi-Paradeep (d) Kochi-Kandla-Haldia-Paradeep 222. Which one among the following is a primary rock? (a) Sedimentary (b) Igneous (c) Metamorphic (d) None of these 223. Which one among the following is not a reason for practicing tank irrigation in the Peninsular India? (a) The undulating relief and hard rocks (b) Little percolation of rain water due to impervious rock structure (c) most of the rivers of Peninsular India are perennial (d) There are many streams which become torrential during rainy season 224. Which one among the following is not a source of renewable energy? (a) Hydroelectricity (b) Solar energy (c) Fuel Cell (d) Wind energy 225. Which one among the following is the best reason for the marked increase in the agricultural production in India in the past decades?

(a) Increases in the area under cultivation (b) Conversion of barren land into agricultural land (c) Use of improved agricultural land use of improved agricultural methods and technologies (d) Priority status given by the successive governments to agricultural sector over the industry sector 226. Which one among the following is the largest temperate desert of the world? (a) Patagoniam desert (b) Taklamakan desert (c) Iranian desert (d) Turkmen desert 227. Which one among the following peaks is the highest? (a) Dhaula Giri (b) Namcha Barwa (c) Nanda Devi (d) Nanga Parvat 228. Which one among the following statements relating to an anticyclone is correct? (a) Anticyclone is a wind system with a high pressure center (b) In anticyclone the movement of wind is inward (c) The contribution of an anticyclone towards determining weather of an area is quite significant (d) The movement of wind is clockwise in an anticyclone of Southern hemisphere 229. Which one among the following states has more population than the other three? (a) Gujarat (b) Madhya Pradesh (c) Odisha (d) Rajasthan 230 Which one of the following African countries in not landlocked? (a) Benin (b) Chad (c) Lesotho (d) Mali 231. Which one of the following does not characterize the Himalayas? (a) Various parallel ranges of the Himalayas form a convex arc (b) There exist syntaxial bends at both the terminals of the Himalayas (c) Indus, Sutlej and Brahmaputra rivers are examples of antecedent drainage (d) The Himalayas are wider in the East than in the West 232. Which one of the following gases, present in the air near the surface of the Earth, has maximum concentrations? (a) Oxygen (O2) (b) Hydrogen (H2) (c) Nitrogen (N2) (d) Methane (CH4) 233. Which one of the following is a low cloud? (a) Cirrocumulus (b) Cirrostratus (c) Altocumulus (d) Nibostratus 234. Which one of the following is correct? Benguela current flows (a) Along Brazilian coast (b) Off Southwest African coat (c) Off Spanish and North African cost (d) Along West Coast of United States

Geography 235. Which one of the following is different in local time between the places located at 165º East and 165º West? (a) 0 h (b) 12 h (c) 22 h (d) 24 h 236. Which one of the following is generally found in sedimentary rocks? (a) Basalt (b) Silica (c) Shale (d) Magnesium 237. Which one of the following is produced by rain water action? (a) Gorge (b) cliff (c) Gully (d) Dome 238. Which one of the following is the correct order in which the gases from the atmosphere disappear as one moves away from the surface of the earth? (a) Carbon dioxide – Oxygen – Nitrogen (b) Oxygen – Nitrogen – Carbon dioxide (c) Carbon dioxide – Nitrogen – Oxygen (d) Nitrogen – Oxygen – Carbon dioxide 239. Which one of the following is the correct sequence of the four stages of water movement in a hydrological cycle? (a) Evaporation – Condensation – precipitation – Infiltration (b) Evaporation – Precipitation – Condensation – Infiltration (c) Infiltration – Evaporation – Condensation – precipitation (d) Condensation – Precipitation – Evaporation – Infiltration 240. Which one of the following is the correct sequence of the hills of Meghalaya, from East to West? (a) Garo – Khasi – Jaintia (b) Jaintia – Khasi – Garo (c) Khasi – Jaintia – Garo (d) Jaintia – Garo – Khasi 241. Which one of the following is the highest mountain peak? (a) Nanga parbat (b) Nanda Devi (c) Kanchanjunga (d) K2 242. Which one of the following is the time required for the Earth to return to a given point in its orbit with reference to fixed star, called? (a) Lunar year (b) Solar year (c) Tropical year (d) Sidereal year 243. Which one of the following pairs is not correctly matched? Month

Position of Sun

(a)

June

Midday Sun overhead at Tropic of Cancer

(b)

December

Midday Sun overhead at Tropic of Capricorn

(c)

March

Midday Sun overhed on Equator

(d)

September

Midday Sun overhead on Arctic Circle

244. Which one of the following pairs of oceans and currents is not correctly? (a) North Atlantic Ocean : Canaries Current (b) Eastern Pacific Ocean : Kuroshio current

▌ 2.207

(c) South Atlantic Ocean : Falkland Current (d) Indian Ocean : Agulhas Current 245. Which one of the following paris is correctly matched? Rivers

Major Tributaries

(a)

Godavari

: Hasdo, Tel

(b)

Maha

: Puma, Penganga, Wainganga

(c)

Krishna

: Manjra, Dudhana, Indravati

(d)

Cauvery

: Kabani, Hemavati, Amaravati

246. Which one of the following processes of weathering belongs to both mechanical and chemical weathering? (a) Crystatllization (b) Exfoliation (c) Hydration (d) Carbonation 247. Which one of the following regions on the surface of Earth has Horse Latitudes? (a) Equatorial Low pressure belt (b) Sub-tropical high pressure belt (c) Sub-polar low pressure belt (d) Polar high pressure belt 248. Which one of the following river flow between Vindhyan and Satpura ranges? (a) Narmada (b) Son (c) Mahe (d) Netravati 249. Which one of the following separates North and South Islands of New Zealand? (a) Foveaux Strait (b) Bass Strait (c) Cook Strait (d) Torres Strait 250. Which one of the following sets of States does not receive most of its rainfall from SW monsoon during the months of June to September? (a) Arunachal Pradesh and Gujarat (b) Assam and Karnataka (c) Himachal Pradesh and Rajasthan (d) Tamil Nadu and Jammu and Kashmir 251. Which one of the following soils is most suitable for cotton cultivatin? (a) Red Soil (b) Black Soil (c) Loamy Soil (d) Laterite Soil 252. Which region of the Earth’s surface is called doldrums? (a) Equatorial low pressure belt (b) Sub-tropical high pressure belt (c) Between 10º to 23.5 North and South Latitudes (d) Sub-polar low pressure belt 253. Who among the following was first to measure the circumferences of the Earth? (a) Eratosthenes (b) Aristotle (c) Hecataeus (d) Herodotus

2.208

▌ General Knowledge

ANSWER KEy 1. 2. 3. 4. 5. 6. 7. 8. 9.

(d)

10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32.

(a)

(d) (c) (b) (a) (a) (b) (c) (b) (a) (a) (b) (a) (a) (b) (d) (a) (c) (d) (d) (b) (d) (d) (a) (c) (a) (d) (a) (a) (d) (b)

33. 34. 35. 36. 37. 38. 39. 40. 41.

(a)

42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64.

(b)

(a) (b) (d) (d) (d) (c) (a) (a) (b) (a) (c) (a) (a) (a) (c) (d) (b) (b) (a) (a) (c) (b) (a) (b) (d) (a) (d) (c) (b) (d)

65. 66. 67. 68. 69. 70. 71. 72. 73.

(a)

74. 75. 76. 77. 78. 79. 80. 81. 82. 83. 84. 85. 86. 87. 88. 89. 90. 91. 92. 93. 94. 95. 96.

(a)

(c) (a) (c) (a) (c) (c) (b) (d) (c) (b) (c) (c) (c) (d) (d) (a) (c) (b) (c) (d) (d) (c) (a) (d) (d) (b) (a) (b) (d) (a)

97. 98. 99. 100. 101. 102. 103. 104. 105.

(d)

106. 107. 108. 109. 110. 111. 112. 113. 114. 115. 116. 117. 118. 119. 120. 121. 122. 123. 124. 125. 126. 127. 128.

(a)

(b) (c) (c) (c) (d) (n) (a) (b) (c) (c) (c) (a) (d) (d) (d) (b) (a) (a) (c) (c) (a) (b) (a) (b) (c) (a) (b) (d) (a) (c)

129. 130. 131. 132. 133. 134. 135. 136. 137.

(c)

138. 139. 140. 141. 142. 143. 144. 145. 146. 147. 148. 149. 150. 151. 152. 153. 154. 155. 156. 157. 158. 159. 160.

(a)

(a) (d) (d) (a) (d) (d) (a) (a) (b) (d) (d) (b) (c) (c) (a) (a) (b) (a) (a) () (a) (a) (c) (b) (c) (c) (b) (b) (c) (a)

161. 162. 163. 164. 165. 166. 167. 168. 169.

(a)

170. 171. 172. 173. 174. 175. 176. 177. 178. 179. 180. 181. 182. 183. 184. 185. 186. 187. 188. 189. 190. 191. 192.

(c)

(b) (c) (b) (c) (d) (d) (c) (d) (c) (a) (a) (c) (c) (a) (d) (c) (a) (a) (b) (c) (c) (c) (d) (a) (c) (b) (c) (d) (a) (c)

193. 194. 195. 196. 197. 198. 199. 200. 201.

(d)

202. 203. 204. 205. 206. 207. 208. 209. 210. 211. 212. 213. 214. 215. 216. 217. 218. 219. 220. 221. 222. 223. 224.

(d)

(b) (b) (d) (b) (b) (c) (b) (a) (d) (a) (b) (d) (c) (b) (a) (a) (c) (c) (a) (*) (b) (b) (c) (a) (c) (a) (a) (b) (b) (d)

225. 226. 227. 228. 229. 230. 231. 232. 233.

(c)

234. 235. 236. 237. 238. 239. 240. 241. 242. 243. 244. 245. 246. 247. 248. 249. 250. 251. 252. 253. 254.

(d)

(c) (a) (a) (a) (b) (a) (d) (c) (b) (b) (b) (c) (c) (a) (a) (d) (d) (d) (b) (c) (b) (a) (c) (d) (b) (a) (a) (d)

C

H

3

A

P

T

E

R

Indian Polity The IndIan ConsTITuTIon salient Features 1. It is the longest written constitution in the world consisting of 24 chapters, 395 articles and 12 schedules. 2. It proclaims India a Sovereign Democratic Republic. 3. Fundamental Rights are guaranteed to all citizens of India. 4. Directive Principles of State Policy are incorporated. 5. It established the parliamentary system of government, that is, the president of the Union is the constitutional head, the council of ministers or the Union Cabinet is the real executive and is responsible to the Lok Sabha. 6. It is Quasi-federal, that is, federal in form (in normal times) but unitary in spirit (in emergencies). 7. Article 368 provides that the Constitution is neither too rigid (as some provisions can be amended by a simple majority) nor flexible (as some provisions require special majority for amendment). 8. It declares India a secular state. 9. It guarantees single citizenship to all citizens. 10. It introduced adult franchise, that is, every adult above 18 years has the right to vote (prior to the 61st Amendment 1989, the age limit was 21 years); and the system of joint electorates. 11. It established an integrated as well as independent judiciary; the Supreme Court acts as a guardian of the Constitution in place of the Privy Council.

to all its citizens: JUSTICE, social, economic and political; LIBERTY of thought, expression, belief, faith and worship; EQUALITY of status and of opportunity; and to promote among them all FRATERNITY assuring the dignity of the individual and the unity and integrity of the Nation; IN OUR CONSTITUENT ASSEMBLY this twenty-sixth day of November, 1949, do HEREBY ADOPT, ENACT AND GIVE TO OURSELVES THIS CONSTITUTION.” The key words mentioned in the Preamble are sovereign, socialist, secular, democratic, republic (defining the nature of the Constitution), justice, liberty, equality and fraternity (describing the nature of the Indian Constitution). The Preamble points out that the source of authority of the Constitution and, therefore, that of the state, are the people of India. Thus, the basic principle of democracy that sovereignty rests with the people, is enshrined in India’s Constitution.

The Preamble The draft of the Preamble was moved by Pt Jawaharlal Nehru on 13 December 1946, and is based on ‘Objectives Resolutions’ picked from the American model. It was adopted by the Constituent Assembly on 22 January 1947. The edifice of our Constitution is based upon the basic element mentioned in the Preamble. If any of these elements are removed, the structure will not survive and it will not be the same Constitution or it cannot maintain its identity. The Preamble has been amended only once so far, by the 42nd Amendment in 1976, which added three words “secular, socialist and integrity” and now the Preamble reads as follows: “WE, THE PEOPLE OF INDIA, having solemnly resolved to constitute India into a SOVEREIGN SOCIALIST SECULAR DEMOCRATIC REPUBLIC and to secure

Under the Citizenship Act 2003, those eligible to become citizens of India as on 26 January 1950, could apply for dual Indian citizenship. The government has extended dual citizenship to all those who were holding the Person of Indian Origin Card (PIOC) and who had migrated from India after the formation of the Indian Republic. Persons of Indian origin who were citizens of Australia, Canada, Finland, France, Greece, Ireland, Israel, Italy, the Netherlands, New Zealand, Portugal, Cyprus, Sweden, Switzerland, the United Kingdom and the United States of America were eligible to apply for dual citizenship. A person who has been at any time a citizen of Pakistan, Bangladesh or any other country that the government may notify in future is not entitled to dual citizenship.

Citizenship in the Indian Republic Part II (articles 5–11)

The Constitution provides for only single citizenship and there is no separate citizenship of states. Thereby, at the commencement of the Constitution every person domiciled in the territory of India, born in the territory of India, or who has been a resident of India for not less than five years immediately preceding the commencement of the Constitution (26 January, 1950) is a citizen of India. Dual Citizenship 2005

2.210

▌ General Knowledge

Citizenship act 2003 The Act has amended the Citizenship Act, 1955. It simplifies the procedures to facilitate the reacquisition of Indian citizenship by persons of full age who are children of Indian citizens and former Indian citizens. It provides citizenship of India to persons of Indian origin, belonging to specified nations. The Act makes acquisition of Indian citizenship and naturalization more straight and prevent illegal migrants from becoming eligible for Indian citizenship. The Act is guided towards the exigencies of the global world. It assumes that the intellectual and financial capabilities of the PIOs would contribute towards country’s development.

Fundamental Rights (FRs) Part III (articles 12–35) Part III of the Constitution embodies fundamental rights of the people. The Constitution affirms the basic principle that every individual is entitled to enjoy certain rights as a human being and the enjoyment of such rights does not depend on the will of any majority or minority. These rights include all basic liberties which make life significant and democracy fruitful; which are essential for the proper, moral and material uplift of the people. Classification of Fundamental Rights Originally, seven FRs were listed in the Constitution. However, after the 44th Amendment Act, 1976 there are now only six FRs, viz., 1. Right to equality (Articles 14–18) Right to equality guarantees following five rights: (i) Equality before law (Article 14) (ii) Prohibits discrimination by the State on the grounds of religion, race, caste, sex or place of birth (Article 15) (iii) Equality of opportunity in matters of public appointment (but some posts may be reserved for backward classes) (Articale 16) (iv) Abolition of untouchability (Article 17) (v) Abolition of titles, excepting military and academic (Article 18) Right to Information What does Right to Information mean? It includes the right to—(i) inspect works, documents, records: (ii) take notes, extracts or certified copies of documents or records: (iii) take certified samples of material: (iv) obtain information in form of printouts, diskettes, floppies, tapes, video cassettes or in any other electronic mode or through printouts.[S.2(I)] What does information mean? It means any material in any form including records, documents, memos, e-mails, opinions, advices, press releases, circulars, orders, logbooks, contracts, reports, papers, samples, models, data material held in any electronic form and information relating to any private body which can be accessed by a public authority under any other law for the time being in force but does not include “file notings” [S.2(f)]. When does it come into force? It comes into force on the October, 12 2005. Some provisions have come into force with immediate effect viz. obligations of public authorities [S.4(1)], designation of Public Information Officers and Assistant Public Information Officers [S.5(1) and 5(2)], constitution of Central Information Commission (S.12 and 13), constitution of State Information

Commission (S.15 and 16), non-applicability of the Act to Intelligence and security organizations (S.24) and power to make rules to carry out the provisions of the Act (S.27–28). Who is covered? The Act extends to the whole of India except the state of Jammu and Kashmir. (S.12) 2. Right to freedom (Article 19–22) Under Article 19(1), there are now six rights available, which are as follows: (i) Freedom of speech and expression (Article 19(1)(a)) (ii) Freedom to assemble peacefully and without arms (Article 19(1)(b)) (iii) Freedom to form associations and unions (Article 19(1)(c)) (iv) Freedom of movement throughout India (Article 19(1)(d)) (v) Freedom to reside and settle in any part of India (Article 19(1)(e) Table: Parts and Articles of the Constitution Part/Article Part I/Articles 1–4 Part II/Articles 5–11 Part III/Articles 12–35 Part IV/Articles 36–51 Part IV-A/Article 51A Part V/Articles 52–151 Part VI/Articles 152–237 Part VII/Article 238 Part VIII/Articles 239–242 Part IX/Articles 242–243O Part IX–A/ Articles 243P–243ZG Part X/Articles 244–244A Part XI/Articles 245–263 Part XII/Articles 264–300A Part XIII/Articles 301–307 Part XIV/Articles 308–323 Part XIV-A/323A–323B Part XV/Articles 324–329A Part XVI/Articles 330–342 Part XVII/Articles 343–351 Part XVIII/Articles 352–360 Part XIX/Articles 361–367 Part XX/Article 368 Part XXI/Articles 369–392 Part XXII/Articles 393–395

Deals with Territory of India, admission, establishment or formation of new states Citizenship Fundamental Rights Directive Principles of State Policy Duties of a citizen of India. It was added by the 42nd Ammendment in 1976 Government at the Union level Government at the Union level Government at the State level Repealed by 7th Amendment, 1956 Administration of Union Territories The panchayats The muncipalities Scheduled and tribal areas Relations between the Union and states Finance, property, contracts and suits Trade, commerce and travel within the territory of India Services under the Union and states Added by the 42nd Articles Amendment in 1976 and deals with administrative tribunals. Election and Election Commission Special provision to certain classes SC/STs, OBCs and Anglo Indians Official languages Emergency provisions Miscellaneous provisions Amendment of Constitution Temporary, transitional and special provisions Short title, commencement and repeal of the Constitution

Indian Polity ▌ 2.211

Appendix in the Constitution Appendix I

The Constitution (Application to Jammu and Kashmir) order, 1954.

Appendix II

Re-statement, with reference to he present text of the Constitution, of the exceptions and modifications subject to which the Constitution applies to the state of Jammu and Kashmir.

Appendix III

Extracts from the Constitution (44th Amendment) Act, 1978.

Appendix IV

Extracts from the Constitution (86th Amendment) Act, 2002.

Appendix V

Extracts from the Constitution (88th Amendment) Act, 2003.

(vi) Freedom to practice any profession, or to carry on any occupation trade or business (Articles 19(1)(g) and 19(6)) It has also been provided that no person can be convicted unless he has violated the law in force, nor subjected to penalty greater than permissible under the law; nor prosecuted and punished for the same offence more than once. He cannot be compelled to be witness against himself, nor deprived of his life and personal liberty without authority of the law unless detained under the Preventive Detention Act in which case also, no person can be detained beyond a period of three months without being advised by the prescribed authority (Articles 20, 21 and 22). 3. Right against exploitation (Articles 23–24) Right against exploitation bans traffic in human beings, forced labour and the employment of children below the age of 14 years in a factory, mine etc. 4. Right to freedom of religion (Articles 25–28) According to right to freedom of religion, except when it is not in the interest of public order, morality, health, etc., every citizen is entitled to freedom of conscience and right to profess, practice and propagate any religion freely. 5. Cultural and educational rights (Articles 29–30) Every community has full freedom to run its own institutions, to preserve its own language, script and culture; to receive education (in state-owned institutions) and administer educational institutions of its choice. 6. Right to constitutional remedies (Articles 32–35) Every citizen can move the Supreme Court or High Courts or any other court for enforcement of his fundamental rights, through judicial writs of habeas corpus, mandamus, quo warranto, certiorari, etc. (These are explained separately under ‘writs’) and the like under Part III. This right, however, can be suspended during the operation of emergency proclaimed by the president of India.

Right to Property (Article 31) This used to be a Fundamental Right but has now become only a legal right. Originally, as per this right, the state cannot acquire any property without compensation and no person can be deprived of his property except by the authority of law. Earlier, in the event of acquisition for public purpose, compensation was to be paid but according to the Constitution (25th) Amendment Act 1971, instead of compensation, only a specified amount is to be paid. The Janata Government on 20 June 1978 omitted the Right to Property by Constitution (44th) Amendment Act 1978. Hence, the right of a citizen to acquire, hold or dispose of property ceases to be a Fundamental right. However, under Article 300A, right to property is a constitutional right. It has, therefore, remained now, an ordinary legal right. Suspension of FRs Fundamental Rights (FRs) are not absolute. Individual rights, however basic, cannot override the country’s security and general welfare. Therefore, in case emergency is declared the President may suspend all or any of the fundamental rights and may also suspend right of the people to move High Courts or the Supreme Court for restoration of these rights.

directive Principles Part IV (articles 36–51) Part IV of the Constitution deals with the directive principles of State policy. These principles command the State and every one of its agencies to follow certain fundamental principles while they frame their policies. These principles are assurances to the people, on the one hand, as to what they can expect from the State and, on the other, are directives to central and State governments to establish and maintain a social order. Originally, unlike the fundamental rights, these were only directive principles which could be enforced by the judiciary. These principles are now enforceable through law courts, vide the 42nd Amendment Act. Main Directive Principles 1. Provision of adequate means of livelihood to all. 2. Equitable distribution of wealth among all classes of people. 3. Protection of children and youth. 4. Equal pay for equal work to both men and women. 5. Prevention of cow slaughter. 6. Free and compulsory education for children up to age of 14. 7. The right to work, to education, to public assistance in case of unemployment and old age, sickness and disability. 8. Prohibition of liquor. 9. Establishment of village panchayats. 10. Protection of historical and national monuments. 11. Separation of Judiciary from executive to secure for all citizens a uniform civil code throughout the country. 12. Promotion of international cooperation and world security. 13. Free legal aid from the state to weaker sections of society. 14. State to protect natural environments, forests, wildlife.

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In a nutshell, these principles can be classified into four categories: (i) Economic principles (ii) Gandhian principles (iii) Principles for the promotion of international understanding (iv) Social principles

Duties of a Citizen of India Duties of a citizen of India were not included in the original Constitution. These have been added by the 42nd Amendment in 1976 as Article-51A in Part IV-A of the Constitution. There are 10 fundamental duties, viz., 1. To abide by the Constitution and respect its ideals and institutions, the national flag and national anthem. 2. To cherish and follow the noble ideas which inspired our national freedom struggle. 3. To uphold and protect the sovereignty, unity and integrity of India. 4. To defend the country and render national service when called upon to do so. 5. To promote harmony and spirit of common brotherhood among all the people of India, transcending religious, linguistic and regional sectional diversities; to renounce practices derogatory to the dignity of women. 6. To value and preserve the rich heritage of our composite culture. 7. To protect and improve the natural environment, including forests, lakes, rivers, wildlife and to have compassion for living creatures. 8. To develop a scientific temper, humanism and the spirit of inquiry and reform. 9. To safeguard public property and to abjure violence. 10. To strive towards excellence in all spheres of individual and collective activity so that the nation constantly rises to higher levels of endeavour and achievements. Another duty was added by the 86th Constitutional Amendment Act, 2002, i.e., to provide opportunities for education to his child or ward between the age of 6 and 14 years.

election Elected by an electoral college consisting of: (i) elected members of Parliament (both houses); (ii) elected members of the State legislatures; (iii) elected member of the legislative assemblies of the UTs of Delhi and Pondicherry (added by 70th Amendment Act, 1992 w.e.f. 1st June 1995). The election is conducted by secret ballot in accordance with the system of proportional representation by means of a single transferable vote Qualifications for Election as President 1. He must be a citizen of India. 2. He must not be less than 35 years of age. 3. He must be qualified to be an elected member of the Lok Sabha but shall not be a member. 4. He must not be holding any office of profit under the government of India or any other government. A governor of a state, or a minister of the Union or State or vicepresident of the Union will not, however, be considered as persons holding office of profit and as such they are eligible for election. Tenure Elected for five years but is eligible for immediate reelection and can serve any number of terms. Procedure for Removal of President (Article 61) May be removed from office for violation of the Constitution before the expiry of his term by impeachment. Such impeachment can be moved in either house of Parliament by means of a resolution after 14 days’ notice in writing, duly signed by at least one-fourth of the total membership of the House concerned, is given. The charges must be investigated. He has the right to be heard and defend himself during investigation. If the charges are sustained by a resolution adopted by a two-thirds majority, the President can be removed from office. Presidential Immunity He cannot be questioned by any court for the action taken by him in the discharge of his duties. No criminal proceedings can be launched against him during his tenure. Powers He is the chief executive head of the Indian union and supreme commander of the armed forces. His powers can be classified as under:

There shall be a president of India (Article 52), the executive power of the Union vested in him, exercised by him directly or through officials subordinate to him in accordance with the Constitution of India (Article 53).

executive and administrative powers He appoints the senior officials of the State like the prime minister, ministers, attorney general, comptroller and auditor general, judges, chief justice, governors, financial commissioner, UPSC members and chief election commissioner. All executive action is taken in his name. All Union Territories are under the president of India. He can appoint commissions to investigate into conditions of SCs, STs and OBCs. He can declare any area as schedule area and is empowered with the administration of scheduled areas and tribal areas. He can also appoint inter-state councils to promote centre-state and inter-state cooperation. He directly administers UTs through administrators appointed by him.

Position He is the constitutional head of parliamentary system of government. He represents the nation but does not rule it. The council of ministers is vested with the real power. The president is the supreme commander of the armed forces. He is liable to act on the advice of ministers who are responsible to the legislature (Parliament).

Legislative powers (a) Appoints 12 members to the Rajya Sabha and two Anglo-Indian members to the Lok Sabha; (b) Summons and prorogues the Parliament, dissolve House of People, can also summon joint sitting of both houses; (c) Decides on disqualification of members of the Parliament, in consultation with the Election

Machinery of the Union Government Part V The Union executive of India, as in the parliamentary type of government, consists of the president and the council of ministers.

The President

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PRESIDENT

(Legislative) Parliament

(Executive) Council of ministers

(Lok Sabha) House of people

(Rajya Sabha) Council of states

Not more than 552 members (545 in 14th Lok Sabha)

Not more than 250 members (At present 245)

Not more than Not more than 530 representatives 20 representatives of States: of UTs (At present 530) (At present 13)

Not more than 2 nominated Anglo-indians (At present 2)

12 nominated by the president (At present 12)

Not more than 238 representatives of states and UTs (elected indirectly) (At present 229 from states) (At present 4 members from UTs)

Figure: An Overview of the Indian Parliament Commission; (d) Assents, or withholds his assent to any Bill passed by the Parliament; (e) Rejects Bills for reconsideration; (f) Issues ordinances; (g) Sends messages to the Parliament. Financial powers (a) Causes the budget to be laid before the Parliament; (b) Sanctions introduction of money bills; (c) Constitutes a finance commission every five years to apportion revenue from income-tax between the centre and the states; (d) Can use contingency fund of India to meet any unforeseen expenditure; (e) No demand for any grant can be made except on his commendation. Judicial powers Empowered to grant pardons, reprieve, remit the sentences or suspend, remit or commute punishment. Appoints the Chief Justice and judges of the Supreme Court (in consultation with the Chief Justice) and High Courts. diplomatic powers (a) He represents India in international forums and affairs; (b) International treaties and agreements are negotiated on behalf of the president (subject to approval from the parliament); (c) He receives international diplomats and sends Indian diplomats to foreign countries. Military powers (a) He is supreme commander of the defence forces of India; (b) Appoints chiefs of the Army, the Navy and the Air Force; (c) empowered to declare war or conclude peace (subject to Parliament’s approval). emergency powers Proclamation of emergency: The President of India can proclaim emergency, if he is satisfied that it is justified and can take under his direct charge the administration of any state. He has to get the recommendation of the cabinet in writing before proclaiming emergency. Pardoning powers Article 72 empowers President to grant pardons where, (a) Punishment or sentence is for offence against a Union Law; (b) Punishment or sentence is by a court martial (military

court); or (c) Sentence is a death sentence. The President’s pardoning powers includes pardon, commutation, remission, respite and reprieve. The pardoning powers of president are independent of judiciary and is an executive power, but, while exercising this power, the president does not sit as a court of appeal. These are only theoretical powers of the president. In fact, all these powers are exercised by the president on the advice of the cabinet and the advice of the cabinet is binding on the president. As such, he is only a ‘titular head’.

Vice-President of India election The vice-president is elected by members of an electoral college consisting of the members of both Houses of Parliament. A secret ballot is cast in accordance with the system of proportional representation by means of a single transferable vote. However, his election is different from that of the president as the State legislatures have no part in it. Term of Office The vice-president is elected for a term of 5 years and is eligible for immediate reelection. However, his office may be terminated earlier than the field term of 5 years, either by: (i) resignation to be in writing addressed to the president; or (ii) removal by a resolution of the council of states passed by majority of its members and agreed to by the Lok Sabha (Article 67). Functions (i) Acts as ex-officio chairman of the Rajya Sabha. (ii) Officiates as president in case of death, resignation or removal of the latter till a new president is elected. This period can be extended to a period of six months. (iii) The vice-president functions as the president when the latter is unable to discharge his functions due to illness, absence or any other cause. It is to be noted that the office of the vice-president of India was created with a view to maintain the political continuity

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of the Indian State, though the Constitution has otherwise not assigned any significant function to the vice-president in that capacity Qualifications for Vice-President 1. He must be a citizen of India. 2. He must not be less than 35 years of age. 3. He must be eligible for election as a member of the Rajya Sabha. 4. He must not be a member of either House of Parliament or any state legislature. 5. He should not hold any office of profit in the government.

The Prime Minister As per the Constitution of India, the prime minister is de facto executive (real executive) while the president is de jure executive (nominal authority executive). The prime minister of India heads the government while the president heads the state. He is the leader of the party that enjoys a majority in the Lok Sabha. Functions (i) He occupies a unique position in the council of ministers. He selects other ministers, who are appointed by the president on the advice of the prime minister. (ii) He presides over cabinet meetings. As chairman of the council of ministers, he exercises a casting vote. (iii) He is the principal link between the president and the cabinet. It is the prime minister who keeps the president informed of the decisions of the council of ministers (Article 78). (iv) He guides the ministers and coordinates the policies of various departments and ministries and exercises general supervision over all the departments. (v) In the Parliament, he is leader of the Lok Sabha and chief spokesman of the government. (vi) He is the chairman of the Planning Commission, Inter-State Council, National Development Council and National Integration Council. (vii) He is the chief confidential adviser to the president and plays a major role in structuring foreign policy. The prime minister is assisted by the Prime Minister’s Office (PMO), which is a staff agency meant for providing secretarial assistance and crucial advice to the prime minister. It is an extra-constitutional body, has no attached and subordinate office, headed politically by the prime minister, and administratively, by the principal secretary. Term of Office The term ordinarily does not exceed five years, but the prime minister can be reappointed if his leadership is explicitly recognized in the Lok Sabha. He holds office with the consent of the president till a new Lok Sabha is formed. Even when the Lok Sabha is dissolved by the president, he can ask the prime minister to hold office till new elections are held. Resignation If the government is defeated in the Rajya Sabha, it is not obligatory for the prime minister to resign. However, if defeated in Lok Sabha—the cabinet and the prime minister both have to resign as they are responsible and answerable to the Lok Sabha.

Appointment of the Prime Minister Appointment by the President (Article 75), on the basis of his being leader of the party that enjoys a majority in the Lok Sabha. If, however, no party gets an absolute majority in the Lok Sabha; or when a prime minister resigns or dies and there is no one to take his place, the President has an effective choice (because the Constitution is silent) of choosing a prime minister for a period of six months and such a person must get himself elected to the Lok Sabha within this period.

Council of Ministers The Constitution provides for a council of ministers headed by the prime minister to aid and advise the president in the exercise of his functions. It is the policy-making body of the country and the government in the real sense. Constitution A council of ministers is composed of all the union cabinet ministers, ministers of State and deputy ministers. A cabinet minister is the head of one or more departments. The ministers of state, again, are of two categories. Some ministers of State have independent charge of their ministry, while other act under the supervision of a cabinet member. Sometimes they may be called to attend meetings of the Cabinet when an important issue pertaining to their department is discussed. Deputy ministers work under the supervision of either a cabinet minister or a minister of state. Their main function is to assist the cabinet minister or minister of state, as the case may be, in performing their functions. There is another extra-constitutional body called ‘inner cabinet’ or ‘kitchen cabinet’ consisting of the prime minister as head and some most important ministers in whom the prime minister has faith. It advises the prime minister on important political and administrative issues and help him in taking critical decisions. Tenure They hold office with the sanction of the president, which in fact means the sanction of the prime minister. The prime minister can thus ask any minister to resign; he can also advise the president to dismiss a minister; and he can also reconstitute the ministry and drop any minister. Powers and Functions of the Cabinet (i) It formulates the policies of the country. (ii) It introduces all important bills and resolutions in Parliament. (iii) It exercises controls over higher appointments like constitutional authorities and senior administrators. (iv) It determines the foreign policy of the country and also approves international agreements. (v) Cabinet ministers of the council render advice to the president regarding proclamation of emergency. Hence, the function of the cabinet is executive, legislative and financial. It is the real government, working under the leadership of the prime minister.

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Appointment of the Council of Ministers The prime minister is appointed by the President and other ministers are appointed by the president on the advice of the prime minister. The portfolios are given by the prime minister. Any person from outside the legislature can also be appointed as a minister, but he cannot continue in that capacity for more than six months unless he secures a seat in either house of Parliament (by election or nomination, as the case may be). Qualification of Members 1. He must be a citizen of India. 2. He must not be below 30 years of age. 3. He must be a Parliament elector in state in which he is seeking election. 4. He should not hold office of profit under the government. 5. He should not be of unsound mind. 6. He should not be an undischarged insolvent. 7. He should not be disqualified under any law made by the Parliament. Cabinet vs Council of Ministers Cabinet is a small body of ministers who occupy a prominent position in the party and hold important portfolios. It meets frequently and takes all important policy decisions of the government. Whereas, the council of ministers is a wider body, which does not meet as a body to transact government business and therefore has no collective functions. It implements decisions taken by the cabinet. It is collectively responsible to the Lok Sabha, whereas cabinet enforces the collective responsibility of the council of ministers to the Lok Sabha. Collective Responsibility vs Individual Responsibility As per the Constitution, the council of ministers is collectively responsible to the legislature (House of people) and therefore, shall be under constitutional obligation to resign as soon as the house is dissolved. Individually, however, they are responsible to the executive and shall be liable to dismissal by the president, even if they enjoy the confidence of the legislature.

The Parliament The Parliament is the union legislature of India which comprises: 1. The president 2. The council of states (Rajya Sabha) 3. The house of people (Lok Sabha)

Rajya sabha It is the Council of States which is also known as the upper house of Parliament. Constitution It is made up of representatives from the states or the constituent units of the Indian Union and members nominated by the president, who have distinguished themselves in literature,

arts, science or social service. The rationale behind the practice of nomination is to provide eminent persons a place in the Rajya Sabha without going through the process of election. The Fourth Schedule of the Constitution deals with the allocation of seats in the Rajya Sabha. strength (article 80) The total strength of the Rajya Sabha does not exceed 250 members which comprises 238 members representing the States and UTs, who come through election and 12 members who are nominated by the President. Formation 12 members are nominated by the president and the remaining seats are distributed between the states and UTs. The State representatives are elected by members of the Legislative Assembly by method of a single transferable vote. The representatives of UTs are chosen in such a manner as the Parliament may prescribe. Chairman of Rajya Sabha The vice-president of India is the ex-officio chairman. The deputy chairman is elected from the members of the Rajya Sabha. Tenure The Rajya Sabha is a permanent body, not subject to dissolution; one-third of its members retiring after every two years. Thus, every member enjoys a 6-year tenure. Functions The Rajya Sabha shares with the Lok Sabha the power of amending the Constitution. It can originate any bill other than a money bill; refer the charge of impeachment against the president. The elected members of the Rajya Sabha take part in the election of the president and the vice-president along with the members of the Lok Sabha.

Lok sabha Also called the house of people or the lower house of Parliament. Its members are elected on the basis of universal adult franchise. Anyone who is a citizen of India, has attained 18 years of age and whose name is in the voters’ list is entitled to cast a vote in the election of the Lok Sabha. Constitution The Lok Sabha consists of members elected by direct election from territorial constituencies in states and UTs and two nominated members (Anglo-Indians) by the president. strength The maximum strength of the Lok Sabha has now been fixed at 550, out of which 530 represent the states and 20 represent the UTs. In addition, the president can nominate 2 members (Anglo-Indians), if in his opinion this community is not adequately represented. Chairman (Presiding Officer) The presiding officer of Lok Sabha is the speaker, who is elected by its members. The speaker of the house elects a deputy speaker, to discharge the duties of speaker in his absence. Qualification of Lok Sabha Member 1. 2. 3. 4. 5. 6.

He must be a citizen of India. He must not be below 25 years of age. He must not hold office of profit under the government. He must not be of unsound mind. He must not be an undischarged insolvent. He must not be disqualified by or under any law made by the Parliament.

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Supreme Court of India

Functions of the Parliament (Both Houses)

The Supreme Court stands at the apex of the judicial system of India and functions from New Delhi.

1. 2. 3. 4.

Enact laws for the good governance of the country. Control the finances of the Union. Elicit information by asking questions and supplementaries. Move adjournment motions and thus criticize the government for any failure. 5. Can impeach the President by resolution for violation of the Constitution. 6. Can pass a no-confidence motion to dismiss the government.

Composition The Supreme Court, the highest court of justice in the land, consists of one chief justice and 25 other judges.

Tenure The normal term of the house is five years, subject to dissolution even prior to the expiry of its term. During an emergency the term may be extended. It was raised to six years during an emergency with the 42nd Amendment Act 1976 but status quo of five years has been maintained with the 44th Amendment Act of 1978. Functions (i) It can make laws on subjects contained in the Union and Concurrent lists. (ii) In certain cases it can also make laws mentioned in the State lists. (iii) It has vast financial powers. It passes the budget and authorizes income and expenditure of the government. (iv) It exercises control over the executive. (v) The Lok Sabha also has a share in the election of the president and vice-president along with Rajya Sabha. Rajya Sabha vs Lok Sabha The members of Rajya Sabha have a tenure of six years whereas members of the Lok Sabha usually have a term of five years, depending on the consent of the prime minister. The Rajya Sabha is a permanent body which cannot be dissolved, whereas the Lok Sabha can be dissolved by the president or is automatically dissolved with the fall of the government or resignation of the prime minister. Of the two houses, the Lok Sabha is more powerful as it has been given the sole power to approve the expenditure of the government and ministers. The Rajya Sabha, on the other hand, has no power over money bills (or budget). In case of other bills, both houses need to approve and any difference between the two is settled at a joint sitting of both Houses.

appointment The chief justice of India is appointed by the president. Other judges of the Supreme Court are appointed by the president in consultation with the chief justice. He may also consult other judges of the Supreme Court and High Courts while appointing a judge to the Supreme Court. Tenure Judges of the Supreme Court can hold office up to the age of 65 years. Qualifications Any citizen of India who has been a judge of a High Court for 5 years, or in the opinion of the president, is an eminent jurist or who has been a practising advocate in a High Court for a period of 10 years can be nominated as a judge of the Supreme Court (Article 124(3)). seat The Supreme Court normally sits at New Delhi. However, it can hold its meetings anywhere in India. The decision in this regard is taken by the chief justice of India in consultation with the president. Removal of a Judge A judge of the Supreme Court can only be removed from the office by an order of the president after an address by each House of Parliament; supported by a majority of the total membership of the houses and by a majority of not less than twothirds of the members present and voting. He can be removed only on the grounds of: 1. Proved misbehaviour. 2. Incapacity to act as a judge (Article 124(4)). Functions of the Supreme Court of India 1. It decides disputes between the Union Government and the States. 2. It hears certain appeals in civil and criminal cases from the High Courts. 3. The President can refer any question of law or fact of sufficient importance to the Supreme Court for its opinion. 4. It can issue directions or writs for the enforcement of any of the Fundamental Rights referred by the Constitution. THE JUDICIARY

Supreme Court of India (Chief justice and 25 other judges) High Court (in each state) (Over 517 Judges in 18 High Courts in India) District and Sessions Judge’s Court (In districts) (Over 2067 D&S Judges and 1375 Senior Civil Judges and Chief Judicial Magistrates)

Subordinate Judge’s Court (Civil) Munsiff’s Court (About 4307 Munsiffs/ Subordinate Judges)

(In Metropolitan areas) Metropolitan Magistrate's Court

City Civil and Sessions Courts

Provincial Small Cause Court

Court of Session (Criminal)

Subordinate Magistrate’s Court

Nayay Panchayats Judicial Magistrates

Figure: Judicial System in India

Presidency Small Cause Court

Panchayat Adalats Executive Magistrates

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The state executive The executive at the State level has been modelled on the central pattern. It consists of (i) The governor (ii) The chief minister and council of ministers GOVERNMENT OF STATES

Governor

Council of Ministers (Executive)

(Legislatives)

Legislative Council (Vidhan Parishad)

Legislative Assembly (Vidhan Sabha)

Figure: An Overview of State Executive

The Governor The governor is the nominal executxive head of the State and all executive authority of the State is vested in him. appointment Governors of states are appointed by the president of India. Tenure A governor is appointed for a term of 5 years and holds office with the consent of the president. Remuneration The governor draws a monthly salary of Rs 1,00,000 in addition to free residence, medical facilities, etc. 1. 2. 3.

4.

Qualifications of the Governor Must be a citizen of India Must have completed 35 years of age. Should not be a member of either House of parliament or the state legislature, and if he is member either of Parliament or a state legislature, he ceases to be so on assuming charge of governorship. Must possess the qualification prescribed for membership of the state legislature Must not hold any office of profit.

Financial powers He causes the annual and supplementary budgets to be presented in the legislature each year. No demand for a grant can be introduced in the legislature without his approval. He is empowered to sanction advances to the government out of the contingency fund of the State to meet unforeseen expenses of the State pending formalities by the legislature. Governors are Appointed and not Elected There are several considerations which prompted the constituent Assembly to have the Governor appointed and not elected; important among them being: 1. An elected governor may be considered superior to the chief minister because the latter is elected only from one constituency; 2. To have a strong central government to counter the separatist tendencies in the country, it is better to have the governor appointed by the president and not elected. Judicial powers He is empowered to grant pardon, reprieve or remit the sentence or to suspend and commute the sentence of any offence covered by the executive powers of the state. discretionary powers Article 183(1) empowers him to exercise certain functions without the advice of the chief minister insofar as he is required by the Constitution to exercise his functions in his discretion. Article 162(2) enjoins that in case of a controversy as to whether or not a particular matter pertains to his discretionary powers, the decision of the governor in his discretion shall be final.

President vs Governor The governor has no powers to appoint judges of the State High Courts but he is entitled to be consulted by the president in the matter. Unlike the president he has no emergency powers to meet the situation arising from external or internal aggression.

The Chief Minister The leader of the party that commands a majority in the legislative assembly, and is invited by the governor to become the chief minister and form his cabinet.

Powers of the Governor His powers can be classified as:

appointment He is appointed by the governor of the concerned State on the basis of his leadership of the majority party in the Legislative Assembly.

executive powers He is the executive head of the state. He appoints the council of ministers to aid and advise him to the discharge of his functions. He frames rules for the convenience and efficient administration of the state. His executive powers are wide enough to cover all the subjects included in the State list.

A Non-Member as Chief Minister A person, who is not a member of the State legislature, can be appointed as chief minister. In case of such an appointment, the person concerned is required to get himself elected as a member of the State legislature within six months of his appointment, failing which he is removed from office.

Legislative powers He accords assent to bill passed by the State legislature without which the bill cannot become an Act. He may withhold his assent to a bill, may reserve it for consideration of the president or return it (not Money Bill) to the legislature for reconsideration. He has also the right to address or send messages to either house or both houses. He is empowered to issue ordinances during the recess of the State legislature.

Appointment of Ministers The chief minister recommends the names of ministers together with proposed portfolios for them to the governor who then appoints them.

5.

Functions The chief minister is the real executive authority in the State administration. Ministers of his cabinet are appointed and removed on his recommendations. He convenes and presides over

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cabinet meetings and exercises general supervision and control over all the ministers and coordinates their working. If the chief minister resigns, the entire council of ministers is deemed to have resigned. He formulates administrative and executive policies of the state. He is the main link between the governor and the State cabinet and leader of the House. Term As the term of the legislative assembly is 5 years, so is the term of the chief minister but he can be removed sooner than the expiry of his term if a no-confidence motion is passed against his government. Removal of Chief Minister The chief minister of a State can be removed from his office if: (i) his government is defeated in the State assembly; or (ii) after his defeat in the State assembly, the chief minister refuses to resign; or (iii) in case of his not being a member of the State assembly, he fails to get himself elected to the State assembly within six months of his appointment; or (iv) the president proclaims emergency in the State on account of failure on the part of the State government to carry on the administration in accordance with the provisions of the Constitution. The State legislature consists of the governor and one or two houses, as the case may be. If the State has only one house, it is

known as legislative assembly. The other is the legislative council. The states having one house are called unicameral and the states having two houses bicameral. State Government Structure: Salient Features 1. India contains 28 self-governing states, each with a Governor (appointed by the president for five years), a legislature (elected for five years) and a Council of Ministers headed by the Chief Minister. Five states (Bihar, J&K, Karnataka, Maharashtra and Uttar Pradesh) have bicameral legislatures, the other states’ legislatures being unicameral. 2. Each state has its own legislative, executive and judicial machinery, corresponding to that of the Indian Union. 3. In the event of the failure of constitutional government in a state, presidential rule can be imposed by the Union. 4. There are also six union territories and one national capital territory, administered by lieutenant-governors or administrators, all of whom are appointed by the president. 5. The territories of Delhi and Puducherry also have elected chief ministers and state assemblies.

THE STATE LEGISLATURE Legislative Council (Vidhan Parishad) Also known as the upper house, it is like Rajya Sabha at the centre. It is also a permanent house and cannot be dissolved. It enjoys a term of six years with 1/3rd of its members retiring every two years. Qualifications for Membership To become a member of the legislative council, a person should possess the same qualifications as per Lok Sabha membership, except the age which has been fixed at 30 years. Election of Members Elected indirectly in this manner: one-third of the members of a legislative council are elected by local bodies, one-third by the legislative assembly, one-twelfth by university graduates of at least 3 years standing, similar proportion by teachers of at least 3 years standing and one-sixth nominated by the governor from among those persons who distinguish themselves in literature, science or social science. Chairman The council elects a chairman and a vice-chairman from among its members. Strength of the Legislative Council Each State legislative assembly consists of not more than 525 members and not less than 60 members. The strength varies according to the population of the State concerned. However, the legislative assembly of Arunachal Pradesh, Sikkim and Goa has only 30 members, while that of Mizoram is 40 and Nagaland is 46. The governor can nominate one member from Anglo-Indian community. This provision was originally operational for 10 years only, but it is being continuously

extended and as per the 79th Amendment Act of 1999, this provision of nominated member will last until year 2010.

Legislative assembly (Vidhan sabha) Also known as the lower house, it is like the Lok Sabha at the centre. The legislative assembly has a term of five years but it can be dissolved by the governor sooner than its term. Its term can also be extended by one year during national emergency. Constitution It consists of directly elected representatives of the people. The Constitution provides for the reservation of seats for SC/STs in the assembly of each State on the basis of population ratios. Elections Members are chosen by direct election from the territorial constituencies of the state. The candidate should possess the same qualifications as are fixed for the Lok Sabha or legislative council. The minimum age for becoming a member is 25 years. Functions The council of ministers is collectively responsible to the assembly. The assembly chooses its own speaker and deputy speaker who can be removed by the council of ministers. The chief minister of the State is the leader of the house which is responsible for the administration, executive and legislative policies of the state. Strength of the Legislative Assembly The total strength does not exceed one-third of the strength of the Legislative Assembly, subject to a minimum of 40 members. The strength varies as per the population of the state concerned.

Indian Polity ▌ 2.219

The sTaTe JudICIaRY high Court

Functions of the High Courts

Each State has a High Court, which is the highest judicial organ of the state. However, there can be a common High Court for two or more states. For example, the Punjab, Haryana and the Union Territory of Chandigarh have a common High Court. At present there are 21 High Courts in the country.

Judicial A High Court has original appellate and revisory jurisdiction with respect to revenue and its collection as also for enforcement of the fundamental rights. It is a ‘Court of Record’ and its decisions are referred to in all future cases.

Constitution The State judiciary consists of a chief justice and such other judges as the president of India may deem necessary to appoint. The strength of High Courts is not identical. For example, the Allahabad High Court has 37 judges against five in Jammu and Kashmir High Court. Term A judge of the High Court holds office till the age of 62 years. His term can be cut short due to resignation or removal by the president. He vacates his office either on his appointment as judge of the Supreme Court or when he is transferred to another High Court. Removal The president can remove a judge of the High Court only if the Parliament passes a resolution by a two-third majority of its members present and voting in each house, requesting him to remove a judge. Restriction on Legal Practice A person who has held office of a judge of the High Court is not allowed to practise law before the authority of the same court except the Supreme Court and a High Court, other than the one in which he served as a judge. High Court’s Power of Superintendence Each High Court has power of superintendence over all courts within its jurisdiction. It can call for any returns from such courts, make and issue general rules, and prescribe forms to regulate their practice and proceedings and determine the manner and form in which book entries and accounts shall be kept.

administrative It supervises the working of all subordinate courts and frames rules and regulations for the transaction of business. It can examine the records of subordinate courts. However, it does not have any power of superintendence over any court or tribunal constituted under any military law.

Appointment of Judges Every judge of a High Court including the chief justice is appointed by the president. The appointment of the chief justice is made after consultation with the chief justice of India and the governor of the State concerned. In case of appointment of a judge, the chief justice of the High Court concerned is also consulted in addition to the chief justice of India and the governor of the State concerned. Appointment of all judges, is, however, done by the president only. On October 6, 1993, the Constitution bench of the Supreme Court held that the opinion of the chief justice (of the concerned High Court and the Supreme Court) will have primacy on both the appointment as well a transfer of senior judges. Qualification For appointment as a judge to the High Court, the person: (i) must be a citizen of India; (ii) should have been an advocate of a High Court or of two such courts in succession for at least 10 years; or should have held judicial office in Indian territory for a period of at least 10 years.

MIsCeLLaneous ConsTITuTIonaL PRoVIsIons List of Schedules First schedule deals with 25 states 7 union territories that comprise the Union of India. Second Schedule deals with salaries, allowances, etc., payable to the president of India, governors of states, chief justice of India, judges of the Supreme Court and High Courts and comptroller and auditor general of India. The revised salaries now are (per month) President of India Rs 1,50,000 Vice-President Rs 1,25,000 Governor of a state Rs 1,00,000 Chief Justice of the Supreme Court Rs 1,00,000 Judges of the Supreme Court Rs 90,000 Chief Justice of a High Court Rs 90,000 Judges of a High Court Rs 80,000 Comptroller and Auditor General Rs 90,000

Third Schedule prescribes forms of oaths and affirmations for new incumbents assuming public office. Fourth schedule allocates seats to each State and union territory in the Rajya Sabha. Fifth Schedule contains provisions as to the administration and control of scheduled areas. sixth schedule deals with provisions regarding administration of tribal areas in the states of Assam, Meghalaya and Mizoram. seventh schedule gives three lists of powers and subjects to be looked after by the Union and states as follows: (i) Union List comprises 97 subjects of all-India importance like Defence, International Affairs, Railways, Post and Telegraph, Income-tax, etc. The Parliament has exclusive power to legislate on these subjects. (ii) State List contains 66 subjects of local

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importance. Normally State legislature alone legislates on these subjects. (iii) Concurrent List contains subjects on which the Parliament as well as the State legislature enjoy authority.

eleventh schedule contains 29 subjects, on which the panchayats have been given administrative control. Inserted in the constitution by 73rd Amendment Act, 1992.

eighth schedule gives a list of 22 regional languages recognized by the Constitution. Originally, there were only fourteen languages in this schedule, the fifteenth language, Sindhi, was added by the 21st Constitution Amendment in 1967, and three languages, Konkani, Manipuri and Nepali, were added by the 71st Amendment in August 1992. Four more languages Bodo, Dogri, Maithili and Santhali were added by the 92nd Amendment Act, 2003. The languages are: 1. Assamese 2. Bengali 3. Bodo 4. Dogri 5. Gujarati 6. Hindi 7. Kannada 8. Kashmiri 9. Malayalam 10. Maithili 11. Marathi 12. Oriya 13. Punjabi 14. Sanskrit 15. Sindhi 16. Tamil 17. Telugu 18. Santhali 19. Urdu 20. Konkani 21. Manipuri 22. Nepali

Twelfth Schedule contains 18 subjects, on which the municipalities have been given administrative control. Inserted in the constitution by 74th Amendment Act, 1992.

ninth schedule contains certain Acts and regulations of the State Legislature dealing with land reforms and abolition of the Zamindari system. It contains 257 Acts. This Schedule was added to the Constitution in 1951 by the First Amendment Act. Tenth schedule contains certain provisions regarding disqualification of members on grounds of defection. Inserted in the constitution by the 52nd Amendment Act, 1985.

States and Union Territories of India The Union of India now consists of 28 states and seven union territories (for details please see the chapter on Indian Geography). The union territories of Mizoram, Arunachal Pradesh and Goa were conferred statehood in 1987 by amending the Constitution of India. (1) Mizoram became India’s 23rd State on 20 February 1987 after Mizo National Front (MNF) leader Laldenga signed an accord with the former Prime Minister Mr Rajiv Gandhi on 30 June 1986. (2) Arunachal Pradesh became the 24th State by attaining statehood on 20 February 1987. (3) Goa became India’s 25th State on 30 May 1987. (4) In 2000, three new states were created out of existing states of MP, Bihar and UP. The three new states were— Chhattisgarh, Jharkhand and Uttaranchal (which is now renamed as Uttarakhand).

PRaCTICe eXeRCIse 1. An interpretation of the Indian Constitution is based on the spirit of the _____ (a) Preamble (b) Directive Principles (c) Fundamental Rights (d) Fundamental Duties 2. The Indian Constitution is regarded as _____ (a) federal (b) unitary (c) parliamentary (d) federal in form and unitary in spirit 3. “India, that is Bharat, shall be a union of States,” occurs in which of the following Articles of the Indian Constitution? (a) 8 (b) 7 (c) 6 (d) 1 4. Fundamental Duties were incorporated in the Constitution of India by the _____ (a) 32nd Amendment (b) 42nd Amendment (c) 15th Amendment (d) 46th Amendment 5. The President of India is a _____ (a) real executive (b) titular executive (c) de facto executive (d) None of these 6. The president of India is elected indirectly by the electoral college consisting of the elected members of (a) only Lok Sabha (b) only Rajya Sabha (c) Lok Sabha and Rajya Sabha (d) Lok Sabha, Rajya Sabha and State legislative assemblies

7. The salary of the president of India has been raised to (a) Rs 35,000 (b) Rs 25,000 (c) Rs 50,000 (d) Rs 1,50,000 8. What is the minimum age prescribed for a candidate to be elected to the office of the president of India? (a) 25 years (b) 21 years (c) 30 years (d) 35 years 9. The president may, for violation of the Constitution, be removed from office by (a) impeachment (b) electoral college consisting of members of Lok Sabha, Rajya Sabha and legislative assemblies (c) prime minister (d) a no-confidence motion 10. Legislative powers are vested in the _____ (a) President (b) Parliament (c) Prime minister (d) Governor 11. The Supreme Court judges are appointed by the (a) Prime Minister (b) President (c) Chief Justice (d) Lok Sabha 12. President can dissolve the Lok Sabha on the (a) advice of the prime minister (b) advice of the chief justice of India (c) recommendation of the Lok Sabha (d) recommendation of the Rajya Sabha

Indian Polity ▌ 2.221 13. The President of India is not a member of Parliament because (a) he has to be impartial (b) it is not required under the Constitution (c) his position will be lowered (d) he holds an office of profit under the Government 14. Who has authority to proclaim Emergency in a state? (a) Prime Minister (b) President (c) Lok Sabha (d) Parliament 15. The financial powers of the president of India are (a) No Money Bill can be introduced without his prior consent (b) He is responsible for the preparation of budget (c) He appoints the Finance Commission (d) Only (a) and (c) 16. Who was the first president of India? (a) Dr Rajendra Prasad (b) Dr S Radhakrishnan (c) N. Sanjiva Reddy (d) Mahatma Gandhi 17. The following have held office of the president of India. The chronological order in which they held office is: (l) V. V. Giri (2) N. Sanjiva Reddy (3) Dr Zakir Hussain (4) Fakhruddin Ali Ahmed (a) 4, 3, l and 2 (b) 3, 1, 4 and 2 (c) 2, 3, 4 and 1 (d) 1,3,2 and 4 18. The ex-officio chairman of the Rajya Sabha is (a) President (b) Vice-President (c) Prime Minister (d) Speaker 19. After retirement, a judge can undertake private practice of law (a) in the same court (b) only in a lower court (c) where ever he/she wishes (d) in another court of the same level or higher level 20. The vice-president of India is elected by an electoral college consisting of (a) elected members of Parliament (b) elected members of Parliament and State assemblies (c) members of both the houses of Parliament (d) members of Lok Sabha and legislative councils 21. The vice-president acts as the president of India when (a) president is sick (b) president is unable to discharge his duties (c) president is absent from the country (d) all the above 22. Who was the first vice-president of India? (a) Dr Zakir Hussain (b) N. Sanjeeva Reddy (c) V. V. Giri (d) Dr S. Radhakrishnan

23. The minimum age (in years) required to become the prime minister of India is (a) 25 (b) 30 (c) 40 (d) 35 24. In a parliamentary system to whom is the executive responsible? (a) President (b) Legislature (c) House of People (d) Judiciary 25. India has a parliamentary democracy system as the (a) members of Lok Sabha are elected by the people (b) council of ministers is responsible to Parliament (c) powers distributed between centre and states (d) Constitution has a single framework 26. India is a federal State because its constitution provides for (a) dual citizenship (b) dual judiciary (c) sharing powers between states and centre (d) written constitution 27. Who was the first prime minister of India? (a) Mahatma Gandhi (b) Jawaharlal Nehru (c) S. C. Bose (d) Dr S. Radhakrishnan 28. First mid-term elections were held in (a) 1996 (b) 1971 (c) 1999 (d) 1977 29. The prime minister is responsible to the (a) Council of Ministers (b) President (c) Rajya Sabha (d) Lok Sabha 30. The council of ministers has to resign if a no-confidence motion is passed by majority members of (a) the Lok Sabha (b) the Rajya Sabha (c) both the houses (d) joint session of both the houses 31. Money Bills can only be introduced in (a) the Lok Sabha (b) the Rajya Sabha (c) Joint session of both the houses (d) both the houses (e) None of these 32. Which of the following types of bills can become law despite disapproval by the Rajya Sabha? (a) Finance Bill (b) Bills relating to social reforms (c) Bills relating to Concurrent List (d) Constitutional Amendment Bill

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33. Under the Constitution the speaker of the Lok Sabha may be removed from office by (a) the majority party in the house adopting a no confidence motion (b) a resolution passed by not less than half of the total membership (c) a resolution passed by at least two-thirds of the total membership of the house (d) a resolution passed by a majority of all the members of the house 34. The members of the Rajya Sabha are elected by (a) the people (b) Lok Sabha (c) elected members of the legislative assembly (d) elected members of the legislative council 35. If the speaker of the lower house wants to resign, he should send his resignation to the (a) prime minister (b) president (c) deputy speaker (d) chief justice 36. The Constitution of India provides for the nomination of two members to the Lok Sabha, by the president to represent (a) men of eminence in arts, science, etc. (b) Parsis (c) the Anglo-Indian Community (d) None of these 37. Which Article of the Constitution provides the Parliament the power to amend the Constitution? (a) 370 (b) 368 (c) 390 (d) 376 38. The maximum gap between sessions of the Parliament can be (a) 4 months (b) 10 months (c) 9 months (d) 6 months 39. Who among the following has a right to speak and otherwise take part in the proceedings of either House of Parliament and to be a member of any Parliamentary Committee but is not entitled to vote? (a) The chief election commissioner (b) The comptroller and auditor general of India (c) The attorney general (d) Chairman of finance commission 40. What is the minimum age in number of years prescribed for membership to the Lok Sabha? (a) 21 yrs (b) 25 yrs (c) 30 yrs (d) 35 yrs 41. Which part of Constitution deals with citizenship? (a) I (b) II (c) III (d) IV

42. The president of India can declare emergency if there is (a) war or threat of war (b) failure of Constitutional machinery (c) financial instability (d) any of these three 43. By which Constitution Amendment Bill was the voting age reduced from 21 years to 18 years? (a) 49th (b) 57th (c) 61st (d) 63rd 44. Which Union Territory became the 25th State of India? (a) Mizoram (b) Sikkim (c) Goa (d) Gujarat 45. Indian States with the largest electorate is (a) Bihar (b) UP (c) Karnataka (d) Rajasthan 46. Judges of a High Court are appointed by the president with the consent of the (a) governor of the state concerned (b) chief justice of High Court concerned (c) governor and the chief justice of Supreme Court (d) governor, the chief justice of the High Court concerned and the chief justice of India 47. Judges of the Supreme Court can hold office up to (a) 56 years (b) 54 years (c) 60 years (d) 65 years 48. To be eligible to become a judge of a High Court, the candidate should be a (a) practising advocate of 5 years in a Sessions Court (b) practising advocate of 5 years in a High Court (c) practising advocate of 10 years in a High Court (d) practising advocate of 5 years in Supreme Court 49. What is the minimum age in years prescribed for candidature to the office of governor of a state? (a) 21 (b) 25 (c) 30 (d) 35 50. What is the term of office of the governor of a state? (a) Depends on consent of the prime minister (b) 5 years (c) 6 years (d) Consent of the president and the prime minister 51. How many states in India have two houses, viz., legislative assembly and legislative council? (a) 6 (b) 8 (c) 5 (d) 10 52. Which of the following languages have been added to the Eighth Schedule of the Constitution in 1992? (a) Konkani, Manipuri and Sindhi (b) Konkani, Manipuri and Dogri (c) Dogri, Nepali and Manipuri (d) Konkani, Manipuri and Nepali

Indian Polity ▌ 2.223 53. After retirement a judge can undertake private practice of law (a) in the same court (b) only in a lower court (c) wherever he/she wishes (d) in another court of the same level or higher court

63. According to the 44th Amendment Bill (a) states and union territories (b) salaries of president, vice-president etc. (c) union list, state list and concurrent list (d) allocation of seats in the Rajya Sabha

54. A political party is recognized as a national party (a) when it contests elections in all states (b) when it secures at least 5% of the total votes (c) if it is recognized as a political party in four or more states (d) if it captures power in at least three states

64. When was the Constitution of India first amended after adoption? (a) 1954 (b) 1959 (c) 1950 (d) 1951

55. The deputy chairman of the Rajya Sabha can be removed by a resolution (a) passed by a 2/3rd majority of its total strength (b) passed by a simple majority of its total membership at that time (c) moved by Rajya Sabha and passed by Lok Sabha (d) None of these 56. The 7th Schedule of Constitution of India contains (a) states and union territories (b) salaries of president, vice-president etc. (c) union list, state list and concurrent list (d) allocation of seats in the Rajya Sabha 57. Which of the following languages are included in the 8th Schedule of the Constitution of India? (a) Sindhi and English (b) English and Kashmiri (c) Kashmiri and Gorkhai (d) Sindhi and Kashmiri 58. How many languages are recognized as regional languages in the Constitution? (a) 12 (b) 13 (c) 14 (d) 18 59. The 8th Schedule of the Constitution deals with (a) The list of 18 regional languages (b) Fundamental Rights (c) Directive Principles of State Policy (d) All the above 60. Who decides whether a Bill is a Money Bill or not? (a) Speaker of the Lok Sabha (b) The President (c) The Prime Minister (d) Joint session of both the houses 61. As a non-member who can participate in the proceedings of either House of Parliament? (a) Vice-President (b) Chief Justice (c) Attorney-General (d) None of these 62. The Lok Sabha Secretariat comes under direct supervision of the (a) Ministry of Home Affairs (b) Ministry of Parliamentary Affairs (c) President of India (d) Lok Sabha Speaker

65. Which Amendment to the Constitu tion gave primacy to directive principles over fundamental rights? (a) 44th (b) 42nd (c) 40th (d) 25th 66. Which Amendment Bill extended life of Lok Sabha to 6 years and which Bill restored it to 5 years? (a) 42nd and 43rd (b) 44th and 47th (c) 42nd and 44th (d) 42nd and 43rd 67. Which Amendment Bill inserted the 10th Schedule in the Constitution of India? (a) 50th (b) 25th (c) 44th (d) 52nd 68. According to the 42nd Amendment, two words have been added in the Preamble. These are: (a) Federal and Democratic (b) Secular and Sovereign (c) Secular and Socialist (d) Federal and Socialist 69. Which List contains the subjects over which both the Centre and States have jurisdiction? (a) Union List (b) State List (c) Concurrent List (d) Central List 70. When did India have its first General Elections? (a) 1947 (b) 1949 (c) 1950 (d) 1951 71. If the positions of President and Vice-President are vacant, who officiates as the President of India? (a) The Prime Minister (b) Chief Justice of India (c) Speaker of Lok Sabha (d) Both (a) and (b) 72. What is the elective strength of the Lok Sabha? (a) 525 (b) 450 (c) 545 (d) 548 73. Elective strength of Lok Sabha was increased by the (a) 42nd Amendment (b) 44th Amendment (c) 31st Amendment (d) 40th Amendment 74. A Parliamentary Bill passed by the Lok Sabha is sent to the president for his assent. The president sends it back to the Lok Sabha for reconsideration. If the Lok Sabha again passed the Bill and sends it to the President for assent, the president (a) must approve it (b) can again send the Bill for reconsideration (c) can take the help of the Supreme Court (d) can arrange for a referendum

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75. What is the maximum strength of the State Legislative Assembly? (a) 250 (b) 350 (c) 525 (d) 455 76. The total strength of the Rajya Sabha does not exceed (a) 200 members (b) 250 members (c) 450 members (d) 525 members 77. Who presides over the sessions of the Lok Sabha? (a) President (b) Prime Minister (c) Lok Sabha Speaker (d) Vice-President 78. How many members can the president of India nominate to the Rajya Sabha? (a) 2 (b) 10 (c) 12 (d) 8 79. First national emergency proclaimed in India was in (a) 1962 (b) 1965 (c) 1978 (d) 1966 80. What is the salary of the chief justice of India? (a) Rs 25,000 (b) Rs 30,000 (c) Rs 33,000 (d) None of these 81. The Constitution of India consists of (a) The Preamble (b) Parts I–XXII, Covering Articles 1–450 (c) Schedules 1–12 and the Appendix. (d) All of these 82. The source of authority of the constitution are (a) the houses of the Parliament (b) the people of India (c) the President (d) Only (a) and (b) 83. The draft of the Preamble was prepared by (a) Rajendra Prasad (b) Jawaharlal Nehru (c) Dr B. R. Ambedkar (d) Gandhiji 84. Each standing committee consists of (a) 25 members (b) 30 members (c) 45 members (d) 55 members 85. Sovereign character of Preamble denotes: (a) India is free from any type of external control (b) India can acquire foreign territory (c) India can cede a part of the territory in favour of a foreign state. (d) All are correct 86. Secular character of preamble denotes that: (a) India has no official religion (b) Full opportunity to everybody to profess, practice and propagate religion of their choice (c) Single citizenship is assured to all persons, irrespective of their religion (d) All are correct

87. The 42nd Amendment came into effect on: (a) 3 January 1976 (b) 20 April 1977 (c) 3 January 1978 (d) 20 April 1978 88. Various amendments made in 42nd Amendment? (a) It laid down that amendment to the constitution made in the part or those likely to be made in future could not be questioned in any court on any ground. (b) It removed all amendments to fundamental rights from the scope of judicial review. (c) It removed all limits on parliament’s power to amend the constitution under Article 368. (d) All the above 89. The socialist character of the preamble denotes: (a) All citizens have right to vote. (b) State will secure all citizens, social, economic and political justice. (c) State has no official religion. (d) India is free from any type of external control. 90. The provision in the preamble that confers adult citizens with the right to vote is called (a) Secularism (b) Democracy (c) Socialism (d) Republic 91. Right to Freedom of Religion is guaranteed under the Constitution in Articles ____ (a) 12–19 (b) 19–21 (c) 25–28 (d) 21–28 92. Right to Property is a ____ (a) Fundamental Right (c) Legal Right

(b) Directive Principle (d) Social Right

93. India is a secular State because ____ (a) it is a socialist country (b) it treats all religions alike (c) there is no religion of the people (d) religious beliefs and practices are discouraged 94. Fundamental Rights are enshrined in the Constitution of India in ____ (a) Part I (b) Part III (c) Part IV (d) Part V 95. How can a citizen protect his Fundamental Rights? (a) By approaching the President of India (b) By a writ in the Supreme Court of India (c) Through police action (d) They are already protected 96. Which of the following is not a Fundamental Right? (a) Right against exploitation (b) Equal pay for equal work (c) Equality before law (d) Right to freedom of religion 97. The Directive Principles of State Policy are incorporated in (a) Part III (Articles 26–51) (b) Part IV (Articles 36–51) (c) Part V (Articles 19–21) (d) Part II (Articles 36–56)

Indian Polity ▌ 2.225 98. The verdict by the Supreme Court that the original structure of the Preamble, as given in the Constitution, cannot be changed was given in which case? (a) Sajjan Singh (b) Minerva Mills (c) Golaknath (d) Keshavananda (e) None of these 99. The Gandhian principles have been enumerated in the (a) Fundamental Rights (b) Preamble (c) Directive Principles (d) Fundamental Duties

100. The difference between Fundamental Rights and Directive Principles is (a) Fundamental Rights are guaranteed under the law but Directive Principles are only moral laws (b) Directive Principles can be enforced in a court of law whereas Fundamental Rights cannot be enforced (c) Directive Principles are in the nature of directions to the legislature and executive and assurance to the people what they can expect from the state, whereas Fundamental Rights are basic rights of the citizens (d) None of these

answeR keY 1. (a)

14. (b)

27. (b)

40. (b)

53. (d)

66. (c)

79. (d)

92. (c)

2. (d)

15. (d)

28. (d)

41. (b)

54. (c)

67. (e)

80. (d)

93. (b)

3. (d)

16. (a)

29. (d)

42. (d)

55. (b)

68. (c)

81. (d)

94. (b)

4. (b)

17. (b)

30. (a)

43. (c)

56. (c)

69. (c)

82. (b)

95. (b)

5. (b)

18. (b)

31. (a)

44. (c)

57. (d)

70. (d)

83. (b)

96. (b)

6. (d)

19. (d)

32. (a)

45. (b)

58. (d)

71. (b)

84. (b)

97. (b)

7. (d)

20. (c)

33. (d)

46. (d)

59. (a)

72. (c)

85. (d)

98. (c)

8. (d)

21. (d)

34. (c)

47. (d)

60. (a)

73. (c)

86. (d)

99. (c)

9. (a)

22. (e)

35. (c)

48. (c)

61. (c)

74. (a)

87. (a)

100. (c)

10. (b)

23. (a)

36. (c)

49. (d)

62. (d)

75. (c)

88. (a)

11. (b)

24. (c)

37. (b)

50. (b)

63. (c)

76. (b)

89. (b)

12. (a)

25. (b)

38. (d)

51. (c)

64. (d)

77. (c)

90. (b)

13. (d)

26. (c)

39. (c)

52. (d)

65. (b)

78. (c)

91. (c)

nda BoX 1. Which one of the following sets of Bills is presented to the parliament along with the budget? Lower case/global options (a) Direct taxes Bills and indirect taxes Bill (b) Contingency Bill and Appropriation Bill (c) Finance Bill and contingency Bill (d) Finance Bill and Appropriation Bill

3. The two political parties which shared a common election symbol in the general election of 1996 were (a) Telgu Desam Party and Samajwadi Party (b) Bahujan Samaj Party and Samata Party (c) Dravida Munnetra Kazhagam and Karnataka Congress Party (d) Communist Party of India and Maharashtrawadi

2. Central-State financial distribution takes place following the recommendation made of the (a) inter-state Council (b) Finance Commission (c) Planning Commission (d) Parliament

4. The Constituent Assembly formed the Drafting Committee of India Constitution on (a) 29th August, 1947 (b) 30th August,1948 (c) 29 August, 1949 (d) 29th August,1950

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5. Who is the Chairman of Lok Sabha ? (a) The President (b) The Minister of Parliamentary Affairs (c) The Speaker (d) The Prime Minister 6. The name of the candidate for the office of the President of India may be proposed by (a) any five citizens of India (b) any one member of the Electoral College (c) any five members of the Parliament (d) any ten members of the Electoral College 7. Which one of the following Articles empowers the Presidents to appoint Prime Minister of India? (a) Article 74 (b) Article 75 (c) Article 76 (d) Article 77 8. The Supreme Court of India dispenses justice its reliance on the principles of (a) Rule of law (b) Due pocess of law (c) Procedural due process (d) Procedure established by law 9. Which of the following pairs are correctly matched? (a) K. Ramamurthy Congress (b) Shanker Singh Rashtriya Janta Dal (c) Navin Patnaik Biju Janta Dal (d) Jyoti Basu Communist Party Select the correct answer using the codes given below: (a) 1, 2, 3 and 4 (b) 2, 3 and 4 (c) 1 and 4 (d) 3 alone 10. The terms of a member of the Rajya Sabha is (a) five years (b) six years (c) the same as that of that of the Lok Sabha (d) two years 11. Who decides whether a bill is a Money Bill or not? (a) The President (b) Chairman of the Rajya Sabha (c) Speaker of the Lok Sabha (d) Minister for parliament Affairs 12. Who among the following did not serve as the Vice President before becoming the President of India ? (a) Dr S. Radhakishan (b) Dr Zakir Hussain (c) Neelam Sanjeeva Reddy (d) R.Venkataraman

13. Fifty second Constitutional Amendment Act contains several provisions regarding disqualification of members of State Legislatures in the event of defection. Which one of the following is NOT one of the provisions? (a) Giving up the membership of the party on whose ticket the member was elected to the House. (b) Being elected as presiding officer of the House and giving up his membership of the party to become non-partisan (c) An independent member joining any political party after election. (d) Abstaining from voting or voting contrary to any direction of the political party to which the member belongs. 14. A political party is regarded as a national party if it has secured (a) three per cent of votes polled in three or more states (b) four per cent of votes polled in four or more states (c) five per cent of votes polled in five or more states (d) six per cent of votes polled in six or more states 15. Match List I with List II and select the correct answer using the codes given below the lists: List – I (Political Leader)

List – II (Political Party)

Upper case (A)

Chandra Shekhar

1. Bhartiya Janata Party

(b)

Purno Agitok Sangma

2. Lok Janshakti

(c)

Ram Vilas Paswan

3. Samajwadi Janata party

(d)

Syed Hussain Shahnawaz

4. Nationalist Congress Party

Codes: A B

C D

(a) 3

4

1

2

(b) 4

3

2

1

(c) 3

4

2

1

(d) 4

3

1

2

16. Consider the following states: (1) Andhra Pradesh (2) Bihar (3) Maharashtra (4) Punjab Which of these states have Bicameral Legislature? (a) 1 and 2 (b) 2 and 3 (c) 2 and 4 (d) 1 and 4

Indian Polity ▌ 2.227 17. Consider the following statements regarding the declaration of Emergency in India: (1) During the Emergency, the Parliament can make laws on the subjects mentioned in the State List. (2) The fundamental Rights except the Right to Constitutional Remedies are denied to the citizens. (3) In an Emergency, the normal federal system is modified and the country is governed as if it was a unitary state (4) The declaration of Emergency by the President does not need the approval of the Parliament. Which of these statements are correct? (a) 1 and 2 (b) 2, 3 and 4 (c) 1 and 3 (d) 3 and 4 18. Consider the following statements: (1) A money Bill can be introduced in the Rajya Sabha. (2) If the President resign or dies, a new President must be elected within six months. (3) The President causes the budget to be laid before the Lok Sabha. Which of these statements is/are correct? (a) 1 and 2 (b) 2 only (c) 2 and 3 (d) 3 only 19. How many member are nominated to the Rajya Sabha? (A) 18 (b) 15 (c) 12 (d) 9 20. In the Union Government, the Council of Ministers is collectively responsible to the (a) President (b) Prime Minister (c) Lok Sabha (d) Parliament 21. Consider the following statement: (1) The resignation of the Prime Minister of India means the resignation of the whole Union Cabinet. (2) The President of India can dismiss the Prime Minister on the grounds of the bad performance even when he enjoys the support of the majority in the Parliament. (3) It is on the advice of the Prime Minister that the President of India summons and prorogues the Parliament and dissolves the Lok Sabha. Which of these statements is/are correct? (a) 1 only (b) 2 and 3 (c) 1 and 3 (d) 3 only 22. With reference to the Indian Parliament, consider the following statement: (1) All legislation requires the consent of both Houses of Parliament. (1) All financial legislation must be recommended by the President. (1) Both the houses of Parliament have the power to vote the demands for grants presented by the government. Which of these statements is/are correct? (a) 1 only (b) 2 and 3 (c) 1 and 2 (d) 1, 2 and 3

23. The member of the Rajya Sabha are elected for a period of (a) 3 year (b) 4 year (c) 5 year (d) 6 year 24. Who among the following discharges the functions of the President of India when the offices of the President and the Vice President are vacant? (a) The Speaker of Lok Sabha (b) The Deputy Chairman of Rajya Sabha (c) The Prime Minister (d) The chief Justice of India 25. Which one of the following States is represented by only one member in the Lok Sabha? (a) Manipur (b) Meghalaya (c) Sikkim (d) Tripura 26. With reference to the Indian States and the Constitution, which one of the following statements is correct? (a) All the laws passed by the State Legislature require the assent of the President (b) In emergency, the term of the State Assembly can be extended for a period of one year (c) State Legislative Council can be dissolved by the Governor if he deems it necessary in the public interest (d) The Council of Ministers of the State are collectively responsible to the Governor of the State 27. Who among the following former Presidents of India did not hold the office of the Vice-Presidents of India before becoming the Presidents of India? (a) K.R. Narayanan (b) Neelam Sanjiva Reddy (c) R. Venkataraman (d) Dr. Shankar Dayal Sharma 28. In order to be chosen as Member of Parliament (Rajya Sabha), and a Member of Parliament (Lok Sabha), what is the prescribed minimum age? (a) 35 years and 30 years, respectively (b) 30 years and 25 years, respectively (c) 25 years for each (d) 30 years for each 29. Who among the following constitutes the various consultative committees of Members of both the Houses of Parliament in India? (a) Leader of Opposition in the Lok Sabha (b) Ministry of Parliament Affairs (c) Speaker of the Lok Sabha (d) The Vice President of India in his capacity as the ex officio Chairman of the Rajya Sabha 30. For the Members of Parliament in India, who determines the salaries and allowances from time to time? (a) Union Cabinet (b) Ministry of Parliament Affairs (c) Cabinet by law (d) Parliament by law

2.228

▌ General Knowledge

31. Consider the following statements: (1) The chief Secretary of a state is appointed by the Governor of that state. (2) The chief Secretary of a state has a fixed tenure or till his superannuation whichever is earlier. Which of the statement given above is/are correct? (a) Only 1 (b) Only 2 (c) Both 1 and 2 (d) Neither 1 nor 2 32. Who was the Prime Minister of India when a constitutional status for the Panchayati raj and urban self-governing institution was accorded? (a) Indira Gandhi (b) Rajiv Gandhi (c) P.V. Narasimha Rao (d) Atal Behari Vajpayee 33. Which one of the following statement is correct? The representatives of each States in the Council of States are elected by (a) The elected members of the Legislative Assembly of the state in accordance with the system of proportional representation by means of the single transferable vote (b) The elected members of both the Legislative Assembly and Legislative Council of the State of proportional representation by means of the single transferable vote (c) The Elected member of the House of people from the State in accordance with system of means of the single transferable vote (d) The Council of States consists of not more than two hundred and twenty representatives of the States and of the Union territories. 34. Which one of the following pairs is not correctly matched? State or Union Territory Seat of High court (a) Uttarakhand (Uttaranchal) : Nanital (b) Daman and Diu : Ahmadabad (c) Arunachal Pradesh : Guwahati (d) Chhattisgarh : Bilaspur 35. Consider the following statements: (1) The administrator of Andaman and Nicobar Islands is designated as Lieutenant Governor. (2) Delhi and Puducherry (Pondicherry) have their own Legislative Assemblies and High Courts. Which of the statements given above is/are correct? (a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2 36. Among the following Presidents of India. Who was the speaker of Lok Sabha before becoming President? (a) K.R. Narayana (b) R. Venkataraman (c) N. Sanjiva Reddy (d) V.V. Giri

37. Consider the following statements; (1) In the Lok Sabha proceedings, when the voting is done, the Speaker of Lok Sabha can never cast his vote. (2) In the warrant of precedence, the Deputy Chairman of Rajya Sabha is equal to the Speaker of Lok Sabha in rank. Which of the statements given above is/are correct? (a) 1only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2 38. Consider the following statements: (1) The States Reorganization commission appointed by Pandit Nehru in its report opposed the splitting of the Bombay and Punjab States. (2) The State Reorganization Act passed in the year 1956 provided for 18 States and 7 Union Territories. Which of the statements given above is/are correct? (a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2 39. The 74th Amendment of the Constitution of India relates to which of the following? (a) Public Sector Undertakings (b) Compulsory Universal Educations (c) Urban Local Bodies (d) Rural Local Bodies 40. Which one of the following is the correct statement? In the Parliament of India, a money bill can be introduced only with the recommendation of (a) The President of India (b) The Prime Minister of India (c) The Speaker of Lok Sabha (d) The Union Finance Minister 41. According to the provisions of the Constitution of India, who inquires into all doubts and decides dispute arising out of or in connection with the election of Vice-President of India? (a) The President of India (b) The supreme Court of India (c) The Election Commission of India (d) The Attorney General of India 42. Under whose direct supervision is the Lok Sabha Secretariat? (a) The Cabinet Secretary (b) The Minister of Parliament Affairs (c) The Speaker of Lok Sabha (d) The Prime Minister of India 43. If the annual Union Budget is not passed by Lok Sabha, (a) the President of India asks the Finance Minister to resign (b) the Prime Minister asks the Finance Minister to resign (c) the Speaker of Lok Sabha asks the Finance Minister to modify it and present it again (d) the Prime Minister submits the resignation of his cabinet

Indian Polity ▌ 2.229 44. As per the Constitution of India, how many members can be nominated to the Rajya Sabha by the Presidents of India? (a) 10 (b) 12 (c) 14 (d) 18

53. As per the Constitution of India, what is the limit prescribed for the number of members in the Legislative Assembly of a State? (a) 350 members (b) 400 members (c) 450 members (d) 500 members

45. Which one of the following deals with the Women’s Reservation Bill that seeks to provide for one-third reservation of seats for women in Parliament and State Assemblies? (a) 98th Amendment of the Constitution (b) 100th Amendment of the Constitution (c) 105th Amendment of the Constitution (d) 108th Amendment of the Constitution

54. Consider the following statements 1. Muhammad Hamdulla sayeed, elected from Lakshadweep parliamentary constituency, is the youngest MP in the 15th Lok Sabha. 2. Agatha K Sangma is the youngest Union Minister in the 15th Lok Sabha who represents Shillong parliamentary constituency.

46. The parliament of India consists of: (a) The Lok Sabha, the Rajya Sabha and the office of the President of India only (b) The Lok Sabha and the Rajya Sabha only (c) The Lok Sabha, the Rajya Sabha and the office of the Prime Minister of India only (d) The Lok Sabha. The Rajya Sabha, the office of the President of India and the office of the Prime Minister of India. 47. In the Union Government, under whose charge is the Cabinet Secretariat? (a) The Prime Minister of India (b) The union Minister of Parliamentary Affairs (c) The Union Home Minister (d) Any Cabinet Minister to whom the Prime Minister allocates the charge 48. Russia has proposed the concept of a strategic triangle consisting of Russia, (a) UK and USA (b) India and China (c) China and Pakistan (d) India and Pakistan 49. How was the National Development Council constituted? (a) By an Act of Parliament (b) As a voluntary Association of States and Union Territories (c) By a resolution of the Union Cabinet (d) As an attached office of the PMO 50. The Presidency of the Security Council of United Nations is alternated every (a) Month (b) Six month (c) Year (d) Two year 51. According to Karl Marx, attainment of Communism is possible only after (a) Resolution of tension between the leader and led (b) Completion of permanent revolution (c) Emergence of exploitation – free society (d) Disappearance of difference between urban and rural areas 52. Area-wise, which one of the following is the largest Lok Sabha Constituency? (a) Kutch (b) Kangra (c) Ladakh (d) Patna Sahib

Which of the statements given above is/are correct? (a) 1 Only (b) 2 Only (c) Both 1 and 2 (d) Neither 1 nor 2 55. Consider the following statements: 1. Report of the Administrative Reforms Commission, headed by late Morarji Desai, recommended the establishment of Lokpal and Lokayukta institutions. 2. Lokpal is the highest institution in India to investigate corruption at higher places in Government. Which of the statements given above is/are correct? (a) 1 Only (b) 2 Only (c) Both 1 and 2 (d) Neither 1 nor2 56. Consider the following statements: 1. No person is eligible for appointment as Governor unless he has completed the age of thirty years. 2. The same person can be appointed as Governor for three states. Which of the statements given above is/are correct: (a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2 57. Consider the following statements 1. A Bill pending in the Legislature of 3 State shall not lapse by reason of the prorogation of the House or Houses thereof. 2. A bill pending in the Legislative Council of a State which has not been passed by the Legislative Assembly shall not lapse on dissolution of the Assembly. Which of the statements given above is/are correct? (a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2 58. Consider the following statements: 1. Salary and allowances of the Speaker of Lok Sabha are charged on the Consolidated Fund of India. 2. In the Warrant of Precedence, the Speaker of Lok Sabha ranks higher than all the Union Cabinet Ministers other than Prime Minister. Which of the statements given above is/are correct? (a) 1 only (b) 2only (c) Both 1and 2 (d) Neither 1 nor 2

2.230

▌ General Knowledge

59. Directions: The following question consists of two statements, Statement I and Statement II. You are to examine these tow statements carefully and the answers to these items using the codes given below: Codes: (a) Both the statements are individually true and statement II is the correct explanation of statement I. (b) Both the statements are individually true but statement II is not the correct explanation of statement I. (c) Statement I is true but statement II is false. (d) Statement I is false but statement II is true. Statement I A pubic expression of dissatisfaction with democracy indicates the success of the democratic Project: It transforms People from the status of a subject into that of a citizen. Statement II The fact people are complaining is itself an affirmation of the success of democracy: it shows that people have developed awareness and the ability to look critically at the government. 60. Directions: The following question consists of two statements, Statement I and Statement II. You are to examine these tow statements carefully and the answers to these items using the codes given below: Codes: (a) Both the Statements are individually true and statement II is the correct explanation of statement I. (b) Both the statements are individually true but statement II is not the correct explanation of statement I. (c) Statement I is true but statement II is false. (d) Statement I is false but statement II is true. Statement I Sarkaria Commission recommended that Governor of a state should be a non-political person appointed after consultation with the Chief Minister of the State. Statement II This could be achieved through amending Article 165 of the Constitution of India. 61. Directions: The following question consists of two statements, Statement I and Statement II. You are to examine these tow statements carefully and the answers to these items using the code given below: Codes: (e) Both the statements are individually true and statement II is the correct explanation of statement I. (f) Both the statements are individually true but statement II is not the correct explanation of statement I. (g) Statement I is true but statement II is false. (h) Statement I is false but statement II is true. Statement I Coalition in India is always a Post-poll phenomenon. Statement II The United Progressive Alliance was formed after the Lok Sabha Election of 2004.

62. Directions: The following question consists of two statements, Statement I and Statement II. You are to examine these tow statements carefully and the answers to these items using the code given below: Codes: (a) Both the Statements are individually true and Statement II is the correct explanation of statement I. (b) Both the statements are individually true but Statement II is not the correct explanation of Statement I. (c) Statement I is true but Statement II is false. (d) Statement I is false but Statement II is true. Statement I The constitution of India is a liberal Constitution. Statement II It provides Fundamental Rights to individuals. 63. Directions: The following question consists of two statements, Statement I and Statement II. You are to examine these tow statements carefully and the answers to these items using the codes given below: Codes: (a) Both the statements are individually true and statement II is the correct explanation of statement I. (b) Both the statements are individually true but statement II is not the correct explanation of statement I. (c) Statement I is true but statement II is false. (d) Statement I is false but statement II is true. Statement I In order to create linguistic Provinces in India, the Congress Party constituted Motilal Nehru committee in 1948. Statement II The Motilal Nehru Committee suggested creation of linguistic provinces in India. 64. Freedom of the press in India is (a) Available to the people under the law of the Parliament (b) Specifically provided in the Constitution (c) Implied in the right of freedom of expression (d) Available to the people of India under executive order. 65. If any question arises whether a Bill is a Money Bill or not, whose decision hall be final? (a) The Supreme Court of India (b) The President of India (c) The Speaker of the Lok Sabha (d) Joint Parliamentary Committee 66 In a democracy, individual differences are encouraged because (a) Individuals will Place the good of the nation ahead of their own preferences (b) Individual will tend to act and think alike in the long run (c) Individual differences make for diverse beneficial contributions to the common cause (d) The Constitution guarantees the right of the individual to act as he pleases 67. In the Union Government, under whose charge is the Cabinet Secretariat? (a) The Minister of Parliamentary Affairs (b) The President of India (c) The Prime Minister of India (d) The Union Home Minister

Indian Polity ▌ 2.231 68. In which among the following cases, the joint session of both the Houses of Parliament can be summoned? 1. To amend the constitution 2. When a Bill has been pending with one House for more than six Months after it was passed by the other 3. When both the Houses disagree on the amendments to be made in a Bill 4. When a bill is passed by one House and is rejected by the other Select the correct answer using the codes given below: (a) 1, 2 and 3 (b) 2, 3 and 4 (c) 2 and 3 only (d) 1 and 4 69. In which of the following years was the Planning Commission set up in India? (a) 1947 (b) 1948 (c) 1949 (d) 1950 70. On which of the following grounds can a Judge of the Supreme Court or a High Court be impeached? 1. Violation of the Constitution 2. Proved misbehavior 3. Incapacity Select the correct answer using the codes gives below: (a) 1only (b) 2 only (c) 1, 2 and 3 (d) 2 and 3 71. Public Interest Litigation (PIL) May be linked with (a) Judicial review (b) Judicial activism (c) Judicial intervention (d) Judicial sanctity 72. Right to Information in India is a (a) Fundamental Right (b) Legal Right (c) Both Fundamental and Legal Rights (d) Neither Fundamental nor Legal Right 73. Socialism is opposed to (a) Social security scheme (b) Equal distribution of wealth (c) Unrestricted competition (d) Collective ownership and management 74. Socialism refers to (a) State-controlled economy (b) liquidation of the bourgeoisie (c) Removal of peasantry from administration (d) Establishment of military dictatorship 75. The constitution (93rd Amendment) Act deals with (a) Local self-government (b) Extension of reservation in educational institutions (c) Basic structure of Constitution of India (d) Appointment of judges in the Supreme Court of India 76. The entry ‘Public health and Sanitation’ is included in the Constitution of India in (a) Union List (b) State List (c) Concurrent List (d) None of these

77. The Government Bill means a bill introduced by a (a) Member of the Treasury bench in the Lok Sabha (b) Member of the Parliament who is not a Minister (c) Minister in the Lok Sabha (d) Minister in any House of the Parliament 78. The idea of including the Emergency Provisions in the constitution of India has been borrowed from the (a) Constitution of Canada (b) Weimar Constitution of Germany (c) Constitution of Ireland (d) Constitution or USA 79. The Instrument of Instructions on the Government of India Act, 1935 has been incorporated in the Constitution of India in the year 1950 as (a) Fundamental Rights (b) Directive Principles of State Policy (c) Fundamental Duties (d) Emergency Provisions 80. The Legislative Council in a State in India can be created or abolished by the (a) Parliament on the recommendation of a Governor of the State (b) Parliament alone (c) Parliament after the state assembly passes the resolution of that effect (d) Governor of the state on the recommendation of the Council of Ministers 81. The Speaker of the Lok Sabha can resign his office by addressing his resignation to (a) The President (b) The Prime Minister (c) The Deputy Speaker of the Lok Sabha (d) The Chief Justice of India 82. The Supreme Court of India has expanded the meaning of the Right to Life to include the following derivative rights. Which one among the following is not included in the Court’s definition? (a) Right to food and adequate livelihood (b) Right to education (c) Right to live in a healthy environment (d) Right to health and information 83. The Union Executive of India Consists of (a) The President, Vice-President , Lok Sabha Speaker and the Council of Ministers (b) The President, Deputy Chairman of the Rajya Sabha, Lok Sabha Speaker and the Prime Minister (c) The president and the Council of Ministers only (d) The President, Vice-President and the Council of Ministers only

2.232

▌ General Knowledge

84. What is the number of Judges (including Chief Justice) in the Supreme Court of India as Provided in the Constitution of India? (a) 20 (b) 24 (c) 26 (d) 28 85. When an Ordinary Bill is referred to a joint sitting of both the Houses of Indian parliament, it has to be passed by a (a) Simple majority of the total number of members of both the Houses present and voting (b) Two-third majority of the total number of members of both the Houses (c) Simple majority of the total number of members of both the Houses (d) Two-third majority of the total number of members of both the Houses present and voting 86. Which among the following is an extraconstitutional Growth in Indian democracy? (a) Attorney General of India (b) Autonomous Districts (c) Political Parties (d) Deputy Chairman of the Council of States 87. Which among the following is not a feature of part IX of the Constitution of India? (a) Five Year tenure for panchayats (b) Reservation of seats for Schedule Castes and Schedule Tribes for Panchayat membership (c) Indirect election for all panchayat seats (village/ intermediate level) (d) Reservation for not less than one – third of the seats for women 88. Which among the following is the exclusive jurisdiction of the State Government? (a) Corporation Tax (b) Customs Duty (c) Sales Tax (d) Income Tax 89. Which among the following is/are the feature (s) of a Federal State? 1. The powers of the Central and the State (Constituent Unit) Governments are clearly laid down. 2. It was unwritten constitution. Select the correct answer using the codes given below: (a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 and 2 90. Which among the following statements with respect to the comptroller and Auditor General of India is/are correct? I. The Procedure and grounds for his removal from the office are the same as of a Judge of Supreme Court. II. He prescribes the form in which accounts of the Union and the States are to be Kept. Select the correct answer using the code given below: (a) I only b) II only (c) Both I and II (d) Neither I nor II

91. Which of the following are the Principles of the Panchsheel? 1. Peaceful co-existence 2. Mutual protection of the environment 3. Mutual protection of indigenous population 4. Mutual non-aggression Select the correct answer using the codes given below (a) 2 and 3 (b) 1 and 2 (c) 1, 2 and 4 (d) 1 and 4 92. Which of the following Constitutional Amendments have added Article 15 (5) in the Constitution of India Providing for reservation in educational institutions in the private sector also? (a) 81st Amendment (b) 86th Amendment (c) 91st Amendment (d) 93rd Amendment 93. Which of the following is/are instance (s) of violation of human rights? 1. A person was detained by the security forces while going for casting vote in parliamentary election. 2. A civilian was killed by the army while undertaking combing operation. Select the correct answer using the codes given below: Codes: (a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2 94. Which of the following is/are the function/functions of the Finance Commission of India? 1. Making recommendations as to the Principles governing grant-in-aid of the revenues of the States. 2. To Suggest measures to augment Consolidates Fund of India? Select the correct answer using the codes given below: (a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2 95. Which of the following regarding Revised Macro Management of Agriculture Scheme (2008 -2009) is/are correct? 1. Allocate funds to States and Union Territories on a criterion based on gross cropped area and area under small and marginal holdings. 2. The assistance is provided to the States and Union Territories as 100% grant. Select the correct answer using the codes given below: Codes: (a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2

Indian Polity ▌ 2.233 96. Which of the following statements is/are correct with regard to the Vice-President of India? 1. He must be Member of Parliament. 2. He is elected by Proportional representation. 3. He is Ex-officio Chairman of the Rajya Sabha. Select the correct answer using the codes given below: (a) 1 only (b) 1 and 3 (c) 2 and 3 (d) 1 and 2 97. Which of the following statements is/are not violative of the principle of federalism? 1. The President of India takes over administration of provinces under the emergency provisions. 2. The Parliament of India has exclusive power to make any law with respect to any matter not enumerated in the Concurrent List or State List. 3. The distribution of powers between the Union and Provinces is done through three different lists enumerated in the Constitution of India. Select the correct answer using the codes given below: (a) 1 and 2 (b) 2 and 3 (c) 3 only (d) 1 and 3 98. Which of the following statements with regard to the Federal System is/are correct? 1. In a federation, two sets of governments co-exist and there is distribution of power. 2. There is a written Constitution. Select the correct answer using the codes given below: (a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1nor 2 99. Which one among the following features of Constitution of India is indicative of the fact that the real executive Power is vested in the Council of Ministers headed by the Prime Minister? (a) Federalism (b) Representative legislature (c) Universal adult franchise (d) Parliamentary democracy 100. Which one among the following has the power the regulate the right of citizenship in India? (a) The Union Cabinet (b) The Parliament (c) The Supreme Court (d) The Law Commission 101. Which one among the following pairs is not correctly matched? (a) Union List : Banking (b) State List : Agriculture (c) Concurrent List : Marriage (d) Residuary List : Education 102. Which one of the following Amendments of the Constitution of India deals with the issue of strengthening of the Panchayati Raj? (a) 42nd (b) 44th (c) 73th (d) 86th

103. Which one of the following Articles of the Constitution of India Makes a specific mention of village panchayats? (a) Article 19 (b) Article 21 (c) Article 40 (d) Article 246 104. Which one of the following pairs is not correctly matched? (a) Power of parliament Creating a New state (b) Power of State Legislature Altering the name of a State (c) Equality before the law Both Indian and nonIndian citizens (d) Equality of opportunity Indian citizen 105. Which one of the following political theories advocates the withering away of the State? (a) Capitalism (b) Fascism (c) Marxism (d) Democratic 106. Which one of the following statements about Trusteeship’s is not correct? (a) ‘Trusteeship’ Provides a means of transforming the capitalist order of society into an egalitarian one (b) It is based on the faith that human nature is never beyond redemption (c) It does not recognize any right of private ownership (d) It excludes legislative regulation of the ownership and use of wealth 107. Which one of the following theories of Communism is not a doctrine of karl Marx? (a) Dialectical Materialism (b) Historical Materialism (c) Class Struggle (d) Theory of New Democracy 108. Who among the following appoints the judges of a High Court? (a) The President of India (b) The Chief Justice of India (c) The Governor of the State concerned (d) The Union Minister of Law 109. Who among the following chooses the Speaker of the House of People? (a) The Prime minister of India (b) The Union Minister of Parliamentary Affairs (c) The leader of the Opposition the Lok Sabha (d) The House of People 110. Who among the following first imparted a mass character to the Indian National Congress? (a) Jawaharlal Nehru (b) Mahatma Gandhi (c) Subhash Chandra Bose (d) Lala Lajpat Rai 111. Who among the following was the advisor of the Drafting Committee of the Constituent Assembly? (a) B Shiva Rao (b) Dr DR Ambedkar (c) Sachindananda Sinha (d) BN Rao

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▌ General Knowledge

112. Who among the following was the Chairman of The High level Committee on competition Policy and Law? (a) Bimal Jalan (b) C Rangarajan (c) YV Reddy (d) SVS Raghavan

114. Who is the ex-officio Chairman of the Planning Commission of India? (a) Vice-President of India (b) The Prime Minister of India (c) The Finance Minister of India (d) The Law Minister of India

113. Who can initiate impeachment proceedings against the President of India? (a) Only Lok Sabha (b) Only Rajya Sabha (c) Either House of the Parliament (d) Any Legislative Assembly

115. Whose duty is it to recommend to the President of India on the issue of the distribution and allocation of the net proceeds of taxes in the context of Centre-State fiscal relations? (a) Planning Commission (b) National Development Council (c) Union Ministry of Finance (d) Finance Commission

answeR keY 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

(d) (b) (c) (a) (c) (d) (a) (d) (b) (b) (c) (c) (c) (b) (c)

16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30.

(b) (b) (c) (c) (c) (c) (b) (d) (d) (c) (b) (b) (b) (c) (d)

31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45.

(b) (c) (a) (b) (a) (c) (d) (d) (c) (a) (b) (c) (d) (b) (d)

46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60.

(a) (a) (b) (c) (a) (c) (c) (d) (c) (c) (b) (c) (a) (d) (c)

61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75.

(a) (d) (c) (d) (c) (c) (c) (b) (d) (d) (b) (b) (c) (a) (b)

76. 77. 78. 79. 80. 81. 82. 83. 84. 85. 86. 87. 88. 89. 90.

(b) (c) (b) (d) (c) (c) (b) (d) (c) (a) (d) (d) (c) (a) (c)

91. 92. 93. 94. 95. 96. 97. 98. 99. 100. 101. 102. 103. 104. 105.

(d) (c) (c) (d) (d) (c) (b) (c) (d) (b) (d) (c) (c) (b) (c)

106. 107. 108. 109. 110. 111. 112. 113. 114. 115.

(d) (b) (a) (d) (b) (d) (d) (c) (c) (d)

C

H

4

A

P

T

E

R

indian Economy Planning Commission Year of Constitution In March 1950, Government of India constituted a statutory body, with the Prime Minister of India as its chairman, called the Planning Commission of India. The Planning Commission was set up in accordance with Article 39 of the directive principles of the Constitution of India. First Chairman Pt Jawaharlal Nehru was the first Chairman of the Planning Commission by virtue of his being the Prime Minister of India.

Functions (i) Assessment of the material, capital and human resources of the country, including technical personnel, and formulation or proposals for augmentation of such resources. (ii) Formulation of plans for effective and balanced utilization of resources. (iii) Defining stages in which the plan should be carried out. (iv) Determination of nature of the machinery necessary for implementation of the plans. (v) Appraisal from time to time of the progress achieved. (vi) Public cooperation in national development. (vii) Perspective planning. national Planning Council Is an advisory body attached to the Planning Commission and was established in 1965. It includes experts representing a cross-section of the Indian economy. constitute the National Development Council. The Prime Minister of India presides over the Council.

Five-Year Plans The development plans drawn by the Planning Commission to establish India’s economy in phase of five-years periods are

called Five-Years Plans. The organizations set up to formulate basic economic policies, draft plans and watch its progress and implementation consist of: (a) Planning Commission of India (b) National Planning Council (c) National Development Council and State Planning Commissions In India, we have opted for a five-year medium term plan, anchored in a rolling prospective plan normally, of 15 years as an instrument for bringing about planned socioeconomic development. So far, nine five year plans have been implemented and we are in the midst of the Tenth Five Year Plan. A Five-Year plan is an indicative plan of action over the ensuing five years reflecting largely the intent of the Government for that period at the national, regional and the sectoral level. The provisioning of the indicated plan outlay and the annual break-up of the physical targets in respect of various programmes and schemes, if any, are operationalised through allocations made in each of the annual plans constituting that period. Plan outlays by heads of development for Centre, States and UTs for the successive five year plans are followed by the underlying financing pattern of the public sector plans. Keeping in view the large-scale imports of food grains in 1951, and inflationary pressures on the economy, the First Plan (1951–56) accorded the highest priority to agriculture as well as irrigation and power projects. About 44.6 per cent of the total outlay of Rs 2,069 crore in the public sector (later raised to Rs 2,378 crore) was allotted for such development. The Plan also aimed at increasing the rate of investment from five to about seven percent of national income.

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▌ General Knowledge

Table: Five Year Plan Outlays (in crores) Five Year Plan

Period

Outlays Centre

States

UTs

Total

First Plan

1951–56

1,241.00

828.00

—

2,069.00

Second Plan

1956–61

2,559.12

2,240.88

—

4,800

Third Plan

1961–66

3,600.00

3,725.00

175.00

7,500.00

Fourth Plan

1969–74

8,870.00

6,606.47

425.00

15,902.16

Fifth Plan

1974–79

19,954.10

18,265.08

634.06

38,853.24

Sixth Plan

1980–85

47,250.00

48,600.00

1,650.00

97,500.00

Seventh Plan

1985–90

95,534.00

80,698.00

3,768.00

180,000.00

Eigth Plan

1992–97

247,865.00

179,985.00

6,250.00

434,100.00

Ninth Plan

1997–2002

489,361.00

369,839.00

*

859,200.00

Tenth Plan

2002–2007

8,93,83.00

6,32,456.00

5,25,639.00

1,212,802

Eleventh Plan

2007–2012

21,56,571.00

1,48,88,47.00

36,44718.00

•

The state outlay for 9th Plan includes the figure of UTs.

Table: Annual Plan Outlays (in crores) Annual Plan

Period

Outlays Centre

States

UTs

Total

Annual Plan

1966–67

1,089.37

931.72

60.45

2,081.54

Annual Plan

1967–68

1,172.00

1,010.00

64.00

2,246.00

Annual Plan

1968–69

1,320.00

974.00

65.00

2,359.00

Annual Plan

1979–80

6,411.81

8,910.33

227.49

12,549.63

Table: Five Year Plans Period

Plan

Remarks

1951–52 to 1955–56

First Plan

Priority to agriculture and irrigation

1956–56 to 1960–61

Second Plan

Development of basic and heavy industries

1961–62 to 1965–66

Third Plan

Long term development of India’s economy

1966–67 to 1968–69

Annual Plan

Plan holiday period—China and Pakistan wars

1966–70 to 1973–74

Fourth Plan

1974–75 to 1977–78

Fifth Plan

Terminated a year earlier, by the Janata Government which introduced ‘Rolling Plan’concept

1978–79 to 1979–80

Annual Plan

Launched by the Janata Government

1980–81 to 1984–85

Sixth Plan

Originally launched by the Janata Government. However, abandoned by the new Government and a revised plan for 1981–85 was approved

1985–86 to 1989–90

Seventh Plan

Food, work and productivity were the basis priorities

1990 to 1991–92

Annual Plan

The Eighth plan was to begin from April 1990 but successive governments (V. P. Singh and Chandrashekhar) were unable to finalize the plan document. The two-year shift of the five-year plan by treating 1990–91 and 1990–92 as Annual Plan periods was necessitated, and the Eighth Plan was finally launched on April 1992 and covers the period 1992–93 to 1996–97.

1992–93 to 1996–97

Eighth Plan

Envisages overall growth rate of 5.6 per cent per annum over the period 1992–93 to 1996–97.

1997–2002

Ninth Plan

Priority to agriculture and rural development

2002–2007

Tenth Plan

Rapid economic growth, differntial development strategy, reforms in governance, policy and institutional reforms.

Indian Economy

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Period

Plan

Remarks

2007–2012

Eleventh Plan

Faster and more inclusive growth: An easy access for all to the basic physical infrastructure, health and education with a projected growth rate of 10 per cent by the end of the plan period and an expected growth of 4 per cent in the agricultural sector; reduction in disparities aeross all strata and fast employment creation.

2012-2016

Twelth Plan (Objectives from the initial white paper presented on April 21, 2011)

Basic objective is faster, more inclusive, and sustainable growth. (Could aim at 9.0 to 9.5 per cent which will need strong policy action. Major sectoral challenges are Energy, Water and Environment which need to be addressed without sacrificing growth. To find resources to create a world class infrastructure in the country. Require better performance in agriculture, for growth to be more inclusive. Plus, faster creation of jobs, especially in manufacturing, and stronger efforts at health, education and skill development are given importance. Also, need to improve effectiveness of programmes directly aimed at the poor, special programmes for socially vulnerable groups, and special plans for disadvantaged/backward regions.)

Thrust areas of the Eleventh Plan Eleventh Year Plan Sectors

Thrust Areas

Education

Quality upgradation in Primary Education, Expansion of Secondary Education, major emphasis on upgradation of Higher Education including Technical Education, ICT throughout education system.

Health, Nutrition, Drinking Water, and Sanitation

Major upgradation of rural health infrastructure, Medical education, Nutritional support to children and pregnant and lactating women through ICDS, health insurance based urban health facilities, Health care for elderly, achieving sustainability, improvement in service levels and moving towards universal access to safe and clean drinking water.

Agriculture and Irrigation

Ensuring Food Security, Supporting State-specific agriculture strategy and programmes, Better seed production, Focused agricultural research, Extension, Development of modern markets.

Rural Development, Land Universalization and improvement in programme delivery of NREGP, Integrated Watershed anagement Resources and Panchayati Raj including management of underground water level. Social Justice and Empowerment

Special attention to the needs of SCs, STs and minorities and other excluded groups through pre and post-metric scholarship, Hostels for boys/girls, Income and employment generation opportunities, Multisectoral development programmes for minorities in minority concentration districts.

Physical Infrastructure

Emphasis on the public-private partnership in investment, Initiate policies to ensure time-bound creation of world-class infrastructure, especially in remote and inaccessible rural areas and NE, Hinterland connectivity through improved rail and road infrastructure.

Energy

Electrification of all villages and extending free household connections to all 2.3 crore BPL households through RGGVY, Nuclear power development.

Scientific Departments

Development of satellite launch capabilities to GSLV-Mk-III, Development of new energy systems, viz, advance heavy water reactor and nano-technology.

Eleventh Plan Experience as Guiding Force to Making of Twelfth Plan

GDP growth likely to average 8.2% over 11th Plan: short of the 9% target, but remarkable given the global crisis and drought. In the 10th Plan GDP growth averaged 7.7 % Also, progress was seen on inclusiveness: Agricultural growth, Poverty Reduction, Education, Health, Upliftment of SCs/STs, Minorities etc. However progress on inclusiveness was less than expected and there is a possibility of missing the Millennium Development Goals (MDG), except perhaps on poverty. Inflation has accelerated in the last two years of 11th Eleventh Plan. Current international environment is very uncertain Global pressure on food, oil and other commodity prices. Financial conditions & exchange rates are likely to be volatile due to sovereign debt related problems in Europe/US, and readjustment of extra-ordinary monetary/fiscal easing.

Key Messages from Strong demand from all sectors of society to improve Implementation, Accountability and Service Consultations During Delivery. Citizens Groups broadly support the stated objectives of existing government programmes. Drafting of 12th Five Year Plan However, the design and institutional arrangements are weak. Greater devolution and empowerment needed. Government programmes need a new architecture: greater localisation, break-down of silos, feedback from citizens, and mechanisms for learning and sharing of best practices. A major contribution to economic growth now comes from the private sector. A policy environment that supports this dynamism is therefore important. Create environment for nurturing enterprise, improving markets, supporting innovation, providing access to finance and inculcating respect for common pool resources

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Twelth Five Year Plan (2012-2017) On April 2011 the Prime Minister, Dr Manmohan Singh, addressed the Full Planning Commission meeting in New Delhi concerning the twelfth Five Year Plan of India. Highlights of the white paper on the Twelfth plan are:

12th Plan Resources for the Centre (as per cent of gDP) • Resources for the Plan are being worked out in collaboration with the Ministry of Finance • A preliminary picture (as % of GDP) is:

1 Tax Revenue (Net) 2 Non-Tax Revenue incld.

11th Plan 12th Olan 2011–12 2016–17 Realization Projection BE 7.7 8.0 7.2 8.8 2.4

1.9

2.2

1.6

3 Fiscal Deficit

4.9

3.3

4.6

3.0

4 Total Resources (11213) 5 Non-Plan Expenditure

15.0

13.2

14.0

13.4

10.3

8.0

9.1

7.2

6 Centre’s GBS (425)

4.7

5.4

4.9

6.2

(a) Assistance to States

1.2

1.3

1.2

1.4

(b) GBS for Centre

3.5

4.1

3.7

4.8

7 IEBR of PSUs

2.9

3.3

2.9

3.6

8 Central Plan {6(b) 1 7}

6.4

7.4

6.6

8.4

Resource allocation Priorities in 12th Plan • Health and Education received less than projected in Eleventh Plan. Allocations for these sectors will have to be increased in 12th Plan. • Health, Education and Skill Development together in the Centre’s Plan will have to be increased by at least 1.2 per cent point of GDP. • Infrastructure, including irrigation and watershed management and urban infrastructure, will need additional 0.7 percentage point of GDP over the next 5 years. • Since Centre’s GBS will rise by only 1.3 percentage points over 5 years, all other sectors will have a slower growth in allocations Must reduce the number of Centrally Sponsored Schemes (CSS) to a few major schemes. For the rest, create new flexi-fund which allow Ministries to experiment in other CSS areas. • Use of PPP must be encouraged, including in the social sector, i.e., health and education. Efforts on this front need to be intensified. • Distinction between plan and non-plan being reviewed by Rangarajan Committee.

Poverty line Poverty line In 1980, the Planning Commission in its Sixth FiveYear Plan document defined poverty line on the basis of nutritional standards. The people living below the barest desirable nutritional standards of daily calorie intake of 2,400 calories per person in

rural and 2,100 calories per person in urban areas are said to be living below the poverty line. In terms of money, the poverty line was computed at Rs 76 per capita per month in rural and Rs 88 per capita per person in urban areas. Committee on Poverty Line

The Planning Commission has set up a committee to study whether nutritional requirement of people especially in urban areas have gone down over the years. Committee will also discuss whether the definition of poverty should add greater emphasis on expenditure on items other than food which have gained significance in the consumption pattern. The present definition is in terms of the calorific intake, and anyone who has a daily consumption of less than 2400 in rural and 2100 in urban areas, is classified as poor. In 1993–94, the population below poverty line was 36 per cent, which came down to 27 per cent during 2000–01. In 1994–95, the poverty line was revised upwards in terms of monetary expenditure but the basis of calorie intake remained unchanged, viz.:

For Rural Population

For Urban Population

• Calorie intake per person per 2,400 calories 2,100 calories day below • Per capita expenditure per Rs 107 p.m. Rs 122 p.m. month below • Annual household consumption Rs 6,420 p.a. Rs 7,320 p.a. expenditure average family of 5 persons) below • Percentage of population living below the above figures 25.8 percent (out of which 75 percent is tribal)

Table: Population below Poverty Line Period Poverty ratio percent Rural Urban Total

No. of people below poverty line in million Rural Urban

Total

1972–73

54.1

4.12

51.5

244.2

47.3

291.5

1977–78

51.2

38.2

48.2

253.1

53.7

306.8

1984–85

39.9

27.7

36.9

222.2

50.5

272.7

1989–90

28.2

19.3

25.8

168.6

42.2

210.8

1990–2000

27.1

23.6

26.1

170.5

49.6

220.1

2007 (estimates) 21.1

15.1

19.3

—

—

—

Poverty line at Current Prices (1990–91) Keeping the calorie intake unchanged, in terms of monetary expenditure, the poverty line stands revised as follows:

Calorie intake per person per day below Per capita expenditure per month below Annual household consumption expenditure (avg family of 5 persons)

For Rural Population

For Urban Population

2400 calories

2100 calories

Rs 152 p.m.

Rs 133 p.m.

below Rs 9120 p.a.

Rs 7980 p.a.

Indian Economy at 1991–92 prices According to figures released to the Rajya Sabha in December 1992, there are approximately 24 crore people below the poverty line, out of which about 19.6 crore are in rural and more than four crore are in urban areas. The poverty line has been fixed at Rs 11,000 per annum per family at 1991–92 prices during the Eighth Plan.

Performance of social sectors medical Health Care and Family Welfare Health programmes are primarily the responsibility of the state government, but the Union government provides finance for improvements in public health services. The structure of the health system is based on a network of primary health centres. Till 2003–04, there were 1,63,196 Sub-centres/ Primary Health Centres/ Community Health Centres (SC/ PHC/CHC), 38,031 Dispensaries and Hospitals (all), 9,14,543 Beds (Private and Public), 8,36,000 Nursing Personnel,6,25,131 Doctors (Modern System) in the country. Over the years, there has been an overall improvement in the health. The substantial improvement is the result of many factors including improvement in public health, coupled with infectious disease prevention and control, as also application of modern medical practices in diagnosis and treatment of various ailments.

national Health Policy (nHP), 2002 The basic objective of achieving an acceptable standard of good health amongst the general population of the country is set out in NHP–2002. Improvement in the general levels of health through larger allocations and more effective implementation of communicable disease programmes, changes in pattern of assistance and implementation, and greater focus on tertiary health care have received special emphasis. Despite overall improvement in socio-economic stalin and increasing life expectancy, the prevalence of infectious diseases continues to be an area of major concern in Health sector. Poverty For the country as a whole, the poverty ratio has declined from 54.9 percent in 1973–74 to 36 percent in 1993–94. The poverty ratio has declined in the last two decades as follows: Rural Areas

Urban Areas

1993–1994

37.3 percent

32.4 percent

1999–2000

27.1 percent

23.6 percent

Although reduction in the poverty levels in the last two decades is significant in both rural and urban sectors, but overall results are weak as compared to many other East-Asian countries like China, Indonesia, Korea, Malaysia, Philippines and Thailand, etc., due to their faster economic growth. In general, the faster the rate of overall growth, the faster is the rate of poverty reduction. The sustained solution to the problem of poverty depends upon the creation of opportunities for broad based economic development and higher growth. Together with the overall economic growth, the antipoverty and employment generation programmes have helped in reducing the incidence of poverty over the long run.

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The Prime Minister’s Integrated Urban Poverty Eradication Programme (PMIUPEP) Scheme, was introduced in January1996. The Scheme was envisaged to attack the urban poverty in the targeted urban areas in an integral manner. The scheme had two components, Self Employment Scheme (SES) and Shelter Upgradation Scheme (SUS), and is being implemented in over 400 urban agglomerations. People with income below Rs 11,850 per annum were eligible for assistance under this scheme. Mainly target customers were the urban poor including slum/pavement dwellers and street/destitute children, etc. Under the SES scheme, the borrower is financed for a project cost of Rs 1 lakh only, of which the bank is to finance 95 per cent and only 5 per cent will have to be met by the borrower himself. The loan repayment had an initial moratorium of 6 to 18 months and the installments are spread over 3 to 7 years of time period. No collateral is required under SES scheme. Under SUS, the borrower is given a monetary assistance up to Rs 10,000 for unit/renovation/repair of the household—75 per cent of the cost of project work will be financed and 25 per cent of the amount will form as subsidy. The repayment period is up to 10 years and appropriate guarantees are required for getting the loan approved. Employment The number of persons on the live registers of the employment exchanges gives an idea of the trend of unemployment subject to certain limitations. Employment exchanges cover mainly urban areas and not all the unemployed register their names in exchanges. Further, some already employed get registered for better employment.

Poverty alleviation and Employment generation Programmes India’s anti-poverty strategy comprises a wide range of poverty alleviation and employment generation programmes, many of which have been in operation for several years and have been strengthened to generate more employment, create productive assets, impart technical and entrepreneurial skills, and raise the income level of the poor. Under these schemes, both wage employment and self-employment are provided to the people below the poverty line. In 1998–99, government proposed to unify the various poverty alleviation and employment generation programmes under two broad categories of ‘Self-Employment Scheme’ and ‘Wage Employment Schemes’. Funding and organizational patterns will also be rationalized to achieve maximum beneficial impact of these programmes. The salient features of some of the major employment and anti-poverty programmes are given below: (a) Integrated Rural Development Programme (IRDP) and its allied programmes of Training Rural Youth for Self-Employment (TRYSEM) and Development of Women and Children in Rural Areas (DWCRA)are major self-employment programmes for poverty alleviation. The basic objective of IRDP is to enable identification of rural poor families to augment their incomes and cross the poverty line through acquisition of credit-based productive assets. Assistance is given in the form of subsidy by the government and term credit by the financial institutions for income generating activities.

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(b) Swarnjyanti Gram Swarozgar Yojna (SGSY) launched in April 1999 after restructuring of the IRDP and allied programmes. It is the only self-employment programmed being implemented for the rural poor. The objectives of the SGSY is to bring the assisted swarozgaris above the poverty line by providing them income generating assets through bank credit and Govt. subsidy. The scheme is being implanted on cost sharing basis of 75:25 between the centre and states. (c) The Training of Rural Youth for Self Employment (TRYSEM) aims to train rural youth from the target group of families in various skills to enable them to take up self/wage employment. It has been laid down that the coverage of youth from the SC and ST communities should be at least 50 per cent of the rural youth trained. Out of the total beneficiaries, at least 40 per cent should be women. (d) The Programme of Development of Women and Children in Rural Areas (DWCRA) aims to improve the socio-economic status of poor women in the rural areas through the creation of a group of women for income generating activities on a self-sustaining basis. Approximately 2 lakh women benefited during 1998– 99 under this scheme. (e) Jawahar Rozgar Yojana (JRY) is a wage employment programme with the main objective of generation of employment in the lean agriculture season to the unemployed and underemployed rural people, both men and women, living below the poverty line. The significant aspect of the scheme is that it is implemented by the Panchayats at the village, block, and district levels in the ratio of 70:15:15 respectively. (f) The Employment Assurance Scheme (EAS) has been universalized so as to make it applicable to all the rural blocks of the country. It aims at providing 100 days of unskilled manual work to up to two members of a family in the age group of 18 to 60 years normally residing in villages in the lean agriculture season, on demand, within the blocks covered under EAS. (g) The Million Wells Scheme (MWS) which was earlier a subscheme of JRY, is funded by the Centre and states in the ratio of 80: 20. The objective of the MWS is to provide open irrigation wells, free of cost, to poor, small, and marginal farmers belonging to SCs/STs and freed bonded labour. (h) The National Social Assistance Programme (NSAP) recognizes the responsibility of the Central and State Government for providing social assistance to poor households in case of maternity, old age, and death of the breadwinner. NSAP is a centrally sponsored programme with 100 percent central funding to the states/UTs that provide benefits under its three component schemes—(i) National Old Age Pension Scheme (NOAPS); (ii) National Family Benefit Scheme (NFBS); and (iii) National Maternity Benefit Scheme (NMBS). On the basis of the suggestions made by the Central Advisory Committee on NSAP, the government has since approved changes relating to enhancement in the rate of benefits of NFBS and NMBS. (i) National Rural Employment Guarantee Scheme (NREGS) NREGS was launched in February 2006, in 200 most backward

districts in the first phase, was expanded to 300 districts in the second phase during 2007-08. The remaining 266 districts was notified in September 2008 and the scheme has now been extended to will the districts of India. More than 4.47 crore households were provided employment in 2008-09. (j) Sampoorna Grameen Rozgar Yojana (SGRY) was launched in September 2001. The schemes of Jawahar Gram Samridhi Yojana (JGSY) and Employment Assurance Scheme (EAS) have been fully integrated with SGRY. The objective of the scheme is to provide additional wage employment along with food security; creation of durable community, social and economic assets; and infrastructure development in the rural areas. The scheme envisages generation of 100 crore man-days of employment in a year. The cost of the programme is to be shared between the Centre and the state on a cost sharing ratio of 87.5:12.5 (including food-grains component). (k) Pradhan Mantri Gramodaya Yojana (PMGY) PMGY was launched in 2000–01 in all the states and UTs in order to achieve the objective of sustainable human development at the village level. The PMGY envisages the allocation of Additional Central Assistance to the states and UTs for selected basic minimum services in order to focus on certain priority areas of the government. PMGY initially had five components viz., primary health, primary education, rural shelter, rural drinking water, and nutrition. Rural electrification has been added as an additional component from 2001–02. The Planning Commission directly implements this programme. (l) Pradhan Mantri Gramodaya Yojana (Gramin Awas) It seeks to achieve the objective of sustainable habitat development at the village level. (m) Pradhan Mantri Gram Sadak Yojana (PMGSY) Launched in December 2000 as a 100 percent centrally sponsored scheme, it is a programme to provide road connectivity through good all-weather roads to1.60 lakh unconnected habitations with a population of 500 persons or more in the rural areas by the end of the Tenth Plan period (2007) at an estimated cost of Rs 60,000 crore. The programme is being executed in all the states and six Union Territories. While the focus of the programme is on providing road connectivity to unconnected habitations of stipulated population size, connectivity is being provided to all panchayat headquarters and place of tourist interest under the PMGSY, irrespective of the population size. The present source of funding for PMGSY is the diesel cess, 50 percent of which is earmarked for PMGSY. Efforts are underway to raise additional resources for the programme with financial assistance from the World Bank and the Asian Development Bank. (n) Antyodaya Anna Yojana (AAY) It was launched by the Prime Minister in December 2000. Under the scheme, 2 crore of the poorest families out of the BPL families covered under the Targeted Public Distribution system are identified. To each eligible family, 25 kg of food grain is made available at a highly subsidized rate of Rs 2 per kg for wheat and Rs 3 per kg for rice. This quantity had been enhanced from 25 to 35 kg with effect from April 2002 for a period of one year, that is, up to 31 March 2003.

Indian Economy (o) Annapurna It was launched on 1 April 2000, as a 100 per cent centrally sponsored scheme. It aims at providing food security to meet the requirement of those senior citizens who, though eligible for pension under the National Old Age Pension Scheme, are not getting the same. Ten kilogram of food grains per person per month are supplied free of cost. (p) Indira Awaas Yojana (IAY) It aims at providing dwelling units, free of cost, to the poor. The Ministry of Rural Development (MORD) provides equity support to the Housing and Urban Development Corporation (HUDCO) for this purpose. (q) Rural Employment Generation Programme (REGP) It was launched in 1995 with the objective of creating self-employment opportunities in the rural areas and small towns, and is being implemented by the Khadi and Village Industries Commision (KVIC). Under REGP, entrepreneurs can establish village industries by availing of margin money assistance from KVIC and bank loans for projects with a maximum cost of Rs 25 lakh. (r) Prime Minister’s Rozgar Yojana (PMRY) It was started with the objective of making available self-employment opportunities to the educated unemployed youth by assisting them to set up any economically viable activity. While the REGP is implemented in the rural areas and small towns (population up to 20,000) for setting up village industries without any cap on income, educational qualification or age of the beneficiary, PMRY is meant for educated unemployed youth with family income of up to Rs 40,000 per annum, in both urban and rural areas, for engaging in any economically viable activity.

▌ 2.241

the responsibility for educational planning with the states. A 2% Education less was levied on all major central taxes through the finance (No.2) Act, 2004 to help finance Government’s commitment to quality basic education. The number of secondary and higher education schools in 2006–07 was 1,68,900 and total enrolment in secondary and higher secondary stage was 39.44 Mn in 2006–07. Gross Enrolment Ratio (GER) which shows total enrolment in secondary stage (9k–12k) as a percentage of total population in the relevant age group was 40.62 in 2006–07.The 86th Constitutional Amendment Act, 2002 led to insertion of a new Article 21-A in part III of the constitution on that made free and compulsory Education to all children of 6 to 14 years of age a fundamental Right. However, a suitable follow-up legislation is necessary to give effect to the 86th Constitutional amendment. The target of universalizing elementary education has been divided into three broad parameters: Universal access, universal retention and universal achievement. Various programmes/projects have been envisaged by the central and state governments to achieve Universalization of Elementary Education (UEE): (i) District Primary Education Programme, (ii) The National Bal Bhavan Institute under Deptt of Education, (iii) Bihar Education Project, (iv) Lok Jumbish, (v) National Programme of Nutritional Support to Primary Education (‘Mid-Day Meal Scheme’).

(u) Valmiki Ambedkar Awas Yojana (VAMBAY) It was formally launched by the Prime Minister on 2 December 2001. The Scheme seeks to ameliorate the conditions of the urban slum dwellers living below the poverty line who do not possess adequate shelter. The scheme has the primary objective of facilitating the construction and upgradation of dwelling units for slum dwellers and providing a healthy and enabling urban environment through community toilets under Nirmal Bharat Abhiyan, a component of the scheme. The Central Government provides a subsidy of 50 per cent, the balance 50 percent being arranged by the state government with ceiling costs prescribed both for dwelling units and community toilets.

Sarva Shiksha Abhiyan (SSA) Launched in November 2000 as an umbrella programme, it continued to be implemented to support and build upon other primary and elementary education projects. The programme aims to ensure 5 years of primary education for all children in the age group 6–14 years by 2007 and 8 years of schooling by year 2010. National Programme for Education of Girls at Elementary Level (NPEGEL) It was launched as an amendment to SSA for providing additional support for education of under privileged girls at the elementary level. The scheme is implemented in Educationally Backward Blocks (EBBs) where the level of female literacy is below, and the gender gap is above the national average in blocks of districts which are not covered. Kasturba Gandhi Balika Vidyalaya (KGBV) It is a new scheme approved during 2004–05 for setting up 750 residential schools with boarding facilities at elementary levels for girls belonging predominantly to the SC, ST, other backward classes (OBC), and minorities in difficult areas. Prathmik Siksha Kosh (PSK)) An education cess of 2 percent on all direct and indirect central taxes has been imposed through the Finance (No. 2) Act, 2004 soon after the relevant Bill was initiated for creation of a seperate, dedicated, non elapsable fund to be named as Prathmik Shiksha Kosh and maintained by the Ministry of Education and Literacy. The proceeds would be available on a rollover basis for the schemes of basic education and the Mid-day Meal Scheme.

literacy and Education Education is primarily the responsibility for the individual state governments. But the Union has a clear responsibility regarding the quality and character of education in the nation. In addition to policy formulation, Ministry of Human Resource Development Department of Education shares

Secondary education series as a bridge between primary and higher education and prepares youngsters between the age group of 14–18 years for entry into higher education and work, various programmes under Secondary Education are: (i) Integrated Education f or Disabled Children, (ii) National Population

(s) Jai Prakash Rozgar Guarantee Yojana JPRGY It seeks to provide guaranteed employment to the unemployed in the most distressed districts of the country. Operational modalities for launching of the scheme are being worked out. (t) Swarna Jayanti Shahari Rozgar Yojana (SJSRY) The Urban Self-Employment Programme and the Urban Wage Employment Programme are two special schemes of the SJSRY, initiated in December 1997, which replaced various programmes operated earlier for urban poverty alleviation. This is funded on a 75 : 25 basis between the Centre and the states.

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Education Project (School Education), (iii) National Open School, (iv) Kendriya Vidyalaya, (v) Navodaya Vidyalaya National Talent Search Scheme (NTSS), (vi) National Council for Educational Research and Training (NCERT). According to University Grant Commission (UGC), enrolment in various courses at all levels in universities/colleges and other higher education institutions was 11.61 in 2006–07. Out of this, 4.7 million students were female constituting 40.55%. Universities and university level institutions in India include 20 central Universities, 215 State Universities, 100 Deemed Universities, 5 institutions under state Act, and 13 institutes of National importance apart from around 17000 colleges including 1800 women colleges. Various institutions/ organizations active in University and Higher Education sector are as follows: (i) University Grants Commission, (ii) Indian Council of Historical Research (New Delhi),

(iii) Indian Council of Philosophical Research (New Delhi and Lucknow), (iv) Indian Institute of Advance Study (Shimla), (v) Indian Council of Social Science and Research (New Delhi), (vi) National Council of Rural Institutes, (vii) Indira Gandhi National Open University (IGNOU) Adult Education programmes in the country have been the most important development that has taken place under the National Literary Mission (NLM) which was launched in 1998, with the objective of imparting functional literacy to 800 lakh adult illiterates in the agegroup of 15–35 by 1995. The target was revised to be 100 million by the end of March 1999. In 1995, according to UNESCO estimates, the rate of adult illiteracy in India averaged 48.0 per cent (males 34.5 per cent; females 62.3 per cent). With the success of Ernakulam and Kerala experiments for achieving full literacy, Total Literacy Campaigns have been launched to consolidate the gains of literacy.

CuRREnCY sYsTEm Historical Background • First gold coins were introduced during the reign of the Guptas AD 390–550. • Rupee was first minted in India during the reign of Sher Shah Suri around AD 1542. It was a silver coin weighing around 179 gm and it replaced the gold coins. • In 1873, when the price of silver fell in the world market, the silver coin lost its metallic value. The exchange value of the Indian rupee was Rs 10 per pound sterling till 1873. • Paper currency in India was introduced in 1882 by the British government. • With the establishment of the Reserve Bank of India in 1935, the Indian rupee became an independent currency, although for exchange purposes it continued to be dependent on Sterling. • In 1947, India became a member of the International Monetary Fund (IMF) and exchange value of the rupee came to be fixed by IMF standards.

The Decimal system The Indian currency system was converted into decimal system by Indian Coinage (Amendment) Act 1955 which was brought into force from 1 April 1957. The old system of rupee, annas and paise (1 rupee = 16 annas and 1 anna = 12 paise) was replaced by Rupee and Paise system. The first one-paisa coin under the decimal system was issued in March 1962 and the first one rupee coin in July 1962. issue and Distribution of Currency All coins and one rupee notes are issued by Government of India and, therefore, the one rupee note does not bear the signature of the Governor of the Reserve Bank of India. Currency Denominations At present currency notes of denominations of Rs 1, 2, 5, 10, 20, 50, 100, 500 and 1,000 are in circulation. The

current series which began in 1996 is called the Mahatma Gandhi Series. The currency notes of Rs 500, bearing a portrait of Mahatma Gandhi and the Ashoka Pillar emblem were issued by RBI from 3 October 1987. All notes above one rupee denomination are issued by the Reserve Bank of India and, therefore, currency notes above one rupee denomination bear the signatures of the Governor of Reserve Bank of India. These notes are also called bank notes. Each bank note has its amount written in 15 languages (English, Hindi and 13 others on the back), illustrating the diversity of the country. Distribution and administration of all currency is done by the Reserve Bank of India as an agent to the Government of India. Demonetization of Currency Demonetization refers to the withdrawal of currency from circulation which is done to ambush black market currency and unaccounted money. So far, demonetization has taken place twice, viz., 1. First Demonetization was done in 1946 (World War II) which called for declaration of notes of Rs 100 and above. All notes of Rs 500 denomination and above were demonetized. 2. Second Demonetization was done in January 1978, through which currency notes of higher denominations of Rs 1,000, Rs 5,000 and Rs 10,000 were demonetized. 3. Devaluation of Currency Devaluation refers to reducing value of the Indian rupee in comparison to the US dollar in the world market. In 1947, India became a member of the International Monetary Fund (IMF) which necessitated fixing of exchange value of the Indian rupee as per IMF standards. As a result, India was obliged to devalue the rupee and so far, the following devaluations have taken place: First Devaluation In June 1949, the Indian rupee was devalued by 30.5 per cent. Dr John Mathai was the Finance Minister. second Devaluation In June 1966, whereby the Indian rupee was further devalued by 57 per cent. Sachindra Chaudhury was the Finance Minister.

Indian Economy Third and Fourth Devaluations On 1 July 1991, the Indian rupee was devalued by 9 per cent and again further devalued by 11 per cent on 3 July 1991, bringing the total devaluation to 20 per cent. This was during the finance ministership of Dr Manmohan Singh. Since 20 August 1994, the rupee has been made a freely convertible currency on current account.

small note of Coins system in india The Government of India has the sole right to mint coins. The responsibility for coinage vests with the Government of India in terms of the Coinage Act, 1906 as amended from time to time. The designing and minting of coins in various denominations is also the responsibility of the Government of India. Coins are minted at

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the four India Government Mints at Mumbai, Alipore (Kolkata), Saifabad (Hyderabad), Cherlapally (Hyderabad) and NOIDA (UP). The coins are issued for circulation only through the Reserve Bank in terms of the RBI Act. Coins in India till 2011 were issued in denominations of 10 paise, 20 paise, 25 paise, 50 paise, one rupee, two rupees and five rupees. Coins of denomination of 25 paise and below will cease to be legal tender from 30 June 2011. These will not be accepted for exchange at bank branches and RBI Issue Offices from 30 June 2011. Coins upto 50 paise are called ‘small coins’ and coins of Rupee one and above are called ‘Rupee Coins’. Coins can be issued up to the denomination of Rs 1000 as per the Coinage Act, 1906. Government has in the recent past, imported coins to augment the indigenous production. Notes in denomination of Rs 5 have been reintroduced to supplement the supply of coins.

PuBliC FinanCE Power to raise and disburse public funds has been divided under the Constitution between the Union and the state governments. Ministry of Finance is responsible for administration of finances of the government. This Ministry comprises three departments: (i) Department of Economic Affairs; (ii) Department of Expenditure, and (iii) Department of Revenue. These are concerned with all economic and financial matters affecting the country as a whole. It also regulates expenditure of the government including transfer of resources to the states and other sorts of resource mobilization for development and other purposes. The Department of Economic Affairs is responsible for preparation of the Budget of Union of India as well as the state governments and union territory administrations with legislature when under the President’s rule. The department consists of eight main divisions, (i) Economic, (ii) Banking, (iii) Insurance, (iv) Budget, (v) Investment, (vi) External Finance, (vii) Fund Bank, and (viii) Currency and Coinage. The department also interalia monitors current economic trends and advises the government on all matters of internal and external economic management including working of commercial banks, term-lending institutions, investment regulations, external assistance, etc.

sources of Revenue Almost all the sources of revenue for the Union and the state government are mutually exclusive apart from the shareable taxes and duties. The Union has two types of revenues, tax-revenues and non-tax revenues. The main sources of Union tax revenues are Customs duties, Union excise duties, Corporate tax and Income tax, whereas Union nontax revenues are from interest receipts, including interest paid by the Railway and Telecommunications, dividend and profits. The taxes and duties levied by the state government and share of taxes levied by the union constitute the revenues in the state. Apart from their share of taxes and duties, state governments receive statutory and other grants as well as loans for various development and non-development purposes, and local finances like octroi, terminal taxes, property taxes. All the receipts and disbursements are kept under three separate accounts: (i) The Consolidate Funds of India—All the revenue received, loans raised

and interest receivables are from this fund and any disbursement there from are subject to the vote of the Parliament.(ii) Public Accounts of India—All other receipts by the government, such as deposits service funds, and other remittances go into Public Accounts. Money can be withdrawn from this account without any authority of an Act of the Parliament. (iii) Contingency Fund of India—This fund was established under Article 267(1) of the Constitution which is for meeting the unforeseen requirements not provided in the Annual Appropriation Act. The Indian Constitution provides for the establishment of a Consolidated Fund, a Public Account and a Contingency Fund for each state. Apart from this the Indian Constitution also provides that: • No tax can be levied or collected except by an authority of law • No expenditure can be incurred from public funds except as mentioned in the Constitution of India • Executive authority must spend public money only in the manner sanctioned by the Parliament in case of the Union and by the state legislature in case of a state. The Railways, the largest Public Undertaking, present their budget separately to the Parliament. Appropriations and disbursements under Railway budget are subject to the same form of parliamentary control as other appropriations and disbursements. However, as the Railways have no separate cash balance of their own, total receipts and disbursements of the Railways are incorporated in the Budget of the Union as part of the General Budget.

annual Financial Budget All the receipts and disbursements of the Union for the ensuing financial year are presented for general discussion in both the Houses of Parliament. This is known as presentation of the ‘Budget’ and it covers Central Government’s transactions occurring during the preceding year, the year in which the statement is prepared as well as the ensuing year or the ‘Budget Year’ as it is known. Tax proposals as new sources of revenue for the Union Government are embodied in a Bill and are passed as ‘Finance Bill’ or ‘Finance Act’ of the year.

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Estimates of expenditures from the Consolidated Fund of India are placed before Lok Sabha in the form of ‘Demand of Grants’. All withdrawals of money from the Consolidated Fund are thereafter authorized by an Appropriation Act passed by the Parliament every year. Estimates of receipts and expenditure are similarly presented by the state government in their legislatures before the beginning of the financial year and legislative sanctions for expenditure are secured through a similar procedure.

Public Debt Public debt is divided under two separate headings, namely, external debt and internal debt. The external debt comprises loans

from foreign countries, international financial institutions, etc. Internal debt includes borrowings inside the country like market loans, compensation and other bonds; treasury bills issued to the Reserve Bank of India, state governments, commercial banks and other parties. It also includes non-negotiable non interest bearing rupee securities issued to the international financial institutions. There is another head which comes under public debt called ‘other liabilities’ of the government. It includes outstandings against the various small saving schemes; provident funds; securities issued by Industrial Development Bank of India, Unit Trust of India and nationalized banks; deposits under the special deposit schemes; reserve funds and deposits.

Banking sYsTEm Development of Banking system in india The first bank in India managed by Indians was Oudh Commercial Bank, founded in 1881. It was a bank of limited liability. However, many institutions undertook banking business under British regime as agency houses which carried on banking business along with their trading business. The second Indian bank to be established was Punjab National Bank in 1884, and with the onset of ‘Swadeshi Movement’ in 1906, a number of commercial banks surfaced. In 1921, three presidency banks operating in India were amalgamated into Imperial Bank of India following serious financial troubles. In 1940s, a need for regulating and controlling commercial banks was felt, and in January 1946, the first Banking Act, The Banking Companies (Inspection Ordinance) was introduced, which was followed by another, The Banking Companies (Restriction of Branches) Act in February 1946. After independence, the Banking Companies Act was amended in 1949, and its name was also changed to The Banking Regulation Act. In 1993, new private sector banks were allowed to be set up in Indian banking system as government recognized need to introduce greater competition which can play an important role in ushering in a more efficient and competitive economy. However, the new bank had to satisfy the following requirements: (i) It should be registered as public limited company (ii) The minimum paid-up capital should be more than Rs 100 crore (iii) Should list its shares in the stock exchange (iv) The headquarters of the banks should be preferably located at a place where no other banks have headquarters representations (v) The bank should be subject to the prudential norms in respect of banking operations, accounting and other polices as laid down by the RBI (vi) It is to achieve a minimum capital adequacy of 8 percent from the very beginning In December 1997, another high-level committee, under the chairmanship of M. Narasimham was constituted by Government of India to review the implementation of financial system reforms

recommended in 1991. Also, it was desired from the Committee to analyse the present situation and propose the reforms necessary in years ahead to make the banking system stronger and better equipped to compete effectively in the international economic environment. The Committee submitted its report in April 1998.

narasimham Committee Report on Banking sector Reforms The Narasimham Committee on banking sector reforms has made a series of sweeping recommendations which could be used as a launching pad to take Indian banking into the next century. The Committee’s report, which was presented to finance minister, Yashwant Sinha, on 23 April 1998, covered an entire gamut of issues, ranging from bank mergers and the creation of global-sized banks to bank closures, recasting bank boards and revamping banking legislations. Making a case for a stronger banking system in the country, especially in the context of capital convertibility, the committee stated that CAR will result in large inflows and outflows with attendant implications for exchange rate management and domestic liquidity. Among other things, the committee has recommended merger of strong banks, as this would have a ‘multiplier effect’ on industry. However, it has also recommended not to merge strong banks with weak banks as it will have negative impact. Some other recommendations were: • • • • • • • • •

Free bank boards from day to day management Let some large banks have global character Try narrow banking to rehabilitate weak banks Confine small, local banks to states or cluster of districts Re-examine whether real autonomy is consistent with public ownership Review functions of boards and managements Refresh capital adequacy prescription Update RBI Act, Banking Regulation Act, Nationalization Act and SBI Act Put into place appropriate system for asset, liability and risk management.

Indian Economy

Reserve Bank of india (RBi) Creation The Reserve Bank of India (RBI) came into existence on 1 April 1935 as a privately owned bank with only 5 percent shares of the Government of India having a share capital of Rs 5 crore (share capital is still Rs 5 crore). nationalization From 1 January 1949, the Reserve Bank of India became a state-owned bank by acquiring private shareholdings by the government. administration General superintendence and direction of affairs vests with the Central Board of Directors which consists of 14 directors besides the Governor and four Deputy Governors. The Governor is the Chairman of the Board and the Chief Executive of the Bank.

governors 1. The First Governor Sir Obsborne Arkall Smith (1935–37) 2. The First Indian Governor C. D. Deshmukh (1943–49)

Functions 1. The RBI regulates issue of bank notes above one rupee denominations. Government of India issues one rupee notes/coins of all denominations below one rupee. The Reserve Bank of India undertakes distribution of all notes and coins on behalf of the government. 2. It acts as the banker of Government of India and the state governments, commercial banks and state cooperative banks. 3. It formulates and administers the monetary policy. 4. It maintains exchange value of rupee. 5. It represents India at the International Monetary Fund (IMF). Note: No personal accounts are maintained and operated in the Reserve Bank of India.

imperial Bank of india Before nationalization of RBI, the functions being performed by RBI were performed by the Imperial Bank of India. It was created in January 1921, by amalgamation of three presidency banks, viz., (i) Bank of Bengal, (ii) Bank of Bombay, and (iii) Bank of Madras. After nationalization in 1955, the Imperial Bank of India was named the State Bank of India.

scheduled Banks Reserve Bank of India maintains a schedule of banks which undertake to maintain the minimum percentage of their liability with the Reserve Bank of India and the paid up capital of Rs 5 lakh. The banks covered under this schedule are called scheduled banks.

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Functions The State Bank of India provides long-term credit and acts as the agent of the Reserve Bank of India in the country. It advances loans, deals in foreign exchange and other government’s banking operations. Subsidiaries of State Bank of India Apart from the main State Bank of India, there are seven subsidiaries: 1. 2. 3. 4. 5 6. 7.

State Bank of Bikaner and Jaipur State Bank of Hyderabad State Bank of Indore State Bank of Mysore State Bank of Patiala State Bank of Saurashtra State Bank of Travancore

other Banks industrial Credit and investment Corporation of india limited (iCiCi) Industrial Credit and Investment Corporation of India Limited (ICICI) was established in 1955, as a public limited company to encourage and assist industrial units of the nation. It was aimed to provide assistance in the creation, expansion and modernization of industrial enterprises, encouraging and promoting participation of private capital in such enterprises. It was also focused to promote industrial development and help development of capital markets. Nationalization of Banks Objective To use financial institutions as the instrument for promoting economic and social development in a more purposeful manner. To overcome the monopoly over financial resources, the Government of India nationalized 20 commercial banks during the tenure of Prime Minister Indira Gandhi. First Nationalization of 14 Banks On 19 July 1969, the first nationalization of 14 banks took place whereby the following banks were nationalized: 1. Bank of India 2. Union Bank of India 3. Bank of Baroda 4. Bank of Maharashtra 5. Punjab National Bank 6. Indian Bank 7. Indian Overseas Bank

8. Central Bank of India 9. Canara Bank 10. Syndicate Bank 11. United Commercial Bank 12. Allahabad Bank 13. United Bank of India 14. Dena Bank

Second Nationalization of Six Banks On 15 April 1980, the second nationalization took place whereby the following six banks were nationalized:

Public sector Banks

1. Andhra Bank 2. Corporation Bank 3. New Bank of India

4. Oriental Bank of Commerce 5. Punjab and Sind Bank 6. Vijaya Bank

state Bank of india The largest public sector bank of India which was created after nationalization of Imperial Bank of India in 1955. It is now the largest commercial bank in India and in terms of branches (10,836 branches), largest in the world.

In October 1993, the New Bank of India was merged with the Punjab National Bank. Therefore, at present there are only 19 nationalized banks in the country besides the Reserve Bank of India.

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small industries Development Bank of india (siDBi) SmallIndustries Development Bank of India (SIDBI) was established on 2 April 1990, as a wholly-owned subsidiary of IDBI under Small Industries Development Bank of India Act 1989. It is the principal finance institution for promoting, financing and assisting in development of industries falling under the small scale sector. indian Banks abroad In over 23 countries, nine of the Indian banks are operating through 97 branches (including the offshore branches and mobile agencies). The branches are located at all the major international trade centre including: London, Singapore, Amsterdam, Bahrain, New York, Hong Kong, Tokyo, Frankfurt, Paris, etc. These branches provide services in international banking including financing of foreign trade. The largest number of branches are in UK (21), followed by Fiji Islands (9), USA (8), Singapore (6), Mauritius (7), Sri Lanka (6), Hong Kong (7), UAE (6) and Japan (4). Apart from these branches, Indian commercial banks have also opened representative offices in USA, Brazil, Indonesia, Iran, Egypt, Russia, Italy, Zimbabwe, China, Uzbekistan, Philippines, and Vietnam. These commercial banks also have wholly owned subsidiaries and joint ventures in USA, Canada, Zambia, Nigeria, Uganda, Bhutan, Nepal, Keyna and Mauritius. All these cater to the needs of Indian exporters and importers and to that extent, they form an integral form of the domestic banking system of the country in which these are present.

national Bank of agriculture and Rural Development (naBaRD) The NABARD was set up by a government notification issued on 12 July 1982, and was inaugurated on 5 November 1982. Its primary objective is to promote agriculture and rural development through credit facilities to the farmers. Export-import Bank of india (EX-im) The EX-IM bank was set up on 1 January 1982. Among other things it grants deferred payment credit of medium and long-term duration for exports and renders financial assistance/guarantees to Indian exporters in order to operate in the international market. industrial Development Bank of india (iDBi) The IDBI, established in 1964, is an apex institution for industrial finance. Regional Rural Banks (RRB) The RRBs were brought into existence by a Presidential Ordinance in September 1975, and the first regional rural bank in the country, called ‘Prathama Grameen Bank’ sponsored by the Syndicate Bank was set up in October 1975, at Moradabad (UP). At present, the largest number of regional rural banks are in Uttar Pradesh. Lead Bank Scheme The Nariman Committee recommended the Lead Bank Scheme in 1969. Under this scheme each bank is assigned the major role of assisting in the development of banking and credit in the district allocated to it—by undertaking surveys to assess the credit gap in a particular district and determining the number of new branches required to meet the requirement of the area with the cooperation of other banks.

Functions (i)

To survey the number of industrial and commercial units and theis forms in the district to determine the potential of banking. (ii) To examine the facilities for the marketing of agricultural and industrial products, and linking credit with marketing. (iii) To assists other primary leading agencies. (iv) To maintain contact with the government and quasigovernment agencies.

new Banks in Private sectors In 1993, in recognition of the need to introduce greater competition, new private sector banks were allowed to be set up in India. These new banks had to satisfy certain requirements. On 19 November 2000, the government of India relaxed entry level for the private sector by reducing the government holding in nationalized banks from 51 percent to 33 percent. Further, revised guidelines for entry of new banks in private sector were issued on January 3rd 2001. The applications for setting up new banks received within the stipulated period were scrutinized by RBI and“ on-principle” approvals were issued to two new entities on 7 February 2002, and one of which “Kotak Mahindra Bank” on satisfactory completion of other formalities, was granted banking licence on 6 February 2003. The bank commenced its operation w.e.f. 22 March 2003, and subsequently, it has been included in the Second Schedule to the Reserve Bank of India Act, 1934 on 12 April 2003. On satisfactory completion of all formalities, licence was granted to “Yes Bank Ltd,” on 24 May 2004. List of Private Sector Banks Old Private Sector Banks 1. Bharat Overseas Bank Ltd 2. City Union Bank Ltd 3. Development Credit Bank Ltd 4. Lord Krishna Bank Ltd 5. SBI Commercial and International Bank Ltd 6. Tamilnad Mercantile Bank Ltd 7. The Bank of Rajasthan Ltd 8. The Catholic Syrian Bank Ltd 9. The Dhanalakshmi Bank Ltd 10. The Federal Bank Ltd 11. The Ganesh Bank of Kurundwad Ltd 12. The Jammu and Kashmir Bank Ltd 13. The Karnataka Bank Ltd 14. The Karur Vysya Bank Ltd 15. The Lakshmi Vilas Bank Ltd 16. The National Bank Ltd 17. The Nedungadi Bank Ltd 18. The Ratnakar Bank Ltd 19. The Sangli Bank Ltd 20. The South Indian Bank Ltd 21. The United Western Bank Ltd 22. The Vysya Bank Ltd

Indian Economy

New Private Sector Banks 23. 24. 25. 26. 27. 28. 29. 30. 31. 32.

Bank of Punjab Ltd Centurion Bank Ltd Global Trust Bank HDFC Bank ICICI Bank IDBI Bank Ltd IndusInd Bank Ltd UTI Bank Ltd (AXIS Bank) Kotak Mahindra Bank Yes Bank Ltd Foreign Banks in India

RBI has been issuing licenses to various foreign banks to operate in India. More than 50 foreign and multinational banks are working in India today. Important Foreign Banks in India 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34.

ABN-AMRO Bank N.V. Abu Dhabi Commercial Bank Ltd American Express Bank Ltd Arab Bangladesh Bank Ltd Bank International Indonesia Bank of Muscat S.A.O.G. Bank of America NA Bank of Bahrain and Kuwait B.S.C. Bank of Ceylon Barclays Bank PLC BNP Paribas Chinatrust Commercial Bank Chohung Bank Citibank N.A. Commerzbank AG Credit Agricole Indosuez Credit Lyonnais Deutsche Bank AG Development Bank of Singapore Ltd Dresdner Bank AG HSBC Ltd ING Bank N.V. JPMorgan Chase Bank (The Chase Manhattan Bank) K.B.C.Bank NA KrunThai Bank Public Co.Ltd Mashreq Bank Mizuho Corporate Bank Ltd (The Fuji Bank Ltd) Oman International Bank S.A.O.G. Oversea Chinese Banking Corporation Ltd Societe Generale Sonali Bank Standard Chartered Bank State Bank of Mauritius Ltd Sumitomo Mitsui Banking Corporation (The Sumitomo Bank Ltd)

35. 36. 37. 38. 39.

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The Bank of Nova Scotia The Bank of Tokyo-Mitsubishi Ltd The Siam Commercial Bank The Toronto-Domonion Bank Ltd UFJ Bank Ltd (The Sanwa Bank Ltd)

4.9 a Brief on Banking sector Reforms in india The banking sector reforms in India, initiated since 1992 in the first phase has provided necessary platform to the banking sector to operate on the basis of operational flexibility and functional autonomy, thereby enhancing efficiency, productivity and profitability. The reforms brought out structural changes in the financial sector, eased external constraints in their working, introduced transparency in reporting procedures, restructuring and re-capitalization of banks, and have increased the competitive element in the market. The salient features of these reforms include: • Phasing out of statutory pre-emption The SLR requirement have been brought down from 38.5 per cent to 25 per cent and CRR requirement from 7.50 per cent to 5.75 per cent. • De-regulation of interest rates All lending rates except for lending to small borrowers and a part of export finance have been de-regulated. Interest on all deposits are determined by banks except on savings deposits. • Capital adequacy CRAR of 9 per cent prescribed with effect from 31 March 2000. • Other prudential norms Income recognition, asset classification and provisioning norms have been made applicable. The provisioning norms are more prudent, objective, transparent, uniform and designed to avoid subjectivity. • Debt Recovery Tribunals 22 DRTs and 5 DRATs have already been set up and 7 more DRTs will be setup during the current financial year. Comprehensive amendments in the Act have been made to make the provisions for adjudication, enforcement and recovery more effective. • Transparency in financial statements Banks have been advised to disclose certain key parameters such as CRAR, percentage of NPAs, provisions for NPAs, net value of investment, Return on Assets, profit per employee and interest income as percentage to working funds. • Entry of new private sector banks 9 new private sector banks have been set up with a view to induce greater competition and for improving operational efficiency of the banking system. • Functional autonomy The minimum prescribed Government equity was brought to 51 per cent. Nine nationalized banks raised Rs 2855 crores from the market during 1994–2001. Banks Boards have been given more powers in operational matters such as rationalization of branches, credit delivery and recruitment of staff. • Hiving off of regulator y and super visor y control Board for financial supervision was set up under the RBI in 1994 bifurcating the regulatory and supervisory functions. India has made significant progress in payment system by

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introducing modern payment media viz., smart/credit cards, electronic funds transfer, debit/credit clearing, e-banking, etc. RBI would soon put in place Real Time Gross Settlement System (RTGS) to facilitate efficient funds management and mitigating settlement risks. Indian banking has made significant progress in recent years. The prudential norms, accounting and disclosure standards, risk management practices, etc. are keeping pace with global standards. The financial soundness and enduring supervisory practices as evident in our level of compliance with the Basle Committee’s Core Principles for Effective Banking Supervision have made our banking system resilient to global shocks. The need for further refinements in our regulatory and supervisory practices has been recognized and steps are being taken by RBI to move towards the goal in a phased manner without destabilizing the system. Success of the second phase of reforms will depend primarily on the organizational effectiveness of banks, for which the initiatives will have to come from banks themselves. Imaginative corporate planning combined with organizational restructuring is a necessary pre-requisite to achieve desired results. Banks need to address urgently the task of organizational and financial restructuring for achieving greater efficiency.

Reforms in the Rural and Cooperative Banking sector • All public sector banks, private sector banks and foreign banks as a group has achieved the over all target of priority sector for the last 3 years. • NABARD has sanctioned and disbursed Rs 19,849 crores and Rs 10,078 crores respectively to various State Governments under RIDE I to VII, as on 30 September 2001. • Share of commercial banks, cooperatives and RRBs in the production credit amounted to 38 per cent, 55 per cent and 7 per cent respectively. • There are 196 RRBs functioning in 26 States (including 3 newly created states) covering 495 districts with a network of 14,311 branches. Number of profit making RRBs increased from 44 to 172 during 1996–1997 to 2000–2001 and the amount of profit of RRBs increased from Rs 69.68 crores to Rs 681 crores during the same period.

monetary and Credit Policy Monetary policy is conducted by rule and/or discretion: By rule if policy markers reveal in advance the policy behaviour such as how it will respond to the changing market situations, and by discretion, if policy makes enjoy the freedom to size up the situation case by case and choose the most appropriate policy to deal with the changing situations. It has been customary for RBI to announce a set of measures both of short-term and structural nature in two bi-annual statements on monetary and credit policy released in April and October of each year. Keeping in view the significant changes in the framework for financial markets, the government has decided

with effect from the current financial year to give more importance to structural measures in monetary and credit policy for the current year. The short-term measures like the Bank Rate, Cash Reserve Ratio (CRR), Repo Rates, etc., will be included in the first half of the statement only if they coincide the timing of its release. The second statement of the year, the October statement, covers the mid-term review of the monetary and credit developments.

insurance The First Insurance Company was established in September 1870, at Mumbai

life insurance Corporation (liC) Established 1 September 1956 Head Office Mumbai Zonal Offices 7 (Mumbai, Kolkata, Delhi, Chennai, Kanpur, Hyderabad and Bhopal) LIC operates through 101 divisional offices including one salary savings scheme division at Mumbai and 2048 branches. As on 31 March 2004, LIC had 10,98,910 agents spread all over country. The corporation also transacts business abroad and has offices in Fiji, Mauritius and UK. Major Private Sector Insurance Companies • Life Insurance • • • • • • • • • • • •

HDFC Standard Life Insurance Co. Ltd ICICI Prudential Life Insurance Co. Ltd Tata AIG Life Insurance Co. Ltd SBI Life Insurance Co. Ltd Max New York Life Insurance Co. Ltd Om Kotak Mahindra Life Insurance Co. Ltd Allianz Bajaj Life Insurance Co. Ltd ING Vysya Life Insurance Co. Ltd Aviva Life Insurance Co. Ltd Sahara India Insurance Co. Ltd Met Life India Insurance Co. Ltd AMP SANMAR Assurance Co. Ltd

• General Insurance • • • • • • • • •

IFFCO Tokyo General Insurance Co. Ltd Royal Sundram Alliance Insurance Co. Ltd HDFC Chubb General Insurance Co. Ltd TaTa AIG General Insurance Co. Ltd ICICI bombard General Insurance Co. Ltd Cholamandalam General Insurance Co. Ltd Bajaj Allianz General Insurance Co. Ltd Reliance General Insurance Co. Ltd Export Credit Guarantee Corporation (ECGC) Ltd

general insurance Corporation (giC) Established 1 January 1973 Before the government took over, there were 107 insurance companies (both Indian and foreign) operating in India. By a

Indian Economy presidential proclamation issued on 13 May 1970, the government took over control of the general insurance. With effect from 1 January 1973, the erstwhile 107 Indian and foreign insurers which were operating in the country prior to nationalization, were grouped into four operating companies namely; (i) National Insurance Company Limited; (ii) New India Assurance Company Limited; (iii) Oriental Insurance Company Limited; (iv) United Insurance Company Limited. All these four subsidiaries of GIC operated all over the country, competing with one another and underwriting various classes of general insurance business except for aviation insurance of national airlines and crop insurance which is handled by the GIC. However, GIC has been formally delinked from its erstwhile subsidiary companies w.e.f. November 3rd 2000. The GIC having been notified as Indian insurer has also ceased to carry on Direct insurance Business and Crop Insurance. In its exclusive role as a reinsurance company, GIC’s business volume is growing up and it has also opened two representative offices at London and Moscow to expand its foreign reinsurance business. subsidiary Companies There are four subsidiary companies of General Insurance Corporation, viz., (i) National Insurance Co. Limited, Kolkata (ii) The New India Assurance Co. Limited, Mumbai (iii) The Oriental Fire and General Insurance Co. Limited, New Delhi (iv) United India Fire and General Insurance Co. Limited, Chennai Major Policies of GIC (i) Personal Accident Insurance Social Security Scheme for Poor Families (introduced in 1985) covers all the earning members of the families between age group 18–60 years whose annual income from all sources does not exceed Rs 7,200. (ii) Hut Insurance Scheme for Poor Families in Rural Areas covers the huts and belongings of the landless labourers, small farmers, artisans, and other poor families in rural areas. (iii) Mediclaim Insurance Policy provides for reimbursement of the medical expenses incurred by individual towards hospitalization/domicillary/hospitalization for any illness, injury or disease contracted or sustained during the period of insurance. (iv) Jan Arogya Bima Policy (introduced in 1996) is for target segments of populations who cannot afford the high cost of medical treatment and it provides for reimbursement of the medical expenses incurred by individual towards hospitalization/ domicillary/hospitalization for any illness, injury or disease contracted or sustained during the period of insurance. (v) Overseas Mediclaim Policy offers emergency medical expenses cover to overseas travellers which also includes Inflight Personal Accident cover up to US$10,000, and Loss of Passport cover up to US$ 150. (vi) Videsh Yatra Mitra (introduced from 1 January 1998) provides for medical coverage, in addition to Personal Accident cover, Loss of Personal Baggage cover, Delayed baggage cover, Personal Liability of US$200,000.

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(vii) Crop Insurance Scheme (introduced in 1995) covers for the crop for the Kharif and Rabi seasons. The scheme is administered by GIC on behalf of Central Government. The claims and premium are shared by the central and state governments in the ratio 2:1 respectively. (viii) Jald Rahat Yojana is introduced with an objective of expediting payment of compensation to road accident victims. Under this scheme claimant is not required to go to the Motor Accident Claims Tribunal for claiming compensation and can directly approach concerned insurance company.

new Pension system (nPs) NPS, regulated by PFRDA (Pension fund Regulatory and Development Authority), is a milestone in the development of a sustain all and efficient voluntary defined contribution pension in India. It has following broad objectives: i) To provide old age pension ii) Removable market based returns over the long term; iii) Extending old age security coverage to all citizens. NPS offers the following important features to help some for retirements. The pensioner under this scheme will be allotted a unique Permanent Retirement Account Number (PRAN) which will remain the same for entire life and one will be able to use this account number from any location in India.

main Features and architecture of the new Pension system • The new pension system would be based on defined constitutions. It will use the existing network of bank branches and post offices etc., to collect contributions. There will be seamless transfer of accumulations in case of change of employment and/or location. It will also offer a basket of investment choices and Fund managers. The new pension system will be voluntary. • The system would, however, be mandatory for new recruits to the Central Government service (except the armed forces.) The monthly contribution would be 10 per cent of the salary and DA to be paid by the employee and matched by the Central Government. However, there will be no contribution from the Government in respect of individuals who are not Government employees. The contributions and returns thereon would be deposited in a non-withdrawable pension account. The existing provisions of defined benefit pension and GPF would not be available to the new recruits in the central Government service. • In addition to the above pension account, each individual can have a voluntary tier-II withdrawable account at his option. Government will make no contribution into this account. These assets would be managed in the same manner as the pension. The accumulations in this account can be withdrawn anytime without assigning any reason. • Individuals can normally exit at or after age 60 years from the pension system. At exit, the individual would be required to invest at least 40 per cent wealth to purchase an annuity. In case of Government employees, the annuity should provide for pension for the lifetime of the employee

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and his dependent parents and his spouse at the time of retirement. The individual would receive a lump sum of the remaining pension wealth, which she would be free to utilize in any manner. Individuals would have the flexibility to leave the pension system prior to age 60. However, in this case, the mandatory annuitisation would be 80% of the pension wealth. • There will be one or more central record keeping agency (CRA), several pension fund managers (PFMs) to choose from which will offer different categories of schemes. • The participating entities (PFMs, CRA etc.) would give out easily understood information about past performance and regular NAVs, so that the individual would able to make informed choices about which scheme to choose.

social security group insurance schemes (ssgis) In 1989–90 Social Security Fund came into being, administered by LIC, to meet the requirement of the weaker and vulnerable sections of the society. Under the scheme, people in age-group18–60 years are covered for a sum of Rs 5,000 in case of natural death and Rs 25,000 in case of death by accident. Integrated Rural Development Programme (IRDP) under Social Security Group Scheme is being operated by LIC and is now known as ‘Swarnajayanti Gram Swarozgar Yojana’ (SGSY). KrishiShramaik Samajik Suraksha Yojana was introduced on 1 July2001, to provide Life insurance protection, periodic lump sum survival benefit and pension to agricultural workers. The Janashree Bima Yojana (JBY) was launched on 10 August 2000, to provide insurance cover on natural death. JBY replaced Social Security Group Insurance Scheme (SSGIS) and Rural Group Life Insurance Scheme (RGLIS).

insurance Regulatory and Development authority (iRDa) Insurance sector has been opened up with the establishment of Insurance Regulatory and Development Authority (IRDA) in April 2000. Functioning from Hyderabad, Andhra Pradesh, IRDA at present, consists of a Chairman, 3 whole time and 4-part time members. The major function of IRDA is registration of Indian Insurance companies under IRDA Act, 1999. Since its constitution, it has granted registration to 21 new insurance companies which comprise 13 life insurance and 8 general insurance companies in private sector. With a view to developing the insurance sector and protect the interest of the policyholders, the Authority has also issued regulations. Cooperatives and brokers have been allowed in the insurance sector with the enactment of Insurance (Amendment) Act, 2002. Pradhan mantri swasthya suraksha Yojana (PmssY) It has been introduced under 10th Plan to reduce various imbalances in the availability of tertiary cure, hospitals, Medical Colleges providing speciality/super-speciality services. Under this scheme, institutes similar to AIIMS (All India Institute of Medical Sciences), New Delhi, shall be set up in six backward states and another six existing institutions in other six states shall be upgraded to the level of AIIMS, New Delhi.

national surveillance Programme for Communicable Diseases (nsPCD) It was launched in 1997–98 as a pilot project with overall goal of improving health status of the people. NSPCD is being operated with the objectives of strengthening the surveillance system of communicable diseases and developing capabilities at state and district level to detect outbreaks early and to avert morbidities and mortalities. Yaws Eradication Programme (YEP) It was launched as a centrallysponsored scheme in 1996–1997 in Koraput district of Orissa, which was subsequently expanded to cover all 49 yaws epidemic districts in 10 states during 9th Plan. The program basically aims to reach the unreached tribal areas of the country. Reproductive and Child Health Programme (RCH) It was launched in 1997 and is currently operational in the entire country. It is the flagship programme of family welfare, which combines trinity of objectives, viz., reproductive health, child survival and fertility regulation with a policy and programme operation markedly different from previous programme. universal immunization Programme (uiP) It was launched in 1985 to reduce infant, child and maternal morbidity and mortality by protecting against six-vaccine preventable disease, viz. tuberculosis, diphtheria, pertussis, tetanus, polio, and measles. Pulse Polio Programme The National Technical Advisory Group on Immunization (NTAGI) was constituted in August 2001, to prepare a strategic framework on immunization. Based on the recommendation of NTAGI, multi-year Strategic Plan document has been prepared. Significant progress has been made in eradication of poliomyelitis since the adoption of supplemental immunization activity (Pulse Polio Programme). Under this programme, all children of 0–5 age are given two doses of Oral Polio Vaccine (OPV) at an interval of six weeks. national aiDs Control Programme It is implemented by the government throughout the country, with an objective; (i) to reduce the spread of HIV infection; and (ii) to strengthen the capacity of Central/State governments to respond to HIV/ AIDS on a long-term basis. All states and UTs have constituted state AIDS Control societies, and the central government is providing through the National Aids Control Organization(NACO), technical, managerial and financial resources to state AIDS control societies for the implementation of the programme components. Rashtriya arogya nidhi (Ran) Previously known as National Illness Assistance Fund (NIAF), it was set up under the Ministry of Health and Family Welfare in 1997, with initial contribution of Rs 5 crore. States/UTs have been released grant-in-aid as admissible under the scheme. These funds are replenished from time to time. Janani suraksha Yojana (JsY) It is proposed by way of modifying the existing NMBS (national Maternity Benefit Scheme), by providing an integrated package of obstetric care services to expectant mothers throughout their pregnancy period. Like NMBS, it would be focusing both maternal and child health.

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DisinvEsTmEnT The disinvestment of government equity in public sector was made through public offer in small lots from 1991–92 to 1999–2000. The disinvestment through strategic sale started from 1999–2000. They reflect the policies of respective governments at the centre.

Policy Framework All privatization will be considered on a transparent and consultative case-by-case basis. The existing “navaratna” companies will be retained in the public sector while these companies raise resources from the capital market. While every effort will be made to modernize and restructure sick PSUs and revive sick industry, chronically loss-making companies will either be sold-off, or closed, after all workers have got their legitimate dues and compensation. The government will induct private industry to turn around companies that have potential for revival. The 11 PSUs included in the category of ‘Navaratna’ are: • Steel Authority of India Ltd (SAIL) • Bharat Heavy Electrical Ltd (BHEL) • Indian Oil Corporation (IOC)

• • • • • • • •

Videsh Sanchar Nigam Ltd (VSNL) Hindustan Petroleum Corporation Limited (HPCL) Bharat Petroleum Corporation Ltd (BPCL) Oil and Natural Gas Commission (ONGC) National Thermal Power Corporation (NTPC) Indian Petrochemical Corporation Ltd (IPCL) Gas Authority of India Ltd (GAIL) Mahanagar Telephone Nigam Ltd (MTNL)

unit Trust of india (uTi) The largest Mutual Fund of India, the UTI has been bifurcated into two parts: (i) UTI-I comprising the guaranteed portion, and (ii) UTI-II comprising all NAV (Net Asset Value) based schemes, and the Unit Trust of India Act, 1963, has been repealed. UTI-I will be managed by an Administrator to be appointed by the Central Government, while UTI-II after being re-christened as UTI Asset Management Co. Pvt Ltd floated a mutual fund by the name of ‘UTI-II Mutual Fund’.

sToCk EXCHangEs There are 23 stock exchanges operating in the country. Out of these, 20 stock exchanges are regional and operate in their allocation areas. The remaining three, the National Stock Exchange (NSE), Over the Counter Exchange of India Limited (OTCEI) and Interconnected Stock Exchange of India Limited (ISE) are for nationwide trading. The network of the stock exchanges provides an organized market for transaction of shares and other securities.

securities and Exchange Board of india (sEBi)

deal with the matters of development and regulation pertaining to securities market and to protect the investors from fraudulent dealings. However, in 1992, SEBI was given the statutory powers through promulgated ordinance which was replaced by an Act of Parliament in April 1992. As per this Act, SEBI has power over corporates in issuance of capital, transfer of securities, and to impose monetary penalties on capital market intermediaries and the participants who violate the issued guidelines.

In April 1988, a non-statutory body, Securities and Exchange Board of India (SEBI) was constituted by the government to

minTs anD PREssEs The main functions of government mints are: (i) Minting of the coins to meet the domestic requirements, (ii) Gold and silver assaying, and (iii) Medal production. There are four government mints which are situated at Mumbai, Kolkata, Hyderabad and Noida, producing coins of all denominations and are equipped with the most modern assaying instruments. The Mumbai and Kolkata Mints manufacture commemorative coins. There are five Presses whose functions are as follows: (i) India Security Press (Nashik Road): It consists of two units—a Stamp Press (it prints postal stationery, postal and nonpostal stamps, judicial and non-judicial stamps, RBI/SBI cheques, Bonds, National Savings Certificates, Indira Vikas Patra, Kisan Vikas Patra, Postal Orders, Passports, Promissory Notes, and other security documents) and Central Stamp Depot (which takes care of all the finished product’s delivery to the respective indenters). (ii) Bank Note Press (Dewas): It consists of two units—ink factory

(it manufactures the security inks) and a main press (for printing the bank notes of the denomination of rupees twenty, fifty, hundred and five hundred). (iii) The Currency Note Press (Nashik Road): It prints the bank notes of denomination of rupees ten, fifty and hundred. (iv) Security Printing Press (Hyderabad): This press is used for printing postal stationery to meet the demand of southern states of the nation and Central Excise stamps for the use of whole country. Also printed here are Inland Letter Cards, Postcards, Competition Postcards, and embossed envelopes to supplement the production of India Security Press. (v) Security Paper Mills (Hoshangabad): This paper mill manufactures the special paper for printing the bank notes and Non-Judicial Stamp paper of higher denomination. (vi) Modernized Currency Notes Press: Two new modernized currency notes press are under establishment in Mysore (Karnataka) and Salboni (West Bengal).

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inDusTRial FinanCE insTiTuTions To cater to the needs of increasing modern industries in India, the Government of India has set up a number of specialized financial institutions. These institutions not only provide required finance to the large-scale industries, but also promote balanced industrial development. Some of these important institutions which provide long-term finance to industry are : industrial Finance Corporation of india (iFCi) Industrial Finance Corporation of India was set up in 1948, with the sole aim to provide medium and long-term finance to the industry. With an authorized capital of Rs 20 crore it became the first such termlending institution. Since 1948, when it started moderately, IFCI has sponsored many new schemes such as Risk Capital Foundation (RCF), financial assistance to leading and hire purchase concerns, suppliers credit schemes, assistance for the development of technology through in-house research and development efforts, etc. Its RCF scheme was floated with a view to assist new and technocrat-entrepreneurs in meeting a part of the promoters contribution to the equity capital. RCF was converted into Risk Capital and Technology Finance Corporation Ltd (RCTFC) on 12 January 1988. The financing from IFCI can take any of the following forms: (i) Guaranteeing of loans or debentures raised or issued by companies (ii) Underwriting of the issue of debentures (iii) Granting of loans or subscribing to debentures of approved industrial concerns (iv) Subscribing directly to the shares of the company (v) Guaranteeing deferred payment by importers of capital goods (vi) IFCI can act as an agent of the central government and the World Bank for the loans sanctioned by them to the industrial concerns The Fertilizers, Cement, Power Generation, Paper, Industrial machinery Industries have been on the top priority for IFCI which has sanctioned loans for the establishment of new industrial undertakings as well as expansion, modernization and rehabilitation of existing ones. It grants a loan for 25 years. The industrial Credit and investment Corporation of india ltd (iCiCi) The ICICI was set up in 1955, with the basic objectives of financing the private sector industrial concerns to create, expand and modernize, and to encourage the inflow and participation of foreign capital in these units. ICICI was formed with an authorized capital of Rs 60 crore and since then, it has become a leading institution in the country for providing foreign currency loans. ICICI has promoted a few specialized companies such as CRISIL and TDICI. CRISIL (Credit Rating Information Services of India Ltd) was floated along with Unit Trust of India to provide credit rating services to the corporate sector and TDICI (Technology Development and Information Company of India Limited) was promoted by ICICI to finance technology-transfer activities and also, technology upgradation. The chemical and petrochemical

industries have remained ICICI priorities and a major chunk of the assistance by ICICI has been distributed among three states, viz., Gujarat, Maharashtra and Tamil Nadu. The ICICI grants a loan for 15 years. industrial Development Bank of india (iDBi) Established in 1964, IDBI has grown into the largest long-term financial institution in the country with authorized capital of Rs 2,000 crore. The various functions of IDBI involve— (i) Refinancing loans given by various institutions working in India (ii) To accommodate short-term finances to industrial concerns (iii) To grant loans/advances and underwriting the issues of shares/debentures of individual concerns (iv) Direct subscription of the IFCI, SFCS or other financial institutions (v) Granting deferred payments and arranging assistance for promotion management or expansion of any industry To provide wide range of services to its customers, IDBI has devised various new products/schemes such as Venture Capital Fund Technological Upgradation, etc., which are also provided by way of Equipment Finance and Leading Schemes, Asset Credit Schemes, etc. IDBI has also taken special measures to spread industrialization to the non-industry districts through liberal product/services to the small scale industries there. On 7 March1982, the export finance activities of IDBI were transferred to newly constituted Export-Import (EXIM) Bank of India. The textiles, industrial machinery, electrical machinery, fertilizers, pharmaceutical and iron and steel industries have remained high on the priority list of IDBI. state Financial Corporations (sFCs) There as 18 SFCs, which are working across the country with an authorized capital of Rs 50 crore each. The SFCs came into operation after the State Financial Corporation Act 1951, authorized each state at that time to establish an SFC to perform all the functions of IFCI at a state level. The major chunk of their assistance is provided to small scale units, but it also provides services to concerns with sophisticated lines of manufacture. Among many schemes is the Seed Capital Assistance which is given by the SFCs to provide the special share capital to the needy entrepreneur. small industries Development Bank of india (siDBi) SIDBI was established as a wholly-owned subsidiary of IDBI in 1990. It was established as the principal financial institution for promotion, financing and development of industries in the small scale sector. SIDBI is engaged in providing assistance to the small-scale industrial sector through other institutions like SFCs, commercial banks and State Industrial Development Corporations. unit Trust of india (uTi) It was set up in February 1964 with an initial capital of Rs 2 crore. Its resources are the capital generated by issuing units of Rs 10 each to public. Declaration

Indian Economy of higher rates of dividend, reintroduction of Monthly Income Unit Schemes and introduction of Income Unit Schemes were the prime factors responsible for the record-breaking performance of UTI. industrial Reconstruction Bank of india (iRBi) Since its establishment in 1971, IRBI has made special efforts to provide finance and assistance in the form of technical, managerial, marketing

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guidance, service etc., to the sick units located in the backward areas and the units belonging to smalltime entrepreneurs. There are more than 60 development banks, in addition to these institutions, which meet the financial requirements of the industry. Some of them are the National Small Industries Corporation Ltd (NSIC), Life Insurance Corporation of India (LIC), General Insurance Corporation of India (GIC), etc.

inDusTRiEs Background The Industrial Policy (first announced on 6 April 1948) as adopted by the Government of India envisage a mixed economy, that is., coexistence of public sector, with an overall responsibility of the government for the planned development of industries in the country.

large-scale industries The larger-scale industries cover iron and steel, engineering, jute, cotton, textiles and sugar industries. iron and steel industries In 1870, the first steel industry ‘Bengal Iron Company’ was set up at Kulti. The first large-scale steel plant came into being in 1907 when the ‘Tata Iron and Steel Company (TISCO)’ was set up at Jamshedpur, followed by the Indian Iron and Steel Company, (IISCO) at Burnpur in 1919. Private sector steel Plant The first large scale steel plant in the private sector is Tata Iron and Steel Company, Jamshedpur. Public sector steel Plants The public sector steel plants are managed by the Steel Authority of India (SAIL). At present all important steel plants, except TISCO, are under the public sector.

Engineering industries These industries produced, besides machinery, tools, transport equipment and consumer durables. Jute industry Most of the jute mills are in West Bengal. As a foreign exchange earner, it is an important industry in the country. Cotton Textiles It is the oldest industry and it ranked among the seven largest foreign exchange earners in the country. The largest number of workers are employed in this industry. Statewise Distribution of Industries There is a great regional imbalance in location of various industries in the country. The country can be divided into three regions: (1) Industrially advanced states: Maharashtra, Gujarat, Tamil Nadu and West Bengal (2) Middle-level industrial states: Andhra Pradesh, Karnataka, Uttar Pradesh and Punjab (3) Industrially backward states: Rajasthan, Kerala, Haryana, Orissa, Himachal Pradesh

Table: Public Sector Steel Plants Location

Set up with assistance of

1. Rourkela (Orissa)

Germany

Set up under the Second Five-Year Plan at the cost of Rs 171crore. It went into production in 1959.

2. Bhilai (Madhya Pradesh)

Russian Government

Set up under the Second Five-Year Plan at the cost of Rs 8.5 crore. It went into production in 1959.

3. Durgapur (West Bengal)

British Government

Set up under the Second Five-year Plan at the cost of Rs 188 crore. It went into production in 1962.

4. Bokaro (Jharkhand)

Russian Government

The biggest plant in Asia was set-up under the Fourth Five-Year Plan. It went into production in 1973.

5. Burnpur (West Bengal) 6. Visakhapatnam (AP)

Remarks

Acquired by the nationalization of private sector plant Indian Iron & Steel Co. in 1976. Russian Government

Set up under the Sixth Five-Year Plan at the cost of Rs 2,256 crore.

7. Salem (Tamil Nadu)

Set up under the Sixth Five-Year Plan.

8. Bhadrawati (Karnataka)

Nationalized under the Sixth Five-Year Plan.

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various organizations Connected with industries Bureau of indian standards (Bis) This is a quasi-government institution for drawing up of standards for the products of Indian industry. It was established in 1947 and allots quality marks to various products, known as ISI Marks.

in 1958. It has regional offices throughout the country and it endeavours to create productivity consciousness in the country by applying modern methods and techniques for increasing production. To encourage productivity, NPC awards are given annually for highest productivity in various sectors of industry.

national Productivity Council (nPC) This is an autonomous body formed to inculcate productivity in industries. It was established Table: Principal Manufacturing Regions Regions/ Main Industries

Jharkhand–Bengal Industrial Belt

Mumbai–Pune

(a) Hugli-Jute Cotton Electrical Light engineering goods Chemicals

Cotton (over one-third of workers) Chemicals, Engineering Food Process-ing, Printing, Consumer goods

(b) Chhotanagpur Iron and Steel Wire Rollung, Vehicles

Light Engg., Consumer goods

Ahmedabad– Vadodara

Madurai Coimbatore– Bangalore

Cotton, Chemicals, Pottery

Cotton, Light Consumer goods, Light Engineering goods

Table: Important Resources Regions/ Main Industries

Ahmedabad– Vadodara

Madurai Coimbatore– Bangalore

Maharashtra and Madhya Pradesh

Mumbai

Andhra Pradesh

Singhbhum (Jharkhand)

—

—

Small iron & steel works at Bhadravati using ores extracted from Bahuban hills

Damodar Valley

Important supplies from Western Ghats

—

Cauvery, JogNilgiri Project

Cotton Groundnuts on regular soil of plateau to the East

Cotton from urrounding, Some cotton in Madurai sugar, soils, Salt from and Coimbatore salt pans

Jharkhand–Bengal Industrial Belt

Mumbai–Pune

Coal

Damodar Valleyz (Jharkhand), Assam, W. Bengal, Orissa, Arunachal Pradesh

Iron Ore

Power HydroElectricity

Other Raw Bengal Jute by water routes meeting Materials Damodar Coal (but much of the best (Agricultural) Jute has gone to Bangladesh)

Minor Regions

Important Industries

Assam Valley

Processing of local tea, rice and oilseeds, some petroleum

Darjeeling terrain

Processing of local tea

North Bihar adjoining Uttar Pradesh plains

Manufacturing sugar from local sugarcane

Delhi–Meerut

Manufacturing of sugar from local sugarcane, some textiles, chemicals, engineering goods

Indore–Ujjain

Cotton cloth for local markets, handicrafts (patronized by former Royal courts)

Nagpur-Wardha

Small textiles, iron foundries, railway and general engineering goods, glass and pottery works

Dharwar–Belgaum

Cotton textiles for local and other markets, railway and general engineering goods

Godavari–Krishna delta

Local tobacco, sugarcane, rice and oil, cement small textiles

Indian Economy

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Minor Regions

Important Industries

Kanpur

Textile and clothing, large modern tanneries, leather works, shoe manufacturing, all founded on early military needs

Chennai

Textiles, light engineering, light engineering, consumer goods of wide varieties

Malabar–Quilon Trichur

Cashew processing, coconut and oilseeds processing, associated industries (coir manufacturing, soaps), some textiles, numerous handicrafts

Sholapur

Important textiles based on cotton grown in local regular soils, engineering centres

Source: Economic Geography of India, NCERT.

Table: Important Industries in India Industry

Location

Aircraft industry

Bangalore and Kanpur

Aluminium

Alwaye (Kerala), Asansol (West Bengal), Belur (Karnataka), Hirakud (Orissa), Renukoot (UP), Muri (Jharkhand), Korba (Chhattisgarh)

Automobiles

Mumbai, Burnpur (West Bengal), Kolkata, Jamshedpur (Jharkhand) and Chennai

Cables

Rupnarainpur (West Bengal), Rajpura (Punjab)

Cement

Bhadravati (Karnataka), Churk (UP), Dalmianagar (Bihar), Gwalior, Kymor and Katni (MP), Okha (Gujarat), Sawai Madhopur (Rajasthan), Sahabad (Karnataka), Surajpur (Punjab)

Coir goods

Alleppey and Kalavoor (Kerala)

Cotton textiles

Ahmedabad (Gujarat), Bangalore, Mumbai, Kolkata, Coimbatore (Tamil Nadu), Indore (MP), Kanpur (UP), Ludhiana and Amritsar (Punjab), Chennai, Madurai (Tamil Nadu) Nagpur and Sholapur (Maharashtra)

Cycles

Ludhiana

D.D.T.

Alwaye (Kerala) and Delhi

Glass items

(a) Bangles [Firozabad (UP) and Belgaum (Karnataka)]; (b) Lampware [Kolkata, Naini (UP)] (c) Thermos flask [Faridabd (Haryana)]; (d) Glass bottles [Amritsar (Punjab)]; (e) Glass lenses [Jabalpur (Madhya Pradesh)]; (f) Glass sheets [Bahjoi, Balawali, Ghaziabad, Jaunpur (MP), Banga lore, Mumbai, Kolkata, Hyderabad, Chennai]

Fertilizer

Nangal (Punjab), Sindri (Jharkhand), Gorakhpur (UP), Nahorkatiya (Assam), Neyveli (Tamil Nadu) Rourkela (Orissa), Trombay (Maharashtra)

Hosiery goods

Amritsar, Ludhiana and Kanpur

Jute goods

Kolkata, Gorakhpur and Kanpur

Lac

Jhalda, Kossipore (West Bengal), Mirzapur (UP), Bareilly (UP)

Leather goods

Kanpur, Agra (UP), Batanagar (West Bengal), Mumbai, Kolkata, Chennai and Delhi

Locomotives

Chittaranjan (West Bengal), Varanasi (UP), Jamshedpur (Jharkhand)

Match box

Ahmedabad, Bareilly (UP), Mumbai, Kolkata, Chennai, Pune, Raipur (Chhattisgarh), Srinagar

Paper

Bhadravati (Karnataka), Dalmianagar, Jagadhari (Haryana), Lucknow and Saharanpur (UP), Rajahmundry and Sirpur (Andhra Pradesh), Titagarh (West Bengal), Brajrajnagar (Orissa), Dandeli (Karnataka), Amalai (MP), Ahmedgarh and Malerkotla (Punjab), Nepanagar (Madhya Pradesh). In Nepanagar, newsprint paper is produced.

Penicilin

Pimpri (Maharashtra)

Rail coaches

Perambur (Tamil Nadu), Pune (Maharashtra), Kapurthala

Resin industry

Bareilly (UP), Nahan (Himachal Pradesh)

Rubber goods

Ambapur (Tamil Nadu), Mumbai, Trivandrum, Bareilly (UP)

Salt

Kutch (Gujarat), Sambhar lake (Rajasthan)

Sewing machines

Kolkata, Delhi, Ludhiana

Shipbuilding

Visakhapatnam (Andhra Pradesh), Kochi, Mumbai, Kolkata (Continued)

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▌ General Knowledge

Industry

Location

Silk

Bangalore, Bhagalpur (Bihar), Srinagar

Sugar

Gorakhpur, Sitapur, Rampur, Moradabad, Bijor, Saharanpur, Meerut, Muzaffarnagar (UP), Gaya (Bihar), Zira, Jagraon (Punjab)

Sports material

Agra and Meerut (UP), Batalla and Jalandhar (Punjab), Delhi

Tanks

Avadi (Tamil Nadu)

Telephone

Bangalore, Naini and Rai Bareilli (UP)

Watches

Jalahalli (Bangalore), Pinjore (Haryana)

Table: Public Sector Industrial Undertakings Name

Location

Chemical and Pharmaceuticals 1. Hindustan Organic Chemicals Ltd

Rasayani (Maharashtra)

2. Indian Drugs and Pharmaceuticals Ltd

(a) Antibiotics Plant (IDPL) in Rishikesh (Uttaranchal); (b) Synthetic Drugs Projectin Hyderabad (c) Surgical Instruments Plantin Chennai

3 Hindustan Antibiotics Ltd

Pimpri (Maharashtra)

4. Hindustan Insecticides Ltd

Alwaye (Kerala) and Delhi

Fertilizers 1. Fertilizer Corporation of India Ltd

Sindri (Jharkhand); Trombay (Maharashtra); Gorakhpur (UP); Namrup (Assam); Durgapur (W. Bengal)

2. Heavy Water Plants

Neyveli (Tamil Nadu); Nahorkatiya (Assam); Rourkela (Orissa); Trombay (Maharashtra)

Machinery and Equipment 1. Bharat Dynamics Ltd

Hyderabad

2. Bharat Electronics Ltd

Jalahalli (Karnataka); Ghaziabad (UP)

3. Bharat Heavy Electricals Ltd

Ranipur (UP); Ramachandrapuram (AP); Tiruchirapalli (TN); Bhopal (MP)

4. Bharat Heavy Plate and Vessels Ltd

Visakhapatnam (AP)

5. Central Machine Tools

Bangalore

6. Chittaranjan Locomotive Works

Chittaranjan (WB)

7. Cochin Shipyard

Kochi

8. Diesel Locomotive Works

Marwadeeh, Varanasi (UP)

9. Garden Reach Workshop Ltd

Kolkata

10. Heavy Electricals Ltd

Bangalore

11. Heavy Electricals (India) Ltd

Bhopal

12. Heavy Engineering Corp. Ltd

Ranchi

13. Heavy Machine Building Plant

Ranchi

14. Heavy Vehicles Factory

Avadi (Tamil Nadu)

15. Hindustan Cables Factory

Rupnarainpur (WB)

16. Hindustan Machine Tools

Jalahalli (Karnataka) near Bangalore; Pinjore (Haryana); Hyderabad (AP); Kalamassery (Kerala)

17. Hindustan Shipyard

Visakhapatnam and Kochi

18. Indian Telephone Industries

Bangalore; Naini (UP); Rai-Bareilli (UP); Mankapur (Gonda, UP);

19. Instrumentation Ltd

Kota (Rajasthan); Palghat (Kerala)

20. Integral Coach Factory

Perambur (Tamil Nadu); Kotkapur (Punjab)

21 Machine Tool Corporation of India

Ajmer (Rajasthan)

22. Machine Tools Prototype Factory

Ambarnath, Mumbai

23. Mazagon Docks Ltd

Mumbai

Indian Economy

Name

Location

24. Mining and Allied Machinery Corporation Ltd

Durgapur

25. Nahan Foundry

Sirmur (HP)

26. National Instruments Factory

Kolkata

27. Praga Tools Corporation

Hyderabad

28. Triveni Structural Limited

Naini (HP)

29. Tungabhadra Steel Products Ltd

Tungabhadra (Karnataka)

Mining Projects 1. National Mineral Development Corporation

Hyderabad

2. Hindustan Zinc Limited

Udaipur (Rajasthan)

3. Bharat Aluminium Co Ltd

Korba (MP); Ratnagiri (Mah)

4. Hindustan Copper Ltd

Agnigudala (AP); Dariba (Rajasthan); Malanjkhand (MP) Rakha (Jharkhand)

5. Bharat Coking Coal Ltd

Dhanbad (Jharkhand)

6. Bharat Gold Mines Ltd

Kolar (Karnataka)

7. Coal Mines Authority Ltd

Kolkata

8. Neyveli Lignite Corporation

Neyveli (Tamil Nadu)

9. Zinc Smelter

Zawar (Rajasthan)

Paper National Newsprint and Paper Mills Ltd

Nepanagar (MP)

Petroleum 1. Indian Refineries Ltd

Barauni (Bihar); Noonmati (Assam)

2. Cochin Oil Refinery

Kochi (Kerala)

3. Koyali Oil Refinery

Koyali (Gujarat)

Steel Plants 1. Hindustan Steel Ltd

Bhilai (MP)

2. Hindustan Steel Ltd

Durgapur (WB)

3. Hindustan Steel Ltd

Rourkela (Orissa)

4. Bakaro Steel Ltd

Bokaro (Jharkhand)

Others 1. India Explosives Factory

Gomia in Hazaribagh (Jharkhand)

2. Hindustan Photo Films Manufacturing Co Ltd

Ootacamund (Tamil Nadu)

Table: Cottage Industries in India Name of Industry

States and Cities

Handloom Industry 1. Sarees and Dhotis

Tamil Nadu, Madhya Pradesh, West Bengal, Varanasi, Karnataka

2. Prints

Murshidabad, Farrukhabad, Jaipur, Mumbai, Karnataka

3. Carpets, rugs

Mirzapur, Bhadohi, Ellora, Kashmir, Jaipur, Bangalore

Silk 1. Silk Sarees

Bangalore, Kanjivaram, Karnataka

2. Tassar Silk

Sambalpur, Ahmedabad

3. Patola Silk

Baroda

Metal and Brass Wares 1 Brass utensils

Moradabad (with engraving and polishing)

2 Brass wares/metal wares

Jaipur, Kashmir, Varanasi, Madurai, Tanjore

Ivory Works

Andhra Pradesh, Kerala, Karnataka, Tamil Nadu, Rajasthan

▌ 2.257

2.258

▌ General Knowledge

Table: Indian Towns Associated with Industries Town

Industry

Town

Industry

Agra

Shoe and leather goods

Khetri

Copper

Ahmedabad

Cotton textiles

Ludhiana

Alwaye

Rare E arths Factory

Hosiery, sewing machines, cycles

Ambernath (near Mumbai)

Machine Tools Prototype Factory

Trombay

Uranium, thorium factory

Bangalore

Cotton textiles, toys, carpets, motors, aircraft, telephone and machines tools

Moradabad

Utensils, Calico factory

Mysore

Silk

Neyveli

Lignite

Perambur

Railway coach factory

Pimpri

Penicillin factory

Pinjore

Machine tools

Rana Pratap Sagar

Atomic Power station

Raniganj

Coal mining

Rourkela

Steel, fertilizers

Bareilly

Resin industry, wood work

Bhilai

Steel plant

Bokaro

Steel plant

Mumbai

Cotton textile, film

Kolkata

Jute, Electric bulbs and lamps

Chittaranjan

Locomotives

Delhi

Textile, DDT

Dhariwal

Woollen goods

Durgapur

Steel Plant

Firozabad

Glass and bangles

Gwalior

Pottery and textiles

Jaipur

Embroidery, pottery, brassware

Jamshedpur

Iron and steel goods

Jharia

Coal

Katni

Cement

Petroleum and natural gases History In 1867 the first oil well was sunk in India. The first successful well was sunk at Digboi in 1889. This oil field is still in operation after more than one hundred years. Till Independence, Assam was the only oil producing state in India. Now oil has also been struck in Gujarat near Baudo (Ahmedabad, Ankleshwar, Lunej, Kalol, Nagagaon). The recently discovered sea-bed oil fields of Mumbai High are also yielding substantial quantities of oil and are now the richest oilfields of the country. The total known reserves of crude oil are estimated at 51.08 crore million tonnes. areas would be awarded through international competitive bidding in a deregulated scenario. Since 1991 Exploration Oil and Natural Gas Corporation Limited (ONGC) and Oil India Limited (OIL), the two National Oil Companies (NOCs), and other private and joint venture companies are engages in the Exploration and Production (E and P) of oil and natural gas in the country. Since the year 2002, the Government of India has undertaken concerted efforts for enhancing “Energy Security”. The Tenth Plan formulated there under represented a paradigm shift over earlier plans in as much as exploration, the Government of India has been inviting bids on a regular basis with several rounds of bidding carried out till operationalisation of New Exploration Licencing Policy (NELP). After the operationalisation of NELP under the first four rounds (spanning 2000–2004), Production Sharing Contracts (PSCs) for 90 blocks have been signed. Further, the fifth round of

Rupnarainpur

Cables

Sindri

Fertilizers

Singhbhum

Copper

Surat

Textiles

Tiruchirapalli

Cigars

Titagarh

Paper

Trombay

Atomic power station

Visakhapatnam

Shipbuilding

the NELP (NELP-V) was opened in January 2005 for International bidding by offering 20 exploration blocks, that is., 12 on land, 2 shallow off shore and 6 deepwater blocks. organizational Set Up The Department of Petroleum under the Ministry of petroleum, chemicals and fertilizers, deals with the exploration and production of oil and natural gas, and the refineries and distribution work of the same. oil and natural gas Corporation limited In order to explore natural resources of oil and natural gas in the country, the Oil and Natural Gas Commission (ONGC) was set up at Dehradun in 1956. As per the Government’s decision, ONGC was converted into Public Limited Company under the Companies Act, 1956 and named as “Oil and Natural gas Corporation Limited” from 1 February 1994. In March 2004, 10 percent of the company was disinvested for the first time. At present, the Government has a shareholding of only 74.15 percent in ONGC. ongC videsh limited ONGC Videsh Limited (OVL), a wholly owned subsidiary company of ONGC Limited, was established with the purpose to undertake overseas projects for exploration and production of petroleum and other petroleum projects in order to augment the oil security of India and to bring equity oil from its overseas ventures. OVL has already acquired discovered producing properties in Vietnam, Russia and Sudan. OVL is also having participating interest in one exploration block each in Iran, Myanmar, Iraq, Libya, Australia, Cote d’ Ivoire, Syria, Qatar and Egypt (total 15 active projects in 12 nations).

Indian Economy

▌ 2.259

oil india limited It is now a Government of India organization which was incorporated in 1959 with headquarters in Duliajan (Assam) in collaboration with Burmah Oil Company. In 1961, OIL became a joint venture company with equal share of Government of India and Burmah Oil Company. In 1981, government took over the share of Burmah Oil Company marking the OIL entirely a public sector company. The main objectives of OIL are: (i) Exploration for and production of crude oil (including natural gas); (ii) Construction of pipeline to transport crude oil to government owned refineries of Noonmati and Barauni.

off-shore Drilling Bombay High in the deep waters of the western coast where drilling operations have been done with the help of the drilling platform, Sagar Samrat.

gas authority of india limited GAIL (India), a ‘Navratna’ enterprise, was established in the year 1984 and is primarily an integrated Natural gas company, focusing on all aspects of gas supply and value chain including exploration, production, transmission, petrochemicals, processing, distribution and marketing of Natural gas and other related products and services. The current holding of Government of India in GAIL, after disinvestment, is 57.35 percent.

Hindustan Petroleum Corporation Ltd (HPCL) Established in 1974 by amalgamating ESSO and Caltex. The government has acquired its full ownership since October 1976.

important oil-Bearing states/areas Assam, Tripura, Manipur, West Bengal, Ganga Valley, Himachal Pradesh, Kutch, Andhra Pradesh and offshore areas adjoining West Bengal, Orissa, Andhra Pradesh, Tamil Nadu, Karnataka, Maharashtra and Gujarat. main oil Fields Main oil fields where oil-bearing wells have been drilled are at: Gujarat: Cambay, Ankleshwar, Olpad, Sam, K7alori and Weinad Assam: Digboi, Rudrasagar and Sibsagar Punjab: Adampur, Janauri and Jwalamukhi

Corporation Indian Oil Corporation (IOC) Established in 1964 by amalgamating Indian refinery Limited and Indian Oil Company. Bharat Petroleum Corporation Ltd (BPCL) It came into being by the acquisition of Burmah Shell in 1976.

Oil Refining The domestic refining capacity of India is close to 130 million metric tonnes per annum (MMTPA). At present, there are 18 refineries operating in the country (17 Public Sector and one in Private Sector). Mangalore refinery and Petrochemical Limited (MRPL) which was a joint sector company became a PSU subsequent to the acquisition of its majority shares by ONGC. Out of 17 Public Sector refineries, 7 are owned by IOC Limited, two each by Chennai Petroleum Corporation Limited (a subsidiary of IOCL), Hindustan Petroleum Corporation Limited, and INGC; one each by BPCL, Kochi Refineries Limited (a subsidiary of BPCL), Numaligarh Refinery Limited (a subsidiary of Bharat Petroleum Corporation), and Bongaigaon Refineries and Petrochemical (a subsidiary of IOCL). The private sector refinery belongs to Reliance Industries Limited.

PoPulaTion oF inDia India’s population constitutes nearly 17.5 percent of the total world’s population in 2.4 percent of the world’s geographical area. It is the second country in the world, next to China, in terms of population. With this, India became the only second country in the world, after China to cross the one billion mark. India has as per the 2011 census, 1210 million persons.

Census Year

Total Population

1981

685,184,696

1991

843,930,861

2001

1,027,015,247

*2011

1,210,193,422

* As per provisional population totals: Census 2011

Census Population data are mostly collected through censuses in different countries of the world. First Census The first all-India census was conducted in 1872. However, further censuses were not held on a regular basis. Regular Census The first regular all India census was conducted in 1881 and thereafter every ten years. The last census was conducted in 2011, the second census of 21st century and the third millennium Table: Growth of population Census Year

Total Population

1941 (before independence)

318,660,580

1951

361,088,090

1961

439,234,771

1971

548,159,652

The percentage decadal growth during 2001–2011 has registered the sharpest decline since independence. For 2001–2011, this decadal growth has become 17.64 percent, a decrease of 3.9 percentage points from 21.54 percent for the period 1999–2001. Nagaland led the states with the lowest population growth rate at –0.47 percent whereas Meghalaya showed the highest growth rate at 27.82 percent. Density of Population Density of population is expressed as man to land ratio, that is, number of persons per unit of geographical area. The density of population of India is 382 persons/km2 (in 2011). It is higher than that of China (137 persons/km2 superscript). The density of population is, however, not a true reflection of pressure of population on land. While no land is useless, only arable/cultivable land sustains more people. If the arable land is taken into account a different picture emerges. Madhya Pradesh, a thinly populated state emerges as thickly populated because much of the state is hilly, dissected and forested.

2.260

▌ General Knowledge

Table: Census 2011 Total Population

*2011 Census

2001 Census

Persons

1210 million (approx.)

1027 million (1,02,70,15,247 to be exact)

Males

623.7 million

531.3 million

Females

586.4 million

495.7 million

Rural Poupulation

833 million (68.84 %)

742 million (72. 2%)

Urban Population

377 million (31.16%)

285 million (27.8%)

Absolute

181,455,986

180,627,359

Percentage

17.64%

21.34%

2

Average Density of Population (persons/km )

382

324

Sex Ratio (females/1000 males)

940

933

Birth rate (per 1000 persons)

To be released

26.4

Death Rate (per 1000 persons)

Decennial Population Growth (2001–2011)

Literacy Rate

Decadal Growth

Population (0-6 years)

To be released

9.0

Total

778454120 (74.04% of total population)

566714995 (65.38% of total population)

Males

444203762 (82.14% of total population)

339969048 (75.85% of total population)

Females

334250358 (65.46% of total population)

226745945 (54.16% of total population)

Total

(1) 17.64%

(1) 21.34%

Males

(1) 17.19%

(1) 20.93%

Females

(1) 18.12%

(1) 21. 27%

Persons

15,87,89,287 (13.12% of total population)

15,78,63,145 (15.42% of total population)

Males

8,29,52,135 (13.13% of total population)

8,19,11,041 (15.47% of total population)

Females

7,58,37,152 (12.93% of total population)

7,59,52,104 (15.36% of total population)

Most Populous and Least Populous Regions Most Populous State

Uttar Pradesh

19,95,81,477 (16.49% of total polulation)

16,60,52,859 (16.17% of total population)

Most Populous UT

Delhi

1,67,53,235 (1.38% of total population)

1,37,82,976 (1.34% of total population)

Least Populous State

Sikkim

6,07,688 (0.05% of total population)

5,40,493 (0.05% of total population)

Least Populous UT

Lakshadweep

64,429 (0.01% of total population)

60,595 (0.01% of total population)

*As per provisional population totals: Census 2011

Trends of population density

Year

Density

1951

117 persons/km²

1981

216 persons/km²

1991

267 persons/km²

2001

324 persons/km²

*2011

382 persons/km²

2011

2001

Most Densely Populated State

Bihar (1102 persons/km2)

West Bengal (904 persons/km2)

Most Densely Populated UT

Delhi (11,297 persons/km2)

Delhi (6319 persons/km2)

Least Densely Populated State

Arunachal Pradesh (17 persons/km2)

Least Densely Populated UT

Andaman and Nicobar (46 persons/km )

* As per provision population totals: Census 2011

Arunachal Pradesh 10 persons/km2 2

Andaman and Nicobar (34 persons/km2)

Indian Economy Rural and urban Population As per 1981 census, there were only 4,019 towns in the country with a total urban population of about 160 million people. This figure rose to 217.2 million people living in 4,689 towns in 1991 (according to the 1991 census). However in 2011, of the total 1210 million persons of the country, 833.08 million live in rural areas which is slightly less than 70 percent of the total population. However, about 377.10 million people live in urban areas constituting 31.16 percent of total population. Still India is one of the least urbanized nations of the world as nearly 45 percent of the world’s population lives in urban centers. Interestingly, more than half (51.5 percent) of India’s population lives in states of Maharashtra, Uttar Pradesh, Tamil Nadu, West Bengal, and Andhra Pradesh. It is pertinent to maintain that Uttar Pradesh figures in the list, although it is one of the least urbanized states of India. The urban population of India has increased from 25.85 million in 1991 to 377.10 million in 2011, recording an eleven fold increase during the last 100 years. The growth rate of urban population, however, started declining from the 1980s. It grew 31.13 percent only during 1991-2001. Table: Relative Growth of Rural and Urban Population Year 1941 1951 1961 1971 1981 1991

85.9 82.4 81.7 79.8 76.3 74.3

2001 *2011

72.2 68.84

Percent of Total Population Rural Urban 14.1 17.6 18.3 20.2 23.7 25.7 27.8 31.16

* As per provision population totals: Census 2011

sex Ratio Sex ratio is defined as the number of females per 1000 males. As per 2011 census sex ratio increased from 933 in 2001 to 940 in 2011. Sex composition is considered an important indicator of inequality between males and females in a society. The sex ratio at the beginning of the twentieth century was 972, which declined continuously till 1941. In 1951, there was a marginal increase, but thereafter it declined again. It has fluctuated marginally around 930. The causes of declining sex ratio are: • more males are born than females; • more females die at infancy and during the reproductive period; • general neglect of females is largely responsible for high female fatality at childhood; • pre-birth sex determination leads to female foeticide Table: Sex Ratio 1941–2011 Year 1941 1951 1961 1971

Ratio 945 946 941 930

Year 1981 1991 2001 *2011

▌ 2.261

Ratio 934 926 933 940

State with highest female population Kerala (1,084 females/1000 males) State with lowest female population Haryana (877 females/1000 males) * As per provisional populations totals: Census 2011 Table: Per 1000 Population

Year

Birth Rate

Death Rate

1941–1951 1951–1961 1961–1971 1971–1981 1981–1991 1991–2001 2001–2011

39.9 41.7 41.2 37.2 30.5 25.4 To be released

27.4 22.8 19.2 15.0 10.2 8.4 To be released

literacy Rate A literate is defined as “a person who can read and write with understanding” According to the 2011 census, it is 74.04 per cent registering an increase of 8.66 per cent. Males 82.14% Females 65.46% State with highest literacy rate Kerala 93.91% State with the lowest literacy rate Bihar 63.82% Note: On 18th April 1991, Kerala was declared India’s first nearly 100 percent literate state (94.54 per cent men and 86.93 per cent women)

Distribution of Population Population of India is unevenly distributed mainly because of different levels of natural resource base, presence of river basins, coastal plains or degree of urbanization. A look at the distribution of population shows huge variations. Important to note is that five major states (Uttar Pradesh, Maharashtra, Bihar, West Bengal and Andhra Pradesh) have half the Indian population. Also, Rajasthan the biggest state in terms of area, contains only 5.5 per cent of the total population, while Madhya Pradesh–the second largest contains only 5.88 per cent. literary Education is universally recognized as a major component of human development. As such, certain minimum level of literacy seems to be essential for a population to break out of the vicious cycle of poverty. Human development is therefore assessed in terms of literacy and related indicators. The literacy rate in India increased from 18.33 per cent in 1951 to 74.04 per cent in 2011. Only 68.91 per cent of the rural population is literate while 84.98 per cent of the urban population is literate. Also 65.46 per cent of the females are literate, as against 82.14 per cent of males. There has been a remarkable progress in literacy. For the first time since independence the absolute number of illiterates has declined by over 31.19 million in the last decade.

2

1

Andhra Pradesh

Arunachal Pradesh

Assam

Bihar

Chhattisgarh

Goa

Gujarat

Haryana

Himachal Pradesh

Jammu & Kashmir

Jharkhand

Karnataka

Kerala

Madhya Pradesh

Maharashtra

Manipur

Meghalaya

Mizoram

Nagaland

Orissa

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

India

India/State/Union Territory #

State/ UT Code

155.7

16.6

21.1

22.4

22.3

307.7

308.3

38.9

191.8

79.8

222.2

55.7

44.2

196.0

3.7

135.2

94.1

78.4

83.7

275.1

3

Area (per, 000 sq.km)

41,947,358

1,980,602

1,091,014

2,964,007

2,721,756

112,372,972

72,597,565

33,387,677

61,130,704

32,966,238

12,548,926

6,856,509

25,353,081

60,383,628

1,457,723

25,540,196

103,804,637

31,169,272

1,382,611

84,665,533

1,210,193,422

*2011

4 2001

36,706,920

1,988,636

891,058

2,306,069

2,388,634

96,752,247

60,385,118

31,838,619

52,733,958

26,909,428

10,069,917

6,077,248

21,082,989

50,596,992

1,343,998

20,795,956

82,878,796

26,638,407

1,091,117

75,727,541

1,027,015,247

Persons

21,201,678

1,025,707

552,339

1,492,668

1,369,764

58,361,397

37,612,920

16,021,290

31,057,742

16,931,688

6,665,561

3,473,892

13,505,130

31,482,282

740,711

12,827,915

54,185,347

15,954,927

720,232

42,509,881

623,724,248

*2011

5 2001

18,612,340

1,041,686

459,783

1,167,840

1,207,338

50,334,270

31,456,873

15,468,664

26,856,343

13,861,277

5,300,574

3,085,256

11,327,658

26,344,053

685,617

10,452,426

43,153,964

13,787,799

573,951

38,286,811

531,277,078

Males

Total population

Table: Distribution of Population, Sex Ratio, Area, Density, Decadal Growth Rate and Rank of Population: 2011

20,745,680

954,895

538,675

1,471,339

1,351,992

54,011,575

34,984,645

17,366,387

30,072,962

16,034,550

5,883,365

3,382,617

11,847,951

28,901,346

717,012

12,712,281

49,619,290

15,214,345

662,379

42,155,652

586,469,174

*2011

6 2001

18,094,580

946,950

431,275

1,138,229

1,181,296

46,417,977

28,928,245

16,369,955

25,877,615

13,048,151

4,769,343

2,991,992

9,755,331

24,252,939

658,381

10,343,530

39,724,832

12,850,608

517,166

37,440,730

495,738,169

Females

978

931

975

986

987

925

930

1,084

968

947

883

974

877

918

968

991

916

954

920

992

940

*2011

7

972

909

938

975

978

922

920

1,058

964

941

900

970

861

921

960

990

921

932

901

978

933

2001

Sex ratio(females per1000 males)

269

119

52

132

122

365

236

859

319

414

124

123

573

308

394

189

1,102

397

17

308

382

*2011

8

236

120

42

103

107

314

196

819

275

338

99

109

477

258

363

154

880

340

13

275

324

2001

Density (Per sq.km)

13.97

-0.47

22.78

27.82

18.65

15.99

20.30

4.86

15.67

22.34

23.71

12.81

19.90

19.17

8.17

22.59

25.07

16.93

25.92

11.10

17.64

*2011

9

15.94

64.41

29.18

29.94

30.02

22.57

24.34

9.42

17.25

23.19

29.04

17.53

28.06

22.48

14.89

18.06

28.43

18.85

26.21

13.86

21.3

2001

Decadal growth rate

11th

24th

27th

22nd

23rd

2nd

6th

12th

9th

13th

18th

20th

17th

10th

25th

16th

3rd

14th

26th

5th

*2011

10

11th

24th

27th

23rd

22nd

2nd

7th

12th

9th

13th

18th

20th

16th

10th

25th

17th

3rd

14th

26th

5th

2001

Rank

Rajasthan

Sikkim

Tamil Nadu

Tripura

Uttar Pradesh

Uttarakhand

West Bengal

22

23

24

25

26

27

28

Chandigarh

Dadra & Nagar Haveli

Daman & Diu

Lakshadweep

NCR of Delhi

Puducherry

2

3

4

5

6

7

0.5

1.50

0.0

0.1

0.5

0.1

8.2

88.8

53.6

240.8

10.5

131.1

7.1

342.2

50.4

3

Area (per, 000 sq.km)

*2011

1,244,464

16,753,235

64,429

242,911

342,853

1,054,686

379,944

91,347,736

10,116,752

4 2001

973,829

60,595

13,782,976

158,059

220,451

900,914

356,265

80,221,171

8,479,562

166,052,859

3,191,168

62,110,839

540,493

56,473,122

24,289,296

Persons

199,581,477

3,671,032

72,138,958

607,688

68,621,012

27,704,236

*As per provisional population totals: Census 2011

Andaman & Nicobar Islands

1

Union Territories

Punjab

2

1

21

India/State/Union Territory #

State/ UT Code

*2011

610,485

8,976,410

33,106

150,100

193,178

580,282

202,330

46,927,389

5,154,178

104,596,415

1,871,867

36,158,871

321,661

35,620,086

14,634,819

5 2001

486,705

31,118

7,570,890

92,478

121,731

508,224

192,985

41,487,694

4,316,401

87,466,301

1,636,138

31,268,654

288,217

29,381,657

12,963,362

Males

Total population

*2011

633,979

7,776,825

31,323

92,811

149,675

474,404

177,614

44,420,347

4,962,574

94,985,062

1,799,165

35,980,087

286,027

33,000,926

13,069,417

6 2001

487,124

29,477

6,212,086

65,581

98,720

392,690

163,280

38,733,477

4,163,161

78,586,558

1,555,030

30,842,185

252,276

27,091,465

11,325,934

Females

1,038

866

946

618

775

818

878

947

963

908

961

995

889

926

893

*2011

7

1,001

947

821

709

811

773

846

934

964

898

950

986

875

922

874

2001

Sex ratio(females per1000 males)

*2011

2,598

11,297

2,013

2,169

698

9,252

46

1,029

189

828

350

555

86

201

550

8

2,029

1,894

9,294

1,411

449

7,903

43

904

159

689

304

478

76

165

482

2001

Density (Per sq.km)

27.72

20.96

6.23

53.54

55.50

17.10

6.68

13.93

19.17

20.09

14.75

15.60

12.36

21.44

13.73

*2011

9

20.56

17.19

46.31

55.59

59.20

40.33

26.94

17.84

19.20

25.80

15.74

11.19

32.98

28.33

19.76

2001

Decadal growth rate

6th

2nd

5th

3rd

4th

1st

7th

4th

19th

1st

21st

7th

28th

8th

15th

*2011

10

2nd

1st

7th

6th

5th

3rd

4th

4th

19th

1st

21st

6th

28th

8th

15th

2001

Rank

▌ General Knowledge Density of Population, India, States/Union Territories: 2011

Sex Ratio, 2011 (States/Union Territories)

JAMMU AND KASHMIR 124

RAJASTHAN 201

ARUNACHAL PRADESH 17

ASSAM 397

NAGALAND 119 MEGHALAYA 132 MANIPUR 122

UTTAR PRADESH 828 BIHAR 1102 JHARKHAND H414

MADHYA PRADESH 236

R GA

MIZORAM 52

S TI

Diu (D&D) Daman 2169

T HA CH 89 1

MAHARASHTRA 365

DADRA & NAGAR HAVELI 698 ARABIAN SEA GOA 394

SH

BENGAL

PONDICHERRY 2598 TAMILNADU 555

I N D I A N

GUJARAT 918

ANDAMAN & NICOBAR ISLAND 46

MADHYA PRADESH 930

I N D I A N

HIMACHAL PRADESH 906 UTTARAKHAND 886 NCT OF DELHI 886

SIKKIM 944

ARUNACHAL PRADESH 960

ASSAM 957

NAGALAND 944 MEGHALAYA 970 MANIPUR 934

JHARKHAND H 943 AR

MIZORAM 971

SG

CH 64 9

TRIPURA 953 ORISSA 934

MAHARASHTRA 883

WEST BENGAL 950 BAY OF

SH

ADE

KA A PR ATA ANDHR RN KA 943 3 4 9

PONDICHERRY 965 TAMILNADU 946

I N D I A N

ANDAMAN & NICOBAR ISLAND 878

HIMACHAL PRADESH 12.8

PUNJAB 13.7

UTTAR PRADESH 899 BIHAR 933

MADHYA PRADESH 912

LAKSHADWEEP 908 KERELA 959

OF BENGAL

JAMMU AND KASHMIR 23.7

I TT HA

Diu (D&D) Daman 909 DADRA & NAGAR HAVELI 924 ARABIAN SEA GOA 920

WEST BENGAL 947 BAY

Child Sex Ratio in Age Group 0-6, 2011 (States/Union Territories)

JAMMU AND KASHMIR 859

GUJARAT 886

MIZORAM 975 TRIPURA 961

ORISSA 978

Sex Ratio

Child Sex Ratio in Age Group 0-6, 2011 (States/Union Territories)

RAJASTHAN 883

MANIPUR 987

O C E A N

Population Density

HARYANA 830

JHARKHAND 947

NAGALAND 931 MEGHALAYA 986

This map is a sketch only, drawn not to scale, presented for reference and to aid understanding of the concept discussed.

This map is a sketch only, drawn not to scale, presented for reference and to aid understanding of the concept discussed.

PUNJAB 846 CHANDIGARH 867

ARUNACHAL PRADESH 920

ASSAM 954

PONDICHERRY 1038 TAMILNADU 995

LAKSHADWEEP 946 KERELA 1084

O C E A N

SIKKIM 889

UTTAR PRADESH 908 BIHAR 916

Diu (D&D) Daman MAHARASHTRA 618 925 DADRA & NAGAR HAVELI SH 775 ADE A A PR ARABIAN TAK NDHR NA A SEA R GOA KA 992 968 968

WEST BENGAL 1029 BAY OF

LAKSHADWEEP 2013 KERELA 859

RAJASTHAN 926

TRIPURA 350 ORISSA 269

ADE

R KA RA P ATA ANDH RN KA 08 3 319

HARYANA 877

SIKKIM 86

HIMACHAL PRADESH 974 UTTARAKHAND 963 NCT OF DELHI 886

C 99 HH 1 AT TI SG

UTTARAKHAND 189 NCT OF DELHI 11297

PUNJAB 893 CHANDIGARH 818

H

HIMACHAL PRADESH 123

PUNJAB 550 CHANDIGARH 9252 HARYANA 573

GUJARAT 308

JAMMU AND KASHMIR 883

AR

2.264

BENGAL

ANDAMAN & NICOBAR ISLAND 966

O C E A N

CHANDIGARH 17.1

UTTARAKHAND 19.2 NCT OF DELHI 21.0

HARYANA 19.9 RAJASTHAN 21.4

ORISSA 14.0

PONDICHERRY 27.7 TAMILNADU 15.6

I N D I A N

MEGHALAYA 27.8 MANIPUR 18.7

JHARKHAND H AR 22.3 SG

I TT HA CH 2.6 Diu (D&D) 2 Daman MAHARASHTRA 53.5 16.0 DADRA & NAGAR HAVELI H D ES 55.5 PRA ARABIAN KA HRA ATA AND SEA N R GOA KA 11.1 8.2 15.7

LAKSHADWEEP 6.2 KERELA 4.9

ASSAM NAGALAND 16.9 0.5

UTTAR PRADESH 20.1 BIHAR 25.1

MADHYA PRADESH 20.3

GUJARAT 19.2

ARUNACHAL PRADESH 25.9 SIKKIM 12.4

MIZORAM 22.8

WEST BENGAL 13.9 BAY

TRIPURA 14.7

OF BENGAL

ANDAMAN & NICOBAR ISLAND 6.7

O C E A N

This map is a sketch only, drawn not to scale, presented for reference and to aid understanding of the concept discussed.

This map is a sketch only, drawn not to scale, presented for reference and to aid understanding of the concept discussed.

Child Sex Ration in Age Group of 0-6 Years

Growth of Population

▌ 2.265

Indian Economy Table: Literacy: 1941-2011

Literacy percentage

24.95

7.93

1961

24.02

34.44

12.95

1971

29.45

39.45

18.69

1981

36.17

46.74

24.88

1991

52.11

63.86

39.42

2001

65.38

75.85

54.16

*2011

74.04

82.14

65.46

*As per provisional population totals: Census 2011

Adult Female

Female Child

k jr

Table: Number of Literates

dh tux.ku

A

18.33

SU

DI

1951

Hk

7.30

e n t

24.90

r m

16.10

House listing & Housing Census

o w e

1941

Interpreation

k

Females

N

Males

CE

Average

E m p

Year

Census 2011 Logo and its Interpretation

N S OF I

Adult Male

Male Child

Star with 10 points, representing decenial Census

Literates

Person (million)

Males (million)

Females (million)

1991

352.08

224.29

127.79

2001

566.70

339.90

226.79

*2011

778.45

444.20

334.25

% increase

32%

31%

47%

*As per provisional population totals: Census 2011

People’s participation for inclusive growth G. C. 3. 1 Census 2011 Logo and its Interperation

Family Planning India is the first country in the world which has officially sponsored family planning programmes to control population in order to accelerate overall development of the country. The programme was officially launched in 1962. The first two family planning clinics were opened in Karnataka in 1930 which was supposed to be the first in the world.

CommuniCaTion sYsTEms Postal services

mail system

The Indian Postal network is the largest postal network in the world with approx. 155,837 post offices. On an average, a post office serves an area of 21.09 km2 and a population of 6,602.The postal network consists of four categories of post offices— head post offices, sub-post offices, extra-departmental sub-post offices and extra-departmental branch post offices.

The department handled more than 909.4 crore pieces of mail in 2002–03 out of which 782.77 crore was first class mail and rest second-class mail. The first class mail includes postcards, inland letters and envelopes, and are airlifted for speedy delivery without any surcharge between station connected by air. The second-class mail includes book packets, registered newspaper and periodicals and are carried by surface transport. The following new channels were introduced by the department:

First postal service started First postal stamp issued

1837 1852 in Karachi (provincial) First all-India postal stamp issued 1854 Establishment of Postal Department 1854 Money Order service started 1880 Post Office Savings banks started 1882 Airmail service 1911 (mail flown from Allahabad to Naini) No. of Post Offices in India Aprox. 155,015 [Rural Area: 139,144 and Urban Area: 15,871]

(a) Rajdhani Channel For transmission of mail from national capital to six state capitals. Yellow coloured letter boxes have been installed for this purpose. (b) Metro Channel For transmission of metro pin coded letter between Bangalore, Hyderabad, Kolkata, Chennai, Delhi and Mumbai. Blue coloured letter boxes have been installed for this purpose. (c) Green Channel Green coloured letter boxes are installed at selected Post Offices for posting local mail. The main objectives is to segregate the local mail for speedy processing and expeditious delivery.

2.266

▌ General Knowledge

(d) Business Channel The basic aim of this channel is to segregate first-class unregistered mail posted by various corporate bodies in bulk and its exclusive processing so as to decongest the network to avoid delay of household mail. (e) Periodical (Patrika) Channel Separate channel for speedy transmission of newspapers and periodicals. (f) Bulk Mail Channel Business mail posted in bulk is brought to separate offices in presorted condition directly from the companies only to be bagged and forwarded to ensure speedy transmission.

modernization of mail Transmission and Processing satellite network A private network of 227 VSAT (Very Small Aperture Terminals) and 1,350 Extended Stations, a total of 1,977 post offices would be linked through VSAT to handle money order and other financial transactions. automatic mail Processing Centres Induction of letter sorting machines at Chennai, Mumbai, Kolkata and Delhi to cull, face and cancel mail automatically. Post office Computerization 500 portfolios were made fully computerized in March 2002, and another 2,000 were installed with multi-purpose machines to handle the counter operations. ePost Introduced recently at 203 states at Andhra Pradesh, Gujarat, Goa, Kerala and Maharashtra to cater to the customers who do not have access to PC or Internet. Bill Post Web-based application to make the bill data of various government and private services at PO counters available on Internet. The customer can pay for services like electricity, water, mobile phone, telephone, municipality taxes etc. Certifying Authority To provide a secure environment to its customers for electronic transaction, the department intends to become a trusted third party and issue digital signatures. Premium Services (i) Speed Post

:

Introduced on 1 August 1986

(ii) Express Post

:

Introduced on 7 March 1999

(iii) Business Post

:

Introduced on 1 January 1997

(iv) Media Post

:

Introduced in August 1996

(v) Satellite Post

:

Introduced in August 1998

(vi) Retail Post

:

Introduced in September 2000

(vii) Greeting Post

:

Introduced in September 2000

(viii) Data Post

:

Introduced in August 1999

:

Introduced in August 1994

(ix) Post shoppe

Pin Code To facilitate sorting and speedy distribution of mail, PIN Code (Postal Index Number Code) system was introduced in 1972. The PIN code consists of six digits. • First digit indicates region (zone) • Second digit indicates sub-region (sub-zone) • Third digit indicates sorting district

The first three digits together represent a Sorting District. The last three digits stand for Delivery Post Offices in the areas served by the Sorting District.

Quick mail service (Qms) QMS was introduced in 1975 and today there are 45 national and 410 regional QMS centres in the country. All unregistered articles bearing the PIN Code of destination and posted in special QMS letter boxes are covered by the scheme.

speed Post services The Speed Post Services for national and international destinations were inaugurated in 1986. The service was launched for seven national (Delhi, Mumbai, Chennai, Kolkata, Bangalore, Ahmedabad and Hyderabad) and five international (UK, USA, Germany, Japan and Hong Kong) destinations, which is now extended to 74 national cities and 74 countries. Number of Postal Zones: 8

Zones

Area Covered

Zone 1

Delhi, Haryana, Punjab, Chandigarh, Himachal Pradesh and Jammu and Kashmir

Zone 2

Uttar Pradesh

Zone 3

Rajasthan, Gujarat, Daman & Diu, Dadra and Nagar Haveli

Zone 4

Maharashtra, Goa, Madhya Pradesh

Zone 5

Andhra Pradesh and Karnataka

Zone 6

Tamil Nadu, Kerala and Lakshadweep

Zone 7

West Bengal, Andaman and Nicobar, Orissa, Arunachal Pradesh, Assam, Manipur, Meghalaya, Nagaland and Tripura

Zone 8

Bihar

The Indian Post Offices operate the largest saving bank accounts in the country (over 94 million depositors). It is the biggest service department after the Railways. The post office Savings Bank runs Savings Accounts, Recurring Deposit Accounts, Time Deposit Accounts, National Savings Scheme, Monthly Income Scheme, Public Provident Fund, Indira Vikas Patra, Kisan Vikas Patra, and the National Savings Certificates 8th issue on behalf of the Ministry of Finance.

agency Functions Post office savings Bank With more than 1,55,000 post offices performing savings operations, the Post Office Savings Bank remains the largest savings bank in the country. It holds more than 5.5 crore savings bank accounts and a outstanding balance of about 1,25,000 crore along with Public Provident Fund (PPF) amounting to about Rs 2,500 crore. Postal life insurance Postal Life Insurance (PLI) is insurance cover extended to employees of government and semi-government employees. Rural Postal Life Insurance (RPLI) is a low premium insurance cover to the rural masses and weaker sections of the society. ‘Yugal Suraksha’ and ‘10-year Rural PLI’ are famous schemes introduced by Postal Life Insurance.

Indian Economy

▌ 2.267

Types of Policies

PLT

RPLT

1. Suraksha (whole Life Assurance)

1. Gram Suraksha

2. Suvidha (Convertable whole Life Assurance)

2. Gram Suvidha

3. Santosh (Endowment Assurance)

3. Gram Santosh

4. Sumangal (Anticipated Endowment Assurance)

4. Gram Sumangal

5. Yugal Santosh (Joint Life Endowment Assurance)

5. 10-year Rural Postal Life Insurance (Gram priya)

TElECommuniCaTions First telegraph line

1851; between Kolkata and Diamond Harbour

First telephone service

1991 in Kolkata

First telephone exchange

1881 in Kolkata (50 lines only)

First automatic exchange

Shimla, 1913 (capacity of 700 lines only)

First international telephone line

1870; between London and Mumbai

No. of telephone exchanges

About 37,313 (all electronic type)

internet service Providers (isP) Internet services were opened for private participation since November 1988, and as on 31 March 2004, 385 ISP licences were issued out of which 121 have been permitted to offer Internet and Telephony service. Internet exchanges are set up in STPI, Noida, Mumbai, Chennai and Kolkata (proposed) to ensure switching of Internet traffic within the country.

other service Providers (osPs)

Now most of the cities (216) and towns (3,029), in addition to two-third of the villages in the country are covered with telephone facility.

Services like Tele-education, Tele-medicine, Tele-banking, Call Centres etc., are covered under OSPs. Call Centres/ Business Process Outsourcing (BPOs) are permitted internet and IPLC connectivity on the same Local Area Network(LAN). Interconnectivity of call centers of the same group of companies is also permitted for redundancy and load balancing.

No. of telephone subscribed in Inidia

325.78 million (includes 266.86 million mobile cases)

Cellular mobile Telephone service

Averge teledensity in India

28.33 (as on 30th June 2008)

indian Telecom sector India is the 10th largest telecom network in the world, measured in the terms of number of phones. As on 31 May 2004, the network comprises 77.93 million telephone connections and over 1.79 million Public Call Offices (PCOs). There were 31 private licences and two public sector licences at the end of March 2004. 27 licences of those 31 licenses were converted to unified Access Service Licences, while 18 more licences were issued for Unified Access Services during 2004–05. Fully automatic Subscriber Dialing (ISD) service is available to almost all the countries. The total number of stations connected to National Subscriber Dialing (NSD) is 31,686.In the field of international communication, tremendous progress was made by use of satellite communication and submarine optical fibre limits. The voice and non-voice telecom services, which include data transmission, facsimile, mobile radio, radio paging and leased line service, cater to a wide variety of needs of both residential and business customers. Integrated Services Digital Network (ISDN) facility is available in a number of cities. A dedicated Packet Switched Public Data Network with international access for computer communication services also exists.

India is divided into 19 Telecom Circle Services and 4 Metro Service areas for the Cellular Mobile Telephone Service. There are 3 private and one state-owned operator in each service area. There are over 27 million cellular subscribers and growing at the rate of over 1 million/month. FDI received in the telecom sector from August 1991 to January 2004 is Rs 987,205 crore. The FDI approved in telecom sector during this period is Rs 57,260.14 crore which is the second highest, next only to power and fuel sectors by 2004. Telecom Regulatory Authority of India (TRAI) was established in 1997 to regulate telecommunication services and for matters concerned there with or incidental thereto. The establishment of the Regulator was considered necessary in context of liberalization and private sector participation in Telecom Sector, and to provide a level playing field for all operators. PSUs in Telecom Sector (i) Bharat Sanchar Nigam Limited (BSNL) (ii) Mahanagar Telephone Nigam Ltd (MTNL) (iii) Indian Telephone Industries (ITI) (iv) Telecommunications Consultant India Ltd (TCIL) *VSNL was divested and TATA through m/s Penatone Finvest Ltd controls its management; the government has only 26.12 percent equity in VSNL.

2.268

▌ General Knowledge

inDian RailWaYs Table: Administrative Zones

Important Facts

Zone

Date when Created Headquarters

First train in India (also in Asia)

On 16 April 1853; Mumbai to Thane (32 km rail track)

1. Southern

14 April 1951

Chennai

Fastest train

Shatabdi Express New Delhi/ Bhopal (150 km/hr)

2. Central

5 November

1955 Mumbai CST

First electric train in India

Deccan Queen, introduced in 1929 between Kalyan and Pune

3. Western

5 November

1951 Mumbai

4. Northern

14 April 1951

New Delhi

Number of employees (1991)

Approximately 1.7 million

5. North-Eastern

14 April 1952

Gorakhpur

Capital investment (1991)

Rs 22,200.48 crore

6. Eastern

1 August 1955

Kolkata

Route covered (track length)

64,015 km

7. South-Eastern

1 August 1955

Kolkata

Railway bridges (total length)

About 112,445 km

8. North-East-Frontier

15 January 1958

Maligaon-Guwahati

Number of trains running per day

About 11,275

9. South-Central

Number of railway stations Passenger traffic (1991) Fleet strength

2 October

1966 Secunderabad

10. East Central

1 October 2002

Hajipur

7,030

11. East Coast

1 April 2003

Bhubaneswar

Over 386 crore

12. North Central

1 April 2003

Allahabad

7,681 locomotives, 39,852 passenger service vehicles, 4,904 coaching vehicles and 214,760 wagons (as on 31 March 2003)

13. North Western

1 October 2002

Jaipur

14. South East Central

1 April 2003

Bilaspur

15. South Western

1 April 2003

Hubli

16. West Central

1 April 2003

Jabalpur

Other Facts • The Indian Railway system is the largest in Asia, and third largest in world after USA and Canada. • It is the biggest employer in the world and the largest single undertaking in the country. • It is the second biggest electrified railway system in the world after Russia.

Rail Tracks

Traction 1. Steam locomotives: Runs on coal. However, production of new steam locomotives has now been discontinued as the entire system is envisaged to be dieselized or electrified. 2. Diesel locomotives: Introduced in 1957. 3. Electric locomotives: Introduced in 1929. Table: High Speed Trains

According to distance between the rails, there are three systems operating in the country, they are:

Name of Train

Runs between

Introduced

Max. Speed

1. Rajdhani Express New Delhi/ (Broad gauge) Howrah

1969

130 km/h

2. Rajdhani Express New Delhi/ (Broad gauge) Mumbai

1972

130 km/h

Rail Tracks

Distance between Rails

Route (km)

Running Track (km)

Total Track (km)

1. Broad Gauge

1,676 mm

52,808

75,808

99,416

2. Metre Gauge

1,000 mm

8,473

8,973

10,656

2,734

2,736

3,043

New Delhi/ Jhansi

140 km/h

762 mm/ 610 mm

3. Shatabdi Express (Broad gauge)

1988

3. Narrow Gauge

64,015

86,937

1,13,115

4. Pink City Express New Delhi/ (Metre gauge) Jaipur

1977

100 km/h

Total

administration Responsibility for the overall control and administration of the railways vests in the Railway Board set up in 1905. The Railway Board consists of a Chairman, who is an ex-officio SecretaryGeneral of the Railway Ministry, a Financial Commissioner and three members each in charge of mechanical, traffic and personnel who hold the status of secretaries to the Railway Ministry. Besides this there are seven additional members.

• First high speed train introduced in 1969 running between New Delhi/Howrah running at a speed of 120 km/h which was subsequently increased to 130 km/h. • Shatabdi Express introduced in 1988 between New Delhi/ Jhansi is the fastest train in India Stock fleet of India Railways as on 31 march 2009 stood at 8592 locomotives (out of which there are 43 steam, 4963 diesel and 3586 electric locomotives), 49,110 passenger service vehicles, 5,985 other coaching vehicles and 2,11,763 wagons.

Indian Economy Table: Railway Manufacturing Units

▌ 2.269

Total No. of Units: 6

Name

Location

Estbd in Items Manufactured

1. Chittaranjan Locomotive Works*

Chittaranjan

1950

• Originally manufactured steam engines, which were discontinued from December 1971 • Now manufactures electric and diesel shunting engines

2. Diesel Locomotive Works

Varanasi (UP)

1964

• B.G. and M.G. diesel engines (shunters) and electric shunters

3. Integral Coach Factory

Perambur (Chennai)

1955

• B.G. and M.G. coaches, AC coaches

4. Wheel and Axle plant

Yalahanka (Bangalore)

1983

• Wheels and axles

5. Diesel Component Works

Patiala (Punjab)

6. Rail Coach Factory

Kapurthala (Punjab)

• Components of diesel engines 1988

• Rail coaches

As per the Budget 2004–05, a new wheel manufacturing unit is to be set up at Chhapra. M.G. = Metre Gauge B.G. = Broad Gauge *First locomotive was turned out in November 1950 at Chittaranjan Locomotive Works Project unigauge On 3 July 1992, the then Prime Minister P. V. Narashimha Rao dedicated Project Unigauge of the Indian Railways to the nation. The project envisaged to convert a total of 28,000 km meter gauge rail track to broad gauge under the Eighth Plan. Jeevan Rekha Jeevan Rekha or Life Line Express, the world’s first hospital on wheels was flagged off from Mumbai on 16 July 1991. The Express caters to the medical needs of the people in rural areas.

Public undertakings There are nine undertakings under the administrative control of the Ministry of Railways: (i) Rail India Technical and Economic Services limited (RITES) (ii) Container Corporation of India Limited (CONCOR)

(iii) (iv) (v) (vi)

Indian Railways Construction Company Limited (IRCON) Indian Railways Finance Corporation Limited (IRFC) Konkan Railway Corporation Limited (KRCL) Indian Railway Catering and Tourism Corporation Limited (IRCTC) (vii) Railtel Corporation of India Ltd (Rail Tel) (viii) Mumbai Rail Vikas Nigam Ltd (MRVNL) (ix) Rail Vikas Nigam Ltd (RVNL) The Centre for Railway Information System (CRIS) has been set up as a registered society to design and implement various railway computerization projects. The Research, Design and Standards Organization (RDSO) at Lucknow is the R&D wing of Indian Railways.

sHiPPing shipping Corporation of india (sCi) Established 2 October 1961 The Shipping Corporation of India came into being with the merger of the Eastern Shipping Corporation and the Western Shipping Corporation. The Mughal Lines Limited, another public sector shipping company, was merged in June 1986. The status of SCI has been changed from a private limited company to public limited from 18 September 1992. The SCI has been conferred ‘Mini Ratna’ status by government of India on 24 February 2000. The present authorized capital of SCI is Rs 450 crore and paid up capital is Rs 282.30 crore. The SCI operates on almost all the maritime routes and is one of the biggest shipping lines in the world. It has a merchant fleet of 86 vessels and accounts for about 40 percent of the total tonnage of the country. SCI manages 40 ships on behalf of the government organizations/departments.

As on 1 April 2004, there are about 150 shipping companies in the country, out of which about 35 are engaged in overseas trade, about 95 exclusively in coastal trade, and the remaining in both overseas and coastal trade. Some of the important private sector shipping companies are given below: 1. The Scindia Steam Navigation Co. Ltd 2. Great Eastern Shipping Co. Ltd 3. India Steamship Co. Ltd 4. South India Shipping Corporation Ltd and Surinder Overseas Ltd 5. Ratnakar Shipping Corporation Ltd 6. Chowgule Steamship Ltd 7. Damodar Bulk Carriers Ltd 8. Essar Shipping Company Ltd 9. Varun Shipping Company Ltd

4.2 shipping Companies

10. Sannar Shipping

India has the largest merchant shipping fleet among the developing countries and ranks 17th in the world in terms of shipping tonnage.

12. Century Shipping

11. Tolani Shipping Company

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shipyards There are four major and four medium-sized shipping yards in the country. In addition, there are also 32 small shipping yards in the private sector. The major shipyards are: 1. Garden Reach Shipbuilders and Engineers Ltd, Kolkata (GRSEL) 2. Mazagaon Dock Limited, Mormugao (MDL) 3. Hindustan Shipyard Limited, Visakhapatnam (HSL) 4. Cochin Shipyard, Kochi (CSL) 5. Hooghly Dock and Port Engineers Limited, Kolkata (HDPEL)

Training institutions There are four training institutes established by the government of India.(1) Training ship ‘Chanakya’; (2) Marine Engineering and Research Institute (MERI), Kolkata, (3) Marine Engineering and Research Institute, Mumbai, (iv) Lal Bahadur Shastri (LBS) College of Advance Maritime Studies and Research, Mumbai. In addition to above, there are more than 130 training institutes in private sector approved by the Director General of Shipping, impacting para-sea and post-sea training in various disciplines. In wake of coming into force of the ‘STCW95’ and the regulations there under stipulated by the International Maritime Organization in 1997, the maritime training in India has undergone a sea change. The government is considering the formation of a full-fledged Maritime University.

and edible oils. It serves the 1,600 km coastline between Mumbai and Karachi (Pakistan). 2. Mumbai (Maharashtra) It is the biggest port of the country. It is a natural harbour and handles more than one-fifth of the total traffic of all the ports. It handles over 40 percent of India’s trade. 3. Mormugao (Goa) It is second largest port in term of total traffic tonnage, bulk of which is iron ore export. It is the finest natural port and an important naval station. 4. Mangalore (Karnataka) It was declared a major port in 1974, and is a tidal port. 5. Kochi (Kerala) It is a natural harbour. 6. Tuticorin (Tamil Nadu) It is a shallow but big port on the east coast, almost at the tip of the Indian peninsula. Declared a major port in 1974. 7. Chennai (Tamil Nadu) It is one of the oldest ports in India. It is an artificial harbour and the third largest in India. 8. Visakhapatnam (Andhra Pradesh) It is the deepest landlocked and protected port. It has a big shipbuilding unit and is a major naval base. 9. Paradip (Orrisa) It was declared a major port in 1966. 10. Kolkata (West Bengal) Situated on the Hooghly river, some 144 km from the Bay of Bengal. It is the largest terminal port in South Asia.

major Ports in india

11. Nhava Sheva ( Mumbai) Located some 10 km from Mumbai harbour, it is a newly constructed port completed in 1987–88 and at a cost of Rs 878.64 crore.

1. Kandla (Gujarat) A tidal port situated on Kandla creek, a port of the Rann of Kutch was declared a major port in 1955. It largely handles petroleum products, fertilizers, food grain, cotton, cement

12. Ennore (Tamil Nadu) The newest and youngest port of the millennium, used primarily to cater bulk trade like coal, iron-ore, naptha and the likes. In addition to 12 major ports, there are 225 minor and intermediate ports in the country. The greatest problem with many Indian ports is their not being natural in their setting. Indian Civil Aviation completing 100 years on 18th February, 2011.

Geographically, from the northern-most point on the Arabian seaboard in the west to the northern-most point on the Bay of Bengal coastline in the east, the major ports are:

aiR TRansPoRT The year 2011-12 will be declared as the Civil Aviation Centenary Year, starting from 18th February, 2011 and ending on the same date in 2012. On this date in 1911, the first commercial plane flew in India between Allahabad and Naini. Since then, aviation in India has grown from strength to strength. Today India is the 9th largest civil aviation market in the world and this forward march is likely to culminate in India becoming one of the three largest markets in the world by 2020. There are 449 airports/airstrips in the country, out of which Airports Authority of India (AAI) manages a total of 125 Airports, which include 11 International Airports, 08 Customs Airports, 81 Domestic Airports and 25 Civil Enclaves at Defence Airfields. AAI also provides Air Traffic Management Services (ATMS) over entire Indian Air Space and adjoining oceanic areas with ground installations at all Airports and 25 other locations to ensure safety

of Aircraft operations. The Airports at Ahmedabad, Amritsar, Calicut, Guwahati, Jaipur, Trivandrum, Kolkata & Chennai, which today are established as International Airports, are open to operations even by Foreign International Airlines. Besides, the International flights, National Flag Carriers operate from Coimbatore, Tiruchirappalli, Varanasi, and Gaya Airports. Not only this but also the Tourist Charters now touch Agra, Coimbatore, Jaipur, Lucknow, Patna Airports etc. AAI has entered into a Joint Venture at Mumbai, Delhi, Hyderabad, Bangalore and Nagpur Airports to upgrade these Airports and emulate the world standards. All major air-routes over Indian landmass are Radar covered (29 Radar installations at 11 locations) along with VOR/DVOR coverage (89 installations) co-located with Distance Measuring Equipment (90 installations). 52 runways are provided with ILS installations with Night Landing Facilities at most of

Indian Economy these Airports and Automatic Message Switching System at 15 Airports. Air transport in India operates at three levels: 1. Air India (International level); 2. Indian (domestic level); 3. Private Airlines and charters such as 8 private scheduled operators, Jet, Sahara, Deccan, Go Airways, Kingfisher, Paramount, Go Airlines and Inter Globe (Indigo). However, recently Indian and some private airlines have been allowed to fly across some select international destinations. First solo Flight J. R. D. Tata was the first to make a solo flight from Mumbai to Karachi (now in Pakistan) in 1932. Today, the Civil Aviation Sector of India has three main functional divisions— regulatory, infra-structural and operational. On the operational side, Indian Airlines, Alliance Air (subsidiary of Indian Airlines), private scheduled airlines and air-taxis provide domestic air services, while Air India provides international air services. Many private airlines and charters such as NEPCDamania, Jet Airways, Sahara, Air Deccan, SpiceJet, etc., have acquired significant market share in domestic aviation sector. ‘Pawan Hans’ renamed ‘Pawan Hans Helicopters Ltd provides helicopter services to ONGC in its off shore operations, and to inaccessible areas and difficult terrains. Indian Airlines operations also extend to neighbouring countries, South East Asia and Middle East. Two private scheduled airlines—Sahara and Jet Airways, have been permitted to operate to Sri Lanka and Nepal.

air india Air India (formed in 1946) took over Tata Airlines and Air India International (formed in 1948), amalgamated them and was nationalized on 15 June 1953 as Air India Corporation. It operates in all the five continents with a fleet of about 22 aircraft. It has been denationalized now. Air India has four subsidiary companies— Hotel Corporation of India (HCI) Ltd, Air India Charters Ltd (AICL), Air India Air Transport services Ltd (AIATSL) and Air India Engineering Services Ltd (AIESL). Under AICL, Air India has now established a Low Cost Carrier (LCC).

indian The merger of Air India (erstwhile) and Indian Airlines was approved by the Government on 1 March, 2007 and a new company viz., National Aviation Company of India Limited (NACIL) was incorporated on 30 March, 2007 with the brand name of ‘Air India’. The merger of two airlines was envisaged to provide, among other benefits, an integrated international/ domestic footprint which will significantly enhance customer proposition and allow easy entry into one of the three global airline alliances. Also, it will enable optimal utilization of existing resources through improvement in load factors and yields on commonly services routes as well as deploy ‘freed up’ aircraft capacity on alternate routes

Pawan Hans Helicopters limited (PHHl) Incorporated in 1985, PHHL commenced its operations within one year to provide helicopter services to the oil sector, operate in

▌ 2.271

hilly and inaccessible areas and make available charter flights for promotion of travel and tourism. PHHL has a fleet of 35 helicopters and operates slights from 39 destinations covering 64 sectors by 148 weekly flights under the aegis of the State Governments of Meghalaya, Tripura, Sikkim and Arunachal Pradesh.

airport authority of india (aai) The Airport Authority of India (AAI) formed by the merger of IAAI and NAA through Airports Authority Act (No. 55 of 1994), came into existence on 1 April 1995. There are 449 airports/ airstrips in the country, out of which 92 airports and 28 civil enclaves at defence airfields provide air traffic services over the entire Indian airspace and adjoining oceanic areas. However, airlines currently utilize only 61 of these airports. main Functions The main functions of AAI are: (i) To control and manage the entire Indian airspace (excluding the special user airspace) extending beyond the territorial limits of the country, as accepted by ICAO. (ii) Provisioning of Communication and Navigational aids viz. ILS, DVOR, DME, Radar, etc. (iii) To design, construct, operate and maintain international airports, domestic airports, civil enclaves at defence airports. (iv) Development and Management of International Cargo Terminals. (v) Provisioning of Passenger Facilitation and Information System. (vi) Expansion and Strengthening of Operational areas viz. Runways, Apron, Taxiways, etc. (vii) Provisioning of Visual Aids. Airport Classification The airports in India are presently classified as: 1. International Airports: The airports are available for scheduled international operations by Indian and foreign carriers. 2. Custom Airports: Custom Airports have customs and immigration facilities for limited international operations by national carriers and for foreign tourist and cargo charter flights. These include the airports at Bangalore, Hyderabad, Ahmedabad, Kozhikode, Kochi, Goa, Varanasi, Patna, Agra, Jaipur, Amritsar and Tiruchirapalli. 3. Model Airports: Model Airports are domestic airports which have a minimum runway length of7500 feet and adequate terminal capacity to handle Airbus A320 type of aircraft and can cater to limited international traffic, if required. These include the airports at Lucknow, Bhubaneswar, Guwahati, Nagpur, Vadodara, Coimbatore, Imphal and Indore. 4. Other Domestic Airports: All other airports are covered in this category. 5. Civil Enclaves in Defence Airports: There are 28 civil enclaves in Defence airfields.

major airports of india 1. Amritsar International Airport (Rajasansi Airport), Amritsar 2. Indira Gandhi International Airport, New Delhi

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▌ General Knowledge

3. Lokpriya Gopinath Bordolio International Airport, Guwahati 4. Sardar Vallabhbhai Patel International Airport, Ahmedabad 5. Netaji Subhash Chandra Bose International Airport, Kolkata 6. Chhatrapati Shivaji International Airport, Mumbai 7. Rajiv Gandhi International Airport, Shamshabad (Hyderabad) 8. Goa Airport, Dabolim (Goa) 9. Chennai International Airport (Meenambakkam Airport), Chennai 10. Bengaluru International Airport, Devanhalli (Bengaluru) 11. Cochin International Airport, Kochi 12. Trivandrum International Airport, Thiruvananathapuram 13. Kozhikode Airport, Calicut 14. Pune Airport, Pune

major Domestic airlines of india 1. Air India (Domestic) 2. Jet Airways 3. Jet Lite

4. 5. 6. 7. 8.

Kingfisher Spice Jet Paramount Go Air IndiGo

Seaplane Operations: On 27 December 2010, the Seaplane service was operational in Andaman & Nicobar Islands as per MOU signed between Andaman & Nicobar Administration and Pawan Hans Helicopters. This is the first time that Seaplane operations are being introduced in India. The service shall be provided with a Cessna 208A (capacity 2+8 passengers) Amphibian seaplane. Project GAGAN: GAGAN (GPS Aided Geo Augmented Navigation) is an augmentation system to enhance the accuracy and integrity of GPS signals to meet precision approach requirements in Civil Aviations and it is being implemented jointly by AAI and ISRO. The footprint of this satellite will cover a vast geographical area from Africa to Australia and hence would facilitate expansion of the service are of GAGAN far beyond Indian airspace. This ambitious project of the satellite based air navigation system (GAGAN) will also help in the growth of the sector by enabling the handling of more air traffic in a much safer manner.

RoaD nETWoRks in inDia History The planned roads finds their origin during the time of Sher Shah Suri (1540–45). The main highway from Indus to Sonargaon (Grand Trunk Road), which was first built by the Mauryans, was rebuilt by him. He also built a road from Agrato Jodhpur and Chittor. Approximate road network length 3.32 million kilometers Highest road in India Khardungla (Leh–Manali) Longest road in India Grand Trunk Road (G.T. Road)

g.T. Road G.T. Road is India’s longest road which runs from Kolkata to Amritsar, passing through Varanasi, Allahabad, Kanpur, Agra, Delhi and Ambala. Originally, the road connected Lahore (Pakistan) with Kolkata.

national Highways The National Highways are the principal highways running through the length and breadth of the country. They connect major ports, foreign highways, capitals of states and major towns, including highways required for strategic movement for the defence of India. The National Highway Authority of India (NHAI) constituted on 15 June 1989, is responsible for the construction and maintenance of the national highways. Length of national highways network Number of important highways Longest national highway

65,569 km 56 NH–7 (2,369 km)

NH–7, the longest national highway passes through Uttar Pradesh, Madhya Pradesh, Maharashtra, Andhra Pradesh, Karnataka and Tamil Nadu. It connects Varanasi to Cape Comorin (Kanyakumari). The national highways constitute about two percent of the total road length in the country and carry nearly 40 percent of the total road traffic. National Highway Development Project (NHDP) It is the largest highway project ever taken up in India. It aims to upgrade the 14,279 km length of National Highways to four/six lanes to give a boost to the economic development of the country. NHDP has following two components: (i) Golden Quadrilateral (GQ) comprising, national highways connecting four metro cities viz. Delhi, Mumbai, Chennai and Kolkata. Total length of GQ is 5,846 km. (ii) North-South and East-West corridors–comprising the National Highways connecting Srinagar to Kanyakumari including Kochi–Salem spur and Silchar to Porbandar is about 7,300 km. The North-South and East-West corridors is targetted for completion by December 2007.. With a view to strengthen the infrastructure network of India, the National Highway Authority of India is also implementing four laning of 361 km for connectivity to 10 major ports of the country. The government has also undertaken the programmes for improvement of riding quality of National Highways since 1999.

Indian Economy

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Table: Major National Highways NH No.

Route

Length in State (km)

Total Length (km)

Delhi–Ambala–Jalandhar–Amritsar–Indo–Pak Border

Delhi (22), Haryana (180), Punjab (254)

456

1A

Jalandhar–Madhopur–Jammu–Banihal–Srinagar–Baramula– Uri

Punjab (108), Himachal Pradesh (14), Jammu and Kashmir (541)

663

1B

Batote–Doda–Kistwar–Symthan pass–Kkhanabal

Jammu and Kashmir (274)

274

2

Delhi–Mathura–Agra–Kanpur–Allahabad–Varanasi– Mohania– Barhi–Palsit–Baidyabati–Bara–Kolkata

Delhi (12), Haryana (74), Uttar Pradesh (752), Bihar (202), Jharkhand (190), West Bengal (235)

1,465

3

Agra–Gwalior–Shivpuri–Indore–Dhule–Nasik–Thane– Mumbai

Uttar Pradesh (26), Rajasthan (32), Madhya Pradesh (712), Maharashtra (391)

1,161

4

Junction with National Highway No. 3 near Thane–Pune– Belgaum– Hubli–Bangalore–Ranipet–Chennai

Maharashtra (371), Karnataka (658), Andhra Pradesh (83), Tamil Nadu (123)

1,235

5

Junction with National Highway No. 6 near Baharagora– Cuttack–Bhubneshwar–Vishakhapatnam– Vijaywada–Chennai

Orissa (488), Andhra Pradesh (1,000), Tamil Nadu (45)

1,533

6

Hajira–Dhule–Nagpur–Raipur–Sambalpur–Bharagora– Kolkatta

Maharashtra (813), Chattisgarh (314), Orissa (462), Jharkhand (22), Orissa (462), Jharkhand (22) , West Bengal (161), Gujarat (177)

1,949

7

Varanasi–Mangawan–Rewa–Jabalpur–Lakhnadon–Nagpur– Hyderabad–Kurnool–Bangalore–Krishnagiri–Salem–Dindigul– Madurai–Cape Commorin

Uttar Pradesh (128), Madhya Pradesh (504), Maharashtra (232), Andhra Pradesh (753), Karnataka (125), Tamil Nadu (627)

2,369

8

Delhi–Jaipur–Ajmer–Udaipur– Ahmedabad–Vadodara– Mumbai

Delhi (13), Haryana (101), Rajasthan (688), Gujarat (498), Maharashtra (128)

1,428

9

Pune–Sohlapur–Hyderabad–Vijaywada–Machillipatnam

Maharashtra (336), Karnataka (75), Andhra Pradesh (430)

841

11

Agra–Jaipur–Bikaner

Uttar Pradesh (51), Rajasthan (531)

582

12

Jabalpur–Bhopal–Khilchipur-Aklera–Jhalawar–Kota–Bundi– Devli–Tonk–Jaipur

Madhya Pradesh (490), Rajasthan (400)

890

13

Sholapur–Chitradurga–Shimoga–Mangalore

Maharashtra (43), Karnataka (648)

691

15

Pathnkot–Amritsar–Bathinda–Ganganagar–Bikaner– Jaisalmer–Barmer–Samakhiali–(Near Kandla)

Punjab(350), Rajasthan (906), Gujarat (270)

1,526

17

Panvel–Mahad–Panaji–Karwar–Mangalore–Cannanore– Kozhikode–Terokh–Cranganur junction with NH–31 near Edapally

Maharashtra (482), Goa (139), Karnataka (280), Kerala (368)

1,269

28

Junction with NH–31 near Barauni–Muzaffarpur–Pipra– Kothi–Gorakhpur–Lucknow

Bihar (259), Uttar Pradesh (311)

570

31

Junction with NH–2 near Barhi–Bhaktiarpur–Mokameh– Purnea–Dalkola–Silliguri–Sivok–Cooch–Behar–North Salmara–Nalbari–Charali–AminGaon junction with NH-37

Bihar (393), West Bengal (366), Assam (322), Jharkhand (44)

1,125

37

Junction with NH–31–B near Goalpara–Guwahati–Jorabat– Kamargaon–Makum–Saikhoaghat

Assam (680)

680

43

Raipur–Vizianagaram junction with NH–5 near Natavalasa

Chattisgarh (316), Orissa (152), Andhra Pradesh (83)

551

44

Shillong–Passi–Badarpur–Agartala–Sabroom

Meghalaya (184), Assam (111), Tripura (335)

630

47

Salem–Coimbatore–Trichur–Ernakulam–Trivandrum–Cape Commorin (Kanyakumari)

Tamil Nadu (224), Kerala (416)

640

52

Baihata–Charali–Tezpur–Bander Dewa–North Lakhimpur– Pasighat–Tezu–Sitapani Junction with NH–37 near Saikhoaghat

Assam (540), Arunachal Pradesh (310)

850

54

Dabaka–Lumding–Silchar–Aizwal–Tuipang

Assam (335), Mizoram (515)

850

58

Delhi–Ghaziabad–Meerut–Haridwar–Badrinath–Mana pass

Uttar Pradesh (165), Uttarakhand (373)

538

65

Ambala–Kaithal–Hissar–Fatehpur Jodhpur–Pali

Haryana (240), Rajasthan (450)

690

75

Gwalior–Jhansi–Chhatarpur–Rewa Renukut–Garwa– Daltoganj–Ranchi

Madhya Pradesh (600), Uttar Pradesh (110) Jharkhand (245)

955

1

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▌ General Knowledge

76

Pindwara–Udaipur–Mangalwar–Kota–Shivpuri–Jhansi– Banda–Allahabad

Madhypa Pradesh (60), Uttar Pradesh (467), Rajasthan (480)

1,007

78

Katni–Shahdol–Ambikapur–Jashpurnagar–Gumla

Madhya Pradesh (280), Chattisgarh (356), Jharkhand (25)

559

79

Ajmer–Nasirabad–Neemach–Mandsaur–Indore

Madhya Pradesh (280), Rajasthan (220)

500

86

Kanpur–Chhatarpur–Sagar–Bhopal–Dewas

Uttar Pradesh (180), Madhya Pradesh (494)

558

150

Aizwal–Churachandpur–Imphal–Ukhrul–Jessami–Kohima

Manipur (523), Mizoram (141), Nagaland (36)

700

200

Raipur–Bilaspur–Rajgarh–Kanaktora–Jharsuguda–Kochinda– Deogarh–Talcher–Chandhikhole

Orissa (440), Madhya Pradesh (300)

740

217

Raipur–Gopalpur

Chhatisgarh (70), Orissa (438)

508

Total Length of all the 221 National Highways

65,600

mEDia newspapers and Periodicals Oldest Newspapers/Magazines First newspaper of India

Bengal Gazette, published on 27 January 1780 by J. A. Hickey in Kolkata

Oldest existing newspaper

Mumbai Samachar, started in 1822 (in Gujarati)

Oldest existing English daily

The Times of India, started in 1838 from Mumbai

Oldest magazine

Calcutta Review in English, since 1844

Circulation Periodical with highest circulation

Malayala Manorama (over 6.5 lakh copies a day in 1992)

Largest selling English daily

The Times of India (around 6,000,500 copies in 1992)

Largest selling Hindi daily

Punjab Kesari (around 6 lakh copies a day in 1992)

Largest selling weekly

Mangalam in Malayalam (over 11 lakh copies per week in 1990)

Largest selling monthly

Readers Digest

Largest selling yearbook

Manorama Year Book (English)

Newspaper having highest number of single edition

Indian Express (33 editions a day)

No. of newspapers published in India (dailies)

Around 28,491

Number of periodicals (weeklies) published

Around 15,400

Language in which maximum no. of newspapers are published

Hindi

publishing maximum newspapers

Uttar Pradesh (8397 in 2001)

Note: While newspapers are published in all states and Union Territories, there is no newspaper from Arunachal Pradesh and Lakshadweep Table: India’s Important Newspapers (Dailies) Sl. No Newspaper 1 The Times of India

Language English

No. of Editions 9

2

Dainik Bhaskar

Hindi

15

3

Dainik Jagran

Hindi

22

Published From Mumbai, Delhi, Bangalore, Pune, Hyderabad, Manipal, Lucknow, Kolkata, Patna Ajmer, Bhopal, Udaipur, Bikaner, Chandigarh, Gwalior, Indore, Hisar, Jaipur, Jodhpur, Kota, Raipur, Ahmedabad, Jabalpur, Satna Bareilly, Moradabad, Bhopal, Dehradun, Gorakhpur, Jalandhar, Jhansi, Kanpur, Lucknow, Meerut, New Delhi, Patna, Varanasi, Allahabad, Agra, Rewa, Hissar, Ranchi, Jamshedpur, Dhanbad, Aligarh, Panipat

Indian Economy

4 5 6 7 8

9 10

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Malayala Manorama Hindustan Times Gujarat Samachar The Hindu Eenadu

Malayalam

9

English Gujarati English Telugu

5 5 6 21

Ananda Bazar Partika AJ

Bengali

2

Kanpur, Kochi, Killam, Kottayam, Kozhikode, Mallapuram, Pallakad, Thrissur, Thiruvananthapuram New Delhi, Patna, Lucknow, Kolkata and Mumbai Ahmedabad, Surat, Baroda, Mumbai, Rajkot Chennai, Coimbatore, Madurai, Bangalore, Hyderabad, New Delhi Anantapur, Bangalore, Chennai, Cuddapah, Mumbai, Guntur, Hyderabad, Karinmnagar, Khamman, Kurnool, Mahboobnagar, Nellore, Nizamabad, New Delhi, Ongole, Rajamundhry, Srikakulam, Suryapet, Tirupati, Vijayawada, Tadeppali Kolkata and Siliguri

Hindi

5

Varanasi, Patna, Gorakhpur, Ranchi and Allahabad

other important newspapers Besides the top ten newspapers listed above, there are several newspapers being published from various states, important among them are: Sl No.

Newspaper

Language

Published from

1.

Tribune

English

Chandigarh

2.

Lokasatta

Marathi

Chennai and Pune

3.

Rajasthan Patrika

Hindi

Jaipur, Jodhpur, Udaipur, Kota and Bikaner

4.

Navbharat Times

Hindi

Nagpur, Raipur, Bilaspur, Indore, Jabalpur, Bhopal

5.

Indian Express

English

Mumbai, Pune, New Delhi, Chandigarh, Madurai, Chennai, Vijaywada, Bangalore, Kochi, Hyderabad, Vizianagaram, Ahmedabad

6.

Amar Ujala

Hindi

Agra, Bareilly

7.

Mathrubhumi

Malayalam

Kozhikode, Kochi, Thiruvananthapuram

8.

Janasatta

Hindi

New Delhi, Chandigarh

9.

Ajit

Punjabi

Jalandhar

10.

Jag Bani

Punjabi

Jalandhar

11.

Punjabi Tribune

Punjabi

Jalandhar

12.

Hindi Samachar

Urdu/Hindi

Jalandhar

13.

Siasat

Urdu

Hyderabad

14.

Inquilab

Urdu

Mumbai

15.

The Hindustan Times

English

Delhi, Patna

16.

The Statesman

English

New Delhi, Kolkata

17.

Punjab Kesari

Hindi

Jalandhar, New Delhi

18.

Daily Thanthi

Tamil

Bangalore, Chennai, Coimbat ore, Cuddalore, Dindigul, Erode, Thirunelveli, Salem, Puducherry, Tanjavore, Vellore, Tirucherapalli, Tirunelveli

19.

Lokmat

Marathi

Mumbai, Ahmednagar, Akola, Aurangabad, Nagpur, Pune, Kolhapur, Nashik, Jalgoan, Solapur

20.

The Minsif

Urdu

Hyderabad

21.

Asomiya Pratidin

Assamaese

Dibrugarh, Guwahati

22.

Vijay Karnatka

Kannada

bangalore, Gangawati, Mysore, Gulbarga, Hubli, Mangalore, Shimoga

23.

Ajit

Punjabi

Jalandhar

2.276

▌ General Knowledge Table: Important Newspaper Groups

Table: Important Periodicals Sl.No.

Periodical

Language

Periodicity

1.

Saptahik Bartaman

Bengali

Weekly

Name of the Group

Newspaper Printed

No. of Editions

2.

Sharadiya Bartaman

Bengali

Annual

Indian Express (English)

12

3.

Business Today

English

Fortnightly

1. Indian Express (33 editions, 7 languages)

4.

Femina

English

Fortnightly

Lokasatta (Marah)

2

5.

Filmfare

English

Monthly

Dinamani (Tamil)

3

6.

The Week

English

Weekly

Janasatta (Hindi)

2

7.

Competition Success Review

English

Monthly

Lokasatta-Janasatta (Gujarati)

3

8.

Outlook

English

Weekly

Andhra Prabha (Telugu)

5 1 1

India Today

English

Weekly

Kannada Prabha (Kannada)

10.

Manorama Yearbook

English

Annual

Samakaleen (Gujarati)

11.

Chitralekha

Gujarati

Weekly

9.

Financial Express (English)

4

The Times of India (English)

5

Navbharat Times (Hindi)

3

Maharashtra Times (Marathi)

1

The Economic Times (English)

5

Sandhya Times (Hindi)

1

Punjab Kesari (Hindi)

2

Jag Bani (Punjabi)

1

Hind Samachar (Urdu)

1

4. Malayala Manorama (4 editions 1 language)

Malayala Manorama (Malayalam)

4

5. Ananda Bazar Patrika (3 editions, 12 languages)

Ananda Bazar

1

2. The Times of India (14 editions, 3 languages)

12.

Nandan

Hindi

Monthy

13.

Outlook

Hindi

Weekly

14.

Sarita

Hindi

Fortnightly

15.

Nirogdham

Hindi

Quartely

16.

Griha Lakshmi

Hindi

Monthly

17.

Griha Shobha

Hindi

Monthly

18.

Meri Saheli

Hindi

Monthly

19.

India Today

Hindi

Weekly

20.

Saras Salil

Hindi

Fortnightly

21.

Kalnirnay

Hindi

Annual

22.

Griha Shobha

Kannada

Monthly

23.

Taranga

Kannada

Weekly

24.

Mallige Panchanga Darshini

Kannada

Annual

25.

Balarama

Malayalam

Weekly

26.

Vanitha

Malayalam

Fortnightly

27.

Malayala Manorama

Malayalam

Weekly

28.

Chitralekha

Marathi

Weekly

news agencies

29.

Grihashobika

Marathi

Monthly

30.

Kalnirnay

Marathi

Annual

31.

Rani

Tamil

Weekly

32.

Junior Vikatan

Tamil

Twice a week

There are four news agencies, namely: (i) Press Trust of India (PTI)—set up in 1947 (ii) United News of India (UNI)—set up in 1961 (iii) Samachar Bharati (iv) Hindustan Samachar

33.

Mangayar Malar

Tamil

Monthly

34.

Ananda Vikatan

Tamil

Weekly

35.

Kumudam

Tamil

Weekly

36.

Annadata

Telgu

Monthly

37.

Swati Sapari Vara Patrika

Telgu

Weekly

3. Hind Samachar (4 editions, 3 languages)

Patrika (Bengali) Telegraph (English)

1

Hindi news service of UNI is called Univarta which was started in 1982. Similarly, news service by PTI is called Bhasha which was launched in 1986. In addition to the above news agencies, there are several news services which supply news to the media, some of which are: (i) Eastern India News Agency (FINS) (ii) Indian News and Features Alliance (INFA )

Indian Economy (iii) Indian Press Agency (IPA) (iv) Near and Far East News (NAFFN) (v) News Features of India (NFI)

all india Radio (aiR) Started in early 1920s, the first programme was broadcasted in 1923, by the Radio Club of India. This was followed by setting up of two broadcasting services in 1927, with two privately owned transmitters at Bombay and Kolkata. The government took over the transmitters in 1930 and started operating them under the name of Indian Broadcasting Service. It was changed to All India Radio (AIR) in 1936, and it came to be known as ‘Akashwani’ from 1957. AIR has, at present, 223 radio stations spread across the country which includes, 114 regional stations, 77 Local Radio Stations, 14 Relay Centres, 3 exclusive Vividh Bharati Centres and 5 community Radio Stations. The broadcast coverage of AIR is 91.37 percent by area and 99.13 percent by population. AIR also broadcasts a National Channel. Prasar Bharti Corporation Prasar Bharti is the public sector broadcaster in the country with All India Radio and Doordarshan as its two components. The primary objective of the Corporation as laid out in Section 12 of the Prasar Bharati (Broadcasting Corporation of India) Act, 1990 is, amongst other things, to organize and conduct public broadcasting services to inform, educate and entertain the public, and to ensure balanced development of broadcasting of Radio and TV.

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Doordarshan (DD) The national television service of India is one of the largest terrestrial networks in the world. The first telecast in India, originated from a makeshift studio in the ‘Akashvani Bhawan, New Delhi, on Sept 15, 1959. The regular service with a news bulletin became operational in 1965. After 7, years the second TV center commenced service in Bombay (Mumbai). By 1975, TV service was available in Kolkata (Kolkata), Chennai (Madras), Srinagar, Amritsar and Lucknow. The first experiment with satellite technology in India was conducted in 1975–76 under the programme ‘Satellite Instructional Television Experiment’ (SITE). Colour transmission was introduced during the Asian Games held in New Delhi in 1982. The year 1982, also witnessed the introduction of a regular satellite link with Delhi and other transmitters, heralding the arrival of the National Network. Various important DD channels include: DD Today, DD National, DD News, DD Sports, DD Bharati, DD India, Gyandarshan etc. DTH Tv and ku-Band Transmission Direct-to-home (DTH) TV series is a service for direct distribution of television programmes/channels by satellite to the receiving subscriber homes in high frequency ku-band via a small dish and a decoder/set-top box. Unlike the C-band transmission which requires a large dish antenna to receive the signals, ku-band transmission requires a small antenna for reception. Prasar Bharti considers ku-band transmission as a superior technology and the most cost effective method to cover the King areas and sparsely populated regions of the country.

PRaCTiCE EXERCisE Directions: (Q. 1–15 ) Given below are two statements, one of them labeled as Assertion ‘A’ and the other labeled as Reason ‘R’. You are to examine these two statements carefully and decide if the Assertion ‘A’ and Reason ‘R’ are individually true and if so, whether the Reason is a correct explanation of the Assertion. Select your answers to these items using the codes given below: Codes: (a) Both A and R are true and R is the correct explanation of A. (b) Both A and R are true but R is not a correct explanation of A. (c) A is true but R is false (d) A is false but R is true. 1. Assertion A: An important policy instrument of economic liberalization is reduction in import duties on capital goods. Reason R: Reduction in import duties would help the local entrepreneurs to improve technology to face the global markets. 2. Assertion A: Ceiling on foreign exchange for a host of current account transaction heads was lowered in the year 2000. Reason R: There was a fall in foreign currency assets also. 3. Assertion A: Death rate was quite high prior to 1921.

Reason R: There were frequent famines and epidemics prior to 1921. The available medical facilities and health services were very poor. 4. Assertion A: Devaluation of a currency may promote export. Reason R: Price of the country’s products in the international market may fall due to devaluation. 5. Assertion A: Disguised unemployment is a common feature of Indian agriculture. Reason R: India has been relatively slow in adopting farm mechanization. 6. Assertion A: Disguised unemployment is generally observed in Indian agricultural sector. Reason R: More and more people in rural areas are becoming literate. 7. Assertion A: During the year 2001– 02, the value of India’s total exports declined, registering a negative growth of 2.17 percent. Reason R: During the year 2001– 02, negative growth in exports was witnessed in respect of iron and steel, and textiles and marine products. 8. Assertion A: Estimation of national income constitutes sale of shares. Reason R: Sale of shares is a type of financial transaction.

2.278

▌ General Knowledge

9. Assertion A: Fiscal deficit is greater than budgetary deficit. Reason R: Fiscal deficit is the borrowing from the Reserve Bank of India plus other liabilities of the government to meet its expenditure. 10. Assertion A: For the first time, India had no trade deficit in the year 2002–2003. Reason R: For the first time, India’s exports crossed worth $ 50 billion in the year 2002–2003. 11. Assertion A: In Australia, cattle rearing is done more for meat than for milk. Reason R: Australians are traditionally non-vegetarians. 12. Assertion A: India does not export natural rubber. Reason R: About 97 percent of India’s demand for natural rubber is met from domestic production. 13. Assertion A: India’s per capita income does not give a true picture of its economic growth. Reason R: Per capita income of a nation can grow if there is a fall in population of the nation. 14. Assertion A: India’s share in world trade has declined over the period 1950–51 to 1990–91. Reason R: Composition of India’s exports/imports has not changed over the period 1950–51 to 1990– 91. 15. Assertion A: India’s software exports increased at an average growth rate of 50 percent since 1995–96. Reason R: Indian software companies were cost effective and maintained international quality. 16. Match List–I (Type of Inflation) with List–II (characteristic) and select the correct answer using the codes given below: List–I

(A) (B)

List–II (Characteristic)

(Type of Inflation)

1.

Initially, price rises

Demand-pull inflation

2.

Price-rise controlled by rationing and other means

Cost-push inflation

3.

Inflation in which prices increase as a result of increased production costs (labour and raw materials) even when demand remains the same

(C)

Suppressed inflation

4.

Inflation in which rising demand results in a rise in prices

(D)

Creeping inflati on

5.

Fall in price of raw materials

17. Match List–I (Committee/Commission) with List–II (Chairperson) and select the correct answer using the code given below:

List–I

List–II (Committee/Commission) (Chairperson)

(A) FDI

1.

N. K. Singh

(B) Venture Capital

2.

Chandrashekhar

(C) Disinvestment

3.

Tarapore

(D) Capital Account 4. Convertibility

G. V. Ramakrishna

5.

Raja Chellaiah

Codes: A

B

C

D

(a)

4

2

1

3

(b)

1

3

4

5

(c)

4

3

1

5

(d)

1

2

4

3

18. Match List–I (Committee) with List– II (Subject) and select the correct answer using the code given below the lists: List–I (Committee)

List–II (Subject)

(A) Tandon Commitee

1.

Capital Financing

(B)

Rakesh Mohan Committee

2.

Infrastructure Development

(C)

Narasimhan Committee

3.

Financial Sector Reforms

(D) Omkar Goswami Committee

4.

Industrial Sickness

Code: A

B

C

D

(a)

1

3

2

4

(b)

4

2

3

1

(c)

1

2

3

4

(d)

4

3

2

1

19. Match List I with List II. Choose the answer from the codes given below: List I (Committee)

List II (Purpose)

1.

Dutt Committee

A.

Industrial licensing

2.

Chelliah

B.

Tax reforms Committee

3.

Rangarajan Committee

C.

PSU disinvestment

4.

Chore Committees

D.

Cash-credit systems

Indian Economy

22. Match List I with List–II and select the correct answer using the codes given below.

Codes: (a)

1

2

3

4

A

B

C

D

(b) B

C

D

A

(c)

C

D

A

B

(d) D

C

B

A

List–I

20. Match List I with List II and select the correct answer. List I (Committees) A.

List II (Chaired by)

Disinvestment of shares in public sector enterprises

1.

B.

Industrial sickness

2.

Onkar Goswami

C.

Tax reforms

3.

R. N. Malhotra

D.

Reforms in insurance sector

4.

C. Rangarajan

Rajah Chelliah

A

B

C

D

4

2

3

(b) 4

2

1

3

(c) 4

1

2

3

(d) 1

3

4

2

Development Programme

1.

UN India Human Development Report

B.

National Council of Applied Economic Research

2.

India Development Report

C.

Indira Gandhi 3. Development Institute of Research

World Development Report

D.

World Bank Report

Human Development

(a)

List II

2

1

3

3

4

1

(d)

2

1

4

3

A.

List–I WTO

1.

B.

IDA.

2

C. D.

IMF IBRD

3. 4.

B.

budget deficit

2.

Excess of revenue expenditure over revenue receipts

C.

revenue deficit

3

Excess of total expenditure over total receipts less borrowings

(a)

D.

primary deficit

4

Excess of total expenditure over total receipts less borrowings and interest Payments

D

(a)

3

1

2

4

(b)

4

3

2

1

(c)

1

3

2

4

(d)

3

1

4

2

D 3

4

Excess of total expenditure over total receipts

C

C 2

2

1.

B

B 1

(b)

fiscal deficit

A

A 4

(c)

A.

Codes:

4.

23. Match List–I with List–II and select the correct answer using the codes given below.

21. Match List I with List II and select the correct answer using the codes given below. List I

List–II

A.

Codes:

Codes: (a) 1

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List–II provides loans to address short-term balance of payment problems multilateral trade negotiation body sanction of soft loans facilitating lending and borrowing for reconstruction and development

Codes: A

B

C

D

2

3

4

1

(b) 2

3

1

4

(c)

3

2

4

1

(d) 3

2

1

4

24. Match List–I with List–II and select the correct answer using the codes given below. List–I

List–II

A.

boom

1.

business activity at high level with increasing income, output and employment

B.

recession

2.

gradual fall of income output and employment with business activity in low gear

2.280

▌ General Knowledge

List–I C.

List–II depression 3.

D.

recovery

4.

unprecedented level of under employment and unemployment, drastic fall in income, output and employment steady rise in the general level of prices, income, output and employment

Codes: A

B

C

D

(a)

1

2

3

4

(b)

1

2

4

3

(c)

2

1

4

3

(d)

2

1

3

4

32. Passenger carriages for the Indian Railways are built in (a) Chittaranjan (b) Hubli (c) Perambur (d) Varanasi 33. The first telecast in India was in the year (a) 1947 (b) 1952 (c) 1959 (d) 1972 34. The first railway line in India was started in the year (a) 1850 (b) 1853 (c) 1840 (d) 1890 35. Diesel locomotives are built in (a) Bangalore (b) Varanasi (c) Perambur (d) Hubli

25. Which of the following pairs are correctly matched: 1. Dow Jones : New York 2. Hang–Seng : Seoul 3. FTSE–100 : London Codes: (a) 1, 2 and 3 (b) 2 and 3 (c) 1 and 2 (d) 1 and 3 26. Which of the following is wrongly matched? (a) National Institute of Animal Genetics—Karnal (b) Indian Institute of Sugarcane Research—Lucknow (c) Central Institute of Coastal Engineering for Fisheries— Bangalore (d) Central Duck Breeding Farm— Chikmagalur 27. A special economic zone is set up at (a) Positra (b) Kandla (c) Kochi (d) None of these 28. The Sensex is a market capitalization weighted index of … component stocks. (a) 20 (b) 30 (c) 25 (d) 40 29. The core sector includes I. Coal II. Power III. Petroleum IV. Soaps and Detergent (a) I and II (c) I

31. The breadth of the railway broad gauge is approximately (a) 2.00 m (b) 1.83 m (c) 1.67 m (d) 1.33 m

(b) I, II and III (d) II and III

30. Which of the following is the largest employer in India? (a) Food Corporation of India (b) Posts and Telegraphs Department (c) Indian Railways (d) Steel Authority of India

36. The first railway line in India was opened under the GovernorGeneralship of (a) Lord Rippon (b) Lord Dalhousie (c) Lord Bentinck (d) Lord Cornwallis 37. Indian Railways is the world’s ______ railway system in terms of mileage. (a) largest (b) second largest (d) third largest (d) fourth largest 38. The first electric railway was opened in (a) 1853 (b) 1885 (c) 1905 (d) 1929 39. Which one is not a port on the east coast? (a) Kolkata (b) Paradip (c) Kochi (d) Visakhapatham 40. How many trains are run daily by Indian Railways? (a) 5000–9000 (b) 2000–6000 (c) 15000–17000 (d) 10000–15000 41. Which of the following cities lies on the Delhi Mumbai National Highway? (a) Indore (b) Jhansi (c) Jaipur (d) Nasik 42. In which year was the first attempt made to enumerate the population of India? (a) 1945 (b) 1851 (c) 1872 (d) 1951 43 Regular census in India are held every 10 years. In which year was the first regular census held in India? (a) 1921 (b) 1881 (c) 1911 (d) 1931 44. Which year is known as the ‘Year of the Great Divide’ with regard to population, after which there has been a continuous and rapid growth in India’s population? (a) 1911 (b) 1921 (c) 1941 (d) 1951

Indian Economy

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45. Since Independence, how many times has the census been taken? (a) Three (b) Four (c) Five (d) Six

58. The Headquarters of the North- Eastern Railway zone are located at (a) Gorakhpur (b) Lucknow (c) Allahabad (d) Moradabad

46. The last Census in India was held in (a) 2008 (b) 2010 (c) 2011 (d) 2012

59. From when was the production of steam locomotives discontinued in India? (a) 1970 (b) 1972 (c) 1975 (d) 1980

47. Which Union Territory in India has the sex ratio (number of females per 1000 males) above 1000? (a) Puducherry (b) Delhi (c) Chandigarh (d) Lakshadweep 48. According to 2011, Census the population of India is (a) 90-100 crore (b) 100-110 crore (c) 110-120 crore (d) over 120 crore 49. According to the last census, which is the most populated state in India? (a) West Bengal (b) Uttar Pradesh (c) Bihar (d) Karnataka 50. According to the 2011 Census, what is India’s share in total world population? (a) below 10% (b) between 10-15% (c) between 15-20% (d) more than 20% 51. The Rajdhani Express runs between (a) New Delhi and Ahmedabad (b) New Delhi and Guwahati (c) Hazrat Nizamuddin and Madras (d) Hazrat Nizamuddin and Bangalore 52. Which of the following ports is a natural harbour? (a) Kandla (b) Paradeep (c) Tuticorin (d) Mumbai 53. In how many zones the Indian Railways are divided? (a) 6 (b) 7 (c) 8 (d) 9

60. The largest Railways Zone in terms of route kilometerage is (a) Central (b) Northern (c) Western (d) South-Central 61. The first locomotive in India was produced at Chittaranjan Locomotive Works. In which year was it produced? (a) 1960–61 (b) 1950–51 (c) 1951–52 (d) 1961–62 62. Railway locomotives are produced in two places in India. One is Varanasi. Which is the other? (a) Kapurthala (b) Chittaranjan (c) Ambala (d) None of these 63. Nhava Sheva, one of India’s major ports, is in (a) Gujarat (b) West Bengal (c) Maharashtra (d) Andhra Pradesh 64. Assertion A: Indian economy is referred to as a ‘mixed economy’. Reason R: Indian government recognizes the relative importance of both public as well as private sectors in the process of economic development. 65. Assertion A: Indirect taxes contribute a major part of the total taxation revenue in India. Reason R: These taxes cannot be evaded. 66. Assertion A: Industrial Revolution revolutionized the whole industrial system in England in the 18th century. Reason R: Industrial Revolution brought the class conflict to an end.

54. The natural seaport in eastern coast of India is (a) Kolkata (b) Chennai (c) Tuticorin (d) Vishakhapatnam

67. Assertion A: Information technology is fast becoming a very important field of activity in India. Reason R: Software is one of the major exports of the country and India has a very strong base in hardware.

55. Between which of the following stations does the Samjhota Express run? (a) Delhi and Islamabad (b) Kandla and Karachi (c) Howrah and Dhaka (d) Attari and Amritsar

68. Assertion A: Land reforms in India have been successfully implemented. Reason R: Government has been providing fertilizer at subsidized rates.

56. First railway line in India was laid in (a) 1835 (b) 1853 (c) 1917 (d) 1923

69. Assertion A: Sex ratio in India is in favour of males. Reason R: In females, there is a high rate of mortality during maternity.

57. The headquarters of the South- Central Railway zone are located in (a) Kolkata (b) Secunderabad (c) Hyderabad (d) Gorakhpur

70. Assertion A: The EXIM policy is liberal, market-oriented and favours global trade. Reason R: GATT has played a significant role in the liberalization of the economy.

2.282

▌ General Knowledge

71. Assertion A: The population of India has increased rapidly since 1921. Reason R: In India, the birth rate has declined more rapidly than the death rate since 1921. 72. Assertion A: The rate of growth of India’s exports has shown an appreciable increase after 1991. Reason R: The government of India has resorted to devaluation. 73. Assertion A: The United States of America has threatened to ask the World Trade Organization (WTO) to apply sanctions against the developing countries for the nonobservance of ILO conventions. Reason R: The United States of America itself has adopted and implemented these ILO conventions. 74. Match the following pairs. Choose the answer from the codes below: Plans Year 1. First 2. Second 3. Third 4. Fourth 5. Fifth 6. Sixth 7. Seventh 8. Eighth Codes: (a) 1-E, (b) 1-A, (c) 1-A, (d) 1-H,

3-C, 3-C, 3-E, 3-F,

4-G, 4-D, 4-G, 4-E,

5-F, 5-E, 5-H, 5-D,

6-B, 6-F, 6-F, 6-C,

7-D, 7-G, 7-D, 7-B,

8-H 8-H 8-B 8-A

75. Match the following pairs and choose your answer from the codes given below: Laws

Year

1. Water Pollution Control Act

A.

1980

2. Forest (Conservation) Act

B.

1974

3. Air Pollution Control Act

C.

1986

4. Environmental Protection D. Act

1981

Codes:

Trade Union

Year

1. AITUC

A.

1920

2. INTUC

B.

1947

3. HMS

C.

1948

4. UTUC

D.

1949

Codes: 1

2

3

4

(a) A

B

C

D

(b) A

C

B

D

(c) B

D

A

C

(d) D

C

B

A

77. Match List II with List I. List I (Name of the Act)

A. 1956–1961 B. 1980–1985 C. 1961–1966 D. 1985–1990 E. 1951–1956 F. 1974–1979 G. 1969–1974 H. 1992–1997

2-A, 2-B, 2-C, 2-G,

76. Match the following pairs and choose your answer from the codes below:

List II (Year)

1.

The Factories Act

A.

1948

2.

The Minimum Wages Act

B.

1938

3.

The Employment of Children Act

C.

1948

4.

The Maternity Benefit Act

D.

1947

5.

The Industrial Disputes Act

E.

1961

Codes: 1

2

3

4

5

A

C

B

E

D

(b) C

A

E

B

D

(c)

E

C

B

A

D

(d) D

E

C

B

A

(a)

78. Match List I with List II. Choose the answer from the codes below: List I

List II (Nature of work)

1. CART

A.

Small industries

2. ICICI

B.

Agricultural Marketing

3. NAFED

C.

Rural technology

4. NSIC

D.

Financial institution

Codes:

1

2

3

4

(a)

A

B

C

D

(a)

(b)

B

A

D

(c)

D

C

(d)

B

C

1

2

3

4

C

D

B

A

C

(b) A

B

C

D

B

A

(c)

B

C

D

A

A

D

(d) D

A

B

C

Indian Economy 79. The SEBI stands for (a) State Earnings Board of India (b) Securities and Exchange Bank of India (c) Securities and Exchange Board of India (d) State Exchange Bank of India 80. What are gilt-edged securities? (a) Securities issued by the multinationals (b) Securities issued by the government (c) Securities issued by the private companies (d) Securities issued by the joint venture companies 81. Among the following, which is nota credit rating agency operating in India? (a) CRISIL (b) ICRA (c) DOW Jones (d) CARE 82. GDR means (a) ‘gross domestic ratio’ which is a macro aggregate (b) ‘global depository receipt’ which is the mechanism to raise funds from international market (c) global development range’ which has been set up by UNO for developing countries (d) ‘geographically developed regions’ as defined by the UNDP 83. The Industrial Investment Bank of India was earlier known as the (a) IRBI (b) BIFR (c) IDBI (d) IRDA 84. Which of the following committees is related with capital market reforms? (a) Malegam Committee (b) Kelkar Committee (c) Naresh Chandra Committee (d) Swaminathan Committee 85. According to the latest Economic Census of India, the annual growth rate of enterprises since liberalization has (a) declined sharply (b) increased (c) declined slightly (d) remained stagnant 86. Which scheduled bank sponsored the setting up of first regional rural bank (Prathama Gramin Bank) in India? (a) State Bank of India (b) Punjab National Bank (c) State Bank of Patiala (d) Syndicate Bank 87. The latest Bank to be granted licence is (a) Kotak Mahindra Bank Ltd (b) Yes Bank Ltd (c) ANZ Bank Ltd (d) HSBC Ltd 88. How many PSUs are included in the category of ‘Navaratna’? (a) 8 (b) 9 (c) 11 (d) 14

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89. The commodity for which India spends the largest amount to import is (a) iron and steel (b) food grains (c) crude petroleum (d) fertilizers 90. Which of the following commodities earns maximum foreign exchange for India? (a) Jute (b) Iron and steel (c) Tea (d) Sugar 91. The one rupee note bears the signature of (a) Secretary, Ministry of Finance (b) Governor, RBI (c) Finance Minister (d) None of these 92. An ad-valorem duty is tax basis of (a) Price of commodity (b) Value added (c) Advertisement expenses (d) Unit of the commodity 93. Which one of the following is not an affiliate of the Reserve Bank of India? (a) IDBI (b) Agriculture Refinance Corp (c) UTI (d) Deposit Insurance Corp 94. The Indian Financial Year begins on (a) 1 January (b) 1 April (c) 1 July (d) None of these 95. Percentage of Re 1 coins in total currency of India is (a) 1.2 percent (b) 2.2 percent (c) 2.0 percent (d) 2.3 percent 96. Which of the following is a direct tax? (a) Excise (b) Sales tax (c) Income tax (d) None 97. What is the population density in India as per the 2011 Census (No. of persons per km2 of area)? (a) 367 (b) 382 (c) 247 (d) 236 98. During the last decade (2001– 2011) the decrease in percentage decadal growth was approximately registered at (a) Below 4 (b) Between 3 and 4 (c) Between 4 and 5 (d) Above 5 99. Arrange the following four densely populated states in decreasing order: 1. Bihar 2. Kerala 3. Best Bengal 4. Uttar Pradesh (a) 2, 1, 3, 4 (b) 4, 3, 2, 1 (c) 4, 1, 3, 2 (d) 4, 2, 3, 1 100. Which state emerged as the least densely populated state as per 2011 census? (a) Kerala (b) Arunachal Pradesh (c) Nagaland (d) Goa

2.284 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

▌ General Knowledge

(a) (b) (a) (a) (b) (b) (d) (d) (a) (d) (b)

12. (b) 13. (a)

14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.

(c) (a) (c) (d) (c) (a) (b) (a) (a) (b) (a)

25. (d) 26. (c)

27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37.

(a) (c) (b) (c) (c) (c) (c) (b) (b) (b) (b)

38. (d) 39. (c)

40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50.

(a) (c) (c) (b) (b) (d) (c) (a) (d) (b) (c)

51. (d) 52. (d)

53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63.

(d) (d) (d) (b) (b) (a) (b) (b) (b) (b) (c)

64. (a) 65. (c)

66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76.

(c) (c) (d) (a) (b) (c) (a) (a) (a) (b) (a)

77. (a) 78. (a)

79. 80. 81. 82. 83. 84. 85. 86. 87. 88. 89.

(c) (b) (c) (b) (a) (a) (c) (d) (b)

92. 93. 94. 95. 96. 97. 98. 99. 100.

(a) (a) (b) (d) (c) (b) (b) (c) (b)

(c) (c)

90. (c) 91. (a)

nDa BoX 1. Which of the following ports has a free trade zone? (a) Kochi (b) Tuticorin (c) Madras (d) Kandla

8. Which among the following is the fastest train in the country? (a) Taj Express (b) Shatabdi Express (c) Rajdhani Express (d) G.T. Express

2. Which of the following ports does not have a natural harbour? (a) Kochi (b) Mumbai (c) Madras (d) Vishakhapatnam

9. Approximately how many long distance trains (mail/express) are run by the Indian Railways? (a) 650 (b) 750 (c) 850 (d) 900

3. August Kranti Rajdhani Express train runs between which two terminal stations? (a) New Delhi–Howrah (b) New Delhi–Chennai (c) New Delhi–Mumbai (d) New Delhi–Hyderabad 4. Rajdhani Express from Delhi to Howrah passes through (a) Delhi, Rajasthan, Bihar, Bengal (b) Delhi, Punjab, Bihar, Bengal (c) Delhi, UP, Bihar, Bengal (d) Delhi, UP, MP, Bengal 5. The international air route from West Europe to East Asia passes through India because (a) it is the shortest route (b) India is the only country in Asia providing refueling capacity (c) India is a busy centre situated between the two places (d) None of these 6. The international airport in Tamil Nadu is (a) Palam (b) DumDum (c) Santa Cruz (d) Meenambakkam 7. The largest Integral Coach Factory in India is located at (a) Kapurthala (b) Perambur (c) Varanasi (d) Moradabad

10. The electrified railway route-length has been growing annually at the rate of (a) 6% (b) 7.5% (c) 8% (d) 8.5% 11. Which among the following trains covers the longest route in the country? (a) G.T. Express (b) Guwahati–Trivandrum Express (c) Karnataka Express (d) Gorakhpur–Cochin Express 12. Pearl City Express train runs between (a) Delhi–Udaipur (b) Chennai–Tuticorin (c) Delhi–Jodhpur (d) Chennai–Bangalore 13. The Chetak Express runs between (a) Delhi–Udaipur (b) New Delhi–Indore (c) Delhi–Jaipur (d) Jodhpur–Delhi 14. State govts do not levy (a) Income tax (c) excise duty

(b) CST (d) None

15. What is the effect of deficit financing on economy? (a) Inflation (b) Deflation (c) Depression (d) Recession

Indian Economy 16. Temporarily inflation can be checked by (a) Decrease in money supply (b) Increase in wages (c) Decrease in taxes (d) Reduction in taxes 17. Which of the following is not a direct tax? (a) Income tax (b) Wealth tax (c) Sales tax (d) Estate duty 18. Which of the following best defines free trade? (a) Imports are discouraged (b) There are no restriction on exports/imports (c) There are no duties levied on export (d) Imported goods are made duty free 19. Which of following factors contribute to inflation? (a) 20 percent increase in prices of agricultural commodities (b) 20 percent increase in money supply (c) 20 percent fall in the production of industrial products (d) None of the above 20. The largest share of revenue receipts of the State Government comes from (a) General sales tax (b) Land revenue (c) Sharing excise duty (d) Registration fees 21. The largest expenditure being incurred from combined revenue expenditure of centre, states and union territories is on (a) health (b) education (c) interest payments (d) defence services 22. India’s maximum foreign exchange is spent on (a) import of food grains (b) import of iron and steel (c) import of petroleum products (d) import of technical know-how 23. India earns maximum forex through export of (a) leather goods (b) handicrafts (c) electronic goods (d) engineering goods 24. Who among the following was the first to estimate the country’s per capita National Income during the British rule for the year 1888? (a) Pt Jawaharlal Nehru (b) Prof. P. C. Mahalanobis (c) Dadabhai Naoroji (d) None of these 25. Rupee coin was first minted in India during the rule of (a) Shahjahan (b) Sher Shah Suri (c) East India Company (d) Razia Begum 26. The rupee coin was first minted in India in (a) 1542 (b) 1601 (c) 1809 (d) 1677 27. The decimal currency system started in India from (a) January 1955 (b) April 1957 (c) April 1951 (d) April 1955

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28. Paper currency first started in India in (a) 1862 (b) 1542 (c) 1601 (d) 1680 29. Whose signature appears on ten rupee currency notes? (a) Minister of Finance (b) Governor, RBI (c) Secretary, Ministry of Finance (d) None of these 30. How many times has the Indian currency been demonetized so far? (a) Once (b) Twice (c) Thrice (d) Never 31. First demonetization of Indian currency took place in (a) 1946 (b) 1947 (c) 1948 (d) 1949 32. Indian currency was last demonetized in (a) 1951 (b) 1955 (c) 1960 (d) 1978 33. Indian currency has been devalued only _____. (a) Once (b) Four times (c) Twice (d) Never 34. The first devaluation of Indian currency took place in (a) 1947 (b) 1949 (c) 1950 (d) 1951 35. Indian currency was last devalued in (a) 1955 (b) 1956 (c) 1991 (d) 1967 36. Who was the Finance Minister when Indian currency was last devalued? (a) S. Mukherjee (b) Sachindra Banerjee (c) Sachindra Roy (d) Dr Manmohan Singh 37. Where in Indian are coins minted? (a) Delhi, Mumbai, and Kolkata (b) Delhi, Kolkata, and Hyderabad (c) Mumbai, Delhi and Bangalore (d) Mumbai, Kolkata and Hyderabad 38. Currency notes in India are printed and supplied by (a) Security Press, Nashik (b) Security Press, Mumbai (c) Security Press, Noida (d) RBI, New Delhi 39. On which denomination notes has the portrait of Mahatma Gandhi been printed? (a) Rs 100 (b) Rs 500 (c) Rs 50 (d) None of these 40. When did India become a member of International Monetary Fund (IMF)? (a) 1947 (b) 1950 (c) 1951 (d) 1955

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41. Currency notes and coins of one rupee and lower denominations are issued by (a) Reserve Bank of India (b) State Bank of India (c) Ministry of Finance (d) Government of India 42. Bank notes and coins above one rupee denominations are issued by (a) Reserve Bank of India (b) Ministry of Finance (c) State Bank of India (d) Government of India 43. The total number of major banks nationalized so far is (a) 15 (b) 18 (c) 20 (d) 22 44. A personal account cannot be opened in (a) A Cooperative Bank (b) A Scheduled Bank (c) Reserve Bank of India (d) None of these 45. What is the literacy rate in India as per 2011 census? (a) 49.5% (b) 74.04% (c) 42.5% (d) 65.38% 46. As compared to women, men are still ahead in literacy rate with ______ of the male literate population. (a) 57.4% (b) 82.14% (c) 55.7% (d) 60.7% 47. Which state has the highest literacy rate in the country? (a) Goa (b) Kerala (c) Tamil Nadu (d) Karnataka 48. What is the sex ratio (number of females per 1000 males) in India? (a) 856/1000 males (b) 940/1000 males (c) 789/1000 males (d) 890/1000 males 49. Which state has the sex ratio of 1,084 females per 1000 males? (a) Tamil Nadu (b) Kerala (c) Delhi (d) Assam 50. As per Census 2011, the percentage of rural population in India is approximately (a) 50% (b) 67% (c) 72% (d) 80%

indian Economy and misc. Economic 51. In India, public expenditure on education (Center and States) as a percentage of GDP at present is between (a) 2 to 3 per cent (b) 3 to 4 per cent (c) 4 to 5 per cent (d) 5 to 6 per cent

52. In which one of the following Five Year Plans the actual growth performance in India in respect of GDP (at factor cost) was less than the target set? (a) Sixth Five year Plan (b) Seventh Five year plan (c) Eighth Five year Plan (d) Ninth Five year plan 53. In which one of the following periods did India experience a negative population growth? (a) 1901-1911 (b) 1911-1921 (c) 1921-1931 (d) 193-1941 54. India’s Look East Policy’ 1. was articulated during the reign of HD Deve Gowda as India’s Prime Minister 2. is directed towards South-East Asian Countries 3 is about controlling terrorism 4. Seeks to develop a bond among various countries on the basis of economic cooperation Select the correct answer using the codes given below: Codes: (a) 1 and 2 (b) 2 and 3 (c) 2 and 4 (d) 3 and 4 55. Match List I with List II select the correct answer using the code given below the lists. List I (Name of the shipyard)

List II (Location )

A.

Hindustan Shipyard Limited

1.

Kolkata

B.

Garden Reach shipbuilders and Engineers Limited

2.

Vasco-da Gama

C.

Goa Shipyard Limited

3.

Mumbai

D.

Mazagaon Dock Limited

4.

Vishakhapatnam

Codes: A B

C D

(a)

4

2

1

3

(b)

3

1

2

4

(c)

4

1

2

3

(d)

3

2

1

4

56. Planning in India drew on (a) The New Economic Programme of Lenin (b) The Fabian Socialism of Sidney and Beatrice Web (c) The British welfare mechanism (d) New democratic development packages

Indian Economy 57. The ‘Bombay Plan’ drafted by GD Birla and JRD Tata emphasized (a) That the economy Should be left to the dynamic investments by the private sector in heavy industries, etc. (b) The public sector investment in infrastructure and heavy industries (c) Annual planning (d) That the private sector should foot the Bill for intensive and low return investments in the industrial sector 58. The current Economic Census of India, the fifth in the series was conducted in the year (a) 2005 (b) 2006 (c) 2007 (d) 2008 59. The highest body that approves five-year plans in India is (a) Planning Commission (b) National Development Council (c) Interstate Council (d) Union Cabinet. 60. TRIPS (Trade Related aspects of Intellectual property Rights) agreements is administered by (a) United Nations Conference on Trade and Development (UNCTAD) (b) United Nations Organization (UNO) (c) Word Trade Organizations (WTO) (d) Word Bank (WB) 61. Where is the headquarters of the International Bank of Reconstruction and Development located? (a) Geneva (b) Washington DC (c) London (d) Paris 62. Which among the following is not correct with regard to Sampoorna Grameen Rozgar Yojana? 1. The case component of the programme is borne exclusively by the Central Government. 2. Foodgrains are provided free of cost to the States /Union Territories. Select the answer using the codes Given below (a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2

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63. Which among the following statements is/are true with regard to WTO membership? 1. All WTO members automatically receive the ‘Most favoured nation’ status. 2. Over 75% of WTO members are from developing countries, WTO membership allows them access to developed markets at the lower tariff. Which of the statement given above is/are correct? (a) 1only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2 64. Which among the following States/UT is the latest to introduce VAT (Value Added Tax) in India? (a) Gujarat (b) Puducherry (c) Uttar Pradesh (d) Tamil Nadu 65. Which Five Year plan had an objective of ‘Rapid industrialization with particular emphasis on development of basic and heavy industries? (a) First (b) Second (c) Third (d) Fourth 66. Which of the following bodies is responsible for the distribution of revenues between the Centre and the States? (a) Planning Commission (b) Finance Commission (c) Inter-State Council (d) National Development Council 67. Which one among the following statements regarding the Eighth five Year Plan in India is not correct? (a) The Plan was postponed by two years because of political upheavals at the centre (b) It aimed at high growth of both agriculture and manufacturing sectors (c) Its emphasis was on growth in export and import, improvement in trade and current account deficit (d) It set before itself the two principal objectives of ‘growth with stability’ and “growth with justice". 68. Which one of the following is not a member of BASIC bloc of Countries? (a) South Africa (b) China (c) India (d) Australia

2.288

▌ General Knowledge

ansWER kEY 1. (d)

10. (d)

19. (b)

28. (a)

37. (d)

46. (b)

55. (c)

64. (c)

2. (c)

11. (b)

20. (a)

29. (b)

38. (a)

47. (b)

56. (a)

65. (b)

3. (c)

12. (b)

21. (c)

30. (b)

39. (b)

48. (b)

57. (b)

66. (b)

4. (c)

13. (a)

22. (c)

31. (a)

40. (a)

49. (b)

58. (a)

67. (b)

5. (b)

14. (d)

23. (b)

32. (d)

41. (c)

50. (c)

59. (b)

68. (d)

6. (d)

15. (a)

24. (c)

33. (b)

42. (a)

51. (b)

60. (c)

7. (b)

16. (a)

25. (b)

34. (b)

43. (c)

52. (d)

61. (b)

8. (b)

17. (c)

26. (a)

35. (c)

44. (c)

53. (b)

62. (b)

9. (d)

18. (b)

27. (b)

36. (d)

45. (b)

54. (c)

63. (c)

C

5

H

A

P

T

E

R

General Science PHYSICS MECHANICS Matter

Mass

Anything that occupies space and possesses weight is called matter. Matter or material substances consist of atoms and molecules. Atoms combine to form molecules of a substance. Matter can be perceived by one or more senses and possesses mass. Matter can neither be created nor destroyed; it can only be changed from one form to another. Matter exists in three states, solid, liquid and gas, each of which can change into one of the other according to changes in temperature and pressure. There is also a fourth state of matter, the ionized state called plasma. Though this state is not common on the earth, the sun and stars are in this state. Matter can be classified as shown in the figure.

The SI unit of mass is kilogram. It is the measure of inertia possessed by a piece of matter, that is, quantity of matter in a body. Mass is not the same as weight, which is the force with which the earth attracts a body. Mass remains constant while the weight varies from place to place and is zero at the centre of the earth.

Weight Weight is the product of acceleration due to gravity and mass of the body. On the moon, a body will weigh much less than on the earth because the moon’s surface gravity is only 1/6th of that of the earth.

Matter

(1) Physical Classification Solid

Liquid

(2) Chemical Classification Pure substance

Gas

Elements

Metallic

Mixtures

Compounds

Non-metallic

Organic

Homogeneous

Inorganic

Heterogeneous

2.290

General Knowledge

A pair of scales measure mass while a spring balance measures weight. Mass is measured in grams whereas weight is measured in dynes.

Density Density is mass per unit volume of a substance and is expressed in the S.I. unit as kg/m3. Density of water is 1000 kg/m3. Densities are expressed as Relative density, that is, density relative to water, and sometimes, called specific gravity. • A ship floats on the surface of water because the weight of water di weight. • Ice floats on water because its weight is less than the weight of an equal volume of water. • The Lactometer used to test purity (density) of milk is based on the principle that the greater density of a liquid, the lesser will be the immersion of an object.

Force That which makes a body change its state of rest or uniform motion in a straight line—it causes objects to remain stationary, to continue moving steadily or to move faster. Centrifugal Force is the force which appears to act on an object moving in a circular path. Centripetal Force is the force acting on a body constrained to move in a curved path. It acts inwards on an object moving in a circular path. Illustration When a piece of stone tied to a thread is whirled you have to pull the thread inwards. This pull on the stone is called centripetal force and an equal and opposite force is exerted by the, stone on the hand which is called centrifugal force.

Application • Curved rail tracks or curved roads are banked or raised on one side so that a fast moving train or vehicle leans inwards, thus providing required centripetal force to enable it to move round the curve. • A cyclist while turning round a curved road leans inwards so as to provide himself with the required centripetal force which enables him to take the turn.

Friction Friction is a force that resists the movement of one surface over another. Friction is very necessary because without it brakes would not work, belts would not drive the wheels of a machinery, etc. Friction of moving parts causes wastage of energy and in such cases wheels, ball-bearings and lubricants, etc., are used to reduce friction.

Energy The capacity of doing work is called energy. It can exist in a number of forms, for example, mechanical. electrical, potential, chemical, kinetic, nuclear, etc. The energy possessed by a body owing to its position is called potential energy, for example, a wound up spring, a stretched rubber band, a bent bow, or a reservoir of water. The energy possessed by a body due to its motion is called kinetic energy, for example, a bullet shot from a gun or water flowing in downstream. Energy is never lost or gained but only changes from one form to another. Energy can neither be created not destroyed. This is known as Law of Conservation of Energy.

Surface Tension Surface tension is known to be due to intermolecular attraction in the liquid surface and these forces produce a skin effect on the surface. It is surface tension which causes water to climb up a narrow capillary (phenomenon of capillary action). Rain drops are spherical due to surface tension. A needle may be made to float on water. This phenomenon is applicable to the capillary action of a blotting paper absorbing ink. The surface tension of a liquid decreases with increase in temperature and vanishes at the critical temperature.

Motion and Work Motion Almost every event that takes place in the universe involves movement or motion of one kind or the other. Motion is the change of position of a body with respect to its surroundings. Speed

is the rate of change of motion.

Velocity Commonly known as speed, is the rate of change of motion in a particular direction and expressed in metres per second. Acceleration is the rate of change in velocity of a moving body expressed in metres per second square. Negative acceleration is called retardation.

Work When a force creates motion in a body it implies work has been done, that is, work is done by a moving force. It is equal to the product of the force and the distance it moves along its line of action. Power is the rate of doing work and Horsepower (hp) and watts are the units of power.

Elasticity A body that returns to its original shape and size on the removal of the deforming force (after having deformed within elastic limit) is called elastic. Contrary to the conception of elasticity in daily life, in physics it stands for opposition to change. Hence, the more rigid a body, the more elastic it is said to be. That is why steel is more elastic than rubber.

General Science ▌ 2.291

HEAT A form of energy which • raises temperature of matter • increases volume of matter • changes state and physical properties of matter and brings about chemical change

Latent Heat The heat required to change the state of a substance from solid to liquid or from liquid to gas without change in temperature.

Radiation Transmission of heat from one point to another without heating the medium, that is, transmission of heat by means of waves without the help of a medium. Polished surfaces are the best radiators of heat. Hence: • The double-walled glass bottles used in a thermos flask have a silver coating—outside the inner wall and inside the outer wall.

Temperature

• White clothes are preferred in summer. On the other hand, radiation is greater from black surfaces which are very good absorbers of heat. Hence:

The degree of heat of a body/substance which can he measured by various scales like Centigrade, Fahrenheit or Reaumur.

• it is warmer on a cloudy night because the heat radiated by earth is obstructed and sent back to the atmosphere.

Evaporation Is the change of state from liquid to vapour and thereby results in cooling. Rate of evaporation is affected by wind, heat and surface area. Illustration When alcohol is applied on the forehead of a person having high temperature, the temperature reduces because alcohol while evaporating draws the heat out of the body.

Transfer of Heat Conductivity A property enjoyed by metals. The process of heat transfers without visible motion of the atoms or molecules from a region of higher temperature to that of lower temperature. It is the transmission of heat from molecule to molecule. Among metals, silver is the best conductor followed by copper. Why are Metals Good Conductors? In metals, there are a large number of free electrons which transport major amount of heat and move from hotter part of the solid to the colder part transporting heat energy. Illustration In winter, a steel chair appears colder than a wooden chair because steel being a good conductor of heat, when touched absorbs heat from the hand rapidly giving the feeling of coldness. Metals are good conductors while gas and air are poor conductors. Hence: • Two thin blankets are warmer as they enclose air in between • A new quilt is warmer than an old one because it encloses more air • Eskimos make double-walled ice houses and the air in between two ice walls does not allow heat to pass • A thermos flask is made of double-walled glass bottles.

Convection It transmits the heat of heated matter, that is, by actual motion of the heated particles of a liquid or gas. Hence: • Ventilators in a room are on the top portion of the walls because air after being heated rises upwards and passes through the ventilators and cold-air enters the room through doors and windows which are at lower levels.

Laws of Thermodynamics These laws show how heat energy can be passed from one body to another. Zeroth Law No heat will flow between two bodies that are of the same temperature. In other words, a body’s temperature does not depend on the material involved. First Law Internal energy can be added to a body either by heating or doing work on it, and conversely, if work is extracted from the body, its internal energy (and hence its temperature) will fall unless an equivalent amount of heat is supplied from outside. Second Law Heat will not pass spontaneously from a cold body to a hotter body. Third Law It is impossible to cool a body right down to absolute zero, that is, below –273.15°C (see Phenomena and Terms of Physics).

2.292

General Knowledge

LIGHT “Light is an electromagnetic radiation that has a wavelength in about the range of 380–780 nanometers and that may be perceived by the normal unaided human eye”—Reader’s Digest Universal Dictionary. It is a form of energy which travels in straight lines and causes the sensation of vision. If we interpose a small obstacle between the object and our eyes we are unable to see it. Formation of shadows is another proof of the fact that light travels in straight line.

Composition In 1666, Isaac Newton passed a beam of light through a prism and found that it has seven colours, namely, 1. Violet 4. Green 7. Red

2. Indigo 5. Yellow

3. Blue 6. Orange

The band of colours so formed is called a spectrum in which violet and red bands form the extremes. Colour Visualization Different objects have different colours—they absorb or reflect different parts of the spectrum. Thus, a blue object absorbs the red, yellow and green parts of white light and reflects only the colour blue. The colour of an object depends upon the nature of light falling on it and also on the constituent colour of the incident light reflected by it. If all the colours are absorbed the object appears black. It all the colour constituents of light are reflected, it appears white.

Thus, we see objects because of the light they reflect. Most of the objects around us reflect only part of the light that is incident upon them and it is the reflected part which gives the object its colour. When a rose is viewed in white light, it appears red and the leaves appear green, because the petals reflect the red part and the leaves green part. The remaining colours are absorbed. When the same rose is viewed in green light, the petals will appear black and the leaves green. In blue or yellow light both the petals and leaves will appear black. A white paper appears white because it reflects all the colours of light, whereas a print on it appears black because it absorbs all the colours. In red light, green grass will appear black because it absorbs all colours except green and it would absorb the red rays of light falling on it.

Phenomenon of Light Reflection Is the ‘bouncing back’ of light rays. The simple law of reflection is that the angle between the incoming ray and the perpendicular to the surface is equal to the angle between the reflected ray and the same perpendicular. Refraction Is the sudden change of direction of light when passing from one transparent substance into another. For example, a ray of light passing from air into water bends towards the perpendicular. Hence Incident Ray

Incident Ray

Mirror

Angle of Incidence Normal

Air (Normal)

i

Water (Dense) r

i  Angle of Incidence r  Angle of Reflection

Angle of Reflection Reflected Ray FIG. 1 • A stick immersed in water appears bent. • A pond looks shallower than it really is as the rays start from rarer to denser medium, the rays bend towards the normal. • A diamond sparkles because when light enters a piece of diamond and the cut of its faces the light is totally refracted to various faces of the diamond, as diamond has a high refractive index. • A person looks shorter when he is standing in water and if we look at him from a side. • A stone lying at the bottom of pond appears to be at a higher point than it actually is.

Refracted Ray FIG. 2 Diffraction Diffraction is the spreading out of waves of light as it passes through a narrow aperture, because light is a wave of motion. Hence, light rays emerging from a cinema machine spread wide on the screen. Dispersion When white light is passed through glass prism, it is split into the colours of the spectrum. This is because each colour has its own wavelength, which determines the angle at which light is refracted. Red and yellow waves are the longest, and hence are the least refracted. Blue and violet rays are shortest and are the most refracted (See illustration).

General Science ▌ 2.293

Eye Pond Stone Appears Here

Sunlight

ht

ite Wh

Lig

Water Droplet

Red Violet Violet

Stone

Red

FIG. 3

Rainbow The most spectacular illustration of dispersion is the rainbow. When the sun shines after a shower, a rainbow is seen in the sky opposite the sun. The colours of the rainbow are due to the dispersion of sunlight by water droplets suspended in the air. Scattering A rough surface scatters light because each part of the surface reflects the light at different angles which is known as scattering of light. Why is the Sky Blue? Violet and blue light have short wavelength and are scattered by the atmosphere ten times more than red light waves. While the red light, with long wave length, goes almost straight through the atmosphere, blue and violet are scattered by particles in the atmosphere. Thus we see a blue sky. Absorption Some surfaces absorb more of the light falling on them than others. If light of all colours is absorbed, the eyes see black. A black surface absorbs more light than a white surface.

Mirrors Plane mirror Mirrors work on the principle of reflection of light. When a man stands in front of mirror, light from all parts of the body (light which is reflected from the body in the presence of a source of light) is reflected from the mirror back to the eyes and a virtual image appears to be formed behind the mirror. normal

mirror object

image FIG. B.2.4

Some Important Definitions A Real Image is one through which the rays of light actually pass and which can be formed on a screen. A Virtual Image is one throught which the rays, do not actually pass, although they appear to come from it. A consequence of the laws of reflection is that, for a real object, the image produced by a plane mirror is virtual and its distance behind the mirror is the same as the object's distance in vfront of it. Uses of Plane Mirrors Optical level, sextant, kaleidoscope, periscope, telescope flat, dental mirror. Curved Mirrors There are two types of curved mirrors used for specific purposes, viz., concave and convex. In concave mirrors, the light rays are reflected so as to converge to a point called the focus of the mirror. A concave mirror can therefore, concentrate sun’s radiation falling on it at one point. Hence, a concave mirror can be used as a burning glass and is used in solar cookers. The virtual images produced by convex mirror are erect and smaller than the object. The convex mirror is therefore used as a rear-view mirror in vehicles as it has the advantage of a wide field of view.

Some Important Definitions The Principal Axis is a line through the centre of the mirror which passes through the centre of curvature. The Pole of a mirror P is where the principal axis meets the mirror. The Principal Focus F is the point where parallel light close to the axis converges to a focus. The Focal Length (f) is the distance from the principal focus to the pole of the mirror. The Centre of Curvature C is the centre of curvature of the mirror surface. The Radius of Curvature is the distance of the centre of curvature from the pole of the mirror. The Paraxial Ray is one that lies close to the principal axis and may make a small angle with it. Uses of Concave Mirrors Reflecting telescope, dental mirror, headlamp reflectors, shaving and make-up mirrors. Uses of Convex Mirrors Safety viewers at dangerous corners, antishop-lifting devices, car wing mirrors.

2.294

General Knowledge

conave

mirror conave f

c

F

f

P

P

F

mirror (b)

(a) FIG. 5

Lenses Lenses are used in all optical instruments, like microscopes, telescopes. cameras, projectors, etc. Lenses are of the two types: Convex or Converging Lens If a parallel beam of light is incident on a convex lens, all rays, after passing the lens, converge on a point called the principal focus. Convex lens surfaces have a real and therefore positive radius of curvature. It has a real and positive focal length. Uses of Convex Lenses Magnifying glasses, eye (variable focal length), glasses to correct for long sight, microscope, telescope objective, camera (single lens sustem), Projectors. Concave or Diverging Lens In case of a concave lens, the rays of light spread out after passing through the lens. Concave lens

surfaces have a virtual and therefore, negative radius of curvature. It has a real and positive focal length. Uses of Concave Lenses Wide-angle spyhole in doors, glasses to correct for short sight, wide-angle lens on coach rear window, Eye lens in Galilean telesope. Power of the Lens The strength of a lens is described in terms of either its focal length or its power. The power of a lens is defined as: Power of the lens 

1 focal length in metres

Convex lenses have positive power while concave lenses have negative powers. The power of a meniscus lens depends on which face of the lens is the more sharply curved. convex

convex

R

f

F

F C

C

f R

(b)

(a) FIG. 6

SOUND Source of sound is always the vibrations of an object. It is transmitted in the form of waves with alternating increase and decrease in pressure.

Pitch refers to the rate of vibration of sound and is measured in Hertz (Hz). Loudness is measured in decibels (db), a unit based on the weakest sound that can be detected by the human ear.

Characteristics

Phenomena Related to Sound

(i) Pitch or frequency; (ii) Loudness or intensity, and (iii) Quality. On the basis of these characteristics sound can be recognized as we recognize a person from his voice.

Diffraction Like light, sound waves spread as they move past an obstacle or through a narrow aperture. If the aperture is small, the sound waves spread out in all directions. Hence, sound can be heard round the corners of a corridor.

General Science ▌ 2.295 Reflection Deflection of sound waves into new directions by a surface is known as ‘reflection of sound’. Refraction is the change of direction of a sound wave on passing from one medium to another. Echo

How do Bats Fly at Night? Bats can fly in the dark because ultrasonic waves produced by them during flying are reflected back to them from the obstacles in their way and hence they can fly without difficulty.

Repetition of sound by reflection is known as echo.

Oscillation

The term used to describe anything that vibrates.

Sound Waves and Music Sound is generated by a vibrating system setting up waves in the surrounding air. Musical sounds are produced by a vibration with a definite pitch often produced by a vibrating string (for example, sitar) or a vibrating column of air in a tube (for example, sehnai) which does not progress from one point to another but causes the string or column of air to vibrate as a whole with its characteristic frequency.

Ultrasonic and Supersonic Sounds Sounds of frequency higher than 20,000 Hz are known as ultrasonic and are inaudible. The human ear is normally sensitive to sounds whose frequencies are between 16 Hz and 20,000 Hz. The speed greater than the speed of sound is known as supersonic. The sound waves of frequency lower than 16 Hz are called infrasonic.

Speed of Sound in Different Media Ice-cold water

1,505 m/sec

Bricks

3,542 m/sec

Wood

3,847 m/sec

Glass

5,000–6,000 m/sec

Steel

5,000 m/sec

Air

331 m/sec

Speed of Sound The speed of sound depends upon the nature of the carrier media. In general the speed of sound refers to the speed at which sound travels in air at sea level. In dry air, at 0°C the speed of sound is about 331 m/sec or 750 miles/h. The presence of water vapours in the air increases the speed of sound slightly.

ELECTRICITY It is a form of energy which is made from the charges that make up the atom—negatively-charged electrons and positively-charged protons.

Electric Current Is the rate of flow of an electrical charge. It flows from positive to negative. Batteries supply direct current (DC) and the mains supply alternating current (AC). The ratio and TV require DC and this is obtained from AC mains by means of a rectifier which converts AC into DC.

Magnetic Effect of Electricity It is a phenomenon by which magnetic field is produced by an electric current. A linear current carrying conductor produces a circular magnetic field and a circular current (in a coil) produces a straight magnetic field at the centre of the circular coil.

Types of Electricity

Fuse Wire

Static current Is the electricity produced by friction or rubbing between two dissimilar objects. Depending on the nature of the objects, one acquire a positive charge and the other negative charge because of transfer of electrons. For example, when a glass rod is rubbed with a silk cloth, some electrons from the rod are transferred to the silk cloth. Thus by losing electrons the glass rod becomes positively charged and by gaining the number of electrons the silk acquires an equal negative charge. Similarly, by rubbing an ebonite comb on hair, a magnetic property is produced which can attract small pieces of paper.

Fuse wire is made of material having low melting point. If it is inserted in an electric circuit, it will not allow excess electricity to flow through it. When current exceeds its limit, the fuse wire gets heated, melts and breaks the circuit.

Current electricity Is the electricity which brings light and power to our homes. It is of two types: Alternating current which regularly reverses its direction around the circuit. Direct current travels without reversing its direction.

Heating Effect of Electricity When electricity is passed through thin metallic wires of high resistance, they become hot and glow. Metals like platinum or tungsten offer resistance to the passage of electric current. They become white, hot and glow and light is produced by the electric current. The filament of electric bulbs and heater elements is made of such metals which have high resistance.

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Household Wiring System In household wiring systems, the various electrical devices are connected in parallel across the power line, which consists of a pair of conductors, one "hot" and the other "neutral". An additional

"ground" wire is included for safety. The maximum permissible current in a circuit is determined by the size of the wires and the maximum temperature they can tolerate. Protection against excessive current and the resulting fire hazard is provided by fuses or circuit breaker.

MAGNETISM Magnets It is a substance which has the property of attracting pieces of iron, cobalt and nickel and when freely suspended it points towards north and south. The attracting power of a magnet appears to be concentrated at definite regions called poles. When a magnet is broken into pieces, each piece becomes an independent magnet. Two poles of the magnet have equal strength. Like poles repel and unlike poles attract each other. Types of Magnets Artificial magnets These magnets are created by artificial means. Iron ore (minor) which we know as magnetite (Iron oxide) has magnetic properties of attracting iron, cobalt and nickel. These are called Lodestones. Types of Magnetic Substances (i) Ferro-magnets These are special substances which are strongly magnetized by relatively weak magnetic fields. Iron, nickel and cobalt are the familiar examples of this class. (ii) Paramagnetic Substances These substances when placed in a strong magnetic field get magnetized in the same sense as the external field. Aluminium, chromium,

copper sulphate and liquid oxygen are familiar examples. (iii) Diamagnetic Substances These substances when placed in a magnetic field get weakly magnetized in a sense opposite to the applied field. Examples are bismuth, atimony, gold, water, alcohol and hydrogen.

Magnetic Induction When a piece of soft iron or steel is brought close to strong bar magnet, it is magnetized. When an electric current is passed through an insulated copper wire coil the steel or soft iron placed in the coil becomes a magnet. This is due to the magnetic field which is produced in the coil. The electric bell works on the above mentioned principle.

Magnetic Field of Earth Earth has its own magnetic field which is similar to the field produced by a huge magnet at its centre. Its north pole lying towards the geographical south pole and south pole towards the geographical north pole.

IMPORTANT LAWS OF PHYSICS Archimedes’ Principle States that when a body is partially or totally immersed in a fluid, it experiences an upward thrust equal to the weight of the fluid displaced by it, i.e., its apparent loss of weight is equal to the weight of liquid displaced. The principle was discovered in 3rd century BC by the Greek mathematician, Archimedes.

Avagadro’s Law Equal volumes of all gases under the same conditions of temperature and pressure contain equal number of molecules. This was an inspired guess in 1811 by the Italian Scientist Amedeos Avagadro that turned out to be correct.

Newton’s Laws (1642–1727) Law of Gravitation Objects attract each other with a force directly proportional to the product of the masses of the objects and inversely proportional to the square of the distance between them. Hence, for objects on or near the earth, the mass of the earth is very much greater than the object, and so the gravitational force between them makes objects fall towards the earth. That is why, lead and feather fall at the same rate in a vacuum.

Newton’s First Law of Motion A body continues in its state of rest, or of uniform motion in a straight line, except in so far as it be compelled by external impressed forces to change that state. It is also called Law of Inertia. Newton’s Second Law of Motion The rate of change of momentum is proportional to the impressed force, and takes place in the direction of the straight line in which the force acts. In other words, “Force is equal to mass multiplied by acceleration”. Newton’s Third Law of Motion To every action there is an equal and opposite reaction. This is the principle behind the recoil felt on pulling the trigger of a gun. Newton’s Law of Cooling The rate at which a body cools or loses its heat to its surroundings is proportional to the excess of mean temperature of the body over that of the surroundings, provided this temperature excess is not too large.

Coulomb’s Law (1738–1806) The force between the two electric charges reduces to a quarter of its former value when the distance between them is doubled. The SI unit of the electrical charge, coulomb, is named after Charles Augustin de Coulomb who established the law.

General Science ▌ 2.297

Stefan’s Law (1835–1883) The total energy radiated from a black body is equal to the fourth power of its absolute temperature.

Pascal’s Law (1623–1662) (i) When pressure is applied to a fluid, the pressure change is transmitted to every part of the fluid without loss. Hydraulic machines like the hydraulic press work on this principle. (ii) Atmospheric pressure decreases with increase in height. The SI unit of pressure, pascal, is named after Pascal who established this law.

Hooke’s Law (1635–1703) States that the extension of a spring is proportional to the tension stretching it: Doubling of the tension results in the doubling of the amount of stretch.

Lenz’s Law (1805–1865) States that when an electric current is induced by a change in magnetic field, the induced current is always in such a direction that its magnetic field opposes the change of field which causes the induction.

Dalton’s Law (1766–1844) • States that the total pressure of a mixture of gases (or vapours) is equal to the sum of the partial pressures of its components, i.e., the sum of the pressures that each component would exert if it were present alone and occupied the same volume as the mixture.

• In 1803, Dalton formulated his atomic theory stating that matter is made up of particles or atoms which are not subdivided during chemical change. It said that atoms of the same chemical element are similar and equal in weight, that atoms of different elements have different properties and different weights, and that chemical compounds are formed when atoms of different elements combine in simple proportions.

Faraday’s Laws (1791–1867) (i) Law of Electrolysis (a) The amount of chemical change during electrolysis is proportional to the charge passed. (b) The masses of substances liberated or deposited by the same quantity of electric charge are proportional to their chemical equivalents. (ii) Law of Electromagnetic Induction (a) an electromagnetic flux is induced in a conductor when the magnetic field surrounding it changes. (b) The magnitude of the electromagnetic flux is proportional to the rate of the change of the field. (c) The direction of the induced electromagnetic flux depends on the orientation of the field.

Ohm’s Law (1787–1854) States that the amount of current flows in an electric circuit is governed by the voltage of the battery or dynamo which powers it. In other words, the current through a conductor is directly proportional to the potential difference across the conductor and inversely proportional to its resistance. The SI unit of electrical resistance, the ohm is named after George Simon Ohm, who established this law.

PHENOMENA AND TERMS OF PHYSICS Absolute Temperature and Absolute Zero A temperature scale based on Charles Law of expansion of gases. It measures temperature from absolute zero (that is, –273.15°C, below which matter cannot be cooled) in kelvin, (the kelvin degrees being of the same magnitude as the degree centigrade). To convert an absolute temperature to centigrade simply add 273.15. Armature

The coils of an electric motor or dynamo.

Calorie It is the unit of heat, which is equal to the amount of heat required to raise the temperature of 1 gm of water by 1°C. Curie Point Temperature above which a strongly magnetizable or ferromagnetic material looses its magnetic properties and acquire paramagnetic properties. The curie point of iron is 870°C. Depression in Freezing Point The lowering of the temperature at which a liquid freezes as is done by adding common salt and saltpeter to ice to make ice cream.

Electrode The name given to the part by means of which electricity is led into or away from a gas or a liquid. The negative electrode is cathode and the positive electrode is known as anode. Electromagnetic Waves Waves that consist of varying electrical and magnetic quantities travelling along at the speed of light. Light, radio waves, X-rays, Gamma rays, etc., are covered under this category. Electromotive Force The electrical pressure developed by a cell, battery or generator which enables it to produce an electrical current in a circuit. It is measured in volts. Fluorescence The property of some substances to absorb light of one wavelength and emit light of a longer wavelength. Insulator Material such as glass, rubber, porcelain, plastic, etc., which have no free electrons and, therefore, will not allow electric current to pass through even when an e.m.f. is applied.

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General Knowledge

Mole The amount of a substance that contains as many particles as there are atoms in 12 g of the isotope carbon 12.

Momentum The quantity of motion of a moving body, a measure of the time it would take that body to stop when force is applied to it.

Pressure The force or weight per unit of area acting upon a surface. It is measured in newtons/m2 (S.I. units) or kg/m2.

Polarization Separation of positive and negative charges to different regions of a molecule, thus causing it to be an electric dipole.

Super-conductivity A property of some metals and alloys to lose electrical resistance below a certain temperature (usually within 20° of Absolute temperature). This means that an electrical current can flow in a loop of the metal indefinitely without generating heat or decreasing in strength.

Raman Effect The change in wavelength that occurs when light is scattered in a transparent medium.

Thermocouple A kind of thermometer made by joining a wire from both ends to another wire of a different material. If one junction is kept cool and the other heated, an electric current flows and can be used as a measure of the temperature of the hot junction. Flux Amount of a magnetic or electrical field or radiation passing through a given area. Heat Capacity Heat energy as Joules required to raise the temperature of a material by 1 kelvin. Induction Temporary magnetization or electrification of material that is placed in a strong magnetic or electrical field. Inertia A body’s resistance to motion, if it is at rest, or to a change in its motion if it is moving. Insulation Material that will not convey electrical current or flow of heat, such as glass, rubber, porcelain, plastics, etc.

Resistance Property of an electric circuit or its part that impedes the flow of current. Rheostat

Resistor whose resistance can be varied.

Spectrum The band of colours produced when light is passed through a prism and is split up into its component wavelengths. Supersonic Description of motion through air or some other medium at a speed greater than the speed of sound through that medium. Thermal

Connected with or produced by heat or temperature.

Thermal Radiation Emission of electromagnetic radiation by a body due to the thermal agitation of atoms and electrons in the body. Thermionic Effect

Emission of electrons from a hot body.

Torque The turning effect of a force on a body that can rotate freely. Vacuum

Space in which there is no matter.

Viscosity

Friction or drag within a liquid.

PHYSICS QUIZ Questions

Scientific Explanation

A man with a load on his head jumps from a high building. What will be the load experienced by him?

Zero, because the acceleration of his fall is equal to the acceleration due to gravity of the earth.

Why is spring made of steel and not copper?

The elasticity of steel is greater than that of copper.

Which is more elastic, rubber or steel?

Steel is more elastic for the same stress produced in rubber is more than that in steel, the elasticity is R 1/strain

Why is it easier to spray water to which soap is added?

Addition of soap decreases the surface extension of water. The energy for spray is directly proportional to surface tension.

A piece of chalk when immersed in water, emits bubbles. Why?

Chalk consists of pores forming capillaries. When it is immersed in water, the water begins to rise in the capillaries and air present there is expelled in the form of bubbles.

Why does a liquid remain hot or cold for a long time inside a thermoflask?

Because of the presence of air, which is poor conductor of heat, in between the double glass walls of a thermoflask.

Why does a ball bounce up on falling?

When a ball falls, it is temporarily deformed. Due to elasticity the ball tends to regain its original shape, for which it presses the ground and bounces up (Newton’s Third Law of Motion).

Why is standing in boats or double-decker buses not allowed, paticularly in the upper floor of buses?

On tilting the centre of gravity of the boat or bus is lowered and it is likely to overturn.

General Science ▌ 2.299

Questions

Scientific Explanation

Why is the boiling point of sea water more than that of pure water?

Sea water contains salts and other impurities with different boiling points which jointly raise its boiling point.

Why is it recommended to add salt in water while boiling grams?

By addition of salt the boiling point of water gets raised which helps in cooking.

Why is soft iron used as an electromagnet?

Because it remains magnetic only till the current passes through the coil and loses its magnetism when the current is switched off (principle of electric bells).

Why is the sky blue?

Violet and blue light have short waves and are scattered more than red light waves. While red light goes almost straight through the atmosphere, blue and violet are scattered by particles in the atmosphere Thus we see a blue sky.

Why does ink leak out of a partially filled pen when taken to a higher altitude?

As we go up the pressure and the density of air go on decreasing. A partially filled pen leaks when taken to a higher altitude because the pressure of air acting on the ink inside the tube of the pen is greater than the pressure of air outside.

On the moon will the weight of a man be less or more than his weight on the earth?

The gravity of the moon is one-sixth that of the earth; hence the weight of a person on the surface on the moon will be one-sixth of his actual weight on the earth

Why do some liquids burn while others do not?

A liquid burns if its molecules can combine with oxygen of the air with the production of heat. Hence, oil burns but water does not.

Oil and water do not mix. Why?

(i) Molecules of oil are bigger than that of water and therefore do not mix easily. (ii) Molecules of water are polar, i.e., they have opposite charges at two ends whereas oil molecules do not; as a consequence they tend to stay away from water molecules.

Why can we see ourselves in a mirror?

We see objects when light rays from them enter our eyes. As mirrors have a shiny surface, the light rays are reflected and come back to us and enter our eyes.

Why does a balloon filled with hydrogen rise in the air?

Its weight is less than the weight of air displaced by it. In balloons normally hydrogen is filled which is lighter than air.

Why does smoke curl up in the air?

Smoke contains hot gases which, being lighter in weight, follow a curved path because of the eddy currents that are set up in the air.

Why do we lean forward while climbing a hill?

In order to keep the vertical line passing through our centre of gravity always between our feet to attain balance or stability.

Why does an electric bulb explode when it is broken?

The bulb is a partial vacuum and as it breaks air rushes in causing a small explosion.

Why does a man fall forward when he jumps out of a running train?

He is in motion while in the train. When he jumps out, his feet come to rest while touching the ground but his upper portion which is still in motion takes him forward.

Why does an ordinary glass tumbler crack when very hot tea is poured in it?

When the inner layer of the tumbler gets heated it expands before the outer layer and an unequal or mild expansion of both layers causes the tumbler to crack.

Why is a compass used as an indicator of direction?

The magnetic needles of a compass under the influence of the earth’s magnetic field lie in a nearly north-south direction. Hence we can identify direction.

Why is water from a hand pump warm in winter and cold in summer?

In winter the outside temperature is lower than that of water flowing out of the pump and, therefore, it feels warm. Whereas in summer the outside temperature is higher than the water of the pump and, therefore, it feels cold.

Why is a rainbow seen after a shower?

After a shower the clouds containing water droplets act like a prism through which the white light is dispersed and produces a spectrum.

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General Knowledge

Questions

Scientific Explanation

Why does a swimming pool appear less deep than it actually is?

The rays of light coming from the bottom of the pool pass from a denser medium (water) to a rarer medium (air) and are refracted (bend away from the normal). When the rays return to the surface they form an image of the bottom of the pool at a point which is little above the real position.

Why does kerosene oil float on water?

Because the density of kerosene oil is less than that of water. For the same reason cream rises in milk and floats on the top.

Why is one’s breath visible in winter but not in summer?

In winter water vapour contained in the breath condenses into small droplets which become visible but in summer they are quickly evaporated and thus not seen.

Why doesn’t the electric filament in an electric bulb burn up?

First because it is made of tungsten which has a very high melting point (3,410°C) whereas the temperature of the filament required to glow is only 2700°C. Secondly oxygen is absent since the bulb is filled with an inert gas which does not help in burning.

Why does ice float on water?

The weight of the ice block is equal to the weight of the liquid displaced by the immersed portion of the ice.

A tumbler is filled to the brim and a piece of ice is placed in it. When the ice melts will the tumbler overflow?

Level of water remains unchanged because the volume of water produced by the melting ice is exactly the same as that of the piece of ice. Therefore, the water form the tumbler will not overflow.

Why does moisture gather outside a tumbler containing cold water?

The water vapour of the air condenses on cooling and appears as droplets of water.

Why is the water in an open pond cool even on a hot summer day?

As the water evaporates from the open surface of a pond, heat is taken away in the process, leaving the surface cool.

Why is it difficult to cook rice or potatoes at higher altitudes?

Atmospheric pressure at higher altitudes is low and boils water below 100°C; because the boiling point of water is directly proportional to the pressure on its surface.

Why is it difficult to breathe at higher altitudes?

Due to low air pressure at higher altitudes the quantity of air is less and so also that of oxygen.

Why are winter nights warmer in a cloudy weather and summer nights hotter in a cloudy weather than when the sky is clear?

Clouds being bad conductors of heat do not permit radiation of heat from land to escape into sky. As this heat remains in the atmosphere, the cloudy nights are warmer.

Why are cloudy days cooler?

Clouds do not permit the radiation of the sun to reach the earth.

Why is a metal tyre heated before it is fixed on wooden wheels?

On heating, the metal tyre expands by which its circumference also increases. This makes fixing the wheel easier and thereafter cooling down shrinks it; thus securing it tightly.

Why is it easier to swim in the sea than in a river?

The density of sea water is higher; hence the upthrust is more than that of river water.

Who will possibly learn swimming faster—a fat person or a thin person?

The fat person displaces more water which will help him float more freely compared to a thin person?

Why is a flash of lightening seen before thunder?

Because light travels faster than sound; it reaches the earth before the sound of thunder.

Why cannot petrol fire be extinguished by water?

Water, which is heavier than petrol, slips down permitting the petrol to rise to the surface and continue to burn. Besides, the existing temperature is so high that the water poured on the fire evaporates even before it can extinguish the fire. The latter is true if a small quantity of water is poured.

General Science ▌ 2.301

Questions

Scientific Explanation

Why does water remain cold in an earthen pot?

There are pores in the earthen pot which allow water to percolate to the outer surface. Here evaporation of water takes place thereby producing a cooling effect.

Why does an ordinary pendulum clock lose time in summer?

In summer, due to heat, the length of the pendulum increases. This in turn results in the increase of duration of each oscillation of the pendulum. Therefore the clock loses time in summer.

Why are mercury thermometers not used to measure very low temperatures?

The freezing point of mercury is only 39°C and it also has non-uniform expansion at low temperatures.

Why do we place a wet cloth on the forehead of a patient suffering from high temperature?

Due to body’s temperature water evaporating from the wet cloth produces cooling and brings the temperature down.

Why do we apply eau-de-cologne to a person having high temperature?

Eau-de-Cologne contains alcohol which is quickly evaporated and takes away much of the local heat from the body of the person.

When a needle is placed on a small piece of blotting paper which is placed on the surface of clean water, the blotting paper sinks after a few minutes but the needle floats. However in a soap solution the needle sinks. Why?

The surface tension of clean water being higher than that of soap solution, it can support the weight of needle due to its surface tension. By addition of soap, the surface tension of water reduces, thereby sinking the needle.

To prevent growth of mosquitoes, it is recommended to sprinkle oil in the ponds with stagnant water. Why?

Mosquitoes breed in stagnant water. The larvae of mosquitoes keep floating on the surface of water due to surface tension. However, when oil is sprinkled, the surface tension is lowered resulting in drowning and death of the larvae.

Why does oil rise on a cloth tape of an oil lamp?

The pores in the cloth tape suck oil due to the capillary action of the oil.

Why are ventilators in a room always made near the roof?

The hot air being lighter in weight tends to rise above and escape from the ventilators on the top.

Why are chimneys of factories using boilers high?

The gases produced in boilers are hot and being lighter in weight tend to go up. The chimneys exhaust these gases above in the atmosphere without polluting the lower layer of atmosphere.

How does ink get filled in a fountain pen?

When the rubber tube of a fountain pen immersed in ink is pressed, the air inside the tube comes out and when the pressure is released the ink rushes into fill the air space in the tube.

Why are air coolers less effective during the rainy season?

During the rainy season the atmospheric air is saturated with humidity. Therefore the process of evaporation of water from the moist pads of the cooler slows down, thereby not cooling the air blown out from the cooler.

Why does grass gather more dew in nights than metallic objects like stones?

Grass being a good radiator enables water vapour in the air to condense on it. Moreover, grass gives out water constantly (transpiration) which appears in the form of dew because the air near grass is saturated with water vapour and slows evaporation. Dew is formed on objects which are good radiators and bad conductors.

Why does a solid chunk of iron sink in water but float in mercury?

Because the density of iron is more than that of water but less than that of mercury.

Why do stars twinkle?

The light from a star reaches us after refraction as it passes through various layers of air. When the light passes through the earth’s atmosphere, it is made to flicker by the hot and cold ripples of air and it appears as if the stars are twinkling.

Why is cooking quicker in a pressure cooker?

As the pressure inside the cooker increases, the boiling point of water is raised, hence, the cooking process is quick.

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General Knowledge

CHEMISTRY ELEMENTS Everything in the universe—living or non-living—is made of a combination of a few basic substances called elements. An element is the simplest form of matter composed of atoms having identical number of protons in each nucleus. An atom is the smallest fragment of an element that can take part in a chemical action. The theory that all matter is made up of small particles (atoms) was put forward by John Dalton. He defined the atom as the smallest neutral particle of matter which may have independent existence. It consists of a central nucleus (made up of protons which are positively charged and neutrons which are neutral particles) surrounded by orbiting electrons. Atoms of different elements differ from one another in the number of protons and neutrons in the nucleus, and the number of electrons surrounding the nucleus. The number of electrons is, however, always equal to the number of protons which is known as the elements’ atomic number. There are around 118 known elements, out of which 98 are naturally occurring and 20 are man-made elements, created by nuclear bombardment. The man-made elements are known as transuranics. Neptunium (Element 93) was the first such element discovered in 1940. Lightest (simplest) element (naturally occurring)

Hydrogen (Element 1)

Most complex element (naturally occurring)

Uranium (Element 92)

Hydrogen has atomic number 1, with 1 proton and 1 electron. Uranium, the heaviest naturally occurring element has atomic number 92, having 92 protons, 92 electrons and 146 neutrons. All elements heavier than uranium are man-made and are produced in nuclear reactors or accelerators or isolated from debris of atomic explosions (transuranics). The chemical elements can be arranged in order of increasing atomic number in such a way that elements with similar properties appear together. Such an arrangement is called a periodic table. The Russian scientist Dimitri Mendeleev devised the periodic table in 1869.

Types of Elements Elements are classified in two main classes, they are, (i) Metals like lead, gold, mercury, etc. (ii) Non-metals like chlorine, bromine, carbon, etc. Some elements behave chemically both as metals and nonmetals, and are called metaloids, for example, boron, silicon, germanium and antimony. There are also some elements which are neither metals nor non-metals. These elements are called noble gases and are present in the atmosphere as helium, argon, neon, krypton, radon and xenon.

New Elements These are four transuranic elements which were discovered in the seventies: Name of Element

Atomic No. Symbol Year

Country

Rutherfordium

104

Rf

1969

Former USSR/USA

Dubnium

105

Db

1970

Germany

Seaborgium

106

Sg

1974

USA

Bohrium

107

Bh

1976

Former USSR

Hassium

108

Hs

1992

German

Meitnerium

109

Mt

1997

German

Darmstadtium

110

Ds

2012

German

Roentgenium

111

Rg

2012

German

Copernicius

112

Cr

2012

German

Element 104 In 1969, the Russians at Dubna synthesized element 104 and named it Kurchatovium after the physicist I. V. Kurchatov. Simultaneously, the Americans at Berkeley synthesized similar elements by a differ ent process and challenged the Russian findings. They named this element Rutherfordium in honour of Lord Rutherford. However, it has now been named Unnilquadium. Element 113, 114, 115, 116, 117 and 118 —Element 113 —Element 114 —Element 115 —Element 116 —Element 117 —Element 118

Ununtrium (2004) Flerovium (2009) Ununpentium (2011) Livermorium (2011) Ununseptium (2012) Ununoctium (2012)

These elements are confirmed as discovered by IUPUC hut a formal introduction of these element is pending till now. Commonest element (lithosphere)

Oxygen

Commonest element (atmosphere)

Nitrogen

Commonest element (universe)

Hydrogen

Rarest element

Astatine

Lightest element (metal)

Lithium

Heaviest element (gas)

Radon

Lightest element (gas)

Hydrogen

A list of common elements, their chemical symbol, atomic number and relative weight is given below:

General Science ▌ 2.303 Table: List of Elements Elements and Symbols

Atomic Number

Atomic Weight

Discoverer

Year

Actinium

(Ac)

89

227.0

A. Debierne

(1899)

Aluminium

(Al)

13

27.0

F. Wholer

(1827)

Americium

(Am)

95

(243)

G. Seaborg and others

(1944)

Antimony

(Sb)

51

121.8

B. Valentine

(1604)

Argon

(A)

18

39.9

W. Ramsay and J. Rayleight

(1894)

Arsenic

(As)

33

74.9

A. Magnus

(1250)

Astatine

(At)

85

(210)

E. Segre and others

(1940)

Barium

(Ba)

56

137.3

H. Davy

(1808)

Berkelium

(Bk)

97

249

S. Thompson

(1949)

Beryllium

(Be)

4

9.0

N. Vauquelin

(1798)

Bismuth

(Bi)

83

209.0

C. Geoffrey the Younger

(1953)

Boron

(B)

5

10.8

H. Davy

(1808)

Bromine

(Br)

35

79.9

A. Balard

(1826)

Cadmium

(Cd)

48

112.4

F. Stromeyer

(1817)

Calcium

(Ca)

20

40.1

H. Davy

(1808)

Californium

(Cf)

98

(251)

S. Thompson

(1950)

Carbon

(C)

6

12.0

...

Cerium

(Ce)

58

140.1

J. Berzelius and W. Hislinger

(1803)

Cesium

(Cs)

55

132.9

R. Bunsen and G. Krichhoff

(1860)

Chlorine

(Cl)

17

35.5

K. Scheele

(1744)

Chromium

(Cr)

24

52.0

N. Vauquelin

(1797)

Cobalt

(Co)

27

58.9

G. Brandt

Copper

(Cu)

29

63.5

...

Curium

(Cm)

96

(248)

G. Seaborg

(1944)

Dysprosium

(Dy)

66

162.5

L. de Boisbaudran

(1886)

Einsteinium

(E)

99

(254)

A. Ghiorso

(1953)

Erbium

(Er)

68

167.3

C. Mosander

(1839)

(c. 1735)

Europium

(Eu)

63

152.0

E. Demarcay

(1896)

Fermium

(Fm)

100

(253)

A. Ghiorso

(1952)

Fluorine

(F)

9

19.0

H. Moissan

(1886)

Francium

(Fr)

87

(223)

M. Perey

(1939)

Gadolinium

(Gd)

64

157.3

J. C. de Marignac

(1880)

Gallium

(Ga)

31

69.7

L. de Boisbaudran

(1875)

Germanium

(Ge)

32

72.6

C. Winkler

(1886)

Gold

(Au)

79

197.0

...

Hafnium

(Hf)

72

178.5

D. Coster and G. De Hevesy

(1923)

Helium

(He)

2

4.0

J. C. P. Janssen and N. Lockyer

(1868)

Holmium

(Ho)

67

164.9

J. Soret and M. Delafontai

(1878)

Hydrogen

(H)

1

1.0

H. Cavendish

(1766)

Indium

(In)

49

114.8

F. Reich and T. Richter

(1863)

Iodine

(I)

53

126.9

B. Courtois

(1811)

2.304

General Knowledge

Elements and Symbols

Atomic Number

Atomic Weight

Discoverer

Year

(Ir)

77

192.2

S. Tennant

(1803)

Iron

(Fe)

26

55.9

...

Krypton

(Kr)

36

83.8

W. Ramsay and M.Travers

(1898)

Lanthanum

(La)

57

138.9

C. Mosanter

(1839)

Lawrencium

(Lr)

103

257.0

A. Ghiorso

(1961)

Lead

(Pb)

82

207.2

...

Lithium

(Li)

3

6.9

Lutetium

(Lu)

71

Magnesium

(Mg)

12

Manganese

(Mn)

Mendelevium

(Mv)

Mercury

(Hg)

Molybdenum

(Mo)

Neodymium Neon

Iridium

A. Arfvedson

(1817)

175.0

G. Urbain

(1907)

24.3

Recognized by J. Block

(1755)

25

54.9

Recognized by K. Scheele

(1774)

101

(256)

A. Ghiorso and others

(1955)

80

200.6

...

42

95.9

K. Scheele

(1778)

(Nd)

60

144.2

C. Von Welsbach

(1885)

(Ne)

10

20.2

W. Ramsay and M. Travers

(1898)

Neptunium

(Np)

93

(237)

E. McMillan and P. Abelson

(1940)

Nickel

(Ni)

28

58.7

A. Cronstedt

(1751)

Niobium

(Nb)

41

92.9

C. Hatchett

(1801)

(Columbium) Nitrogen

(N)

7

14.0

D. Rutherford

(1772)

Nobelium

(No)

102

(254)

Fields and others

(1951)

Osmium

(Os)

76

190.2

S. Tennant

(1803)

Oxygen

(O)

8

16.0

J. Priestly

(1774)

Palladium

(Pd)

46

106.4

W. Wollasten

(1803)

Phosphorus

(P)

15

31.0

H. Brand

(1669)

Platinum

(Pt)

78

195.1

D. de Ulloa

(1735)

Plutonium

(Pu)

94

(242)

G. Seaborg

(1940)

Polonium

(Po)

84

210.0

P. and M. Curie

(1898)

Potassium

(K)

19

39.1

H. Davy

(1807)

Praseodymium

(Pr)

59

140.9

C. Von Welsbach

(1885)

Promethium

(Pm)

61

(147)

J. Marnsky

(1947)

Protactinium

(Pa)

91

231.0

F. Soddy

(1917)

Radium

(Ra)

88

226.1

P. & M. Curie

(1898)

Radon

(Rn)

86

222.0

Rutherford (thoron isotope)

(1899)

E. Dom (radon isotope)

(1900)

Rhenium

(Re)

75

186.2

E. Noddack and others

(1925)

Rhodium

(Rh)

45

102.9

W. Wollaston

(1803)

Rubidium

(Rb)

37

85.5

R. Bunsen and G. Kirchhoff

(1861)

Ruthenium

(Ru)

44

101.1

K. Claus (or Klaus)

(1844 )

Samarium

(Sm)

62

150.4

L. de Boisbaudran

(1879)

Scandium

(Sc)

24

45.0

L. Nilson

(1879)

Selenium

(Se)

34

79.0

J. Berzelium

(1817)

Silicon

(Si)

14

28.1

J. Berzelium

(1824)

General Science ▌ 2.305

Elements and Symbols

Atomic Number

Atomic Weight

47

107.9

...

(Na)

11

23.0

H. Davy

(1807)

(Sr)

38

87.6

H. Davy

(1808)

Sulphur

(S)

16

32.1

...

Tantalum

(Ta)

73

181.0

A. Ekeberg

(1802)

Technetium

(Tc)

43

(99)

E. Segre and C. Perrier

(1937)

Tellurium

(Te)

52

127.6

M. von Reichenstein

(1782)

Terbium

(Tb)

65

158.9

C. Mosander

(1843)

Thallium

(Tl)

81

204.4

W. Crookes

(1861)

Thorium

(Th)

90

232.0

J. Berzelium

(1828)

Thulium

(Tm)

69

168.9

P. Cleve

(1879)

Silver

(Ag)

Sodium Strontium

Discoverer

Year

Tin

(Sn)

50

118.7

...

Titanium

(Ti)

22

47.9

W. George

(1791)

Tungsten (Wolfram)

(W)

74

183.9

G. and F. d Ethuyar

(1783)

Uranium

(U)

92

238.0

E. M. Peligot

(1841)

Vanadium

(V)

23

51.0

A. Delrio

(1801)

Xenon

(Xe)

54

131.3

W. Ramsay and M. Travers

(1898)

Ytterbium

(Yb)

70

173.0

C. Marignac

(1878)

Yttrium

(Y)

39

88.9

J. Gabolin

(1794)

Zinc

(Zn)

30

65.4

...

Zirconium

(Zr)

40

91.2

M. Klaproth

(1789)

METALS There are two types of elements—metals and non-metals. About 80 per cent of known elements are metals. Elements which are hard, ductile, brittle, and malleable; possess lustre and conduct heat and electricity are termed metals. All metals are solids, except mercury, and gallium which are liquids at room temperature. Metals usually have high melting and boiling points.

Chemical Nature of Metals Usually metals have the tendency to lose electrons and while reacting with acids, usually replace hydrogen in dilute nonoxidizing acids like hydrochloric acid (HCl) and sulphuric acid (H2SO4). Exceptions are copper, silver and gold. Chlorides of metals are true salts and oxides of metals are usually basic. Hydrides of metals are ionic, unstable and reactive. Although all the metals are reactive, that is, they are acted upon by common reagents like oxygen (air), hydrogen, halogens, sulphur, water, acids, etc., the extent of reaction is different in the case of each metal. Only gold, platinum and silver, are not affected by air and water under ordinary conditions, and are known as free metals. Various compounds of metal called minerals are found in nature and can be mined. The mineral from which metal can be

extracted economically is called ore and the process of extraction of metals from their ores is called metallurgy, which involves: Calcination The process of heating the concentrated ore in the absence of air. Roasting

The process of heating the ore in excess air.

Smelting The process by which roasted ore is mixed with coke and heated in a furnace to obtain free metal.

Steel and Iron Steel is a form of iron. To make steel from iron, the carbon content is bought down from 5% to 0.5–1.5%. Heat Treatment of Steel Quenching If steel is heated to bright redness and then suddenly cooled in water or oil it becomes extraordinarily hard and brittle. Tempering By controlled heating (250–325°C) of quenched steel, its brittleness can be removed without affecting its hardness.

2.306

General Knowledge

Annealing Steel is heated to a temperature well below red hot and then cooled turns it soft and the process is called annealing.

Non-metals and Their Chemical Symbols Name

Rusting of Iron Majority of metals occur in nature in the combined form and are extracted from their ores. When these metals are exposed to atmospheric conditions they have a tendency to return to their original form. This change is called corrosion of metals and in the case of iron it is known as rusting. Rusting consists of the formation of hydrated ferric oxide. For rusting water and oxygen are essential—in the absence of water or electrolyte rusting does not occur. The process involves addition of hydrogen and oxygen elements and it is found that mass of an iron rod increases by rusting. Rusting is prevented by coating the surface of iron with metals or non-metals, or by alloying. The coating of another metal is known as electroplating or hot dipping. In electroplating chromium or nickel is used. When a coat of zinc is applied on an iron surface by the hot-dipping process, it is known as galvanizing.

Symbol

Arsenic

As

Astatine

At

Bromine

Br

Carbon

C

Chlorine

Cl

Fluorine

F

Hydrogen

H

Iodine

I

Nitrogen

N

Oxygen

O

Phosphorus

P

Selenium

Se

Non-metals

Sulphur

S

Non-metals are electro-negative elements which have a tendency to gain one or more electrons to form negative ions called anions. All non-metals generally exist as powders or gases, except bromine which is liquid under normal conditions. Non-metals are non-lustrous and are bad conductors of heat and electricity. They cannot be hammered into sheets or drawn into wires like metals. The melting point of non-metals is lower than that of metals.

Tellurium

Te

Alloys Alloys are homogeneous mixtures of two or more metals and non-metals and have more commercial utility than some of the constituent elements. The following table lists some of the commercially important alloys.

Table Principal Metal

Alloy

Composition (%)

Commercial Utility

Aluminium

Magnalium

Al = 85–99

Used in making balance

Mg = 1–15

beams and light instruments

Al = 95

Used in the production of airships,

Cu = 4

pressure cookers

Duralumin

Mg = 0.5 Mn = 0.5 Copper

Aluminium

Cu = 88–90

In the manufacture of utensils, decorative

Bronze

Al = 10–12

articles like photo frames, coins and jewellery

Brass

Cu = 60–80

Utensils, condenser sheets and cartridges

Zn = 20–40 Bronze

Cu = 75–90

Utensils, coins and statues

Sn = 10–25 Gun metal

Cu = 87 Sn = 10

Guns, gears, castings

General Science ▌ 2.307

Principal Metal

Alloy

Composition (%)

Commercial Utility

Zn = 3

Lead

Coinage

Cu = 75

alloy

Ni = 25

Type metal

Pb = 82

In the making of coins For manufacturing printing types

Sb = 15 Sn = 3 Solder

Pb = 50

For soldering

Sn = 50 Iron

Stainless

Fe = 73

In the manufacture of utensils,

steel

C=1

automobile parts,

Cr = 18

cycle parts, cutlery

Ni = 8 Invar

Duriron

Fe = 63

In the manufacture of meter

C=1

scales measuring tapes,

Ni = 36

pendulum rods

Fe = 84.7

Laboratory plumbing

C = 0.8 Si = 14.5 Tungsten steel

Fe = 75–81

For making high speed cutting tools

W = 14–20 Cr = 4 C=1

Minerals Minerals are naturally occurring chemical compounds of fixed composition and characteristic, physical form and properties. A few minerals consist of only one element. Example: graphite and diamond (both forms of carbon), sulphur and gold.

Most minerals, however, are a combination of two or more elements as in halite or rock salt (NaCl).The most common group of minerals are: Silicates, oxides, sulphides, halides, carbonate. Minerals are of two types, namely, metallic or ore, and non-metallic, for example, carbon, sulphur, etc. (See table)

Table Common Minerals Name of the Mineral

Composition

Commercial Utility

Albite

Sodium aluminium silicate

Glass, ceramics

Anhydrite

Calcium sulphate

Cement, fertilizers, chemicals

Anorthite

Calcium aluminium silicate

Glass, ceramics

Apatite

Calcium phosphate

Phosphate,

flour-phosphate or

Fertilizers, gemstones

chlorophosphate Aragonite

Calcium carbonate

Azurite

Copper carbonate

Source of copper

2.308

General Knowledge

Name of the Mineral

Composition

Commercial Utility

Bauxite

Hydrated aluminium oxide

Source of aluminium

Calcite

Calcium carbonate

Cement, plaster, paint glass, fertilizers

Calamine

Zinc carbonate

Source of zinc

Cassiterite

Tin oxide or Tinstone

Source of tin

Cerussite

Lead carbonate

Source of lead

Chalcocite

Copper sulphite

Source of copper

Cinnabar

Mercuric sulphite

Source of mercury

Dolomite

Calcium magnesium carbonate

Cement and building stone (marble)

Fluorite

Calcium fluoride

Glass, enamel

Galena

Lead sulphide

Source of lead (main ore)

Gypsum

Hydrated calcium sulphate

Plaster of Paris, glass, fertilizer

Halite

Sodium chloride

Source of common salt

Haematite

Ferric oxide

Important source of iron

Kaolinite

Hydrated aluminium silicate

Porcelain tiles, filters, earthenware

Malachite

Copper carbonate

Source of copper

Mircocline

Potassium aluminium silicate

Glass, ceramics

Pyrite

Iron sulphite

Source of sulphur

Quartz

Silica

Abrasives, cement, glass, electronic equipment

Rutile

Titanium oxide

Source of titanium

Talc

Magnesium silicate

Talcum powder, asbestos

Topaz

Hydrous aluminium flourosilicate

Gemstones

CHEMICAL COMPOUNDS The atoms of an element, the smallest component—seldom exist singly. They usually join up with atoms of other elements to form a molecule of a compound. For example, two atoms of oxygen combine to form a molecule of oxygen written as O2, O being the symbol of oxygen and 2 the number of atoms combined. In the formulation of a compound, 2 atoms of iron (Fe), for example, may combine with 3 atoms of oxygen to form a molecule of ferric oxide (Fe2O3). It has been estimated that there are 4,040,000 described compounds of which 63,000 are in common use.

Chemical Reaction and Chemical Change Chemical change happens everywhere all the time—for example when coal burns, when iron rusts, when beer ferments, when concrete and cement set or when food is digested to give energy, etc. When coal burns, carbon and hydrogen compounds within it combine with the oxygen of the air to form carbon dioxide (CO2) and water vapours. Thus, original constituent chemicals, called

reactants, are converted into different substances, called products, which have different properties.

Characteristics of Chemical Change 1. As seen above, in a chemical change the resultant product has different properties as against that of the reactants. 2. Conservation of Mass For matter can neither be created nor destroyed during a chemical reaction. In the given example, the mass of coal and oxygen which combine during combustion is exactly equal to the mass of carbon dioxide, water vapour and ash produced. 3. When substances are formed in different ways, that is, by different chemical reactions, it always has the same composition. In carbon dioxide (CO2), for example, no matter how it is formed, carbon (C) and oxygen (O) are always in the ratio of 3:8 by mass.

General Science ▌ 2.309 4. In a chemical reaction, energy is given out or absorbed. For example: When coal burns in air, energy in the form of heat and light is given out. On the other hand, when carbon and sulphur are made to combine, heat is absorbed in this process. Chemical Equation Chemical change can be represented by an equation, e.g., the combustion of carbon(C)—in oxygen (O) to form carbon dioxide (CO2) can be shown as: C + O2 → CO2 The subscripts show that there are two atoms in an oxygen molecule. When hydrogen (H) and chlorine (Cl) react to form hydrogen chloride, the equation will be: H2 + Cl2 → 2 HCl Note that two must be added before HCl on the right to balance the equation. The equation shows that one molecule containing two hydrogen atoms plus one molecule containing two chloride atoms react to form two molecules of hydrogen chloride.

Acid-base Reaction One of the most common chemical reactions is double decomposition, a process in which two compounds react together to form two new compounds. For example: Magnesium sulphate (MgSO4) reacts with the solution of caustic soda (NaOH) to form sodium sulphate (Na2 SO4) and antacid magnesium hydroxide (Mg(OH) 2) MgSO4 + 2NaOH → Na2SO4 + Mg(OH)2 Oxidation and Reduction Another common chemical reaction is oxidation. Originally it meant combination of a substance with oxygen. However, now the term covers all analogous reactions in which substances combine with other elements and lose electrons in the process. Oxidation is always accompanied simultaneously with reduction in which electrons are gained. For example: Action of hydrogen (H2) with copper oxide (CuO) CuO + H2 → Cu + H2O The oxide (CuO) is reduced to copper (Cu), the copper gains electrons in the reduction. Chemical reactions may take place either slowly—for example, rusting—or quickly as in an explosion. Rate of chemical reaction can he greatly increased by the presence of a catalyst—a substance which influences the reaction but does not change with it.

Air Is a colourless and tasteless gaseous mixture of nitrogen (78%), oxygen (21%) with lesser amounts (say traces) of argon. carbon dioxide, neon, helium, ozone and other gases. Air also contains water vapour and pollutants enveloping the earth. Being a mixture (not compound) its composition varies from one place to another. Its constituents can be separated and it can be

prepared by mixing oxygen and nitrogen. Air is a bad conductor of heat. Of its constituents, oxygen helps in burning of substances and respiration, and nitrogen dilutes the action of oxygen. Carbon dioxide is added to the atmosphere through burning and also through respiration and water vapour is formed during evaporation from the sea, rivers, ponds, etc. Water Vapour in the Air Air contains about 0.4% of water vapour. If we place a glass containing ice cubes in the open air, the outer surface of the glass gets covered with water droplets. This is due to the condensation of water vapour, from the atmospheric air on the cooler surface of the glass. Carbon Dioxide Air contains about 0.03% carbon dioxide. If we place lime water in the open air, it turns milky due to absorption of carbon dioxide from the air.

Water Water was shown by Cavendish, in the eighteenth century, to be a chemical compound. It consists of hydrogen and oxygen in the ratio of 2 : 1 by volume and 1 : 8 by mass. Hence, when an electric current is passed through acidified or alkaline water, for every one volume of oxygen two volumes of hydrogen evolve. Water can be prepared by combining oxygen and hydrogen with the help of an electric current where, for every one part of hydrogen, 8 parts of oxygen are required. Boiling point of water is 100°C and freezing point is 0°C.

Hard and Soft Water Hard Water

Does not produce lather with soap.

Soft Water Produces lather with soap very easily. Hardness of water is of two types: (i) Temporary hardness is due to the bicarbonates of calcium and magnesium. It can be removed by (a) boiling or (b) addition of lime. (ii) Permanent hardness is due to the sulphates and chlorides of calcium and magnesium. It can be removed by (a) addition of washing soda or (b) distillation.

Rain Water is the purest form of water since it is condensed water vapour of the air. It is soft water because it does not contain salts like bicarbonates, sulphates and chlorides of calcium and magnesium. River Water By flowing over the earth’s surface carries with it soluble minerals of earth and becomes hard water and also contains several pollutants.

Important Gases Oxygen Is a colourless, odourless and tasteless gas, sparingly soluble in water and slightly heavier than air. It does not burn itself but helps in burning of other substances. It is found in

2.310

General Knowledge

abundance in the earth both in the free state and combined state with other elements. Oxygen can be prepared in a laboratory by heating potassium chlorate and manganese dioxide together. It can also be obtained in small quantities by heating oxides or salts rich in oxygen. Oxygen can be separated from the air by passing an electric current through water. It is essential for plant and animal respiration and for nearly all kinds of combustion. Atomic no.—8, Relative atomic mass—15.999, Melting point –218.4°C, Boiling point–183.0°C, Density at 0°C = 1.329 kg/m3, Valency: 2. Hydrogen Is a colourless, highly flammable gaseous element, the lightest of all substances known and in most abundant supply in the universe. In the free state it occurs in volcanic gases. Hydrogen burns with a pale blue flame but does not help combustion and is slightly soluble in water. It is used in the manufacture of vanaspati ghee, alcohol and ammonia. Hydrogen can he obtained from water, acids and alkalies. In a laboratory it is prepared by the action of dilute sulphuric acid on commercial zinc.

Atomic No. 1. Relative atomic mass: 1.008, Melting point: –259.14°C. Boiling point: –252.5°C, Density: 0.08988 kg/m3, Valency 1. Nitrogen A colourless, tasteless and odourless gas constituting nearly four-fifths of the air by volume. It is an almost inert diatomic gas, neither combustible nor helping combustion. Slightly soluble in water. In a laboratory, it can be prepared by heating ammonium nitrite. On a large scale it can be obtained from air. Air is liquefied first and then evaporated, nitrogen evaporates first, leaving oxygen. Nitrogen is used to manufacture nitric acid, ammonia and fertilizers. Atomic No. 7, Relative atomic mass: 14,007, Melting point: –209.86°C, Boiling point: –195°C, Valencies 3 and 5. Carbon Dioxide A colourless, odourless, incombustible gas formed during respiration, combustion and organic decomposition and is heavier than air. Carbon dioxide is acidic and turns lime water milky. It is used in food refrigeration, carbonated beverages, fire extinguishers, etc. Carbon dioxide is prepared by the action of dilute acids on carbonates or by fermentation of sugar. In a laboratory it can be prepared by treating marble pieces with dilute hydrochloric acid.

Table: Useful Gases Name Acetylene, ethylene Ammonia

Use To produce a hot flame for welding/cutting Fertilizer, synthetic fibres, refrigeration

Butane CS gas Cyanogen Ether Ethylene Helium Krypton Laughing gas (Nitrous oxide) Methane Neon Propane Radon Xenon

Cigarette, lighters/domestic fuel Tear gas used in riots Welding, chemical weapons, rocket propellant Anaesthetic and industrial processes Plastics Fluorescent tubes, laser, balloons Flourescent tubes, high speed photography Mild anaesthetic To make chloroform Illuminated signs Fuel and refrigerant Radiotherapy, atomic research Flash lamps and lasers

Table: Natural Acids Name Acetic acid Amino acid Ascorbic acid Citric acid Hydrochloric acid Lactic acid Malic acid Tannic acid Uric acid

Source Vinegar Proteins Vitamin C Lemon/citrus fruit Digestive juices Milk Unripe apples/fruits Tea Urine

General Science ▌ 2.311

INDUSTRIAL CHEMISTRY Soaps

Cement

Soaps are the alkali salts of higher fatty acids. Washing soap is, sodium salt of strearic acid and toilet soap is potassium salt of oleic acid. These soaps contain a charged—COONa+ end and a hydrocarbon end C nH 2n+1. The charged end has a tendency to interact with polar substances like water and fibre, while the hydrocarbon part interacts with non-polar material like oil.

Materials required to manufacture cement are calcium carbonate (limestone, chalk, etc.) aluminium silicate (clay) and a small qantity of gypsum (CaSO4.2H2O). The best cement is Portland cement, the average composition of which is:

Cleansing Action of Soaps Is based on a surface phenomenon. Oil coats the surface of objects (for example, fibre) involving weak interactions between the polar fibres and the non-polar oil. When clothes are soaked with water containing soap then the polar end of the soap orients towards oil. The stronger interaction between the charged end of soap and water overweighs the weak interaction between the fibre and oil. Thus the interfacial contact between the oil (dirt etc.) and the fibre (or any other object) is reduced and oil separates in the form of droplets.

Glass Glass is a mixture of an alkali silicate with the silicate of a base, that is, silica, sodium silicate (Na2SiO3) and calcium or lead silicate. The selected materials, that is, sand (silica), soda ash (sodium carbonate), and lime stone (calcium carbonate) are mixed in the required proportion and broken pieces of previously made glass known as ‘cullet’ are added. These help in easier fusion of the mixture. The mixture is heated up to a temperature of 1400°C in a rotary furnace. When the mass is completely mixed and melted, the glass is made in various shapes by blowing and moulding.

CaO

63%

Fe2O3

3%

MgO

1.5%

Akali

0.5%

SiO2

21%

SO3

1.5%

A12O3

7%

The raw materials are first crushed and mixed together and ground to a fine powder. The powder is then fed in a kiln (Temperature: 1890 K). At this temperature calcium oxide (from limestone) combines the aluminium silicate to form calcium silicate and aluminate. The resultant mixture is mixed with 2–3% gypsum and ground to form cement.

Coal Coal originates from the remains of trees, bushes, ferns, mosses and other forms of plant life that flourished in swamps and marshes million of years ago. Important products are derived from coal by a process called pyrolysis—heating of coal in the absence of air which produces coke (residue) and volatile matter such as coal gas and a liquid known as coal tar.

ORGANIC CHEMISTRY Carbon Compounds

Organic and Inorganic Compounds

Until 1828, scientists believed that organic compounds occur only in living organisms, things that were or had been alive. Therefore, study of those compounds became known as organic chemistry. This was based on the so-called vitalist theory, that is, to produce organic compounds, vital energy is required. However, the vital force theory was disapproved when in 1828, the German chemist, Friedrich Wohler, prepared an organic compound Urea in his laboratory by evaporating a solution of inorganic compound ammonium cyanide:

Most organic compounds can be burned while most inorganic compounds cannot. Most organic compounds are gases, volatile liquids and solids, with relatively low melting points; and most inorganic compounds are solids with high melting points. While most organic compounds are insoluble in water, a great majority of inorganic compounds are soluble.

NH4CNO

Ammonium cyanide (inorganic compound)

CO(NH2)2 Urea (organic compound)

Therefore, now organic chemistry is the study of carbon compounds.

Carbon Found in abundance, carbon ranks twelfth among the elements in the earth’s crust but in importance it ranks first. It is a unique element which readily combines with itself to form large molecules of carbon atoms linked in long chains (rings). In all there are more than a million such combinations.

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General Knowledge

Different forms of Carbon Different forms of carbon are: (a) diamond (c) charcoal (e) coke (g) coal

(b) (d) (f) (h)

graphite lamp black gas carbon animal charcoal

Allotropic forms of Carbon When a substance exists in different crystalline modifications the phenomenon is called allotropy, and different distinct forms of the substance are called allotropes. Carbon shows allotropy because it exists in different forms. There are two allotropic forms of carbon, namely, (i) Diamond and (ii) Graphite. Coke, charcoal, lamp black, etc., were thought to be amorphous forms (without definite shape) of carbon but it is now known that all the amorphous carbons contain microcrystals of graphite. Though these allotropes of carbon have different crystal structures and different physical properties, their chemical symbol is the same and show similar chemical properties. Both diamond and graphite have symbol ‘C’. Both give off carbon dioxide when strongly heated in the presence of oxygen: C (diamond) + O2 (gas) → CO2 (gas) C (graphite) + O2 (gas) → CO2 (gas)

Since 1955, diamonds are also prepared synthetically from carbon compounds at high temperature and very high pressure. The transparent form of the diamond is used as gems while dark coloured diamonds are used for making cutting-tools. The Koh-i-Noor is the world’s most precious and famous diamond mined in India but was taken away by the British. The Cullinan found in 1905 in South Africa is the largest diamond in the world weighing 570 grams and 2.850 carats.

Graphite The name graphite is derived from the Greek word graphein that means ‘to write’. This suggests that this substance has been used to make lead pencils since ancient times. Graphite is dark grey, an opaque solid with a soapy touch, and has a metallic shine. It is a good conductor of electricity and heat. It does not undergo any change when mixed with acids or alkalies. However, when heated with nitric acid graphite acid is formed. Graphite is used as a lubricant, in paints, for making electrodes and lead pencils. Pure graphite is manufactured by heating coke in an electric furnace to a temperature of about 2,500°C in the absence of air.

Petroleum

Diamond Is the hardest substance found in natural form. Its name is derived from the Greek word ‘adamas’ which means invincible or adamant. It is the purest form of carbon. It does not allow heat or electricity to pass through. It is inert as it resists action of chemicals but gives out CO2 when strongly burnt in air. It is insoluble in all solvents.

It is a mixture of hydrocarbons believed to have originated from bacterial decomposition of animal and vegetable fats under high pressure and atmospheric temperature. It is converted into a variety of products by Fractional Distillation based on the principle that lower hydrocarbons boil at a lower temperature than the higher ones.

Table B.3.6 Products Obtained from Petroleum Fraction

Distillation Temperature Range (K)

1. Gaseous hydrocarbons

Approx. %

Uses

< 293

3

2. Light distillates— Petroleum ether Petrol or gasoline Kerosene oil

293–363 343–473 450–560

3 32 18

As solvent in varnish, dry cleaning Motor fuel Fuel and illuminant

3. Intermediate distillates— Gas oil, diesel or heavy oil

525–673

20

Fuel

> 673

—

> 673

—

Used as a Lubricant, in toilet goods ointments Candles, boot polish, etc.

4. Heavy distillates— Lubricating oils, gaseous and petroleum jelly Paraffin (wax) 5. Residue— Asphalt, petroleum coke (Bitumen and coke)

Residue

LPG Domestic gas, also known as LPG or bottled gas or liquefied petroleum gas is a by-product of petroleum refining and also obtained from natural gas. It is a mixture of hydrocarbons such as

» 40

As fuel and for gasoline and rubber manufacture

Fuel for making electrodes

propane, butane and pentane. These gases can be liquefied under moderate pressure at normal temperature. Because of low boiling point (–44°C) these gases are stored under pressure to keep in a liquid state in gas cylinders. Therefore cooking gas cylinders contain the mixture of these gases in liquid form.

General Science ▌ 2.313

Synthetic Rubber Produced by polymerization of certain hydrocarbons, namely, (i) Neoprene—a polymer of chloroprene (ii) BUNA-S—a polymer of styrene and butadiene (iii) BUNA-N—a polymer of butadiene and acrylonitrile Rubber is made hard by vulcanization, a process of heating rubber with sulphur.

Synthetic Fibre Nylon was the first synthetic fibre made as a result of research begun in 1928. It is a polyamide made by polymerizing adipic acid and hexamethylene diamine. Terylene Discovered in 1943. Produced from terephthalic acid and ethylene glycol. Plastics There are certain synthetic materials which are neither rubber nor fibre but are used as a substitute. These are called plastics. Plastics are also polymers. The raw material is basically a polymer of acetylene, the common gas used for welding. The polymer is obtained by treating acetylene gas under pressure in presence of a catalyst. The result is a long-chain molecule.

Radioactivity A phenomenon of spontaneous disintegration, first observed in certain naturally occurring heavy elements like radium, actinium, uranium, thorium, etc., with the emission of alpha, beta and gamma rays. It is the property of the nuclide to disintegrate in which a transformation takes place—of a relatively unstable nuclide to relatively stable nuclide accompanied with the emission of particles or electromagnetic radiation. The nuclide that decays is said to be radioactive. Discovery of Radioactivity The phenomenon was accidentally discovered in 1896 by French physicist Henry de Becquerel. He observed that uranium mineral gave off invisible radiation. He termed this property of uranium radioactivity. Later Pierre and Madam Curie showed similar phenomenon in other metals like poeonium, francium and radium.

Radioactive Emissions (i) Sub-atomic Particles (Radiation) (a) Alpha (α) particles A positively charged helium atom which has very little penetrating power. They can be absorbed by a sheet of paper or stopped by aluminium foil. (b) Beta (β) particles A negatively charged light particle. Its penetrating power is greater than that of alpha-ray. (ii) Penetrating Particles (Radiation) Also called Gamma (γ) emission. These are electromagnetic radiations of low wavelength, high frequency, and high energy.

Their penetrating power is very great as they can pass through several centimeter of lead.

X-rays Are a form of penetrating electromagnetic radiation similar to light but of shorter wavelength and capable of penetrating solids. X-rays are produced when cathode rays fall on anti-cathodes (a metal of high atomic mass like tungsten).

X-Ray Photographs X-ray passes through considerable thickness of matter without being entirely absorbed. although a fraction of the original radiation is always lost. Dense materials, such as metal and bone, absorb X-rays more strongly than materials such as wood or flesh. Therefore, it is possible to produce X-ray photographs for use in medical diagnosis. Nuclear Reaction and Atomic Energy

Nuclear Reaction A nuclear reaction is one in which a nucleus bombarded with an elementary particle (like neutron, proton. etc.) or with another nucleus to produce other products in a very short timespan. The first nuclear reaction was discovered by Rutherford in 1919 when he bombarded nitrogen with alpha particles. Nuclear Fission Nuclear fission is the fragmentation of a large nucleus into two smaller nuclei and the liberation of large amount of energy. In 1939, the German scientists Otto Hahn and F. Steersman observed that when uranium was bombarded with slow neutrons, then two smaller products were obtained with a tremendous amount of heat. The splitting of uranium was called nuclear fission. Types of Nuclear Fission (i) Controlled Nuclear Fission Carried out in nuclear reactors in which rate of fission reaction is reduced and energy produced can be used for constructive purposes. (ii) Uncontrolled Nuclear Fission In an atom bomb, uncontrolled fission takes place. A very large amount of heat is produced and the process continues until the entire amount of fissionable material is exhausted.

First Atom Bomb On 6 August 1945, an atom bomb was dropped on Hiroshima city in Japan. The second bomb was dropped on Nagasaki, another city of Japan on 9 August 1945. The bomb was made of plutonium-239. Nuclear Fusion Is a nuclear reaction in which lighter nuclei fuse to form a nucleus of greater mass. In this reaction also an enormous amount of heat is produced. By carrying on nuclear fusion under controlled conditions, the large amount of energy could be made available for useful purpose. Atomic Energy (Nuclear Energy) Energy produced by nuclear fission or nuclear fusion is called nuclear energy or atomic energy. In nuclear reactions there is loss of mass. This mass is converted into energy. It can be transformed into electrical and mechanical energy which can be used for various peaceful purposes.

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General Knowledge

GENERAL Important Chemical Tests

Important Laws of Chemistry

Brown-ring Test Is used for chemical analysis of nitrates in which the solution to be tested is mixed with iron sulphate solution in a test tube and concentrated H2SO4 (sulphuric acid) is carefully poured along the side of the test tube. In nitrate containing substances a brown ring is formed where the layer of acid touches the solution (FeNO)SO4.

Beer’s Law States that in photo chemistry, the proportion of light absorbed by a solution depends on the thickness of the absorbing layer and on the concentration of the absorbing substance in the solution.

Flame Test Is used to identify certain elements in which a clean platinum wire is dipped into the mixture to be tested and the wire is heated using a busen flame. The presence of certain elements can be detected by the change in the colour of flame. For example, a brilliant orange-yellow will indicate sodium; crimson, strontium; and apple green, barium. Beilstein’s Test Is used for the detection of halogen in an organic compound in which a clean copper wire is heated in an oxidizing flame till the flame is no longer green. The wire is then dipped in a solution of the substance to be analysed and heated again. If Cl, Br or I is present the flame turns a bright green. Fehling’s Test Is used to detect sugars and aldehydes in a solution. Equal amounts of solution of copper sulphate (Fehling A) and sodium tartrate (Fehling B) are mixed in a test tube, On boiling this with a given solution, a red precipitate forms if sugar or aldehyde is present. Kjedahl Method Is used to measure nitrogen in an organic compound. The compound is boiled with concentrated sulphuric acid and copper sulphate (catalyst) to convert nitrogen to ammonium sulphate. An alkali is added to the mixture and boiled to distill of ammonia which is passed into a standard acid solution and estimated by titrating the solution. Molish’s Test Is used to detect carbohydrates in a solution. The solution to be tested is mixed with a small quantity of alcoholic alphanaphthol and concentrated sulphuric acid is slowly poured down the side of the test tube. When the two liquids meet the formation of a deep violet rings indicates presence of carbohydrate. Rast’s Method Is used to determine molecular weight by measuring the depression of freezing point of a camphor by a known weight of the solute. Schiff’s Test Is used to distinguish between aldehydes and ketones. An aqueous solution of rosaniline and sulphurous acid (Schiff’s reagent) is used to test for the presence of aldehydes, which oxidize the reduced form of the dye rosaniline back to its original magenta colour. The aldehydes restore the colour immediately whereas ketones, restore the colour slowly.

Boyle’s Law States that the volume (V) of a given mass of gas at a constant temperature is inversely proportional to its pressure (p), that is, pV = constant. This means that if a gas is compressed threefold its volume is reduced by two-thirds. Boyle (1627–90) was the first to define an element as a substance that cannot be broken down into something simpler by a chemical process. Charle’s Law States that under constant pressure the volume of a fixed mass of gas varies directly with its absolute temperature. The absolute temperature is that measured from absolute zero, about -273° on the Celsius scale. In other words, the pressure of a gas increases by 1/273 of its volume at 0°C for every 1°C rise in temperature. In other words, if the pressure of a gas remains constant, the volume of a gas increases by 1/273 of its volume at 0°C for every 1°C rise in temperature. Alternatively, at constant pressure the volume of a given mass of gas is directly proportional to the absolute temperature. The principle was formulated by the French scientist, Jacques Alexandre Charles. Faraday’s Law of Electrolysis States that (i) The amount of decomposition during electrolysis is proportional to the quantity of current passed, and (ii) For the same quantity of electricity passed through different solutions, the extent of decomposition is proportional to the chemical equivalent of the element or group liberated. The law was formulated by an English chemist, Michael Faraday (1791–1867).

Gay-Lussac’s Law (i) Law of Gaseous Volume States that when gases combine chemically, the volumes of the reactive gases and gaseous products are in simple proportion at the same temperature and pressure. In other words, when gases combine they do so in volumes which are in a simple ratio to each other, and to that of the product, if it is also gaseous. For example: One volume of nitrogen combines with three volumes of hydrogen to form two volumes of Ammonia. (ii) Law of Thermal Expansion It states that at constant pressure all gases expanded by the same amount for the same increase in temperature. These laws were formulated by a French chemist, Joseph Lois Gay-Lussac (1778–1850).

General Science ▌ 2.315 Hess’ Law States that the heat exchange in a chemical reaction is the same, no matter whether the reaction takes place in one stage or more. The principle was formulated by a German chemist Henri Hess (1802–50). Graham’s Law of Diffusion States that the rate at which two gases diffuse is inversely proportional to their densities. It means that the lighter the gas, the faster it will diffuse through any medium. The law was defined by a Scottish chemist, Thomas Graham (1805–60). Henry’s Law States that the mass of a gas which is dissolved in a given volume of liquid at constant temperature is directly proportional to the pressure of the gas. It applies to gases that do not react with the liquid (solvent). The principle was formulated in 1803 by the British chemist William Henry. Lambert’s Law States that layers of equal thickness of homogeneous material for example, coloured filter) absorb equal proportion of light. Raoult’s Law States that the lowering of the vapour pressure of a solvent by a solute (dissolved substance) is proportional to the MOLE fraction of the solute—the proportion of solute molecules to the total number of molecules, solute and solvent present. Since the lowering of vapour pressure causes an elevation of the boiling point and a depression of the freezing point, it is used to determine the molecular mass of a solute. The law is named after the French chemist, Francois Marie Raoult (1840–1901). Law of Conservation of Mass and Matter Matter can neither be created nor destroyed. The sum total of mass or matter for a system always remains constant without any increase or decrease in quantity.

Important Chemical Processes Bessemer Process Is a method of converting pig iron to steel by blowing air through the molten metals to oxidize impurities such as carbon, silicon, phosphorus and manganese normally present in pig iron. Clemmensen Reduction Is a process used to convert aldehydes and ketones to the corresponding hydrocarbons by heating with amalgamated zinc and hydrochloric acid. Gattermann Reaction Is a process used to convert an aromatic amine into the corresponding halogen derivative through diazonium salt formation using copper as a catalyst. Haber Process An industrial process of producing ammonia by the reaction of nitrogen with hydrogen in the presence of a catalyst.

Kolbe Reaction Is used for the preparation of saturated or unsaturated hydrocarbons by the electrolysis of solutions of the alkali salts of aliphatic carboxylic acids. Solvay Process Is a process of snaking sodium carbonate from calcium carbonate and sodium chloride in large scale. The process involves heating of calcium carbonate to give calcium oxide and carbon dioxide which is bubbled into a solution of sodium chloride in ammonia. Sodium hydrogen carbonate is precipitated which on heating gives sodium carbonate. Bayer Process A process used to extract aluminium oxide Al2O3 or aluminia by treating powdered bauxite with hot caustic soda solution under pressure. The process was developed by German chemist, Karl Joseph Bayer in 1888. Bergius Process A process for making lubricants and synthetic fuel for example, petrol, from coal by heating a mixture of powdered coal and heavy oil or tar with hydrogen under pressure in the presence of a catalyst (iron, tin, or lead). The process was developed by German chemist, Friedrich Bergius, who shared the 1931 Nobel Prize. Bosch Process A process used to make industrial hydrogen by passing steam over white-hot coke to produce water gas (a mixture of carbon monoxide and hydrogen) which in the presence of a catalyst (a metal oxide) reacts with more steam to liberate hydrogen and carbon dioxide. The process is named after the German chemist, Carl Bosch (1874–1940). Down Process Is a process of making sodium metal by electrolysis of molten sodium chloride. The molten sodium and calcium formed at the cathode are separated. Frasch Process Is used to extract sulphur from subterranean deposits in which superheated water is forced down the deposits which melts the sulphur under the ground. Molten sulphur is collected by forcing compressed air from another side. The process was developed by German chemist, Herman Frasch in 1891. Hall-Heroult Process (Hall-Heroult) A process used to prepare aluminium by electrolysis in which aluminia (aluminium oxide) is dissolved in cryolite (sodium aluminium fluoride) and electrolyzed. It was developed in 1885 in US by Charles Hall, and in France by P. T. Heroult. Parkes Process A process used for extraction of silver traces from lead ore galena. Molten zinc is added to molten galena and lead is separated leaving zinc-silver which on heating distills off zinc freeing the silver .

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General Knowledge

Table: Common Substances and Their Chemical Compositions Substance

Chemical

Composition

Formula

Alum

Potash

Potassium, Sulphur, Aluminium, Hydrogen and Oxygen

K2SO4Al2(SO4)3

Bleaching Powder

Calcium hypochlorite

Calcium, Chlorine, Oxygen

Blue Vitriol

Copper sulphate

Copper, Sulphur and Oxygen

Calomel

Mercurous chloride

Mercury, Chlorine

Caustic Lotion

Silver, nitrate

Silver, Nitrogen, Oxygen

Candy Fluid

Potassium permanganate

Potassium Manganese, Oxygen

Caustic Potash

Potassium hydroxide

Potassium, Hydrogen, Oxygen

Chalk

Calcium carbonate

Calcium, Carbon, Oxygen

Caustic Soda

Sodium hydroxide

Sodium Hydrogen, Oxygen

Baking Soda

Sodium bicarbonate

Sodium, Hydrogen, Carbon, Oxygen

Common Salt

Sodium chloride

Sodium, Chlorine

Epsom Salt

Magnesium sulphate

Magnesium, Sulphur, Oxygen

Galena

Lead sulphide

Lead and Sulphur

Green Vitriol

Iron sulphate

Iron, Sulphur, Oxygen

Glauber’s salt

Sodium sulphate

Sodium, Sulphur, Oxygen

Gypsum

Calcium sulphate, Calcium, Sulphur, Oxygen

Hypo

Sodium thiosulphate

Sodium, Sulphur, Hydrogen, Oxygen

Laughing gas

Nitrous oxide

Nitrogen, Oxygen

Lime water

Calcium hydroxide

Calcium, Hydrogen, Oxygen

Litharge

Lead monoxide

Lead, Oxygen

Nitre

Potassium nitrate

Potassium, Nitrogen, Oxygen

Plaster of Paris

Calcium sulphate

Calcium, Sulphur, Hydrogen, Oxygen

Quartz

Sodium silicate

Sodium, Silica, Oxygen

Ouick lime Red lead

Calcium oxide Triplumbic tetroxide

Calcium, Oxygen Lead, Oxygen

Sal ammoniac

Ammonium chloride

Ammonia, Chlorine

Soda ash

Sodium carbonate

Sodium, Carbon, Hydrogen, Oxygen

Soda bicarb

Sodium bicarbonate

Sodium Hydrogen, Carbon, Oxygen

Washing soda

Sodium carbonate

Sodium, Carbon, Hydrogen. Oxygen

White vitriol

Zinc sulphate

Zinc, Sulphur, Hydrogen, Oxygen

Terms in Chemistry Acid A wide range of sour-tasting chemical substance, the aqueous solution of which is capable of—turning blue litmus red: dissolving certain metals to form salts: reacting with bases or alkalies to form salt and giving hydrogen (H+) ions. An acid consists of two parts of hydrogen H2 and an acid radical. For example in Nitric acid, HNO3, the acid radical is NO3 and hydrogen H2. The most popular acids are the inorganic or mineral acids, for example. Sulphuric acid H2SO4 Hydrochloric acid HCl Nitric acid HNO3 Phosphoric acid H3PO4 These are classed as strong acids.

CaCl(OCl)4H2O

CuSO45H2O

Hg2Cl2

AqNO3

KMnO4 KOH

CaCO3 HaOH

NaHCO3 NaCl

MgSO4.7H2O PbS

FeSO4.7H2O

Na2SO4.10H2O CaSO4.2H2O

Na2S2O3.5H2O

N 2O

Ca(OH)2 PbO

KNO3

2CaSO4.H2O Na2SiO3

CaO Pb3O4

NH4Cl

Na2CO3.10H2O NaHCO3

Na2CO3.10H2O

ZnSO4.7H2O

Aerosol A colloidal system in which particles are dispersed in a gas as in smoke or mist. Alcohol Is a colourless, volatile and flammable liquid, synthesized or obtained by fermentation of sugars and starches containing the hydroxyl (–OH) group. The simplest alcohol is methyl alcohol or methanol, CH3OH. Typical of the group is ethyl alcohol in wine, or ethanol, C2H5OH, the alcohol in wine and spirit. Aldehyde One of a family of organic chemicals containing the characteristic group—CHO. Like Ketones, they contain the carbonyl group >C = 0. Best known is the gas formaldehyde, HCHO, which when dissolved in water becomes the preservative formaline. Alkali Is a metal hydroxide that is soluble in water and is a strong base. For example, the hydroxides of sodium and potassium

General Science ▌ 2.317 (caustic soda caustic potash). The aqueous solution of an alkali does not change colour of blue litmus but instead turns red litmus blue. Alkalis have a soapy touch and bitter taste.

Catalyst Is a substance that affects the rate of chemical reaction without itself being consumed in the process. It accelerates the chemical action.

Alkali Metal A group of elements that react with water to form alkalies, found in Group IA of the Periodic table. For example, sodium, potassium, rubidium, caseium, etc.

Condensation A process of reduction to a denser form such as vapour to liquid.

Alkane Simple hydrocarbons that are commonly called paraffins. The series of alkane begins with methane, CH4. Allotropy The existence of elements in more than one form in the same physical state, for example, the existence of two or more crystalline or molecular structural forms of an element, e.g., carbon (diamond and graphite). Alloy

See under metals.

Amino Acids An organic compound containing both an amino group (NH2) and a carboxyl group (COOH) which are essential components of proteins. There are about 100 amino acids, of which the simplest is glycine, CH2(NH2)COOH found in gelatine and silk. Anhydride A chemical compound formed from another by the removal of water; a compound that forms an acid or base when water is added to it. Acqua Regia A mixture of concentrated nitric and hydrochloric acids in the ratio 1 : 3. Aromatic Compound An organic compound derived from benzene; has a powerful and pleasant aroma. Atomic Number element.

Number of protons in the atomic nucleus of an

Azeotropic Mixture A mixture of two or more liquids that cannot be separated by distillation. When most liquid mixtures boil, the vapour has a different composition from the liquid, which enables separation of the constituents to be achieved. Alcohol and water in the proportion 96:4 form an azeotrope which boils at 78.l5°C. Base A chemical substance which reacts with and neutralizes an acid to give a salt and water; solution of which turns red litmus blue, for example, calcium hydroxide, magnesium hydroxide, sodium hydroxide. Boiling Point The temperature at which a liquid is converted into vapour. The boiling point of water is 100°C. Carbonate A compound made from carbonic acid (H2CO3). In nature they occur as chalk or limestone, etc. Carbohydrates One of a large group of organic compounds found in most foods, e.g., starch, cereals, sugar, etc. They consist of carbon, hydrogen and oxygen. Chemical Bond

The force that holds atoms or ions together.

Colloid A mixture or suspension of very fine particles within a fluid, as in fog or paint.

Corrosion Destruction of a metallic body by chemical action or electro-chemical attack. It can be prevented by giving the metal a protective coating of paint, bitumen or plastic. Crystal A solid pattern of substance composed of a group of atoms or molecules which is repeated in space to form a very regular structure. For example, common salt, sugar, diamond. etc. Carbide A combination of carbon with a metal or sometimes with a non-metal. Catalysis A process in which chemical reaction is speeded up by the presence of an extra substance (called catalyst) which remains unchanged at the end of the reaction. Decantation The process of separating a solid held in suspension in a liquid by allowing the former to settle down after which the liquid is poured out. Deliquescence A property possessed by some substances of absorbing moisture from the air on exposure. Anhydrous calcium chloride, which possesses this property is widely used as a drying agent. Desiccation substance.

A process of drying or removal of moisture from a

Diffusion The natural process by which one gas is disseminated through another, or a process by which one material moves through another, from a state of high concentration to one of low concentration. Odours, for example, diffuse through air. Dopant An impurity added to a pure substance such as a semiconductor, to alter its properties. Distillation The process of converting a liquid into vapour and then condensing this vapour into the liquid form again, for example, manufacture of distilled water. Double Salt A compound of two salts formed by the crystallization of a liquid solution containing both of them. Electrolysis The process of chemical decomposition of an electrolyte (that is, a compound which in molten state or when dissolved in solution dissociates into oppositely charged ions) by means of an electric current. Electrons nucleus.

A negatively charged particle orbiting the atomic

Emulsion

A colloidal dispersion of one liquid in another.

Ester An organic compound obtained by reaction of an acid with alcohol and has a fruity smell. In nature esters give flower their smell and fruits their flavour.

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General Knowledge

Ether A class or organic compound made from alcohol. It is volatile, heavier than air, inflammable, and is only slightly soluble in water.

Ideal Gas (Perfect Gas) The gas conforming accurately to the gas laws (Boyle’s, Charles’ and Gay-Lussac’ laws), for example, nitrogen and hydrogen which obey the laws fairly well.

Fatty Acids The organic carboxylic acids occurring in fats and oils. They occur not as acid itself but as a glycerol, an ester derived from the alcohol. Most common fatty acids are palmitic, stearic, oleic and lineoleic.

Ion An atom or group of atoms which has either gained or lost electrons from its normal complement.

Fermentation The chemical changes brought about by living organisms (yeast, bacteria, etc.) in which bigger organic molecules are converted into smaller molecules, for example, conversion of glucose and cane sugar into alcohol and carbon dioxide.

Isomer Refers to different organic compounds having the same number of atoms of the same element arranged in a different manner. Isomers have the same basic chemical formula. Isotope An atom of an element having the same number of neutrons, for example, protium, deutrium and tritium are isotopes of hydrogen.

Freezing Mixture Salt of metals (NaCl, MgSO4) which, when mixed with crushed ice, lower its temperature below 0°C.

Litmus Paper A dyed paper that turns red when treated by acids but remains, or reverts, to blue when treated by alkalies.

Galvanization The process in which iron or steel articles are coated with zinc by dipping them in a bath of molten zinc or by electrodes to protect them from corrosion.

Mass Number of an atom.

Glycerine A colourless, thick liquid prepared by the hydrolysis of fats and oils. It is also called glycerol. Grignard Reagent An organic compound of alkylhalide and magnesium used to synthesize organic chemicals. Halide The halogen element compounds like fluorides, chlorides, bromides and iodides. Halogen Group Non-metallic elements belonging to Group VIIA of the Periodic Table, comprising fluorine (F), chlorine (C), bromine (Br), iodine (I) and astatine (At) Hard Water Water that contains salts of calcium and magnesium and does not lather easily with soap. Heavy Water Is a liquid, like water, but it contains deuterium (an isotope of hydrogen) in place of hydrogen (D2O). Helium (He)

Is the lightest of the noble gases.

Homologous Series A group of chemical compounds whose structures differ by a simple unit. Most popular homologous series include the alkanes in which structures differ by methylene (CH2) units. The series begins with methane, ethane, propane and the more complex butane. Hydrocarbon An organic compound of carbon and hydrogen, for example, methane (CH4). Hydrolysis A process of decomposing a compound by the reaction of water. For example, in digestion, starches and sugars are broken down by hydrolysis into more digestible sugars, such as glucose. Inert Gas These are inactive gases of Group O in the Periodic Table. They are now called noble gases.

Total number of protons and neutrons in the nucleus

Metalloid Element having both metallic and non-metallic properties, for example, boron, silicon, arsenic, antimony, etc. Molecular Weight Is the ratio of the weight of one molecule of a compound with that of one atom of carbon-12 taken as 12. Molecule The smallest particle of a compound that is capable of independent existence. The properties of a compound are the properties of its molecule. Neutralization A process in which acid and alkali react together and the resultant solution is neutral to litmus. Neutron atom.

Electrically neutral particles in the nucleus of an

Noble Gas These are inert gasses, for example, helium (He), neon (Ne), argon (Ar), krypton (Kr), xenon (Xe) and radon (Rn). Nucleus The positively charged central core of an atom containing protons and neutrons. Octane gases.

A hydrocarbon of paraffin series occurring in fuel

Osmosis The diffusion of substances from one solution to another through a semipermeable membrane due to a difference in concentration of the two solutions. This is the principle of the process of dialysis, where solution of less concentration passes to the more concentrated solution. Oxidation Adding of oxygen to compound or removal of hydrogen from it; a reaction in which atoms, molecules or the like gain electron. Oxide A compound in which oxygen is combined with another element. Ozone (O3) An allotropic form of oxygen. It is found in the upper atmosphere (15–30 km above the earth’s surface), and is responsible for absorbing a large proportion of the sun’s highly energetic and harmful radiation which would otherwise kill life.

General Science ▌ 2.319 Pasteurization The process of killing bacteria or retarding their growth in milk. Milk is pasteurized by heating it to a temperature of about 67°C, maintaining it at that temperature for about half an hour and then chilling it. pH Value A measure of acidity or alkalinity of a solution expressed in numbers and measure of hydrogen ions by the equation of pH = – log (1/H+). Polymerization The formation of an insoluble chemical compound or residue in a solution as a result of a chemical reaction. The residue so formed is called precipitate. Proton The positively charged heavy particle of the nucleus of an atom, having equal and opposite charge to that of an electron. Radical A group of atoms that stay together when a compound dissociates, but yet not a stable group to qualify as a molecule. Rare Gases Chemically inactive or inert gases, for example, argon, krypton, xenon, radon, etc. Rare Earth Any group of metallic elements with atomic number from 57 to 71. Also called lanthanides. Reagent A chemical commonly used in chemical laboratories for experiments and analyses such as dilute hydrochloric acid, ammonium hydroxide, dilute nitric acid, etc. Real Gases

The gases which do not follow ideal gas laws.

Salt A product formed as a result of acid-base reaction. It is formed when the hydrogen in an acid is replaced by a metal, for example, potassium nitrate and calcium sulphate. Sodium chloride (NaCl) is a salt in the chemical sense but is also called common salt. Saponification The reaction in which esters are hydrolyzed with the help of alkalies into acids and alcohol.

Slag

A non-metallic material obtained during smelting of ores.

Solution mixed.

A mixture of two or more substances not chemically

Solvent A substance which can dissolve other substances in it forming a solution. Sublimation Conversion of a solid directly into vapour, or the process of change from the solid to the gaseous state or from the gaseous to the solid state without becoming a liquid. Sulphonation The process in which one of the hydrogen atoms of the same organic compound is replaced by a sulphonic group (–SO2OH). Suspension A mixture consisting of very small undissolved or partially dissolved particles of a solid, distributed in a liquid dispersion medium. Synthesis Formation of a compound directly from its elements or a simple compound. State of Matter The material substance of any kind exists in three states, they are, solid, liquid and gas—each of which can change into one or the other according to change in temperature and pressure. The so-called fourth state of matter is called plasma where negatively charged electrons and positively charged ions are in flux. Transuranium Those man-made elements that have a higher atomic number (number of protons in the nucleus) than uranium (at No. 92) for example, plutonium (Pu, 94) Valency The combining capacity of an element measured by the number of hydrogen atoms which combine with, or are displaced by, an atom of other elements.

CHEMISTRY QUIZ Table Reaction

Result

What happens when Steam is passed over red hot iron?

Ferric oxide and hydrogen are produce 3Fe + 4H2O → Fe3O4 + 4H2

Steam is passed over magnesium?

Hydrogen is formed along with manganese oxide Mg + H2O → MgO + H2

Steam is passed over white coke?

Water gas is produced C + H2O → CO + H2

Steam is passed over calcium cyanide?

Ammonia is formed CaCN + 3H2O → CaCO3 + 2NH3

Water is treated with calcium phosphide?

Calcium hydroxide and phosphine are produced Ca3P2 + 6H2O → 3Ca (OH)2 + 2PH3

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Reaction

Result

Water is treated with calcium hydride?

Calcium hydroxide and hydrogen are produced CaH2 + 2H2O → Ca(OH)2 + 2H2

Water is treated with calcium nitride?

Calcium hydroxide and ammonia are produced Ca3N2 + 6H2O → 3Ca(OH)2 + 2NH3

Water is treated with calcium carbide?

Calcium hydroxide and acetylene are produced CaC2 + 2H2O → Ca(OH)2 + C2H2

Potassium chlorate is heated?

Potassium chloride and oxygen is produced 2KClO3 heated → 2KCl + 3O2

Lead nitrate is heated?

Lead monoxide nitrogen and oxygen are produced 2Pb (NO3)2 heated → 2PbO + 4NO2 + O2

Hydrocloric acid reacts with zinc?

Hydrogen is formed along with zinc chloride Zn + 2HCl → ZnCl2 + H2

Sulphuric acid reacts with copper?

Copper sulphate and sulphur dioxide are formed Cu + 2H2SO4 → CuSO4 + SO2+ 2H2O

Nitrogen is passed over heated magnesium?

Magnesium nitride is formed

Pieces of zinc are boiled with caustic potash

Zinc will dissolve and hydrogen is formed along with potassium zincate

3 Mg + N2 → Mg3N2 Zn + 2KOH → K2ZnO2+ H2 Mixture of ammonium sulphate and sodium nitrate is heated?

Ammonium nitrate is formed which on decomposition gives nitrous oxide (NH4)2SO4 + 2NaSO3 → 2NH4NO3 + Na2SO4 NH4NO3 → 2H2O + N2O

Ammonia reacts with chlorine?

If ammonia is in excess, ammonium chloride and nitrogen are formed 8NH3+3Cl2 → N2 + 6NH4Cl If ammonia is less than chlorine explosive nitrogen trichloride is formed 2NH3 + 6Cl2 → 2NCl3 + 6HCl

Sulphuric acid is added to barium chloride?

Hydrochloric acid and barium sulphate are formed BaCl2 + H2SO4 → BaSO4 + 2HCl

Mixture of slaked lime and ammonium chloride is heated?

Ammonia gas is formed along with calcium chloride Ca(OH)2 + 2NH4Cl → 2NH3 + 2H2O + CaCl2

Calcium bicarbonate is heated?

Calcium carbonate, water and carbon dioxide are produced Ca(HCO3)2 → CaCO3 + 2H2O + CO2

Ammonia and air are passed over hot platinum?

Nitric acid and water are produced 4NH3 + 5O2 → 4NO + 6H2O

Potassium chlorate is heated?

Potassium chloride and oxygen are produced

If a lighted paper is introduced in a jar of carbon dioxide?

The flame will be extinguished because carbon dioxide does not help in burning

If lime water is kept in air?

It turns milky due to carbon dioxide in the air

Blue litmus is put into a solution of acid?

It turns red

Red litmus is put into a solution of base/alkali why?

It turns blue

Why does a candle blow off when covered?

Because it does not get oxygen which helps in burning

Why is phosphorus kept in water?

Because with air it catches fire and in water it is insoluble

2KClO3 → KCl + 3O2

General Science ▌ 2.321

Reaction

Result

Sugar gets charred on heating. Why?

When sugar is heated above 200°C it decomposes into carbon and water and therefore gets charred

Why does the mass of an iron rod increase on rusting?

Because rust is hydrated ferric oxide (2FeO3 3H2O) which adds to its mass

Why is salt mixed with ice when making ice cream?

Salt causes reduction of temperature and helps to freeze the ice cream and freezing temperature is lowered from 0°C to 5°C

Why does milk curdle?

Lactose (milk sugar) content of milk undergoes fermentation and changes into lactic acid which on reacting with lactose forms curd

Why does milk turn sour?

The microbes react with milk and grow. They turn lactose into lactic acid which is sour in taste

Why doesn’t hard water lather with soap profusely?

Hard water contains sulphates and chlorides of magnesium and calcium which form an insoluble compound with soap. Therefore soap does not lather with hard water

Why does sea water boil at a higher temperature than fresh water?

Sea water contains impurities which raises the boiling point of water

Why is it dangerous to have charcoal fire burning in a closed room?

When charcoal burns its produces carbon monoxide which is suffocating

Why is it dangerous to sleep under trees at night?

Plants respire at night and give out carbon dioxide which reduces the oxygen content of air required for breathing

Why does ENO’s salt effervesce on addition of water?

It contains tartaric acid and sodium bicarbonate. On adding water carbon dioxide is produced which when released into water causes effervescence

GENERAL BIOLOGY (BOTANY AND ZOOLOGY) THE CELL Cell is the smallest unit of life that is capable of independent existence.

Historical Background (i) In 1665 Robert Hook, an English scientist, first saw a tiny compartment in a section of cork under his selfmade microscope and named it cell. (ii) In 1824, R. J. H. Dutrochet concluded that both animals and plants are made up of cells. (iii) In 1831, Robert Brown first observed the nucleus of the cell. Van Mohl and Purkings coined the term protoplasm for the fluid content of the cell. Size of the cell The average cell size is around 3–30 microns (1 micron = 1/1000 mm). The smallest cell size is of bacteria which is around 0.2–0.5 microns. The largest cell is of an ostrich egg (unfertilized) which is about 18 cm (about 7 inches) in diametre.

Structure Every cell is surrounded by a membrane or a living coverage— through which the cell takes in what it needs from the outside atmosphere and discharges what is superfluous. Within the membrane is the protoplasm. Parts of the Protoplasm The nucleus—controls and directs the activities of all the other parts of the cell. The remainder is known as cytoplasm, where many of the vital activities of the cell take place. A cell shows the following structures under a microscope: Cell Wall Found only in plant cells. It consists of non-living substances, for example, lignin, pectin, cellulose, etc. Cell Membrance (Plasmalemma) is the outer membrane of the cytoplasm found both in animals and plants. It consists of living substances, such as proteins. Endoplasm Recticulum (ER) A network of tubes or channels of membrane in the cytoplasm which helps in protein synthesis and conduction of material. Ribosomes Externally small, dense, granular, spherical bodies found in a free state in the cytoplasm, composed of RNA and proteins. They help in synthesis of proteins from amino acids.

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General Knowledge

Golgi Bodies Bag-like structures formed of stacks of membrane. In plants they are called dictysomes. Their functions include secretion of various substances, secretion of pectic material of cell wall in plants, and help in the formation of cell plate during cell division. Vacuoles A fluid-filled sac within a cell. In plants cells they are very big and surrounded by a membrane called tonoplast. Whereas in animals they are tiny. Functions include: Osmo-regulation, maintenance of cell turgidity. Mitochondria Rod-like or spherical semi-solid structures containing DNA in their matrix along with some enzymes which are found in all cells. The synthesize A.T.P. (energy storing molecules). Plastids Small bodies found in the cells of higher plants. They are of two types, they are.,

chromatin

nucleaus

{

{

Chromoplasts are coloured and are of two types: (a) Non-photosynthetic chromoplasts which provide colour to flowers, fruits and leaves. (b) Photosynthetic chromoplasts which in green plants manufacture food from carbon dioxide and water in the presence of sunlight. Nucleus May be round, oval, cylindrical or elongated. Each nucleus is bound by the nuclear membrane which contains nucleoplasm. Nucleoplasm consists of chromatin and nucleoli. Chromatin consists of DNA, RNA and proteins. The nucleus controls all cell activities and its responsible for transfer of heredity characteristics and assists in cell division.

middle lamella free ribosomes rough endoplasmic reticulum chloroplast chloroplast envelope granum mitochondian dictyosome (Golgi apparatus) cell sap vacoule tonoplast

}

microvillus pinocytotic vesicle forming pinocytotic vesicle smooth endoplasmic reticulum

Golgi vesicles microtubule Golgi apparatus microfilaments

rough endoplasmic reticulum endoplasmic reticulum bound ribosomes

lysosome free ribosomes two centrioles at right angles to each other nuclear envelope nucleopore nucleolus heterochromatin

mitochondrian euchromatin plasma membrane

}

nucleus chromatin

smooth endoplasmic reticulum cytoplasm microtubules microfilaments

which are colourless and store starch, protein or

absorptionat the base of microvilli exocytosis of secretory product secretory vesicle

Plasma cell wall membrane nuclear envelope (two membranes) nuclear pore nucleolous heterochromatin euchromatin

Leucoplasts lipids.

}

cytoplasm

FIG. 1 Parts of Typical Cell

Centrosome A rather dense area of protoplasm, lies close to nucleus. Usually found in animals cells. In the middle of the centrosome are two small dot-like bodies called centroils. They form a spindle in lower plants during cell division. Cilia and Flagella These are fine extensions of the cell surface and are similar in structure. Their functions are to help in locomotion or movement and in some animals they also help in feeding. Cell Inclusions These are non-living substances present either in the cytoplasm of the cell or in the vacuoles. In animal cells, the inclusions are in the form of secretory granules as in zymogen granules, haemoglobin in RBC, food material in the form of glycogen in liver cells, pigments as in the cells, of skin, eye and hair. Whereas in plants cells they are in the form of food particles such as starch, oil globules, or excretory products like resin, gum, etc.

Cell Division Growth and development of all living things depend upon the growth of cells in size and division whereby increase in number takes place. There are two kinds of cell division, they are, Mitosis Also called somatic cell division. Literally it means division of nucleus but it is generally described as cell as a whole—both nucleus and cytoplasm. It results in formation of two daughter cells with equal number of chromosomes in each, which occur in somatic cells or organs. It is the most common form of cell division which is essential for cell multiplication and growth. Since the number of chromosomes remains the same during mitosis, this division is also referred to as the equational division. Meiosis Occurs in reproductive cells. It causes a reduction in number of chromosomes in the cell besides, formation of gamets

General Science ▌ 2.323 which contain half the number of chromosomes. Since the daughter cells have half the number of chromosomes as compared to the mother cell, this division is also called reductional division. It causes hereditary variation in a species due to exchange of genetic material. Before cell division, the cell prepares itself for division; and is called interphase in which the cell becomes more active. Each cell division has two steps, such as, division of the nucleus and division of the cytoplasm. During cell division chromatin material of the cell gets organized into chromosomes.

Plants and Animal Cells The basic difference between plants and animal cells are that plant cells are enclosed by a rigid cellulosic cell wall (made of cellulose) surrounding their plasma membrane whereas animal cells have no such wall. Another difference is that the plant cells have a special organelle, the chloroplast with the help of which they are able to photosynthesize. Also plant cells lack centrioles which are concerned with the organization of the mitotic spindle or the flagella and cilia (discussed above). Plant cells have a large sap-filled vacuole.

CLASSIFICATION OF LIVING THE ORGANISMS It has been estimated that between 3 million and 20 million different kinds of organisms are alive in the world today. A large number of these organisms have become extinct, and some of these are preserved as fossils. Most modern schemes of classification of living beings are based upon the pioneering work of Carl Von Linne.

Scientific Nomenclature In the 18th century, a Swedish biologist Carl Von Linne (in Latin, Carolus Linnaeus) developed a scheme of scientific classification and devised a system of scientific nomenclature. According to Linne all organisms, having some important characteristics in common and different from a others in one or more ways are placed in a category termed species, for example, cat, monkey, house-fly, frog, etc. Different species having certain common characters are placed in a genus. A genus, is therefore, a collection

of some related species but they also have some characteristics in common, for example, cat, lion and tiger constitute three different species but they also have some common characteristics and therefore, are placed in the same genus. Similarly, genera are grouped into families and families into orders and orders into classes and classes into phyla. Therefore, the lowest group is species and the highest unit is phylum and various phylum constitute a kingdom, either plant kingdom or animal kingdom. The naming of organisms is also done on a scientific basis. Each organism is given two names, nemely: a generic name and a specific name. The generic name begins with a capital letter and the specific name begins with a small letter. For example: the zoological name of the common house-fly is Musca domestic and that of man is Homo sapien. The botanical name of wheat is Triticum aestivum. The order of this nomenclature is illustrated below:

Table: Classification

Frog

Dog

Man

Kingdom

Animalae

Animalae

Animalae

Sub-Kingdom

Metazoa

Metazoa

Matazoa

Phylum

Chordata

Chordata

Chordata

Sub-phylum

Vertebrae

Vertebrae

Vertebrae

Class

Amphibia

Mammalae

Mammalae

Order

Anura

Carnivora

Primate

Family

Ranidae

Canidae

Hominidae

Genus

Rana

Canis

Homo

Species

Pipiens

Familians

Sapiens

General Classification Traditionally all living organisms were classified under two main kingdoms, they are, (i) the plant kingdom and (ii) the animal kingdom. Certain unicellular organisms like bacteria, fungi, etc., were earlier classified under the plant kingdom. However, taxonomists in the last

few decades have been opposing this classification of unicellular organisms into the plant kingdom and there have been constant endeavours to revise the system of classification. Accordingly, in 1969, R. H. Whittaker, classified all living organisms into five main kingdoms and this system is now receiving wide acceptance. According to this system, the five kingdoms are:

2.324

General Knowledge

The kingdom Prokaryotae or Monera is sometimes given to include the viruses. Other systems describe viruses as being outside the normal system of classification. (i) (ii) (iii) (iv) (v)

Monera (true-bacteria, blue-green algae) Protista (Golden algae, yellow-green algae) Fungi (slime molds, bread molds, sac fungi) Plantae Animalae

A brief description of the classification along with subclassifications is given below: Monera are organisms with prokaryotic cells and diverse nutritional habits, including photosynthesis, chemosynthesis and absorption. Reproduction is a sexual, by simple cell division, for example, true-bacteria, fungus like bacteria, blue-green algae.

Protista are unicellular or colonial eukraryotes with diverse nutritional habits, reproduce both sexual and asexual involving meosis and nuclear fusion. Mobility is by means of more advanced type of flagella, e.g., Euglenophytes, Dinoglagellates, golden algae, yellow-green algae. Fungi are multinucleate organisms, with non-photosynthetic nutrition through absorption, lacking tissue differentitation, e.g., slime molds, water molds, white ruts, bread molds, sac fungi, etc. Plantae Multicellular organisms with cells being enclosed by a rigid cell wall. Nutrition mostly by photosynthesis with exception of a few species being absorptive. Primarily non-motile, often anchored to the substratum. Plantae include all botanical organisms. Animalae Include all protozoa and metazoa zoological organisms. These are multicellular eukaryotic organisms with cells lacking a rigid cell-wall and photosynthetic apparatus.

CLASSIFICATION OF THE PLANT KINGDOM (PLANTAE) The plant kingdom is classified into a number of divisions, they are. Rhodophyta (Red Algae) Are mostly multicellular marine plants (some 2500 species) ranging from simple filaments to large plants (even 10 ft long). Reproduction is mainly sexual through complex reproductive systems. They are found attached to rocks or other algae. Their reddish colour is due to a pigment called phycaerythrin. Phaeophyta (Brown Algae) Are multicellular marine plants growing on rocks or attached to sea shores, ranging from a few millimetre to massive plants up to 200 ft. Reproduction in these plants is both sexual and asexual. Their brownish colour is due to xanthophyll pigment called fucoxanthin. Chlorphyta (Green Algae) Are a group of bright green (some 3700 species) mostly aquatic plant, unicellular, colonial and multicellular. Reproduction in these plants is both sexual and asexual. They are found in abundance in fresh water as well as sea water, on tree trunks, moist rocks, leaf surfaces and soil. Bryophyta Simple, non-vascular small plants grow in moist places. The plant body is thallus-like and remains attached to the soil by rhizoids. They lack roots, flowers and seeds. Reproduction is mainly through vegetative means or sexual, for example, liverworts, hornworts, mosses, etc. Lycopodophyta (Club Mosses) Multicellular terrestrial plants with vascular tissues. Their body is differentiated into roots, stem and leaves. Stem not jointed; spores are produced in the axis of fertile leaves, mostly aggregated into club-like terminal cones. Arthophyta (Horsetails) Are multicellular pants with vascular tissues. The plant body is divided into root, stem and small whorled leaves. They have upright stems which grow from underground branches. Arthopthyta are largely an extinct group represented by a single living genus, the Equisetum (the horsetails).

Pterophyta (Ferns) Are multicellular plants with vascular tissues; the body is differentiated into root, stem and leaves; contain spores for reproduction which are produced usually in the lower surface of the leaves or on their margins. Seeds are not produced. Cycadophyta Are multicllular terrestrial plants with vascular tissues (without vessels). The plant body is divided into root, stem and leaves. The stem is large and woody, leaves are large and fern-like, for example, Cycas (sago palm), Zamia (sago tree), etc. Coniferophyta (Conifers) Are multi-cellular plants with welldeveloped tissues. The main plant body is large and woody, leave are simple, smaller than Cycadophyta and needle-like. Examples: Cedrus (deodar), Taxus (yew). Anthrophyta (Angiosperms) More advanced flowering plants with well-developed vascular tissues, predominantly saprophytic with a well-differentiated body of roots, stem and leaves. They occur in almost all places and make up more than half of all known species of plants—about 200,000 species. They range in size from minute-floating duckweeds to giant eucalyptus and silk-cotton trees, and include plants of a great variety of forms—cacti, water lilies, sunflowers, orchids, pitcher plants, Indian pipe, etc. The reproductory organ of these plants are flowers, and the seeds are enclosed within an ovary which develops into a fruit-wall. The distinct parts of the body of these plants are: Root is the descending and positively geotropic part of the plant, which develops from the radical or the primary root. It anchors the plant to the soil and absorbs water and minerals and salts from the soil. Stem is the ascending and negatively geotropic part of the plant. It bears nodes and internodes and supports leaves, flowers and fruit.

General Science ▌ 2.325 Leaves are generally green and carry out the function of synthesis of food (photosynthesis). Flower is the reproductive organ of angiosperms. It generally consists of four whorls: Sepals, Petals, stamens and pistils. After pollination and fertilization the ovary of the pistil develops into fruit and the ovules into seeds. Some flowers are found in groups or a clusters which are called peduncle. The arrangement of flowers on the peduncle is called inflorescence. The ripened ovary becomes a fruit which generally contains seeds. On the basis of morphology angiosperms are further classified into: (i) herbs, (ii) shrubs and trees. On the basis of age, they are grouped as: (i) Perennials which live for many years. (ii) Annuals which produce flowers and fruit in the course of a single season. (iii) Biennials

which live for two seasons.

There are two major classes of Angiosperms: (i) Dicotyledons These plants have two cotyledons in their seeds and may be annuals, biennials or perennials, for example, gram. (ii) Monocotyledons These are mostly annuals and have only one cotyledon in their seeds. Other Classfications Terrestrial plants which grow on soil. Hydrophytes plants which grow in water. Epiphytes plants which perch on other plants but do not take nourishment from them. They are not rooted in soil but grow upon branches and stems of other plants, For example, urn plant. Xerophytes are adapted to grow in a dry habitat like deserts and can survive without moisture, for exmaple, Cacti. Mesophytes thrive under conditions intermediate between very wet and very dry. The great variety of crops, for example, beans, tomatoes, peas, etc., belong to this category. Parasitic which depend on other plants for their nourishment. They lack chlorophyl and thus cannot make their own food, for example, bacteria and fungi. (mushroom) Carnivorous Plants which trap insects and other small creatures on their sticky leaves and digest them to obtain nitrogen and other material essential for their growth. They are also called insectivorous plants. For example: pitcher plant and bladderwort.

Modification of Plant Parts Sometimes certain parts of a plant are modified to carry out specific functions:

Root Modifications Storage Roots Store food material, for example, carrot, reddish, turnip, sweet potato. Aerial Roots are aerial nodes that develop from the stem and go into the soil. Example: Still roots of sugarcane and maize, prop roots of banyan trees which provide reinforcement to the stout branches. Parasitic Roots Arise from stem and penetrate into the host plant. They are called ‘Haustoria’. For example, Cuscuta. Respiratory Roots have pores and are founding plants that thrive in swamps, for example, Rhizophora. Assimilatory Roots contain chlorophyll and perform the function of carbon assimilation, for example, Trapa roots. Epiphytic Roots are aerial roots which absorb moisture from air, for example, Vanda roots. Stem Modification In some plants, the stem is modified to perform different functions, they are, Rhizomes Underground modification of the stem. They bear nodes and store food material, for example, ginger, grass and canna. Tuber are swollen tips of underground branches that store food, for example, potato. Corm underground stem which bears leaves, for example, saffron, gladiolus. Bulb is a modified underground stem in the form of a disc-like structure that bears fleshy leaves that store food material, e.g., onion, lilies. Cladophyll is a modification of the stem into a large, flat green and fleshy structure that bears leaves into spines. Such stems carry out the function of leaves, for example, cactus. Tendril an aerial modification of the stem where the stem becomes a long, slender, coiling structure serving as an organ of attachment for certain climbing plants. It helps the plant to climb up the support, such as grapes and cucurbits.

CLASSIFICATION OF THE ANIMAL KINGDOM (ANIMALAE) The entire animal kingdom is broadly classified into two main groups , they are,

Protozoa Single-celled microscopic organisms in which all functions are performed within a single cell itself, for example, amoeba.

Metazoa Multicellular organisms in which cells are grouped together to form tissues, organs and systems. All animals, including man, are placed under the Metazoa sub-kingdom. Metazoa are further classified into two main groups,

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General Knowledge

Invertebrates They do not possess a vertebral column, and are classified into various phyla:

Porifera are fixed multicellular organisms, aquatic and have a porus and cylindrical body, e.g., Ascom Sycon (sponges). Coelenterata are sac-like multicellular aquatic creatures with tentacles stinging cells; possess tissues and digestive cavity, for example, Hydra, Obelia, Aurelia (jelly fish). Platyhelminthes are both parasitic and free-living organisms that have a flat body, thin, soft and appear leaf or ribbon-like, for example, liver fluke, pork tapeworm, blood fluke, etc. Nemathelminthes are both parasitic and free-living organisms, with a round body and possess a mouth and anus, for example, roundworms, filaria, hooke worms, etc. Annelida These are mostly free-living wormlike organisms having soft, round and segmented bodies (marked with external rings), for example, earthworm, leech, sea mouse. Arthropoda is a large group of organisms having joined feet, segmented body and an external skeleton (exoskeleton) and are classified into four large classes: (i) Crustacean Arthropods with limbs attached to most segments, for example, Palaemon (prawn) Dophinai (water flea), Branchipus (Shrimp). (ii) Insects Arthropods with three pairs of legs and body divided into three part, for example, Periplanta (cockroach), Pailio (butterfly) Musca (housefly), Culex (mosquito). (iii) Arachnids Arthropods with four pairs of legs and body divided into two parts for example, Buthus (scorpion), Lycosa (wolf spider). (iv) Myriapods Arthropods with body divided into a number of similar segments with one or two pairs of legs to most segments, for example Scolopendra (centipede), Julus (milipede).

Mollusca are shell-bearing organisms having an unsegmented body without appendages. They possess a ventral muscular organ called a foot for locomotion, for example, Pila (apple or pond snail), Limax (slug) Sepia (cuttlefish). Echinodermata include spiny-skinned animals built on a five-radial plan. They move my numerous soft tube-like structures called tube feet, and are marine by habitation, for example, Astropecten (starfish), Antedon (sea-jelly) Exchinus (sea urchin), etc. Largest Known Invertebrate The Giant Squid belonging to Phylum Mollusca, the body of which measures about 6 m and tentacles may measure up to 10 m.

Vertebrates The remaining animals with a vertebral column are grouped under this main class also known as Phylum Chordata. There are however, a few forms that lack backbones and are considered primitive protochordates. Phylum chordata has three unique characteristics, they are. (i) A dorsal, hollow tubular nerve cord; (ii) A notochord which is ventral to the nerve cord and is replaced by cartilage or bone to form a vertebral column in higher vertebrates, and (iii) Gill slits in the phyarynx. Chordata is further divided into the following classes:

Pisces (Fishes) are aquatic cold-blooded, jawed vertebrates. Their bodies are covered with scales, they respire through gills, move with the help of fins (no legs) and have a two-chambered heart, for example, Labeo (rohu), Hippocampus (sea horse), Torpedo (electric fish). Amphibia (Frogs) are cold-blooded vertebrates with four legs and scaleless skin and adapted to live both in water and on land. Respiration is done by gills or the skin or lungs. Fertilization is external, that is takes place outside the body, generally in the water. In most amphibians, there are two stages of life. In one, larval form, they live only in water and the other, that is, adult stage, they live in water as well as on land, for exmaple Rana (frog), Bufo (toad). Reptiles (Creeping Creatures) are cold-blooded vertebrates, adapted to life in dry places. Respiration is by means of lungs. The heart is three-chambered, except crocodiles, which have a four-chambered heart. There is no larval stage in their development. Fertilization is external, for example, Testudo (tortoise), Hemidactylus (house lizard), Naja (cobra). Aves (Birds) are warm-blooded animals with a body-covering of feathers. Their four legs are adapted as wings and are usually used to fly. There are, however, a few birds which cannot or hardly fly, for example ostrich, kiwi, etc. The heart is four-chambered, and the digestive tract has two additional organs, crop which stores food and gizzard which grinds the food. Fertilization, is internal, that is takes place inside the body,for example, Passer (sparrow), Columba (pigeon), Pavo (peacock). Mammalia are warm-blooded vertebrates and usually nourish their young ones with their milk produced in special glands called mammary glands. The heart is four-chambered and respiration takes place through the lungs. There is muscular partition (diaphragm) that separates the thoracic from the abdominal cavity. The embryo receives nourishment and oxygen through the placenta in the female’s womb and disposes of wastes through the maternal blood circulation, for example, Macropus (kangaroo), Felis (cat), Canis (dog), Balaena (whale), Homo (man). Largest Known Vertebrate The whale, measuring up to 33 m and weighing over 1,35,000 kg.

General Science ▌ 2.327

IMPORTANT BIOLOGICAL PHENOMENA Photosynthesis Process by which green plants are able to utilize energy of sunlight in order to convert water and carbon dioxide into food in the form of simple carbohydrates. The energy from sunlight is trapped in plant cells by choloroplasts which are packed with chlorophyll (gives green colour to the plant and leaves). Incoming sunlight generates tine electric currents in them: These currents convert ADP (adenosine diphosphate) into ATP (adenosine triphosphate) and split water molecules in the cells into oxygen and hydrogen—paving the way for the whole complex process of photosynthesis. Thus oxygen is dissipated into the atmosphere and hydrogen combines with carbon dioxide to become glucose, one of the main fuels used by living organisms.

Metabolism All biochemical reactions of processes taking place in the living organisms have two interwoven aspects, viz., Anabolism Synthesis of complex organic molecules from simple molecules by the living organisms. It is the constructive or synthetic phase of metabolism, for example, the formation of proteins from amino acids. Catabolism (also katabolism) Breakdown of complex molecules into simple molecules to release energy, that is, bearing down of hydrocarbons into simpler forms with the release of energy. Metabolism takes place in a controlled manner at body temperature. Enzymes act as bio-catalysts in the process. The energy released in such reactions is stored and used for growth and development. The elimination of waster products generated in the process is called excretion.

Respiration A term with two distinct meanings, such as, breathing and the transport of oxygen and carbon dioxide to and from the cells known as external respiration. Internal respiration or cell respiration occurs within the cell and constitutes the chemical reactions from which an organism derives energy. Internal respiration using oxygen is termed aerobic while anaerobic respiration may occur in the absence of oxygen. Both plants and animal respire. Animals need oxygen to release the energy present in their food and get rid of waste, that is, carbon dioxide.

Skin Breathing animals.

Carried out through the skin by slow-moving

Gill Breathing Fish possess gills which absorb oxygen from water flowing over them taken in through the mouth. Tracheal Breathing System developed by insects which include network of tubes which take oxygen to all parts of the body. Lung Breathing Acquired by evolved animals.

Fertilization Is the union of two gametes or sex cells to produce a zygote or fertilized egg which grows into a new offspring.

External and Internal Fertilization • Fertilization in frogs and toads is external, as it is in fishes, most water creatures and lower plants (e.g., Ferns). By this we mean, the male gamete swims across a watery surface to fertilize the female gamete or egg. The fertilized egg cell is then able to develop into an individual creature. • Fertilization in mammals, birds, reptiles and insects is internal. That is to say, the sperms are shed directly into the female’s body where they fertilize the egg before these are laid as in birds and reptiles whereas in mammals the cell develops into an embryo within the female body. Fertilization in Flowering Plants In higher plants, it is a complex process involving pollen grains which are transferred to the egg-bearing organism by wind or insects. The process is called pollination in plants. On falling on a stigma each pollen grain of the right kind germinates and sends a pollen tube containing a male reproductive cell down through the style and into an ovule where the male cell fuses with an egg cell which can develop into an embryo plant.

Transpiration Evaporation of water from the stem and leaves of plants. It serves the purpose of cooling the leaves on hot days, and protects delicate leaf tissue from drying up. It draws water up from the roots towards the leaves which give out excess water into the atmosphere.

BIOLOGICAL TERMINOLOGY Botanical Terms

Algae

Adventitious roots Roots which grow from the stem or leaf rather than from the root system of a plant.

Anaphase The stage in cell division in which chromatids or chromosomes move towards opposite poles. It begins with the splitting of the sister chromatids. They separate at their disk-like centromeres and are pulled apart by the contracting action of the spindle.

Anabolishm Synthesis of complex organic molecules from simple molecules during metabolism (See Metabolism).

Lowest aquatic green plants of Thallophyte group.

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General Knowledge

Annual rings Concentric rings seen in a cross-section of a tree trunk each representing the age of the plant.

Chloroplast Coloured plastids containing pigments other than chlorophyll. They give colour to flowers and fruits.

Annulus A ring-like structure present on the stalk of a mushroom showing the remnant of the velum.

Chloresis Disorder which affects a plant’s ability to form chlorophyll.

Anthridium Male sperm-bearing organ of lower plant such as ferns, algae and mosses.

Chromatin Hereditary substance of the nucleus. It is a network of threads in nucleus of a cell that shows up when the cell is stained with certain dyes.

Aquatic Plants (Hydrophytes)

Plants grown in water.

Archegonium Female, egg-bearing organ of lower plants like ferns, mosses, etc. Bark

Outermost corky sheath of trees and shrubs.

Basidia The club-shaped cells in the gills of mushrooms which give rise to haploid spores called ‘basidiospores’. Basidioscarp Biennial

The fruity body of the mushroom and other fungi.

Plants which complete their life cycle in two years.

Blight Any of the plant diseases that result in sudden dying of leaves growing tips, or an entire plant. Bryophytes

Refer to ‘Classification of the Plant Kingdom’.

Bud A outgrowth on a stem or branch often enclosed in protective scales, comprising a shortened stem and immature leaves or floral parts; a partially opened immature flower. Bulb A disc shaped condensed underground stem with fleshy leaves, for example, onion or turnip, usually surrounded by scale-like modified leaves and containing stored food for the undeveloped shoots of the new plant enclosed within it. Cactus Desert plants, most of which lack leaves to minimize water loss. They are characterized by thick, fleshy often prickly stems that function as leaves.

Cladophyll Modified flattened stem which functions and resembles a leaf. Also called ‘Cladode’, ‘Phylloclade’. Cilia

Hair like cell prolongations meant for locomotion.

Citrus fruit

Fruit of the citrus (orange) group.

Coennocoeti/Coenocyte The protoplast or cell containing many nuclei. It is found in many fungi and some green algae. Corm A swell on underground stem having buds. It is responsible for vegetative growth. Cotyledon An embryonic leaf in a seed which usually stores food matters. Cryptogam Non-flowering seedless plant which produces spores, for example, algae, mosses, fungi and ferns. Cuticle Non-cellular waxy layer found on the surface of stems and leaves to prevent water loss. Also strip of hardened skin at the base of a finger nail or toenail. Deciduous Plants which shed leaves to eliminate water loss. In tropical regions it is in the hot summer and in temperate areas it is winter, when such plants shed leaves. Decumbent stem A creeping stem in which tips turn upward at or near the apex.

Callus Thick layer of scar tissue which forms over a wounded area of stem or trunk to protect the inner tissues from exposure.

Dicotyledons Plants whose seeds contain two cotyledons,for example, gram.

Protective portion of a flower consisting of sepals.

Dichotomous branching A kind of branching in which the main branch or axis stops growing when lateral branches appear on it.

Calyx

Cambium A layer of cells in between xylem and phloem which increases the growth of the plant. Carnivorous Plants Plants which make food from insects, for example, pitcher plant. Catabolism The part of metabolism in which chemical substances in living things are broken down into simpler substances. The process usually releases energy. Cellulose A kind of carbohydrate, the main component of the cell wall of a plant. Centriole The part of a cell which is important in the formation of a spindle during process of a cell division. Chlorophyll The green pigment found in all plants (see photosynthesis) capable of photosynthesis. The molecules of chlorophyll absorb energy from sunlight and make it available for the synthesis of organic material.

Dyctyosome Plant cell organelles which are formed of small bag-like folds of membrane and vesicles that contain metabolite cell. Ectoderm Layer of tissue in an animal embryo which later develops into cellular material in the epidermis and nerve cell. Endocarp Innermost often hard or leathery layer of the fruit wall. Endodermis Innermost layer of cortex surrounding vascular bundles, found in all roots and stem of certain plants which controls passage of water. Endosperms The nutritive tissue surrounding and absorbed by the embryo in flowering plants. Epidermis

The outermost cell layer of an organism.

Epiphyte Plants whose roots grow upon stems or branches of other plants.

General Science ▌ 2.329 Exocarp Outermost layer of the fruit wall which forms the skin in many fleshy fruits. Ferns A large group of primitive plants of the plant kingdom which grow in moist shady floors in tropical and sub-tropical climates. Fossils

Remains of an organism preserved in rocks.

Fungus A non-flowering plant which lacks chlorophyll and cannot manufacture its own food and therefore depends entirely on either living or dead organic substances. Gamete Mature reproductive cell, that is cell sperm or formed in sexual reproduction or reproductive/germ cell produced by male and female organisms. Gametogenesis

The process of formation of gametes.

Gemma A small cup-shaped organ of vegetative reproduction in bryophytes. Gene

Part of the chromosome that determine inheritance.

Geotropism Response to the stimulus of gravity (Adjective : Geotropic). Germination

Initiation of growth in plant spores or seeds.

Interphase A stage in cell division when the nucleus is not actually undergoing division but very actively involved in the general process. It is the period between two divisions when nucleus is said to be in interface. Insectivorous Plants which attract insects and digest them to obtain nitrogen. for exmaple, Pitcher plants. Isobilateral Leaves on both sides.

Leucocyte Any of the white or colourless nucleated cells occurring in blood. Leucoplast Colurless plastids, in the cytoplasm of plant cell around which starch collects. Lignin Complex organic compound found in the cell wall of the plants which gives strength, particularly in parts of xylem tissue. Lignin forms up to 50 per cent wood in trees. Lipid A fatty compound of carbon, hydrogen and oxygen in which oxygen is in lower proportion than carbohydrates. It includes fats, oils, waxes and related substances. Liverworts Groups of primitive spore-bearing plants which together with mosses comprise the group bryophyta.

Grana A stack of membranes found in choloroplast containing chlorophyll.

Matrix

Gymnosperm Any plant of the class Gymnospermae, Which includes coniferous trees and other plants having seeds not enclosed within an ovary.

Metabolism

Haploid Characteristic of gametes which contain one set of unpaired chromosomes, in the gamets or sex cells of animals. Haustoria Specialized organ by which parasitic plants such as fungi obtain food from a host plant. Herb Term describing any tiny plant that lacks woody stems above the ground.

Non-cellular ground substance of tissue.

Meiosis

Refer to ‘Cell division’. Refer to ‘Biological phenomenon’.

Metaphase Second stage of cell division of cell division of the nucleus during mitosis and meiosis. The membrane breaks down and a fibrous spindle forms. The spindle has two poles at opposite sides of the cell; the chromated pairs align along the spindle’s equator. Mitosis

Refer to ‘Cell division’.

Meristem tissue Mitotically active cells commonly found at the apex of shoot and root or branch.

Inactive chromatin found in the nucleus.

Mesocarp Middle layer of fruit which becomes fleshy sometimes.

The point at which the seed is attached to the fruit.

Mesoderm The embryonic germ layer lying between the ectoderm and endoderm.

Heterochromatin Hilium

The monocot leaves having similar structure

Histone Type of protein found in chromosomes in association with the nucleic acid DNA. Hymenium The outer layer of gill-bearing sports in mushrooms, or the spore bearing layer of the fruiting body of certain fungi. Hypodermis Outermost layer of cortical cells beneath epidermis that acts as water-storing or strengthening tissue. Hypha Fine-breached or unbranched filament that spreads to form a loose network called mycelium. Hybrid The plant produced by crossing plants belonging to different species or genera, also plants and animals that are bred from genetically different parents. Hypocotyl Part of the stem of a young seedling that lies between the cotyledons.

Mesophyll Soft tissue of a leaf in between the upper and lower epidermis, that contains the chloroplasts and is involved in photosynthesis. Microsporophyll Structure in ferns that bears the microsporangia. Microsome Cell particle of the smallest size, typically consisting of a piece of endoplasmic reticulum to which ribosomes are attached. Parthenocarpy The process through which development of fruit takes place without fertilization. Pedicel Peduncle

The stalk of a flower which serves as a support. The axis on which flowers are borne in inflorescence.

Pericarp The cashing of the seed or seeds within a fruit, developed from the ovary wall.

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General Knowledge

Pileus

Cap-like part of mushroom and any other fungus.

Spine A leaf modified into thin pointed thorn-like structures.

Pinna

A small leaflet of a compound leaf.

Spindle

Phloem food.

PhloeA complex tissue found in plants which conducts

Plasmalemma

Outer living membrane of the cell (see cell).

Prophase First stage of cell division in which chromosomes or chromatids get aligned in the middle line to form a spindle, the chromosomes appear as two sister chromatides, which coil and then shorten and thicken. Prothallus

Gametophyte of fern which has sex organs.

Protonema The young gametophyte of bryophyte which develops following spore germination. Protoplasm The living content of cell which includes cytoplasm and nucleus (Rafer to Cell). It is a jelly-like colloidal substance constituting the living matter of plant and animal cells and performing the basic life functions. Rachic The axis of a fern leaf from which the pinna originates. Rhizoid A hair-like short growth found in lower plants which fixes the plant to the soil and also helps it derive nourishment from the soil. Rhizome

Underground stem.

Ribosome Any of the numerous spherical cytoplasmic particles consisting of RNA and protein, that are the sites of protein synthesis in the cell. Rot Term describing the breakdown or decomposition of plant tissue usually caused by bacteria or fungi or sometimes initiated by a drought. Rust Parasitic fungus which infects a number of plants causing red and black spots on leaves and stem. It is a plant disease caused by fungi, characterized by reddish or brownish spots on leaves, stems and other parts. Sap Watery fluid found in plant tissue and cells. It is transported throughout the plant by the vascular tissues or conducting tissues—xylem and phloem. Saprophyte The plant which grows and obtains food from dead organic matter like bacteria or fungus, which lacks chlorophyll and therefore cannot manufacture its own food.

Structure seen in the cell at the time of cell division.

Spore Microscope reproductive body of plants which is the chief means of reproduction among ferns, mosses, fungi and other lower plants. Sporophyte Spore-bearing generation in the life cycle of plants in which each cell has a nucleus with 2 sets of chromosomes and which can produce reproductive spores with only one set of chromosomes by reduction division (meiosis). Stamen Male reproductive part of a flower which produces male sex comprising stalk plus head with pollen sacs that burst to release pollen grains. Stigma Receptive surface area of the carpel or female reproductive organ of a flower. It is the sticky surface in flower on which pollen grains are stimulated to start their own growth. Stipe

Stalk of mushroom.

Stroma The colourless dense material occurring around the grana in a chloroplast. Tendril Modified thread-like stem or leaf designed to cling to a support. Tepals

Collective term applied to petals and sepals.

Thallus A plant body not differentiated into root, stem and leaves, for example, in some algae, bryophyta and fungi. Transpiration a plant.

Evaporation of water from the stem and leaves of

Telophase Last stage of cell division in which the nucleus starts returning to the interphase stage. A nuclear membrane forms around each pair of daughter chromosomes, thus creating two complete daughter cells. Tropism Movement of a plant in response to a particular external stimulus such as gravity or light. It is a change in position of some part of plant in relation to rest to body, which has remained fixed. Tuber Swollen underground stem or root which serves as a storage organ, for example, potato (stem tuber), dahlia (root tuber). Vascular tissue

Term used to describe xylem and phloem.

Seedling A stage in the life of a plant which begins with germination and ends with the emergence of the first true leaves. A protective leaf-like part of a flower (Refer to ‘Flower’).

Vegetative propagation Any type of reproduction which does not involve the formation of spores or seeds, e.g., Reproduction in bacteria, algae, fungi, etc.

Shrub Term used to describe any plant with several woody stems which grow from a point near or below ground level.

Velum A covering of hyphae on the young, fruiting body of mushroom.

Spermatophyte Seed-bearing plants which include both angiosperms and gymnosperms.

Vernation Arrangement and folding of leaves in a bud.

Sorus A reproductive structure found in ferns containing sporangia which are found on the placenta.

Viruses They are nucleoprotein in nature, but a particular virus has either DNA or RNA never both. Thus they possess the characteristics of both the living and non-living, they are at the threshold of life.

General Science ▌ 2.331 Xerophyte Plant adapted to survive long periods without moisture, for example, cati. (See ‘Plant Classification’)

Chitin A non-protein substance which forms an outer exoskeleton of arthropods.

Xylem Complex tissue which conducts water and mineral salts upwards in plants.

Cold-blooded Animals Animals that are unable to regulate their body temperature, which therefore, adopt the temperature of their surroundings. Most animals, other than birds and mammals, are cold-blooded.

Xygot A cell formed by fusion of a male and a female gamete or fertilized egg, that is a fertilized egg or ovum.

Zoological Terms Adipose Animal cellular fat. Adipose Tissue A type of connective tissue in the body that contains stored cellular fat.

Chordata A phylum of animals having anotochord, tubular nerve cord, pharangeal gills and a tail at some stage of their life history. Coelom The body cavity of coelomate animals is lined on all sides by mesoderm, in which gut is suspended.

Agnatha Acquatic vertebrates lacking jaws.

Coleoptera A large order of insects with horny forewings covering part or all of abdomen.

Amoeba Simplest known unicellular animal, the largest species of which is just 1 mm in length and hardly visible to the naked eye.

Commensal An organism living in association with other animals without causing harm.

Amphipoda Group of crustacea, including sandhoppers and fresh water shrimps.

Chondrichthyes Cartilaginous fish the endoskeleton of which is exclusively made of cartilage.

Antennae Paired sensory organs which stick out from the heads of insects.

Crustaceans A large and varied class of anthropods which include crabs, lobsters, prawns, crayfish, etc. They have hard external skeleton and are mostly acquatic.

Anthropoids Highly evolved sub-order of primates, for example, monkey, ape and man. They have grasping hands and feet and nails on fingers and toes. Anura

Group of a Amphibia, for example, frogs and toads.

Aocoelmoate Animals without coelom and body cavity. Apes Family of primates that includes the species closest to man, for example, gorillas. Arachnids Large class of arthopods which include spider, scorpions, etc. Areolar Tissue A type of connective tissue in which cells distributed in the matrix, they contain white and yellow fibres. Bone A connective tissue of special matrix which consists of elastin, impregnated by calcium phosphate, calcium carbonate, magnesium phosphate and calcium flouride. They provide skeletal support to the body and protect delicate organs. Bovidae Family of cloven-hoofed mammals with horns, for example, cattle, sheep and goats. Brachiopods A group of large, mostly extinct marine invertebrates with twin shells. Capillary Extremely fine blood vessels, walls of which are made up of single cells. Caterpillar Insect larva with a slender body, six legs in the front end and a number of soft prolegs, for example, caterpillar of butterfly and moth. Cell sap

Fluid present in the cell.

Diapause Period of suspended development occurring in certain organisms, often at certain seasons. Diploblastic Animals, the body of which has two layers— ectoderm and endoderm. Diploid

Having two sets of chromosomes in cell nucleus.

Ectoderm The outer germ layer of the metazoans which develops into the epiderm and its derivatives. Endoderm The innermost germ layer of the metazoans, or layer of tissue in embryo. Flame cells Excretory cells of helminthes which have a tuft of flickering flagella inside the hollow cavity of the cell. Ganglion A mass of nerve cell bodies. Gill An organ of respiration found in fishes, and other aquatic organisms. (Rafer to ‘Respiration’) Green Glands

Excretory organs of crustaceans.

Larva A stage in the life history of animals which are capable of an independent life. Mesoderm The middle layer of the three, primary germ layers of metazoans. Notochord A solid cylindrical rod-like structure made of vacuolated cells, present in the mid-dorsal body axis of chordates; and is present during some stage of their life-history. Nervecord Hollow, tubular cord composed of neurons and usually with ganglia; forms an important part of the nervous system.

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General Knowledge

Placenta The tube with which the foetus remains attached to the uterus, the foetus gets nourishment from the mother through this. Pseudopodi Temporary outgrowths of protoplasm from the body of the cell. Locomotory organs of an amoeba. Thorax The part of the body between head or neck and abdomen.

Trachea

Breathing tubes of vertebrates and some invertebrates.

Triploblastic Animals having three primary germ layers. Viscera Collective term used for internal organs situated in the body cavity.

HUMAN BODY ANATOMY AND PHYSIOLOGY Blood Blood contained in blood-vessels is a connective tissue in the form of a red viscous fluid.

Functions of Blood (i) Carries oxygen and essential nourishment through arteries to every living tissue in the body. (ii) Carbon dioxide and waste products generated in the body by metabolism (Metabolism = All chemical processes which govern living organisms), such as urea, are removed by the blood through veins. (iii) Controls body temperature. (iv) Defends against infections. Quantity On an average, a healthy man has about 5 litres of blood in the body, while a woman has about 500 ml less than man. Total volume of blood is said to be 60–80 ml per kg of body weight. Constituents There are two main constituents of blood, such as, solid or cellular part called blood cells (35%) and fluid or humoral part called plasma (65%). The blood cells called corpuscles are of three types: Red Blood Corpuscles (RBC) The RBCs, which are called Erythrocytes are disc-shaped cells concave in the middle and visible under a microscope. They have no nucleus and contain a pigment called Haemoglobin which gives blood its red colour. Haemoglobin is composed of an iron-containing pigment called haema and a protein called globin. In a normal man the amount of haemoglobin is 14–15.6 g/100 cm3 of blood, whereas in a woman it is about 11–14 grams/100 cm3. RBCs are produced in the spleen and the bone marrow at the rate of 1.2 million corpuscles per second. In a lifetime the bone marrow creates about half a tonne of RBC. Life of the RBC is about 100–120 days. Function Haemoglobin in the RBC picks up oxygen in the lung tissues by forming a chemical compound with it. The oxygen is carried to the tissues where it is used in chemical reactions with the products of digestion in order to produce energy. It then combines

with carbon dioxide which is produced in these reactions and returns to the lungs via the heart where the cycle starts again. White Blood Corpuscles (WBC) The WBCs are the ‘soldiers’ of the body’s defence system. They are round, semi-transparent cells containing a nucleus and visible only under a microscope. They are a little larger than RBCs. WBCs are far less numerous than RBCs and in normal conditions the proportion of WBCs to RBCs is 1:400–500. WBCs are little larger in size than RBC. Functions Broadly, the WBCs act as a defence system in the body. There are several varieties of WBC performing specific functions, such as,

(a) Red Blood Corpuscles

(b)White Blood Corpuscles

FIG. 1 The Composition of Blood

In an average human being, blood accounts for 7–8% of body weight. Blood consists of: Plasma: Water (90%), proteins (7%), nutrients, salts, nitrogen waste, carbon dioxide, harmones. Red blood cells (erythrocytes): 54% of which is haemoglobin. Normal count = 4–6 million per mm3. White blood cells (leukocytes): Normal count = 4500–11000 per mm3. Platelets (thrombocytes): Normal count = 150000–300000 per mm3. • Neutrophils are by far the most numerous of the white cells, making up to 65–70% of the total. They attack the invading bacteria and engulf them. • Lymphoctes are second to Neutrophils and constitute about 25% of total WBCs. They produce antibodies. Antibody is a substance produced in the blood in response to an antigen. An antigen is a foreign substance like bacteria, viruses, toxins, etc. Antibodies protect the body against antigen and thus provide immunity against infection.

General Science ▌ 2.333 • Basophils are a type of WBC which secrete an anticoagulant called heparin which prevents clots within the blood vessels. • Eosinophils and Monocytes also assist in the defence mechanism of the body of becoming active against specific antigens. Blood Platelets (Thrombocytes) are tiny, circular or oval colourless cells, a quarter of the size of WBC, which clump together to release thromboplasm. This is a substance which sets off blood clotting, a mechanism that seals tears in blood vessels. Blood platelets live only for a few hours. Plasma Plasma is the watery part of the blood making up to 50–60% of the total. It is a clear, yellow fluid which transports blood cells. Plasma contains about 90% water, protein and inorganic salts. Organic substances such as glucose, amino-acids, fats, urea, hormones and enzymes, occur in the plasma either suspended or dissolved. It also transports nutrients from the small intestine to the body tissue, and return the waste materials to the kidney, where they are filtered out. The protein in plasma includes antibodies to assist in the body’s defence system against disease and infection. Blood Grouping K. Landsteiner, in 1900–1902, classified human blood into four groups A, B, AB and O. The cells of these groups contain the corresponding antigens—A, B, AB, except O cells which contain neither antigen A nor B. For this reason the O group can be given to any of the other groups and therefore, a person possessing the O group is known as a universal donor. In the plasma, there are agglutinins which will cause agglutination of any cell carrying the corresponding antigen, for example group A plasma contains anti-B agglutinins, group B plasma contains anti-A agglutinins, group O plasma contains both anti-A and anti-B agglutinins. Group AB, therefore, is called universal recipient and can receive A, B, AB and O blood groups. Table B.5.1

Blood Group

Can donate blood to

Can receive blood from

A

A, AB

A and O

B

B, AB

B and O

AB

Only AB

*AB, A, B and O

**O

AB, A, B and O

Only O

*Universal receiver/recipient. **Universal donor. Blood Clotting Blood clotting is a body’s defence system to combat bleeding. Plasma contains Fibrinogen— a soluble protein of the blood which produces the insoluble protein called fibrin essential for blood coagulation which is formed in the liver.

Process of Clotting Blood platelets break down in the vicinity of an injury and release an enzyme that causes formation of fibrin from fibrinogen. Fibrin forms a tangled mass of fibres which shrink and form a hard clot to plug the opening of the blood vessel to stop bleeding. After clotting occurs, a straw-coloured fluid called serum is left. This portion of blood retains its liquid form indefinitely. Rh Factor A blood antigen discovered in 1940 by Landsteiner and A. S. Wiener. It is another blood grouping which has important bearing on blood transfusion along with the A, AB and O blood grouping. The Rh factor is an agglutinogen found in RBC of most people and is called Rh+. It was initially found in the rhesus monkey and later on found in man. People who do not have this antigen in their blood are called Rh–. The Rh– blood does not carry anti-Rh antibodies naturally but could synthesize them if sensitized through blood transfusion of Rh+ blood. If Rh+ blood is transfused into an RH– patient, the serum will produce anti-Rh agglutinin. If another dose of Rh+ blood is given, the anti-Rh agglutinin will cause clumping of RBC of the donor’s blood as soon as it enters the patient receiving it. Blood Transfusion The injection of blood from one person (the donor) into the circulatory system of another is called blood transfusion. It is used to make up a loss in volume of blood or to improve its composition. A rapid loss of 3–3.5 litres of blood can be fatal unless it is properly replaced by transfusion. Blood transfusion is done after proper matching of blood groups and the Rh factor. The technique was first developed by James Blundell (London) in 1825. Blood Pressure Is the force exerted by the blood beating against artery walls. As blood is ejected from the left ventricle into the aorta, there is an increase in blood pressure, i.e., the pressure of the blood upon the walls of the blood vessels. As the ventricle begins to relax, the pressure falls. The highest point in the pressure range is called systolic pressure (upper reading) and lowest point in the pressure is called diastolic pressure (lower reading). Blood pressure is measured in a large artery of the arm called the brachial artery by sphygmomanometer. The diastolic pressure is always lower than the systolic pressure. The average systolic pressure of a healthy young man of 20 years of age is about 120 mm Hg and the diastolic pressure about 80 mm Hg. that is, 120/80 is the normal blood pressure. Any systolic pressure over 140 or diastolic pressure over 100 is considered abnormal. High blood pressure may be caused by excitement and low blood pressure may result from malnutrition, etc. High blood pressure is called hypertension and low blood pressure hypotension. Blood pressure depends on many other factors including the strength of the heart-beat, volume and thickness of blood, elasticity of the arteries and resistance to blood flow offered by capillaries.

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The Skeletal System (Bones) • There are 206 bones in the skeletal system of an adult. Bones of hands and feet alone constitute 50% of the total bones in the human body. Bones account for about 1/5th of the body’s total weight. • A new born baby has 300 bones, out of which 94 bones fuse together as it grows. • The largest bone of human body is the femur in the thigh which constitutes about 27.5% of a person’s stature. The average length of this bone is about 50 cm. • The shortest bone in the human body is stapes or stirrup bone in the middle ear (there are three bones in the middle ear; the hammer, the anvil and the stirrup.

Functions (i) (ii) (iii) (iv) (v) (vi)

They are connective tissues in the body They stiffen the body They provide levers upon which muscles work They give shape to the body They protect internal organs The bone marrow produces blood cells (see details on ‘Blood’).

1

7

2 4 3 6 5

8 12 9 10 11 FIG. 2

1. 3. 5. 7.

Frontal Bone Ethmoid Zygomatic Process Parietal Bone

Skull 2. 4. 6. 8.

Sphenoid Nasal Bone Maxilla Squamous Part of Temporal Bone 10. Mastoid Process 12. Temporal Bone

9. Occipital Bone 11. Mandible

Important Bones Skull Bones (Cranium) There are eight hones forming the skull. One frontal (forehead), two parietals (side of the head), one occipital, two temporals, one sphenoid and one ethmoid.

Facial Bones Facial bones comprise 14 individual bones, two lachrymals, two Os-Nazale or Zygomalicum (nasal bones), two check bones, one vomer, two sponge bones, two palate bones, two superior maxillary (Maxilla) and one mandible (lower jaw). Upper Extremities There are 32 individual bones forming the shoulder and the arms. Shoulder bones (i) Two scapula (triangular-shaped shoulder blades) and (ii) Two clavicles (collar bones). Bones of the Arm Each arm has 30 individual bones: (i) One humerus (the largest bone of the arm) (ii) one radius (outer bone) (iii) one ulna (inner bone), (iv) eight carpals, (v) five metacarpals (forming middle hand) and (vi) 14 phalanges (forming the fingers). Backbone (the Spine) Also called vertebrae which encloses the spinal cord (see nervous system). In childhood, the backbone consists of 33 separate bones, but in an adult there are only 26 separate vertebrae as several bones become welded (fused) together. The 26 vertebrae are sub-divided in five regions: (i) (ii) (iii) (iv)

Cervical vertebrae (neck bones)—seven Dorsal vertebrae (ribs support)—twelve Lumbar vertebrae (backbones)—five Sacrum (also called ‘sacred bones’) consists of five vertebrae (25th to 29th) fused together in adults, and (v) Coccyx (four tail bones fused together in adults).

(a) FIG. 3

(b)

Shoulder Bones (a) Scapula (b) Clavicle

The spine An adults’ spine consists of 26 bones called vertebrae. It is divided into 4 sections.

Cervical vertebrae The top seven bones of the spine in the nect.

Dorsal These 12 vertebrae are attached to ribs.

Lumbar vertebrae Five more vertebrae are below the ribs. Sacral The sacrum is made of 5 vertebrae and the coccyx of Coccyx four. FIG. 4

General Science ▌ 2.335

Bone cells 2

6

3

3

1

4 5

5 7 FIG. 5 Pelvis

1. 3. 5. 7.

Sacrum Acetabulum (Hip joint) Ishial tuber osity Pubis

2. Iliac Crest 4. Ischium 6. Ilium

Chest Bones (the Thorax) The thorax, which is a protective casing for the lungs and heart consists of one sternum (breast bone) and 24 ribs. Hip Bone (The Pelvis) Each hip bone, originally consists of three bones, the ilium, the ischium (seat bone) and the pubis. These bones become one in a mature adult and called the os coxae or the hip bone. There is a notable difference between male and female pelvis. The female pelvis is proportionately broader (for maternity reasons) and not so deep as that of male, which is designed for greater strength. Bones of the Leg

Each leg consists of 30 bones, namely,

(i) Femur (the thigh bone: The longest bone of the human body)—one (ii) Tibia (shin bone)—one (iii) Fibula (calf bone) (iv) Patela (knee cap) (v) Tarsus (ankle bones)—seven (vi) Phalanges 14 (toe bones)

Types of Bones The bones of the body fall into four general categories: long bones, short bones, flat bones, and irregular bones. Long bones are longer than they are wide and work as levers. The bones of the upper and lower extremities (for example. humerus, tibia, femur, ulna, metacarpals, etc.) are of this type. Short bones are short, cubeshapted, and found in the wrists and ankles. Flat bones have broad surfaces for protection of organs and attachment of muscles (for exmaple. ribs cranial bones, bones of shoulder girdle). Irregular bones are all others that do not fall into the previous categories. They have varied shapes, sizes, and surfaces, features and include the bones of the vertebrae and a few in the skull.

There are five main types of bones cells in bone tissue. Osteogenic cells respond to traumas, such as fractures, by giving rise to bone forming cells and bone-destroying cells. Osteoblasts (boneforming cells) synthesize and secrete unmineralilzed ground substance and are found in areas of high metabolism within the bone. Osteocytes are mature bone cells made from osteoblasts that have made bone tissue around themselves. These cells maintain healthy bone tissue by secreting enzymes and controlling the bone mineral content; they also control the calcium release from the bone tissue to the blood. Osteoclasts are large cells that break down bone tissue.They are very important to bone growth, healing, and remodeling. The last type of cells are bone-lining cells. These are made from osteoblasts along the surface of most bones in an adult. Bone-lining cells are thought to regulate the movement of calcium and phosphate into and out of the bone.

Bone Composition Bones are composed of tissue that may take one of two forms. Compact, or dense bone, and spongy, or cancellous, bone. Most bones contain both types. Compact bone is dense, hard, and fomrs the protective exterior portion of all bones. Spongy bone is inside the compact bone and is very porous (full of tiny holes). Spongy bone occurs in most bones. The bone tissue is composed of several types of bone cells embedded in a web of inorganic salts (mostly calcium and phospohorous) to give the bone strength, and collagenous fibers and ground substances to give the bone flexibility.

Constitution of Bones The bones in the human body consist of 80% solid matter and 20% water. The solid matter in the bone is derived mainly from cartilage. Impregnation of salts including carbonates and phosphates of lime impart hardness to the bones. In old age, the proportion of lime decreases and therefore bones become brittle and are prone to fracture easily. Constitution of Bones

Organic matter

33.30%

Phosphate of lime

51.04%

Carbonate of lime

1.30%

Flouride of calcium

2.00%

Phosphate of magnesia

1.16%

Soda/Chloride of sodium

1.00%

Human Anatomy Terms The following terms are those which are used to identify the location of parts of the human body in medicine and academic study. These terms are often used to describe a specific portion of a structure or to compare the locations of two different structure. “The hand is distal to forearm” or “the medial portion of the frontal bone contains the frontal sinus” are examples of this. We have organized this list of terms by keeping similar pairs or groups of terms together instead of alphabetical order so that you will find them easier to learn and

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General Knowledge

remember. Superior—toward the head. Inferior—away from the head. Anterior—the front of the body or body part. Posterior—the back of the body or body part. Medial—toward the midline that divides left and right. Lateral—to the side away from the midline. Proximal—closer to the torso. Distal—farther away from the torso. Anatomical position—standing erect, facing the observer, arms are at the sides with palms facing forward. A joint, or articulation, is the place where two bones come together. There are three types of joints classified by the amount of movement they allow: immovable, slightly movable, and freely movable. (i) Immovable joints are synarthroses. In this type of joint, the bones are in very close contact and are separated only by a thin layer of fibrous connective tissue. An example of a synarthrosis is the suture in the skull between skull bones. (ii) Slightly movable joints are called amphiarthroses. This type of joint is characterizid by bone that are connected by hyaline cartilage (fibro cartilage). The ribs that connect to the sternum are an example of an amphiarthrosis joint. (iii) Freely Moving Joints: Most of the joint s in the adult human body are freely movable joints. This type of joint is called a diarthrosis joint. There are six types of diarthroses joints. These are: (a) Ball-and-Socket: The ball-shaped end of one bone fits into a cup shaped socket on the other bone allowing the widest range of motion including rotation. Examples include the shoulder and hip; (b) Condyloid: Oval shaped condyle fits into elliptical cavity of another allowing angular motion but not rotation. This occurs between the metacarpals (bones in the palm of the hand) and phalanges (fingers) and between the metatarsals.

Muscular System Muscles are tissues attached to bones; are composed of fibres; are capable of contracting and relaxing to effect body movement. Total Number of Muscles There are about 630 important muscles in the human body which normally account for 40% of the body weight. There are about 100 joints and about 10,000 km of blood vessels in the human body. Water constitutes about 70% of human body tissues (45 litres). Largest Muscle The bulkiest muscle in the body is Gluteus maximum or buttock muscles. Longest Muscle Sartorius (Tailor’s muscles) is the longest muscle in the body. It has its origin in the upper part of the hip bone, crosses obliquely over the upper portion of the thigh, passes behind the femur and is attached to the tibia bone of the leg. Smallest Muscle The smallest muscle in the human body is the Stapedius which controls the stapes, an auditory ossicle in the middle ear. It is less than 1/20 of an inch and 0.127 cm long. The ear also contains one of the few tissues which has no blood supply.

Main Organs Heart The heart is situated on the left side beneath the left nipple and is enclosed in a tough fibrous wrapping called the pericardium. Its average weight is about 340 grams in men and 255 grams in women. The heart consists of two halves, the left half and the right half divided by a wall called septum. Each half, in turn, is divided into an upper chamber called auricle and a lower chamber called

ventricle. The upper half (auricle) receives blood from veins and lower half (ventricle) pumps blood into the arteries. Heartbeats In a normal, healthy adult contraction and relaxation mechanism of the ventricles results in a heart beat of about 70–72/ min in males and 78–82/min in females. Contraction of ventricles is called systole and relaxation is called diastole. The heart beats more than 2,000 million times and pumps more than 500 million litres of blood during the lifespan of a person Functions Circulation of blood in the body is regulated by the heart. The right auricle receives blood from a large vein and the left auricle receives blood from the lungs. The functions of the heart are: (i) to supply pure blood to all parts of the body; (arteries) (ii) to collect impure blood from organs/tissues; (veins) (iii) to pump impure blood into the lungs for purification; (iv) to collect pure blood from the lungs. Blood Circulation The credit of discovering the circulatory system goes to William Harvey (1578–1657). The heart and blood vessels are concerned with blood circulation in the human body. Blood vessels consist of: Arteries

Thick-walled tubes which carry pure blood away from the heart to the tissues at a high pressure. • Largest artery The elastic aorta which begins at the heart; approximately 2.5 cm in diameter, is the largest artery in the human body. • Smallest artery arteries that are known as arteroiles which are approximately 0.1 mm in diameter. Veins Thin-walled tubes which collect impure blood from the tissues and carry the same to the heart for purification in the lungs. Veins have valves to prevent the backward flow of blood, which is carried at low pressure. Human Temperature

Normal human body temperature is about 37°C (98.6°F). Some people have a norm slightly higher or lower-especially young children. Norms change during the day-usually rising a little by mid-afternoon and falling a little during sleep. Aorta To Right Lung Anterior Vena Cava From Right Lung

Left Pulmonary Artery To Left Lung Pulmonary Vein Left Auricle

Right Auricle

Left Ventricle

Posterior Vena Cava

FIG. 6 Heart

General Science ▌ 2.337

Human Pulse Rate Normal resting pulse rates in healthy persons Beats/min Foetus in utero

150

New born (full term)

140

First year

120

Second year

110

5 years

100

10 years

90

20 years

71

50 years

72

70 years

75

> 80 years

78

The pulmonary artery (the only artery which carries impure blood) carries blood to the lungs from the heart. The pulmonary vein (the only vein which carries pure blood) carries pure blood again to the heart from the lungs to be supplied to the entire body tissues. Both impure and pure blood are forced into two ventricles by contraction of the same. The pure blood from the left ventricle goes into the large aorta and impure blood from the right ventricle goes into the pulmonary artery. Thus the aorta takes blood to various parts of the body. The pulmonary artery takes impure blood to the lungs. When ventricles relax, the auricles are again filled with blood and the same process is repeated. Lungs A pair of spongy organs consisting of elastic tissues situated in the chest cavity and separated from each other by the heart and other contents of the mediastinum. The right lung is larger than the left lung. The right lung weighs approximately 620 g while the left lung weighs about 570 g and together they weigh between 1.18–1.19 kg, in a healthy adult. Functions The lungs are concerned with oxynation (purification) of the blood. Their main function is to purify blood and supply oxygen to it. The entire blood supply (4–4.5 litres) washes through the lungs about once a minute.

Liver

(iii) The synthesis, conjugation and transformation of substances (for example, formation of proteins, dioxication of poisonous substances, production of carbohydrates from proteins, etc.) The bile produced by the liver is an important agent to digestion, especially fats. It contains water, bile salts and bile pigments. It does not contain digestive enzymes and as such does not take part directly in digestion. It contains salts like bicarbonate, glycocholate and taurocholate of sodium. The sodium bicarbonate neutralizes the acid and makes the churned food called chyme alkaline, whereas glycocholate and taurocholate of sodium break down the fats of tissues into small globules. See diagram ‘Vital Organs of Human Body’. Kidneys The two kidneys are situated in the upper posterior abdominal cavity, one on each side of the vertebral column. A kidney is approximately 10 cm long, 5 cm wide and 2.5 cm thick. From each kidney emerges a long channel called the ureter by which the urine passes into the urinary bladder. Each kidney has thousands of minute glands in the form of canals known as uriniferous tubules. Refer to the diagram ‘Vital organs of the Human Body’.

Functions They filter nitrogenous waste of the body from the blood and throw them out in the form of urine. Kidneys are responsible for the removal of excess water, salts and waste products from the blood and maintaining its pH level (pH value is a number used to express degrees of acidity and alkalinity in a solution). To achieve this, blood flows through the wide renal vessels in approximately one quarter of the resting cardiac output, that is, 1.2–1.4 litres/min. The urine expelled by a normal adult is about 1.4 litres per day. The human body contains about 45 litres (70%) water. Spleen Is situated near the diaphragm on the left side of the body. It is the largest lymphatic gland in the human body. Along with the rest of the lymphoid tissue in the body, the spleen plays a considerable role in the development of immunity. However, this gland is not essential for life and can be removed. Its removal tends to increase the susceptibility of infection. Functions The main functions of spleen are: (i) (ii) (iii) (iv)

Formation of lymphocytes and antibodies; Formation of red blood cells (RBC); Storage of RBC; Helps to remove worn-out RBC and foreign particles from the blood stream.

Structure The liver, situated on the right side of the stomach is the largest gland in the human body. It is dark brown in colour and divided into two lobes varying in weight from 1.359 g to 1.812 g. The gall bladder is attached to the liver and stores bile produced by it. The gall bladder has the storage capacity of 30–60 ml of bile.

Human Brain Human brain consists of two parts, viz., the brain lodged in the brain case (skull) and the spinal cord lodged in the vertebral column.

Functions The liver secretes bile, forms and stores glycogen and plays as important part in metabolism of protein and fats. The liver is responsible for: (i) The metabolism of the products of digestion (ii) The storage and release of substances (principally glucose) so as to maintain constant level in the blood

• The weight of the average human brain triples between birth and adulthood. The final weight of the brain in an adult male is about 1.4 kg, and 1.3 kg in the case of a woman; which averages about 3% of body weight of a normal person. In both sexes the brain makes up a similar proportion of the total body weight.

2.338

General Knowledge

10

5

6

4

3

1

8

2

7

9

FIG. 7 Circulation (Schematic Diagram) (Black areas indicate blood flow from the heart; White areas indicate flow back to the heart) 1. Heart: Right Ventricle 2. Heart: Left ventricle 3. Right lung 4. Left lung 5. Circulation through head and upper trunk 6. Superior vena cava 7. Inferior vena cava 8. Circulation through abdominal organs including portal circulation 9. Circulation through lower trunk and legs 10. Aorta Trachea Bronchiole

Central Nervous System The brain and the spinal chord along with nerves constitute the nervous system. The brain consists of: (i) Cerebrum The largest part of the brain consists of two hemispheres separated by corpus callosum, a deep median furrow. It controls voluntary actions and is the seat of intelligence, memory association, imagination and will. (ii) Cerebellum The large mass having ridges and furrows, situated above and behind the mediulla and attached to cerebrum (See diagram). It regulates muscular movement of locomotion. (iii) Medula oblongata It is the lowermost part of the brain which continues as the spinal chord in the vertebral column. It controls involuntary action. (iv) Spinal chord is the elongated, nearly cylindrical continuation of the medulla. It is enclosed in vertebrae and runs down the back. The length of the spinal cord in an average man is about one metre (3.3 ft). (v) Nerves are whitish cords consisting of large numbers if exceedingly fine filaments (nerve fibres) of variable diameter, bound together in bundles by fibrous tissues. Function of nerves Nerve fibres transmit messages (nerve impulses) both to and fro from the central nervous system to the various structures of the body. The fibres which carry impulses from the central nervous system are called efferent. Many of these pass to the muscles to make them contract, and are, therefore, called motor nerve fibres. Those which carry impulses to the central nervous system are called afferent fibres. The information which they transmit from the skin and deeper tissues often evokes subjective sensation within the nervous system, hence they are called sensory nerve fibres. Pulmonary Artery Descending Vena Cava

Arch of the Aorta Pulmonary Vein

Ascending Vena Cava

Alveoli

Heart

Lung Liver

Aorta

Duodenum

Stomach Spleen

Left Kidney Gall Bladder Right Kidney

Descending Codon

Pulmonary Heart Vein Bronchioles FIG. 8 The Lungs • The brain uses about 20% of the oxygen a man breathes, 20% of calories a man takes in and about 15% of body blood. • The average brain contains about 10,000 million neurons— microscopic nerves cells.

Appendix

Right Common Iliac Artery and Vein

Left Common Iliac Artery and Vein Rectum

FIG. 9 Vital Organs of Human Body

General Science ▌ 2.339 Classification of Nerves Cranial nerves are attached to the brain and emerge from the skull (cranium)

Spinal nerves are attached to the spinal chord, which is enclosed in the vertebral column. There are 31 pairs of these, named after the groups of vertebrae between which they emerge—eight cervical, 12 thoracic, five lumbar, five sacral and one coccygeal.

Sense Organs The organs that receive external and interal stimuli and convey the same to the central nervous system are called sense organs. These comprise the following. Sense of Touch This is due to touch corpuscles in the skin, which respond to the physical sensation of pressure, heat, cold, contact, and pain. The skin consists of an outer, protective layer (epidermis) and an inner, living layer (dermis). The top layer of the epidermis is composed of dead cells containing keratin, the horny scleroprotein that also makes up hair and nails. Skin is essential to a person’s survival. It forms a barrier that helps prevent harmful micororgaisms and chemicals from entering the body, and it also prevents the loss of life-sustaining body fluids. It protects the vital structures inside the body from injury and from the potentially damaging ultraviolet rays of the sun. The skin also helps regulate body temperature, excretes some waste produces, and is an important sensory organ. It contains various types of specialised nerve cells responsible for the sense of touch. Important Information on Skin Skin covers the muscles of the body and is said to the largest organ of the human body. In an adult man skin cover is about 1.9 m2 (20 sq ft), a woman has about 1.7 m2 (17 sq ft) of skin. The human skin is composed of two main layers, viz.

(i) The outer layer called epidermis which has no nerves of blood vessels and is continuously renewed. There are pigment grains located in the basal layer of the epidermis which give colour to the skin. (ii) Inner layer called dermis is a thick layer below the epidermis which contains nerves, blood vessels, hair follicles, sebaceous and sweat glands

Poison Gland

Hair grows upward from the root. Lengthening fibres of keratin-filled dead cell arount the semihollow medulla, make up the cortex. A living structure called the b as white lump at the end of a plucked hair) surrounds and feeds the root, which is pocket of the epidermis called the follicle. Hair grows fastest when it is short. The hais is due to melanin, produced by melanocytes in the bulb of the hair follicle incorporated into the keratinocytes that form the hair. Dark hair contains true melanin like that found in the skin, while blond and red hair result from types of melanin that contain sulphur and iron. Hair turns grey when melanocytes age and lose the enzymes necessary to produce melanin. White hair occurs when air bubbles are incorporated into the gtrowing hair. The texture of hair results from the shape of the hair shaft. Straight hair appears round in cross section, wavy hair has an oval shape, and the cross section of a curly hair has an elliptical or kidney-shaped appearance. Nails human fingernails and toenails, are made of dead cells containing the protein keratin. Nails have three parts: the concealed root, the body, which is exposed but attached to skin; and the edge. The nail grows out from the addition of new cells at the root. Most of the nail body appears pink because of blood flowing in the tissue underneath, but at the base of the body is a pal,e semicircular area called the lunula. This area appears white due to an underlying thick layer of epidermis that does not contain blood vessels. The part of the nail that is buried under the skin is called the root. Nails grow as epidermal cells below the nail root and transform into hard nail cells that accumulate at the base of the nail, pushing the rest of the nail forward. Fingernails typically grow 1 mm (0.04 in) per week. oe nails generally grow more slowly. Tongue, the muscular organ in the mouth, is the primary organ of taste and important in the formation of speech and in the chewing and swallowing of food. The tongue, which is covered by a mucous membrane, extends from the hyoid bone at the back

Mucous Glands Stratum Corenum

Epidermis Stratum Spongiosum

* * *

* * * * *

*

* * * *

Stratum Compactum

* * * * *

Stratum Malpighian Chromatophores

Blood Vessels Connective Tissue (Vertical Fibres) Connective Tissue (Horizontal Fibres) Blood Vessel

Lymph Sinus

FIG. 10 Cross-section of Skin

Body Muscles

2.340

General Knowledge

of the mouth upward and forward to the lips. Its upper surface, borders, and the forward part of the lower surface are free; elsewhere it is attached to adjacet parts of the mouth. Extrinsic muscles attach the tongue to external points, and the intrinsic muscle fibres, which run vertically, transversely, and longitudinally, allow it a great range of movement. The upper surface is convered with small projections called papillae, which give it a rough texture. The colour of the tongue—usually pinkish-red, but discoloured by various diseases—is an indication of health. The tongue serves as an organ of taste, with taste bud scattered over its surface and concentrated towards the back of the tongue. In chewing, the tongue holds the food against the teeth; in swallowing, it moves the food back into the pharynx, and then into the oesophagus when the pressure of the tongue closes the opening of the trachea, or windpipe. It also acts, together with the lips, teeth, and hard palate, to form word sounds.

the various siuses in the ethmoid, sphenoid, frontal, and superior maxillary bones communicate through narrow apertures. The margins of the nostrils are usually lined with a number of stiff hairs (vibrissae) that project across the openings and serve to arrest foreign substances, such as dust and small insects, which might otherwise be drawn up with the current of air intented for respiration. The skeleton, or framework, of the nose is partly composed of the bones forming the top and sides of the bridge, and partly of cartilage. On either side are an upper lateral and a lower lateral cartilage, to the latter of which are of four small cartilaginous plates, termed seasmoid cartilages. The cartilage of the septum separates the nostrils and, in association posteriorly with the perpendicular plate of the ethmoid and with the vomer, forms a complete partition between the rigfht and left nasla fossae. The nasal fossae, which consititute the interla part of the nose, are lofty and of considerable death.

Sense of Smell This is due to the olfactory receptors in the upper porton of the nose. These respond to the air molecules through which smell is recognised in the brain. The sense of taste and smell are closely linked. Taste and Smell

Taste (gustation) and smell (olfacation) are two senses that work together in humans to determine the flavour and palatability of foods and beverages and to signal the presence of dangerous gases and toxic foodstuffs. In many animals, these senses play a critical role in locating food, detecting predators, and recognizing appropriate mating partners. Despite their close association, taste and smell are anatomically and functionally distinct. In the case of taste, chemicals that evoke sweet, sour, bitte and salyt sensations stimulate taste-bud receptors located in the throat and the tongue and palate. The stimulation triggers nerve cells to send signals to the brain stem located in hte base of the brain. Odoregiseer in the brain when airbone chemicals stimulate receptors located on the olfactory epithelum, a smell patch of tissue position high in the nose. These receptor cells send signal into the brain through a section of skull located between the eyes, therm the cribriform plate. Signals are conducted from overlying ofactory bulb higher brain centers associated with arousal, emotional phenomena, and nition. Fewer factors in taste perception. Crette smoking tends casue hypogeusia, lessened taste function dulling the ability to taster-tasting chemicals. So medications produce geusia,l or distortion of the function, by inducing pleasant or persistent to sensations. The nose, the main organ of smell, is also part of the apparatus responsible for respiration and voice. Cosidered anatomically, it may be divided into an exteranl portion—the visibly projected portion, to which the term nose is popularly restricted—and an internal portion, consisting of two principal cavities, or nasal ffossae (separated from each other by a vertical septum), subdivided by spongy or turbinated bones that project from the outer wall into three passages (meatuses) with which

FIG. 11 Anatomy of the Nose

The uppermost portion of the human respiratory system, the nose is a hollow ari passage that functions in breathing and in the sense of smell. The nasal cavity moistens and warms incoming air, while small hairs and mucous filter out harmful particles and microorganisms. This illustration depicts the interior of the human nose. They open in front through the nostrils and at the rear end in vertical slits on either side of the upper pharynx, above the soft palate a near the orficies of the Eustachian tubes, leadin to the tympanic cavity of the ear. In the olfactory region of the nose, the mucous membrane is very thick and coloured by a brove pigment. The olfactory nerve, or nerve of smell, terminates in the nasla cavity in seven small branches; these ramify in the soft mucous membrane and end in tiny varicose fibres that in turn terminate in elongated epithial cells projecting into the free surface of the nose. Sense of Sight

Eye The eyeball, almost a perfect sphere of 2.5 cm diameter consists of three coatings, viz., sclerotic (upper coating) continuous with the transparent cornea in front. Within the sclerotic is the

General Science ▌ 2.341 middle layer call choroid and the inner line called retina. The membrane covering the cornea is conjuctiva. The cavity of the eyeball is filled with aqueous humour and vitreous humour and in front there is a convex lens. The lens attached to the ciliary body is situated behind the iris. The iris is the coloured portion of the eyeball situated behind the cornea, which automatically adjusts the size of the pupil (the circular opening or diaphragm in the iris through which light passes) to the intensity of light falling on it. In bright light the iris automatically shuts tighter, reducing the amount of light entering the pupil. In order to form an image on the retina, light must pass through all the media of the eye (cornea, aqueous humor, lens, vitreous humor) and be brought to a focus on the retina to form an upside down image on it and the process of interpretation takes place in the brain. Ciliary Body Sclerotic

Suspensory Ligaments

Choroid

Iris

Retina

Lens Pupil Optic Nerve

Cornea Conjunctiva

Suspensory Ligaments

Virteous Humour

Blind Spot

FIG. 12 Eye

Hearing Sound waves or vibrations in the air—are collected by the pinna, which acts as a sound collector, and travel to the ear-drum, which is set into vibration. The ossicles transmit the vibration to the inner ear, where they cause waves in the fluid (called Perilymph) filling the coiled tube of cochlea, a tube coiled like a small, snail shell of 2 1/2 turns. These waves vibrate the basilar membrane which divides the tube lengthways. The membrane it tuned so that each part responds to a different frequency, and along it are some 10,000 sensory structures—the rods or corti. These structures convert the basilar membrane’s vibrations into nerve impulses, which reach the brain along the auditory nerve where they are intercepted into sound. Maintenance of balance The second function performed by the ears, is the maintenance of balance which is achieved by three semi-circular canals and the utricle and saccule, all of which are filled with fluid. The semi-circular canals detect movement in three planes of right angles, and so provide a stream of information for the brain on movements of the head. The utricle and saccule detect tilting of the head and combine with the canals to give rapid control of balance in normal circumstances. Digestive System The process of converting food into energygiving substances is carried out by the digestive system. The digestive system comprises the alimentary canal and the associated digestive glands like liver and pancreas.

Alimentary Canal The entire tube-like structure starting from mouth to the anus, is called alimentary canal. It includes various digestive organs, for example, mouth, gullet, oesophagus, stomach, small and large intestine, rectum and anus (See Fig. 14). Digestive Organs

Pinna

Outer Ear Middle Ear

Inner Ear

External Auditory ar le E ic Meatus nic ss pa e O m n Ty bra em M

Ear Ossicle Eustachian Tube

FIG. 13

Teeth The teeth cut and grind the food with the help of saliva secreted by the mouth.

Hair Cell

r ete Ur anal C

Round Window

Upper Membrane Basilar Membrane Lower Canal Organ of Corti

Auditory Nerve Fibres

Hearing Mechanism

Sense of Hearing The ears have two main functions, viz.: (i) Defection and analysis of sound, and (ii) Maintenance of balance of the body. Ear Broadly, the ear is divided into three main regions—the outer ear, the middle ear and the inner ear.

Milk teeth Appear by the age of 7 months and are 20 in number. Adult teeth Milk teeth drop and 32 permanent teeth begin to replace them from the 6th year onwards. There are four chisel-shaped incisors, two conical premolars and six molars for crushing, and grinding the food. The two wisdom teeth, the rearmost, develop at a later stage and are largely nonfunctional. In the mouth salivary glands secrete saliva which moistens the food and ptyalin ferment contained in it acts on carbohydrates (See Digestive Secretions). Gullet Also called oesophagus, is a 25 cm long duct system covering mouth and the food pipe pharynx. The food is carried to the stomach thereon by peristalsis movement (contraction of the wall of the pipe). Stomach A warehouse where food can be stored to await the main process of digestion. Gastric juices produced in the stomach help in digestion of food.

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Small intestine Measuring 6-7 m long and 2 cm in diameter is arranged in coils. Here the food from the stomach is mixed with bile and pancreatic juice and moves forward through peristalsis movement. Part of the intestine near the stomach is called the duodenum where bile and pancreatic juices digest the food. Large intestine A 1.4 m long tube called colon. It receives undigested material from the small intestine and absorbs water. The remaining waste material is sent to the rectum and discharged from the anus.

Process of Digestion Chewed food is converted into food pulp called chyme. After being processed by bile and other secretions it becomes an emulsion called chyle. The end product is absorbed through the walls of the small intestines and taken into the blood. The undigested waste part is excreted as faeces. Digestive Secretions Saliva Produced by salivary glands in the mouth (situated under the tongue) saliva helps to lubricate the chewed food making it fit for swallowing. It also contains a ferment called ptyalin which acts on starches (carbohydrates) and hydrolyzes the same into maltose. Gastric juice Produced in the stomach, contains hydrochloric acid and enzymes (See section on ‘Secretion from Glands’). Hydrochloric acid kills bacteria present in food and the enzymes help to digest the food. It contain 90% water, 5% enzymes (pepsin, renin and gastric lipase) and 5% hydrochloric acid. Pepsin and hydrochloric acid turn protein into peptones and proteoses. Renin curdles milk to form casein and gastric lipase acts on fat to convert it into glycerol and fatty acids.

1. Oesophagus 2. Liver 3. Gall-bladder 4. Stomach 5. Duodenum 6. Pancreas 7. Transverse colon 8. Ascending colon 9. Descending colon 10. Small intestine 11. Caecum 12. Appendix 13. Rectum About Human Teeth

The approximate times of eruption and shedding of teeth. Milk Eruption Shed Incisor 1 6–10 months 6–7 years Incisor 2 8–12 months 7–8 years Canine 16–22 months 10–12 years Molar 1 13–19 months 9–11 years Molar 2 25–33 months 10–12 years Permanent Eruption Incisor 1 7–8 years Incisor 2 8–9 years Canine 10–12 years Premolar 1 10–11 years Premolar 2 10–12 years Molar 1 6–7 years Molar 2 12 years Molar 3 17–21 years Note: The lower teeth usually appear before the equivalent upper teeth.

Pancreatic juice Produced in the pancreas, meets the food pulp in the duodenum along with bile juice (alkaline) from the liver plus intestinal juice from the small intestine. Bile produced in the liver is stored in the gall bladder. Pancreatic juice contains several enzymes which act on proteins, carbohydrates and fats. Amylase catalyzes hydrolysis of starch, lipase acts on fat which is converted into glycerol and fatty acids.

1

2

2

3

4

4 5

6

6

7

8

8

1 2 4 3 5

6

8

7 9

11

10

1

3

5

7

FIG. 15 Teeth 12

13

FIG. 14 Alimentary Canal (from lower end of oesophagus)

1 . Central incisors 2. Lateral incisors 3. Canines 4. 1st premolars 5. 2nd premolars 6. 1st molars 7. 2nd molars 8. 3rd molars

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Cerebrum Cerebellum

Pituitary Gland

Medulla Oblongata

Ductless glands Also called endocrine glands or internally secreting glands. They secrete hormones (See ‘Secretions from the Glands’) directly into the blood stream in response to instructions from the brain. The blood then carries them around the body where they control internal chemical processes. Hence they are linked with the nervous system and also with the circulatory system and their secretions are known are hormones or enzymes. Secretions from Glands

Fig. 16 View of Exterior Brain Showing Convulsion of the Cerebrum and the Cerebellum

The Reproductive Organs A human reproductive organs or sex organ, or primary sexual characteristic, narrowly defined, is any of those parts of the body (which are not always bodily organs organs according to the strict definition) which are involved in sexual reproduction and constitute the reproductive system in an complex organism; namely: Female Vulva (notably the clitoris), vagina (notably the cervix), labia, uterus, fallopian tubes, ovaries, Skene’s glands, Bartholin’s glands.

Hormones Chemical substances formed in an endocrine gland (ductless glands) and conveyed by the bloodstream to a specific organ or tissue elsewhere (sites remote from that of its production) whose function they modify by means of chemical activity. Hormones are produced in extremely small quantity but their action is very rapid. They control growth and various life processes of the body. For example, hormones produced in the stomach where they regulate digestion, in the kidneys where they control production of red blood cells (RBC) by the bone marrow, in the placenta where they regulate pregnancy, and in the ovaries and testes where female and male hormones are produced, respectively, which are responsible for female and male characteristics like voice, facial and body hair, stature, appearance, etc.

Male Penis (notably the glans penis and foreskin), testicles, scrotum, prostate, seminal vesicles, epididymis, Cowper’s glands. The latin term genitalia is used to describe the sex organs, and in the Englilsh language this term and genital area are most often used to describe the externally visible sex organs or external genitalia: in males the the penis and scrotum, in females the vulva. The other parts of the sex organs are called the internal genitalia. A gonad is a sex organ that produces gametes, specifically the testes or ovaries in humans. Organs of sexual anatomy originate from a common anlage and differentiate into male or female sex organs. Each organ in one sex has a homologous counterpart in the other one.

Hypothalamus Pituitary Thyroid Parathyroids Thymus Stomach Pancreas Adrenals Kidneys Alimentary Canal Pancreas

Glands Organs of the human body which manufacture some liquid products which are secreted from the cells (both internal and external secretions).

Testes Ovaries

Largest Gland

FIG. 17 Position of the Endocrine Glands

The largest gland in the human body is liver. Types of Glands There are two types of glands, namely, (i) Ducted glands and (ii) Ductless glands. Ducted glands Ducted glands secrete their product through well defined ducts, for example, (a) Liver Secretes bile in the stomach (b) Lachrymal Secretes tears in the eyes. (c) Salivary Secretes saliva in the mouth. (d) Sweat glands Are numerous small, tubular glands that are found nearly everywhere in the skin. They secrete sweat, a watery fluid containing sodium chloride and urea through pores. (e) Spleen See under ‘Main Organs’.

Table B.5.2: Important Hormones Hormone

Produced by

Action

Adrenalin

Adrenalin

Blood pressure control

Insulin

Pancreas

Sugar metabolism

Oestrogen

Ovary

Affects female organs

Testoterone

Testes

Affects male reproductory function

Thyroxin

Thyroid

Growth and metabolic rate

Enzymes Enzymes are complex chemical substances (biochemical catalysts) consisting wholly or in large part of proteins (soluble colloidal proteins) produced in the living cells. Enzymes in

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very small amounts hasten (catalyse) change in other chemical compounds, itself being unaltered at the end of the reaction. They are responsible for biochemical reactions in the protoplasm of blood (See ‘Bile Juice’, ‘Gastric Juices’). Any one type of enzyme can only cause such change in a few types of chemical compounds with which it is said to be active. Table: Important Digestive Enzymes Enzyme

Produced by

Converts

Amylase, Salivary glands in Ptylain mouth

Starch to sugars (Maltose)

Peptin

Gastric glands in stomach

Proteins to amino acids (Peptones)

Trypsin

Pancreas in abdomen

Proteins to Peptones

Amylase

Pancreas

Carbohydrates to Maltose

Lipase

Pancreas

Fats to fatty acids and glycerol

Erepsin

Intestinal glands in small intestines

Peptides to amino acids

Maltase

Intestinal glands

Maltose to glucose

Lactase

Intestinal glands

Lactose to glucose

Sucrase

Intestinal glands

Sucrose to glucose

Human Cell, Genes and Heredity Human cell The smallest unit of life that is capable of independent existence. It occurs in all living organisms (both plants and animals). The average diameter of a cell is between l/50 and 1/100 millimetre. The largest human cell is the ovum. Every square inch of skin of the human body contains about a million cells. The brain is made of 30 million cells. Our blood contains about 20 trillion cells. The whole cell is surrounded by a strong membrane called the membrane, which is approximately 500,000th of an inch in thickness. Each cell has a dense kernel called nucleus which controls the activities of the rest of the cell. Outside the nucleus is cytoplasm, a gelatinous substance The nucleus is enclosed in a thin but strong membrane called nuclear envelope, and contains chromosomes. Chromosomes are carriers of genes (hereditary traits). They are threadlike bodies found in the nucleus of the cell. They are always found in pairs and vary in number in each species of living organisms. Human cells have 23 pairs (46 in all) of chromosomes. In every chromosome, there are protein molecules and two kinds of nucleic acid molecules called RNA (Ribose Nucleic Acid) and DNA (Deoxyribose Nucleic Acid). The DNA is concentrated in the chromosomes while RNA is concentrated in the nucleoli, both of which are present in the nucleus (Fig. 20).

Table: Ductless Glands Name Of Gland

Located

Thymus

Gland-like structure situated in front of the trachea behind the top of the breastbone

Secretion

Function

Thyroid

A two-lobed gland found in all vertebrates located in front of and on either side of the trachea

Thyroxine which contains 65% iodine. Rate of production is regulated by the pitiutary gland

Regulates metabolism. Deficiency causes dwarfness in children called cretinism and myxodema or gill’s disease in adults. Its deficiency also causes goitre or Grave’s disease, that is, enlargement of the thyroid gland.

Para-Thyroid

A gland in males surrounding the urethra at the neck of the bladder. The urine passes through it

Para-thyroid hormones also called parathormone

Raises the level of calcium in blood. Deficiency results tetany. It regulates calcium in blood and growth. Removal of these glands may cause death.

Prostrate

A gland in males surrounding the urethra at the neck of the bladder. The urine passes through it

Prostatic secretion which is mixed up with sperms and other fluids to make up the semen

Secretion is essential to the general functions of the body related to blood pressure and sexual potency. In men of 50 and above enlargement of this gland is a common disorder necessitating surgical removal.

Gondas

The reproductive glands, that is, ovaries in female, testes in male

Gonadotrophins

Relates to reproductive system of the body, increases secretion of sex hormo

Adrenal

Two small dissimilar glands, one above each kidney, consisting of cortex and medula.

Cortex secretes corticosteroid and medula secretes sex hormones including adrenaline and cortisone

Causes acceleration of the breath, contraction of small blood vessels, dilation of the pupil and increases metabolic rate. Also causes heightened emotion and a sudden increase in physical strength, as during fear or anger, controls sexual growth and almost all functions of the body. It regulates blood pressure and functions of the heart. Its rise causes Addison’s disease.

In early childhood plays some part in building resistance (immunity) to disease and physical development.

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Name Of Gland

Located

Secretion

Function

Pancreas

Soft irregular-gland lying near the stomach, 15 cm long

Secretes pancreatic juice into the duodenum and contains the Islets of Langerhans: produces insulin, a clear secretion alkaline in nature containing enzymes

Aids in digestion of proteins, carbohydrates and fats. Its failure results in diabetes.

Pituitary

A small oval-shaped gland attached to the base of the vertebrae in the brain, also called hypophysis

Called the Master Gland as it controls the other endocrine glands and influences growth and metabolism. The hormones secreted by it control mental, sexual and physical growth of the body. Deficiency causes dwarfism. An excess results in much increased body growth called Acromegaly.

Genes The gene is a hereditary unit located in chromosomes. Transmission of physical and mental traits from parents to offspring is through the base unit of genes. The gene consists of DNA molecules in different patterns and hereditary traits are determined by these patterns, and they also mastermind the process of life. In 1896, Gregor Mendel proved that certain hereditary factors operate in all biological species and it was Wilhelm Johannsen (a Danish biologist) who gave them the name genes.

Isolation of Pure Gene In 1969, scientists, at Harvard University isolated and photographed the pure gene. Artificial Gene Dr Hargobind Khorana, an Indian-born American scientist (who received the Nobel Prize in 1968) successfully created an artificial gene in a test tube in 1970 by protein synthesis.

to the discover of DNA in 1953 (responsible for carrying hereditary traits) followed by the development of the technique of manipulating DNA in a living cell by recombinant DNA in 1973. The technique is known as recombinant DNA or gene splicing. The genes cut from the donor DNA using chemicals called enzymes that act as molecular knives. The isolated gene is then inserted into the DNA of a bacterium commonly the intestinal bacterium called E. Coli, again using an enzymatic reaction. The hybrid DNA thus produced is put back into E.Colli growing in the culture medium; has the hybrid DNA in it, nucleus and is capable of exhibiting the traits and characteristics coded by the foreign gene. The four DNA bases A, G, T, and C, like the letters of the alphabet, can be used to store information. This genetic information is passed on via RNA or ribonucleic acid (consisting of the four bases adenine, guanine, cytosine, and uracil [u]) which prov-ides a template for the assembly of amino acids in a particular sequence, thereby building a protein. A group of three DNA or RNA bases is known as a triplet of codon, and codes for a particular amino acid. Information is passed from DNA to RNA by complementary pairing: A pairs only with U, and G only with C.

Genetic Code is the code of life by which inherited characteristics pass from generation to generation. It is expressed by the molecular configration of chromosomes of cells in which the code bearing material is DNA. Genetic Engineering Also called bio-engineering, is the artificial manipulation of parental genes to form combinations to give birth to a new organism with predetermined, mental and physical characteristics. The beginning of bio-engineering can be traced back Table:

Genectic code in RNA triplets

1st base

U

C

A

2nd base

3rd base

U

C

A

G

Phenylalanine

Serine

Tyrosine

Cysteine

U

Phenylalanine

Serine

Tyrosine

Cysteine

C

Leucine

Serine

–

–

A

Leucine

Serine

–a

Tryptophan

G

Leucine

Proline

Histidine

Arginine

U

Leucine

Proline

Histidine

Arginine

C

Leucine

Proline

Glutamine

Arginine

A

Leucine

Proline

Glutamine

Arginine

G

Isoleucine

Threonine

Asparagine

Serine

U

Isoleucine

Threonine

Asparagine

Serine

C

a

a

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1st base

G

2nd base

3rd base

Isoleucine

Threonine

Lysine

Arginine

A

Methionine

Threonine

Lysine

Arginine

G

Valine

Alanine

Aspartic acid

Glycine

U

Valine

Alanine

Aspartic acid

Glycine

C

Valine

Alanine

Glutamic acid

Glycine

A

Valine

Alanine

Glutamic acid

Glycine

G

Chain termination.

a

Sex Determination Out of 46 chromosomes (23 pairs) only 44 (22 pairs) are true homologous (similar). The other two, the X and Y are sex chromosomes. Two X chromosomes produce a female, and an X and a Y chromosome produce a male. Hence the sex of a child depends entirely on whether the female egg is fertilized by an X-carrying sperm or Y-carrying sperm. Twins Fraternal Twins If two eggs are released by the ovaries at the same time due to some irregularity and both get fertilized, they develop into two separate babies. Such twins may be of the same sex or of different sex. Identical Twins Develop from a single fertilized egg during the first division of the zygote (egg) into two similar cells that develop into two separate embryos. Since they develop from a single zygote, these twin have the same genetic characters and are always of the same sex. Identical twins who are born joined together, generally at the head, chest or hip are called Siamese twins. Siamese twins occur very rarely and can usually be separated surgically unless a vital organ such as heart is shared by both. The first such twins who received wide publicity came from Siam (now Thailand) in the 19th century.

Human Genetics Human Genetics is that branch of medical science, which deals with the study of hereditary material passing from one generation to another. The study leads to a better understanding of the way by which this hereditary material develops into physical characters under the influence of intrinsic and extrinsic environmental factors. In other words, the science deals with transmission of physical, mental and biochemical characteristics from parents to offsprings. The science also reasons out why an offspring will not be the exact replica of the parents. The tendency on the part of the offsprings to produce parental features is known as heredity. This science is the youngest branch of medical sciences. During the last six or seven decades, a phenomenal growth of this science has occurred and thus opened up biological thinking towards a better understanding of some of the problems of life as far as health of the future generation is concerned. Thus, in the present era Genetics is one of the new and most significant developing branches of medical sciences. There are two main components of this science:

Heredity Study of similar traits passed from parents to their offsprings. Variation Study of traits influenced by the internal or external forces so that no two individuals are exact replicas of each other. For human beings, the knowledge of Genetics can be helpful in many ways: To understand the underlying cause of the disease and the means of transmission; to understand the reasons of normal variations; to apply the knowledge to the possible means of preventing genetic disorders through counselling and antenatal diagnosis; this knowledge can be applied to solve legal problems like disputed parentage or traits of murderer, etc. The study of Genetics can be subdivided into following branches:

Cytogenetics This field gives the knowledge of structure of nucleus of cell and its part. that is, chromosomes, which normally lies in the condensed form in the nucleus. Molecular and Developmental Genetics The field dealing with molecular structure of the Genetic materials, its significance during embryogenesis and functional state of individuals during normal as well as abnormal conditions. Eugenics The field dealing with clinical applications of principles of heredity for betterment of mankind. Physiological Genetics The field elucidates the normal functions of various organelles of a cell governed by the genetic material. Population Genetics The field deals with the distribution and behaviour of genetic material in a particular population so that the genes and the genotype frequencies are maintained or changed. Clinical Genetics The field deals with the application of knowledge of genetic material responsible for certain diseases and their transmission from one generation to another. The field also includes, the investigative and preventive methods adopted for diagnosis and preventing their disorders. Environmental Genetics This field deals with the manipulation of genetic material in living form by recent advanced technologies of genetic engineering and recombinant DNA. Radiation Genetics This field deals with the effects of various types of radiation on genetic material producing various diseases.

General Science ▌ 2.347 Biochemical Genetics The field deals with the biochemistry of genetic material for normal metabolic processes. This also includes immunogenetics and blood groups, that is, genetic material concerned with the mechanisms of producing antigenes and antibodies.

(the enzymes) present in the digestive juices. Along with their absorption, the entry of water, minerals, vitamins and other diffusible molecules of food also take place. The absence of one of the digestive enzymes leads to serious disorders.

Genetic Counselling The branch deals with the immediate and practical preventive and social health promotional measure, in problems of genetic, disorders. The counsellor named as Medical Geneticist can deal with matters pertaining to the causation, incidence, risk and recurrence of inherited defects, radiation hazards, mutant viral strains in biological warfare. Some newer drugs including the anticancer drugs which can cause genetic problems duo to mutation, also come under the purview of the counsellor.

The Chemistry of Respiration During respiration oxygen is taken into lungs and it diffuses into blood across the membranes. It combines with the haemoglobin of the red cells and in this form, it is carried to the tissue where it is released for the oxidation of the food to liberate energy. The carbon dioxide formed is equal in volume to that of oxygen and passes from the tissue to the bloodstream and is exhaled from the lungs.

Medical Biochemistry Biochemistry is important to physiology; both psychology and biochemistry overlap and merge. Pathological conditions are caused by deranged chemical composition and functioning of tissues and many of the problems of pathology occur from the chemical viewpoint. Biochemistry deals with the chemistry of living organisms—both plants and animals. Protoplasm is the basis of all forms of life. Although the protoplasm of each different kind of cell in each kind of animal or plant is different in characteristics yet the chemical composition, organization and chemical processes in these many different forms of protoplasm are similar in many respects. Biochemistry can be divided into four branches: (i) Animal Biochemistry, (ii) Plant Biochemistry, (iii) Biochemistry of Microorganisms, (iv) Medical Biochemistry. Medical Biochemistry deals with the following: The Chemistry of Tissues and Foods Since the foods are largely derived from animal and plant tissues, the study of chemistry of the one is identical with the other. The components of tissues are carbohydrates, fats, proteins and nucleic acids. Carbohydrates provide a greater part of the energy needs of the body. Proteins are necessary for the growth and maintenance of the positive nitrogen balance. Improper utilization of glucose leads to the development of a disease called ‘diabetes mellitus’. Essential fatty acids are required for maintaining normal health. Nucleic acids (DNA and RNA) of nucleus and cytoplasm, respectively, are involved in the regulation of genes. The Chemistry of Digestion and Absorption The focal is chiefly composed of large molecules of starch, proteins and fats. These large molecules are digested into smaller molecules like glucose, amino acids, fatty acids for absorption by organic catalysts

The Chemistry of Blood The blood carries food to the tissues and waste products from them to the excretory organs. The hormones, produced by various glands pass into blood and through its circulation these hormones reach the specific tissues. The blood distributes heat from one part of the body to another and exerts a cooling effect. The blood also distributes water and salts properly and maintains the acid-base balance of the body. It contains substances that combat infection by microorganisms. The composition of blood is normally constant but it is dynamic. When one of the substances present in it, gets increased or decreased it causes pathological conditions. The Chemistry of Tissue Metabolism The oxidation of foodstuffs in tissues occur by chemical processes with the liberation of energy and water. This is a very complicated process in biochemistry. Several diseases occur in the disorder of the metabolism of these foodstuffs. The Chemistry of Glands of Internal Secretion The control through glands of internal secretion is barely dependent on the activities of hormones and nervous system. The hormones of one gland regulate the activities of other glands. Overproduction of hormones also leads to serious disorders. That is why biochemistry of glands of internal secretion is treated with utmost significance. The Chemistry of Excretion The excretory organs—kidney, intestine skin, etc.—remove the decomposition products of tissue and food in order to make the composition of body fluids constant. The most important decomposition products are urea, uric acid, and creatinine formed from proteins; carbon dioxide formed from carbohydrates, fats and proteins. The kidneys and lungs chiefly perform the excretory functions of the body.

HUMAN DIET Food Nutrients

Source Rice, wheat, honey, sugar, potatoes, dry beans, carrots, etc.

Carbohydrates Organic compounds containing carbon, hydrogen and oxygen which constitute sugars (starches, cellulose, glucose, etc.) and supply calories for body activity. Formed in nature by photosynthesis (See ‘Biology section’).

Deficiency of carbohydrates causes weight loss and weakness. More than 50% of the energy content of a common diet comes from carbohydrates. A normal person needs about 400–500 g of carbohydrates in a day.

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Proteins Are highly complex nitrogenous compounds found in all animal fats. Proteins are hydrolyzed in the body to produce amino acids and are essential for growth and repair of the body tissues. They also provide energy when calories are deficient.

Source

Milk, fish, meat, eggs, nuts, dal, etc.

Deficiency causes loss of weight, slow growth and unhealthy swelling on various parts of the body. For a normal person about 70 g of proteins a day are recommended. Vitamins Vitamins are essential food factors, chemical in nature and present in most foods. The body needs only tiny quantities of vitamins ranging from about 30 mg a day of Vitamins C down to 1 microgram of Vitamin B12.

Minerals Some of the important minerals which form part of the food are: Calcium

Function Builds strong bones and teeth, good nail, normal nerve action, rhythmic heart beat and an overall healthy body. Source

Milk, fish, dals, ragi, eggs, cheese.

Requirement

1200 mg per day.

Iron Function Essential for blood formation (haemoglobin), supply of oxygen to cells, healthy complexion, better memory and warmth of hands and feet.

Table: Vitamins Vitamin

Function

Properties

Deficiency Results in

Source

Vitamin A

Anti-infective. Essential for healthy skin and aids night vision. Also assists in body growth.

Within the body can be synthesized from carotene—yellow pigment present in food is stored in the liver. It is soluble in fat.

Poor growth, rough and dry skin and prone to infection of the same. Severe deficiency may cause night blindness.

All animal fat, carrots, eggs, mango, papaya, apricots, tomatoes, spinnach, cod liver oil, whole milk.

Vitamin B Complex Thiamine

Antineurotic, anti-beri-beri, health of the nervous system, steady and continuous release of energy from carbohydrates, antipellagra.

Destroyed by excessive heat. Helps in metabolism of carbohydrates.

Brewer’s yeast, cereals, Poor growth, neuritis, berieggs, fruit, liver, meat, beri, fissures at corners of pulses. mouth and on tongue, skin diseases, Pellagra, diarrhoea.

Phriodoxine Vitamin B6

Protein metabolism.

Relieves postradio therapy, nausea, vomiting.

Nervousness and insomnia.

Folic Acid

Assists in production of red blood cells.

Can only be absorbed Some form of macrocytic anaemia. in the presence of gastric juices, stored in the liver.

Green vegetables, liver.

Cyanocobalamin Vitamin B12

Essential for red blood cell formation.

Destroyed by cooking in air

Perniaois anaemia.

Liver and all other food containing vitamin B Complex e.g., cereals, eggs, pulses.

Vitamin C (Ascorbic Acid)

Formation of bones, teeth and collagen

Produced in the body by action of sunlight on the skin

Sore mouth and gums, capillary bleeding, Scurvy, delayed healing of wounds.

Fresh fruit, (oranges and lemons) cow’s milk, fresh vegetables.

Vitamin D (Calciferol)

Antirachitic, assists in absorption and metabolism of calcium and phosphorus, assists in body growth.

Rickets in children, osteomalacia and osteoporosis in adults.

Oily fish, dairy products, egg yolk, milk, sun’s rays, butter.

Vitamin E Tocopherods

Helps in reproduction and sex, assists cell respiration, metabolism of fats and starches.

Cereals, fruit, meat, pulses.

Wheat germ, milk, cereals, egg yolk, beef, liver, mutton, spinn-ach, soyabeans.

General Science ▌ 2.349

Vitamin

Function

Vitamin K (Menadione)

Antihaemorrhagic, essential for production of prothrombin (blood clotting).

Green leafy vegetables Thought to interfere with reproduction and also cause oats, fish, rye, peas. certain degenerative diseases of the nervous system, liver damage, increased tendency to haemorrhage, also causes abortion, menstruation irregularities.

Vitamin P

Functions closely with Vitamin C and helps in its function, keeps blood

Delayed clotting of blood. liver damage, increased tendency to haemorrhage.

Source

Properties

Green leafy vegetables, liver, egg-yolk.

Balanced Diet

Phosphorus Function Necessary for energy production, glandular secretion, building and maintaining bones and teeth in the form of calcium phosphate. Source

Fish, meat, milk, cereals.

Requirement

1200 mg per day.

Iodine Maintains stamina, endurance and healthy thyroid

Function glands. Source

Iodized table salt.

Requirement Deficiency

In almost all sources of Vitamin C like oranges lemons, and other citrous fruit.

Requirement 3,500 mg of chloride and 3000 mg sodium per day.

causes anaemia and weakness.

Deficiency

Source

Source Table salt.

18 mg per day.

Requirement

Deficiency Results in

0.15 mg per day. causes goitre.

Chlorine Function Helps in digestion of food and transmission of carbon dioxide by red blood cells (RBC).

A balanced diet is one which contains all the necessary nutrients in the right proportion required for growth and maintenance of the human body. It is related to one’s age, health and occupation. For example, a diet which contains carbohydrates, vitamins, mineral salts, enough fat (but not too much), and proteins in appropriate proportion necessary for growth and maintenance of the body according to age and size of the body and occupation of the person. Normally a balanced diet: (i) should be able to provide about 3000 calories per day for a normal adult; (ii) should contain all essential elements in appropriate proportion. An average person requires approximately 400–500 g of carbohydrates, 70 g of proteins and 75 g of fat, that is, the ratio of fat, protein and carbohydrates should be around 1 : 1 : 4. (iii) should be in a form that can be easily digested and properly cooked so as to remove any harmful bacteria. The requirement of carbohydrates, proteins and fat is greater for growing children, nursing mothers, and patients suffering from certain diseases. A sportsman needs two-and-a half times more carbohydrates than an ordinary person.

DISEASES OF THE HUMAN BODY Types of Diseases Deficiency Diseases A disease resulting from dietary deficiency of any substance essential for good health. Table: Various Deficiency Diseases Disease

Deficiency

Symptoms

Anaemia

Haemoglobin (iron)

General weakness and pale complexion

Goitre

Iodine

Painful joints

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Disease

Deficiency

Symptoms

Beri-beri

Vitamin B, (Thiamine)

Weakness, swelling and pain in legs, loss of appetite, enlarged heart

Scurvy

Vitamin C (Abscorbic acid)

Swollen gums, delayed wound healing

Rickets

Vitamin D

Sleeplessness, pale face, diarrhoea, deformed skull, pelvis and limbs in children

Hypokalemia

Potassium

Rise in heartbeat, kidney damage, weakness

Night blindness

Vitamin A

-

Xerophthalmia

Vitamin A

Dryness

Dermatosis

Vitamin A

Skin diseases

Ariboflavinosis

Vitamin B2 (Ribroflavin)

Blurred vision, soreness of eyes and tongue

Pellagra

Nicotinic Acid (Vitamin B Complex)

Diarrhoea, mental lethargy, red skin, itchy hands, feet, elbows and knees

Endemic Diseases A disease prevalent in a particular locality/ region and more or less confined to a population/area because of prevailing conditions that promote the disease; e.g., malaria in marshy areas.

Hereditary Diseases A disease that a child inherits from his parents. For example, AIDS becomes a hereditary disease when a mother suffering from AIDS gives birth to a child. Similarly there are other diseases which a child could get from his parents, like diabetes.

Epidemic Diseases A disease simultaneously affects a large number of people in an area, i.e., a violent outbreak of a disease affecting several people in a given time and place, e.g., outbreak of cholera or malaria.

Infectious Diseases A disease caused by specific pathogenic organisms and capable of being transmitted to another person by direct or indirect contact, i.e., through air, food, water, etc.

Pandemic Diseases A disease which affects two or more countries or even continents. Sporadic Diseases A disease which occurs erratically in an area with apparently no clue to its occurrence. Infectious and Contagious Diseases and Their carriers

Insect-borne diseases (propagated by insects) Malaria Anopheles female mosquitoes Plague Rat flea Relapsing fever Lice Sand fever Sand fly Kala azar Bed-bug Dengue fever Culex mosquitoes Water borne diseases Diseases like cholera, typhoid, dysentery, hook-worm and diarrhoea. Diseases caused by contaminated food Diseases like tuberculosis, cholera, typhoid, diarrohea, dysentery, Malta fever, etc. Air-borne diseases Diseases such as tuberculosis, influenza, small pox. Diseases caused by physical contact Diseases such as small pox, venereal diseases, AIDS are caused by direct or indirect contact. Diseases caused by wounds in skin Anthrax and Tetanus and results from an infected, open wound.

Contagious Diseases A type of infectious disease which spreads from one person to another by physical contact.

Agents of Diseases Bacteria Bacterium is the smallest known living organism (excluding virus which may or may not be a living organism). These are single-celled organisms, more complicated than viruses but simpler than animal or plant cells. There are three common types of bacteria based on their shape and structure: (i) Spherical or Oval (coccus) (ii) Rod-shaped or Cylindrical (Bacillus) (iii) Spiral (spirillum) Bacteria are said to be both useful and harmful and are found everywhere. They can grow both on dead and living cells. Bacteria are necessary for the process of decay of organic matter; in making atmospheric nitrogen available for plants (Nitrogen Fixation). Bacteria which cause diseases are called parasitic or pathogenic bacteria. Diseases such as typhoid, tuberculosis, cholera and food poisoning, etc., are caused by such bacteria. Virus Discovered by a Russian scientist Dimitri Ivanovsky in 1892. They do not exhibit all characteristics of living organisms, but are similar in composition and structure to a gene and appear in various shapes like rods, threads, spheroids, etc. Viruses are inactive when isolated and multiply only on invading a cell of

General Science ▌ 2.351 a bacterium, animal or plant by using the most complicated chemical machinery of the host cell. Many infectious diseases (such as influenza, smallpox, AIDS, polio, etc.) are caused by viruses invading and disrupting their victim’s cell. Other diseases caused by viruses are typhus, measles, mumps, chicken pox and yellow-fever. Fungus A low form of vegetable life (without chlorophyll) including many microscopic organisms capable of producing superficial diseases. They live either as saprophytes or as parasites on other plants and animals. They cause diseases like infection of foot skin, jaws, large intestines. Penicillin (the antibiotic) is developed from a type of fungus and cures fungus diseases. Amoeba A protozoan, one of the elementary unicellular forms of life, is a formless jelly-like cell. One form entamoebia histolytica is a parasitic pathogen that causes amoebic dysentery. Diarrhoea is also caused by a type of amoeba. Parasite An organism which obtains food or shelter from another host organism. They also act as carriers of disease. Germs of malaria and yellow fever are carried by mosquitoes, plague by fleas and typhoid by lice. Table:

Diseases and Causative Agents

Fever Fever (temperature above normal body temperature of 97.4°F) is in fact not a disease by itself but an indication of presence of a disease. The body temperature rises because of the heat that is produced while the body’s mechanism fights the growth of the invading bacteria. Immunization Immunization is the resistance of the body to a specific disease by process of production of immunity in the body. Immunity is present after the patient has recovered from a disease. The diseases cause the body to form antibodies (Refer to section on ‘Human Blood’) against the specific germs. The patient becomes immune to that infection because the blood now carries antibodies which will destroy the germs of that disease, should an infection occur. Immunity is also produced by vaccination and inoculation. Vaccination It is the introduction of a vaccine to the body. Vaccine is a preparation of dead or weakened pathogenic bacteria or germs. When a vaccine is injected into the body, it stimulates the body to produce anti-bodies in the blood, which may persist in the blood for lung periods of time thus making the body immune. Inoculation Inoculation is the introduction of antitoxins or serum to the body. Serum is a preparation from the blood of an animal that has been inoculated with disease-producing germs or bacteria and has recovered from the ailment caused by the inoculated germs or bacteria. This serum contains antibodies that the animal formed as a consequence of the disease it suffered by inoculation. This serum is removed from the animal blood and after sterilization and treatment is injected into the body to confer immunity should the germs or bacteria enter.

Causative Agent

Diseases

Bacteria

Diphtheria, gonorrhoea, meningitis, cholera, leprosy, typhoid, tetanus, tuberculosis, plague, whooping cough, pneumonia

Virus

Chicken pox, small pox, measles, mumps, AIDS, yellow fever, influenza, dengue fever, rabies, poliomeritis, phelebotomus

Vaccine

Discovered by

Protozoan

Malaria, sleeping sickness, kala-azar, leishmaniasis, amoebic dysentery

Small pox

Edward Jenner (1786) of Gloucestershire made the first, successful small pox vaccination.

Fungus

Athlete’s foot, ringworms, madura foot, dhobi’s itch

Cholera

Louis Pasteur (1880) of France prepared the first Cholera vaccine.

Diphtheria and tetanus

Emil Adolf Von Belming and Shibasaburo Kitasato of Germany and Japan respectvily (1891) developed anti-toxins to treat diphtheria and tetanus.

TB vaccine

Leon Calmette and Camille Guerin (1992) Paris, developed the first TB vaccine.

Polio vaccine

Jonas E. Salk (1954) Pittsburgh (US).

Defence Mechanism Against Infection The first line of defence against infection is the skin which serves as an impermeable barrier and bacteria can only enter if there is a cut in the skin. The eyes, the interior of the nose, the throat, stomach and intestine lack this barrier. However, there are secretions like tears, nasal secretion, saliva and gastric juices which kill bacteria. Should these barriers be passed, the next reaction is an increased blood supply to the injured area as the supply of white blood cells (WBC) are brought up to destroy and kill the bacteria. This is marked by redness and swelling at the site of infection. Neutrophils (See section on ‘Human Blood’) engulf the bacteria and destroy it. If, in spite of all these defences, infection succeeds, the lymphatic system comes into operation to prevent the spread of infection. When all these mechanisms fail to combat the infection, the disease finally attacks the human body.

Table:

Important Vaccines

Oval Polio Vaccine Albert Bruce Sabin (1955) US Measles vaccine

John F. Enders (1960) USA

Rabies Vaccine

Typhus Vaccine Charles Nicolle (1909) France.

AIDS AIDS is a killer disease. Its full form is ‘ACQUIRED IMMUNO-DEFICIENCY SYNDROME’. The disease was first reported in early 1981 from the United States and created a worldwide scare. The World Health Organization’s great concern during the decade is to combat the spread of this fatal incurable disease.

Causes AIDS is transmitted by a virus known as HLV-III, which disrupts the entire immunity system of the human body. When AIDS

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General Knowledge

was first detected, it was thought to be a sexually transmitted disease found in homosexuals only. However, later investigations revealed that the principal mode of its transmission was through blood by way of: • shared needles for injecting drugs • transfusion of contaminated blood • anal sex, which often damages blood vessels making entry of the virus easy • sexual relations with an infected person also spreads the virus

Cause Cholera vibrio or Vibriocholerae which attacks during exposure to chill, when stomach is empty for long duration, eating of unripe or overripe fruits and stale food. Symptoms Vomitting, stomach ache, loose stools with high frequency followed by temperature and unconsciousness. Cure/prevention Avoiding consumption of cut fruits exposed to flies, and contaminated water. Anti-cholera drugs are administered.

Tetanus

Symptoms The most common symptoms are a form of severe lung infection known as Pneumocystis carnii pneumonia and cancerous skill tumour known as Caposi’s Sarcoma. Death mainly occurs as a result of skin tumours and other complications including brain damage and severe diarrhoea.

Cause Bacillus tetanus and Clostridum tetani which live in soil, dust, cow and horse dung. It attacks an open wound exposed to dust and soil.

Treatment There is no specific treatment yet available for AIDS. However, encouraging results have been achieved by administration of AZT (Short for 3-azido-3-deoxythy-midine) taken orally four times a day. However, its long term effects are not yet known. It in fact does not cure AIDS but only keeps the multiplication of new virus in check. Efforts are afoot to develop a vaccine against this virus. According to medical circles in the US a vaccine against AIDS, even if successfully developed, would not be available for general use at least till 1999.

Cure/prevention Preventing exposure of wounds to dust and administering of Anti-Tetanus (ATS) injection.

Common Diseases Malaria

An insect borne tropical disease.

Cause Malaria parasite (Plasmodium) which enters the blood through a mosquito bite (female anopheles).

Symptoms Painful contraction of muscles, usually of neck and jaws, followed by paralysis of thoracic muscle.

Diphtheria

An acute infectious disease of the throat.

Cause Acute infection by diphtheria bacillus/coryne-bacterium diphtheria causing infection of throat. Symptoms Inflammation of throat where a grey membrane (a false membrane on mucous surface) is formed. Pain and swelling of throat with fever. Cure/prevention Immunization vaccine/injection of diphtheria antitoxin within 12–24 hours of appearance of symptoms. Typhoid

A food and water-borne infectious disease.

Cause Salmonella typhi bacillus transmitted through contaminated food and water, either directly by sewage or indirectly by hands and faulty hygiene.

Symptoms Shivering, fever. Repeated attacks lead to enlargement of spleen. Also leads to anaemia, pigmentation of the face and general weakness.

Symptoms Temperature, slow pulse, abdominal tenderness, rose-coloured rash.

Cure/prevention Administration of quinine or plaurdine. Prevented by keeping the atmosphere free from mosquitoes.

Cure/prevention Rest and administration of Chloromycetin, proper sanitation, protection of eatables.

Tuberculosis An infectious and endemic disease, both air-borne and caused by food, unhealthy living and working conditions.

Plague A contagious disease which takes the form of an epidemic.

Cause Mycobacterium tuberculosis (detected by Robert Koch in the mid-19th century), a type of bacteria. It attacks a person suffering from malnutrition, weak chest, unhealthy living and working conditions.

Cause Pasteurella pestis spread by infected rats. Transfer of infection from rat to man through flea bite or accidental contact with infected rats.

Symptoms General weakness, regular temperature (generally in the evening and not very high), coughing, blood-stained sputum. Cure/prevention Streptomycin and surgery. Prevented by BCG inoculation and healthy living and working atmosphere. BCG The vaccine was developed in 1906 by Calmette Guerin in Paris and called BCG. Cholera

An acute epidemic, water and food-borne disease.

Symptoms Acute body ache, reddish eyes, sudden rise of temperature, inflammation of neck glands and glands in armpit, and thighs. Cure/prevention Antiplague inoculation, isolation of patient, disinfection of patient’s clothes and utensils, burning of killed rats. Sulpha drugs and streptomycine administration. Typhus

A viral infection

Cause Rickettsia prowazekii, usually caused by poor hygiene and malnutrition.

General Science ▌ 2.353 Symptoms

High fever, skin eruptions and severe headache.

Cure/prevention

Sulphonamides and antibiotics.

Pneumonia Cause

Diseases of the Eyes

Diplococus pneumonia

Symptoms Chills, pain in chest, rusty sputum, rapid breathing, abdominal pain. Cure/prevention Gonorrhoea

Antibiotics.

A venereal disease

Cause Neisseria gonorrhea, through sexual intercourse with infected person Symptoms Redness, swelling, pus discharge through urethra, painful urination. Cure

Penicillin G. Tetracycline.

Syphillis Cause Treponema pallidum transmitted through sexual contact. Symptoms

A hard painless sore on the genitalia, skin eruption.

Cure/prevention

Penicillin, protected sexual intercourse.

Whooping cough

Chiefly occurs among infants and children.

Cause

Nemophilus pertusis transmitted through air.

Symptoms

Haemorrhage Loss of blood from the blood vessels, which may be from external wounds, and occur in the tissues or escape into the body cavity.

Severe cough, usually at night.

Cure/prevention Immunization of infants with immunity serum.

Some Blood Diseases Anaemia A condition where the RBC are seriously reduced in number or else are deficient in haemoglobin. This results in reduced supply of oxygen to tissues which retards normal functioning of the body systems. Leukemia Also called blood cancer is caused by over production of WBC. The WBC overrun the bone marrow and crowd out the RBC causing anaemia. The disease is caused due to malfunctioning of bone marrow and is normally incurable. However, bone marrow transfusion is believed to yield good results. Agranulocytosis A condition where there are two few WBC. This lowers body resistance to disease and causes secondary infections whereby the patient may die from infections he cannot resist. Thrombocytopenia A condition where there are too few platelets in the blood, as a result blood seeps out of the circulatory system making black and blue bruise spots. If not properly treated, it may result in fatal bleeding. Haemophilia A hereditary bleeding tendency due to inadequate clotting of blood. It is a rare disease almost always seen only in males.

Hypermetropia (Far-sightedness) A vision defect when a person cannot see objects near at hand because the light focuses behind the retina. This can be corrected with convex lenses. (See diagram). Myopia (Near-sightedness) A vision defect when a person cannot see distant objects because the light entering the eyes focuses in front of the retina. The lens refracts the parallel rays entering the eye. This can be corrected by using concave lenses. Astigmatism (Distorted vision) A condition in which light focuses on a line instead of at a point (when cornea is not spherical). This line may focus behind the retina in which case it is called Hyperopia astigmatism or Myopic astigmatism when in front of the retina. This defect can be corrected by using cylindrical lenses. Presbyopia Lack of normal focusing power of the eye on objects near at hand and common in most persons after 40–42 years of age. Presbyopia causes difficulty in reading and working with objects close at hand. This defect can be corrected by the use of convex lenses. Conjunctivitis Inflammation of the eyes resulting in itchiness and reddening of eyes making them sensitive to light. For cure, the affected eye is daubed with a simple solution of weak, salt water or boric acid and use of antibiotic eyedrops. Trachoma Caused by a germ called chalmydiae, and spreads by direct contact. It always affects both eyes and causes mild itching and irritation. For cure tetracyline group orally is administered. Cataract A condition of opacity of lens which occurs with ageing, causing progressive blurring of vision. Distant view is mostly affected whereas near vision is often improved in early stages. Cataract can be cured by operation called lens extraction.

Common Body Disorders Allergy A special reaction to a certain substance such as pollen (causes sneezing) or certain foods (causes skin rash). Allergy can be from any material, even by colour of clothes and furnishings, etc. Arthritis

Inflammation of joints.

Asthma A respiratory disorder caused by narrowing of bronchial tubes. It is caused both by infection and due to allergy to dustridden atmosphere. Bronchitis Inflammation of bronchial tubes caused by bacteria or virus. A fatal disease among infants and secondary infection among adults. Cancer An abnormal growth of body cells, often resulting in a malignant tumour. There are several causes for this dreaded disease, and certain cancers are yet incurable.

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Diabetes Excess sugar in the body, when the body is not able to control level of blood sugar due to malfunctioning of the pancreas (digestive gland), that is, when it produces inadequate insulin. A controlled diet (less carbohydrates) and intake of insulin, is the treatment for diabetes. Epilepsy Unwarned and periodic loss of consciousness along with convulsions, due to nervous disorders. Glandular Fever (Mumps) An acute infection probably caused by a virus occurs particularly in children and young adults following swelling of the glands of the neck and high fever. Hepatitis Any infectious or inflammatory disease of the liver commonly identified by its primary symptoms of jaundice. Hernia A weakness of the muscle surrounding an organ allowing it to bulge through, often found in the groin. It is cured by operation. Jaundice Excessive bilirubbin (present in bile juice secreted by liver) in the blood, causes yellowing of the skin, eyes and yellowish urine. Leukemia

Also called blood cancer (see Diseases of the blood).

Measles A contagious disease caused by virus; a red rash appears on the body along with fever. Disappears gradually after about ten days. Migraine A type of a headache followed by disturbed vision and speech accompanied by nausea. Pellagra A disease caused by deficiency of nicotinic acid (Vitamin B Complex). The symptoms are diarrhoea, mental laziness, and symmetrical dermatitis. Skin becomes red and itchy. Osteomalacia A disease caused by shortage of Vitamin D which results in softening of bones, pain causing frequent fractures and bending of the backbone. Pleurisy Inflammation of the membrane that covers the lungs and lines the chest cavity. Rabies A virus disease transmitted by the saliva of infected animals; symptoms include convulsions and revulsion to water (Hydrophobia). Ringworm A skin disease causing circular swelling on the skin. Transmitted through air-borne spores and contact with infected patient.

Small Pox A contagious virus disease, common among children, characterized by a rash of pustules that leave permanent scars on skin. The WHO started a sustained campaign in the 1960s and 1970s to eradicate small pox by mass vaccination. In 1980 WHO declared that small pox was eradicated. It was the first disease to be eradicated from the face of the earth. Thrombosis Formation of a blood clot in a blood vessel or in the heart which results in death of the patient. Ulcer An inflamed open sore on the skin, or the membrane of a body cavity. Peptic ulcer is a state when there are ulcers in the food pipe associated with indigestion and mental tension; a common disease among people working in an industrial society.

Common Drugs Anaesthetics Drugs that block sensory nerves and make a patient fully unconscious to prevent him from feeling pain. In case of local anaesthesia a particular area is made senseless temporarily. Used for surgical operations and includes drugs like chloroform, ether, sodium pentothal, etc. Analgesics Drugs used to prevent or relieve pain like aspirin (acetylsalicyclic acid) or analgesic tablets. Antibiotics Drugs used to prevent growth of body germs and to destroy them as soon as possible. Most common drugs under this category are penicillin, terramycetin, etc. Antihistamines These drugs are used to relieve symptoms of asthma, hay fever and other allergies. Antipyretics

Drugs used to lower body temperature.

Hormones Drugs used to combat hormone deficiency that causes diseases. Drugs like insulin (for diabetics) or adrenalin come under this category. Narcotics Drugs that deaden the nervous system and prevent a person from feeling pain, for example, opium and its derivatives such as codeine, heroin, morphine, etc. Sedatives Drugs used to induce sleep like barbiturates and bromides. Drugs that calm nervous system and prevent worry,

Scarlet Fever Caused by haemolytic streptococcal infection; symptoms are fever, sore throat and a red rash.

Tranquillizers tension, etc.

Slipped Disc A painful condition in which a cartilage disc in the spine is displaced putting pressure on the nearest nerve.

Vaccines Drugs that are injected to help the body to develop resistance to disease or immunization of the body.

Table:

Communicable Diseases

Name

Cause

Transmission

Incubation period

Aids (Acquired Immune Deficiency Syndrome)

Human Immuno-deficiency Virus (HIV)

Sexual relations; sharing of syringes: blood transfusion

Several years

Brucellosis

Brucellus abortus or B meliteusis bacteria

Cattle or goats

3–6 years Continued

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Name

Cause

Transmission

Incubation period

Chickenpox (Varicella)

Varicella zoster virus (US) Herpes zoster virus (UK)

Infected persons. articles contaminated by discharge from mucous membranes

10–21 years

Cholera

Vibrio cholerae bacterium

Contaminated water and seafood

a few hours–5 days

Common cold

Numerous viruses

Respiratory droplets of infected person

1–4 days

Diphtheria

Corynebacterium diphtheriae bacterium

Respiratory secretions and saliva of infected persons or carriers

2–6 days

Encephalitis

Viruses

Bite from infected mosquito

4–21 days

Gas gangrene

Clostridum welchii bacterium

Soil or soil-contaminated articles

1–4 days

Gonorrhoea

Neisseria gonorrhoeae bacterium

Urethral or vaginal secretions of infected persons

3–8 days

Hepatitis A (infectious)

Hepatitis A virus

Contaminated food and water

15–20 days

Hepatitis B (serum type B)

Hepatitis B virus

Infected blood; parenteral injection

6 week–6 months

Infectious mononucleosis (US) Glandular fever (UK)

Epstein-Barr virus

Saliva; direct oral contact with infected person

2–6 weeks

Influenza

Numerous viruses (types A, B, C)

Direct contact; respiratory droplets; possibly airborne

1–4 days

Legionnaires disease

Legionella pneumophila bacterium

Water droplets in contaminated hot water systems, cooling towers, etc.

1–3 days

Leprosy

Mycobacterium leprae bacillus

Droplet infection (minimally contagious)

variable

Malaria

Plasmodium protozoa

Bite from infected mosquito

6–37 days

Measies (rubeola)

Rubeola virus

Droplet infection

10–15 days

Meningitis

Various bacteria (bacterial meningitis) and viruses (viral meningitis)

Respiratory droplets

Varies with causative agent

Mumps

Virus Salmonella bacteria

Direct contact with infected persons: respiratory droplets and oral secretions

14–21 days

Paratyphoid fevers

Salmonella bacteria

Ingestion of conatminated and water

1–4 days

Pneumonia

Streptococcus pneumoniae bacterium

Droplet infection

1–3 days

Poliomyelitis

Polio viruses

direct conact with nasopharyngeal secretions of infected person; vomit

7–21 days

Rabies Rubella (German measles)

Virus Rubella virus

Bite from rabid animal

10 days–6 months

SARS (severe acute respiratory syndrome)

SARS-associated coronavirus

Direct contact or droplet spread of nasopharyngeal secretion

14–21 days

SARS (severe acute respiratory syndrome)

SARS-associated coronavirus

Direct contact with infected persons or respiratory droplets

2–7 days

Scarlet fever

Group A haemolytic Streptococcus bacteria

Direct or indirect contact with infected persons, 1–5 days or droplet infection

Shingles Smallpox (variola)

(See chickenpox) Poxvirus variola

(See chickenpox) Direct contact; droplet

7–14 days

Syphilis

Treponema pallidum bacterium

Sexual relations: contact with open lesions; blood transfusion

10–90 days

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Name

Cause

Transmission

Incubation period

Tetanus (lockjaw) Tuberculosis

Clostridium tetani bacillus Mycobacterium tuberculosis bacillus

Animal faeces and soil Droplet spread; ingestion from contaminated milk Contaminated food and water

3–21 days variable

Typhoid fever

Salmonella typhi bacillus

Droplet spread

7–21 days

Wooping cough (pertussis)

Bordetella pertussis bacterium

Bite from infected mosquito

10–21 days

Yellow fever

Arbovirus

3–6 days

Commonly Abused Drugs Table: Name

Common name

Effects

Comments

Alcohol

Booze, drink

Acts as central-nervous-system depressant, so reduces anxiety, impairs concentration, slows reactions.

Long-term effects include disease (cirrhosis, liver cancer, hepatitis), heart disease, and inflammation of stomach. Alcoholics have above-average chance of developing dementia.

Amphetamines

Uppers, speed bennies

Promote feelings of alterness; increases speech and physical activity.

Barbiturates (Nembutal, Seconal, Amytal)

Barbs, reds, downers

Calm the nerves; induce sleep: have hypnotic effect.

Highly addictive; overdose is lethal; can induce state of coma. Often fatal if taken with alcohol.

Barbiturates (Nembutal, Seconal, Amytal)

Barbs, reds, downers

Calm the nerves; induce sleep: have hypnotic effect.

Highly addictive; overdose is lethal; can induce state of coma. Often fatal if taken with alcohol.

Benzodiazepines

Tranquillizers

Reduce mental activity and anxiety: slow Can cause dependency. Withdrawal symptoms occur body’s reaction: reduce alertness. on stopping the drug-anxiety, insomnia, panic attacks, headaches, and palpitations.

Cocaine

Coke, crack, ice, snow (crack is a blend of cocaine, baking powder, and water)

Increases blood pressure, heart rate, breathing, and body temperature: gives feelings of euphoria, illusions of increased sensory awareness and mental and physical strength, decreased hunger, pain, and need for sleep.

Regular use can cause anxiety, insomnia, weight loss, increased paranoia, and psychosis. Crack is highly addictive and has more intense effects than cocaine. Increased risk of abnormal heart rhythms, high blood pressure, stroke, and death. Long-term consequencces include mental deterioration, personality changes, paranoia, or violent behaviour.

Heroin

Junk, smack

Induces euhoria; relieves pain; often induces sleep

Highly addictive; overdose can result in death; serum hepatitis is common; as are skin abcesses, inflammation of the veins, constipation, and respiratory depression.

Lysergic acid diethylamide

LSD, acid

Causes hallucinations, alters vision, raises temperature and heart-beat; evokes flashbacks.

Long-term use causes anxiety and depression, impaired memory and attention span, difficulty with abstract thinking.

Marijuana

Grass, pot, weed, dope

Increases heartbeat; heightens senses; Reduces the ability to perform tasks requiring gives feelings of euphoria and relaxation. concentration; slows reactions; and impairs coordination.

MDMA

Ecstasy, E

Promotes mental relaxation, increased sensitivity to stimuli, and sometimes hallucinations.

Mescaline

Peyote, cactus buttons

Induces hallucinations; affects sensations Loss of control of normal thought processes; longand perceptions. term depression and anxiety; can induce ‘breaks from reality’.

High doses have amphetamine-like effects. Can produce severe or fatal reactions, sometimes after only one dose.

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Name

Effects

Comments

Methadone

Common name

Induces sleep and feeling of relaxation.

Addictive; overdose can result in death.

Methadone Nicotine

Stimulates the nervous system; increases concentration; relieves tension and fatigue; increases heart rate and blood pressure.

Taken regularly, can cause increase in fatty acids in bloodstream, increases risk of heart disease, and circulatory problems; can also increase risk of peptic ulcers. increased risk of lung, throat, and mouth cancers from tobacco smoke.

Nitrites

Poppers

Give the user a rapid high, felt as rush of energy. Heart rate increases; there are feelings of dizziness and nausea. High doses can cause fainting.

Lasting physical damage, in the form of cardiac problems, can occur.

Phencyclidine

PCP, angel dust

Gives feeling of euphoria; floating sensation; numbness; change in user’s perception of the body; visual disturbances.

Can produce violent behaviour against the user or others; and schizophrenic–like psychosis which can last for days or weeks.

Cause lightheadedness, dizziness, and drowsiness. Large doses can lead to loss of consciousness.

Some products can seriously disrupt heart rhythm or cause heart failure and sometimes death. Aerosols can cause suffocation by coating the lungs. Risk of death also from depression of the breathing mechanism. Long-term misuse leads to kidney and liver damage.

Solvents

MEDICAL SCIENCE Medical Aids (Equipment) Cystoscope bladder.

An instrument for examining the inside of the urinary

CAT Scanner The Computerized Axial Tomography (CAT) Scanner is a machine for photographing a selected plane of the human body using X-rays. It is a ring-shaped X-ray machine that rotates through 180° around the patient, who lies on a horizontal plane, making numerous X-ray measurements at every few degrees. The data is processed by a computer to produce the final image. Artificial Kidney A device containing tubes of chambers of a special membrane immersed in a bath of dialysing solution, used to remove waste material from the blood of patients whose kidneys do not work properly. Bronchioscope An instrument with a lighted tube which is passed down the trachea (windpipe) for examination of the air tubes of the lungs. Defibrillator An instrument for restoration of rhythmical heartbeat by electrical stimulation in case of fibrillation (rapid uncoordinated heart beating) which is a dangerous condition. Electrocardiograph (ECG) An instrument for graphically recording from the body surface, variations in electrical potential produced by the rhythmic contractions of the heart. Used to check heart ailments. Electroencephalograph (EEG) An instrument to record changes in electric potential in various areas of the brain by means of electrodes placed on the scalp or on the brain itself.

Endoscope Used for internal examination by direct vision through a lighted tube fitted with a system of lenses. Gastroscope An instrument fitted with a lighted tube passed down the gullet (oesophagus) to inspect the lining of the stomach. Heart-Lung Machine A machine comprising a system of pumps which takes over the function of the heart during open-heart surgery and circulates the blood around the body. The machine also takes over the function of the lungs in supplying fresh oxygen to the blood. Laproscope An instrument used for the examination of the abdominal cavity. Olfactometer

An instrument for testing of sense organs.

Ophthalmoscope A device to inspect the inside of the eye and diagnoze eye ailments. Oxygenator A device for introducing oxygen into blood during circulation outside the body, as during open heart surgery. Rachiometer

An instrument used to measure spinal curvature.

Sphygmomanometer pressure. Shygmometer

An instrument used to measure blood

An instrument used to measure pulse rate.

Medical Treatment Following are some of the common medical techniques of diagnosis, and treatment of diseased organs.

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Appendctomy abdomen.

Surgical removal of inflamed appendix from the

Cryosurgery A surgical technique of destroying a tissue by extreme cold. Liquid air or liquid nitrogen is used as the coolant which is circulated through specially designed equipment to destroy tissue that is too delicate to be manipulated with ordinary surgical instruments. Cystoscopy The technique of examining the inside of the urinary bladder using an optical instrument called cystoscope which is a narrow tube fitted with a system of lenses and lights. This instrument is inserted through the urethra. Dialysis A method of removing waste products from the blood using a special membrane which allows water and waste products to pass through but retains blood cells and proteins. This is used in case of kidney failure. Diathermy A technique of generating heat in a tissue by means of a high frequency electric current. Used to relieve pain due to muscular disorders. Enterostomy A surgical technique of making an artificial opening in the intestines, usually as part of treatment of ulcers. Gastrectomy Surgical removal of the stomach or part of it in the treatment of tumours or ulcers in the stomach. Hypothermia A technique to lower body temperature to reduce the body’s requirement of oxygen during extended surgery of the brain or heart. Hysterectomy women.

Surgical removal of the uterus or womb in

Laparoscopy Examination of the abdominal cavity using an optical instrument called laparoscope which is fitted with a system of lenses and a light source. The instrument is inserted into the abdominal cavity through a small incision in the abdomen. Lumbar Puncture A technique of obtaining a specimen of the cerebrospinal fluid by inserting a long needle between two vertebrae in the lower spine. Used for the diagnosis of meningitis and other disorders affecting the brain or spinal cord. Nephrectomy

Surgical removal of a diseased or damaged kidney.

Oophorectomy Surgical removal of a diseased ovary or one with a tumour or cyst. Venipuncture The puncturing of a vein to inject a drug, blood or other fluids into the vein or to obtain a blood sample for clinical examination.

Marvels of Medical Science Transplantation of Organs To replace a diseased organ with a healthy one, is one of the highest achievements of medical science in recent years. The difficulties experienced by the doctors in this connection are: First, the body’s immune system which makes it reject any foreign transplant, and second, to find a donor with a

suitable organ for replacement. However, new anti-rejection drugs have proved very helpful in this direction. Organs that can be easily transplanted: (i) Grafting of blood vessels, heart-valves and bone is done using dead tissue taken from a dead body. (ii) The cornea (in the eye) the transparent window in front of the eye has no blood vessels, so it is not linked to the body’s immune system. The cornea can therefore be transplanted with no fear of rejection. Cornea grafting is one of the most simplest and widely used transplant operations. The organs which are transplanted after proper matching since they can be easily rejected by body’s immune system are:

Kidney Kidney transplantation has the best track record so far. The survival rate is 90%. It is relatively easy to get a spare kidney because anyone with two healthy kidneys can donate one and still continue to lead a normal life. Dr R. H. Lawler performed the first homo-transplantation of kidney in 1950 at Harvard Medical School, US. In India, the first kidney transplant was done in 1971 at CMC, Vellore (Tamil Nadu) on Mr Shanmugham. Heart The first whole organ to be transplanted in the human body is heart. Heart transplant has now become a very common operation but the survival rate is only about 45% due to rejection of the new heart by the body’s immune system. The first heart transplant operation was performed on Louis Washkansky, aged about 55 years in South Africa on December 3, 1967 by a team of 30 surgeons headed by Prof. Christian Neething Barnard. The donor was Miss Denise Ann Darwall, aged 25, an accident victim. The longest surviving heart transplant has been of Emmanuel Vitra of France who received the heart of a 20 years old man on November 28, 1968 and lived for another 15 years (upto 1982). Madhava Rao, was the first Indian to have a heart transplant in 1985 at Pennsylvania (USA) at the cost of Rs 12 lakh. Open heart surgery in India was performed for the first time at CMC Vellore in 1989 by Dr N. Gopinath and Dr P. H. Betts on girl, Beulah, who was only 12 years of age. India’s First Heart Transplantation India’s first successful heart transplantation operation was performcd by Dr P. Venugopal at the All India Institute of Medical Sciences (AIIMS) on August 3, 1994 on a 42-year old mechanic with a 34-year-old dead woman serving as the heart donor. The five-hour long operation was successful and the patient is reported to be leading a normal life now.

Artificial heart In view of the difficulty to get suitable donors, attempts have been made, with encouraging results, to replace the diseased heart with a totally artificial heart implant. Bone Marrow Transplantation Is done to treat certain blood-related diseases like Leukemia and Aplastic anaemia. A donor’s bone marrow is removed from the pelvic bone and infused into the

General Science ▌ 2.359 recipient’s arm. The success rate is now about 60% in case of acute leukemia in children. Lungs Replacement of diseased lungs has been tried and the survival period has been only 10 months. Better results have, however, been obtained with the transplant of the heart and lungs unit together in which case survival rate has been up to 4 years. Pancreas First pancreas transplant was attempted in 1966 to treat diabetic patients. It eliminates need of regular insulin injections and prevents other diabetes associated complications. The survival rate so far has been only 25%. Transplants and the Pig A pig’s heart valves have been used as replacement for human ones. In emergencies pig skin has been grafted to deal with severe burns. Because pig tissues are the nearest in chemical composition to those of man, they are readily accepted by the defence system of the recipient’s body unlike tissues from other animals. Test Tube Baby A test tube baby is a baby born by artificial fertilization of the ovum. The main cause of infertility is the blockage of the fallopian tubes that lead from the ovaries to the uterus; consequently spermatazoa are unable to reach the ova for fertilization. In such cases, the ovum is removed and fertilized by the sperm externally (test tube) where the fertilization takes place and cell division commences. The fertilized ovum is then implanted into the womb of the woman where subsequent development of the embryo takes place as in the case of natural fertilization. The technique was developed by Robert Edwards and Patrick Steptoe in 1970. The first test tube baby was a baby girl, Louise Brown, born to Lesley Brown in Lancashire (UK). In India, the first test tube baby Harsha was born on August 6, 1986, to Mani Chawla. The technique was developed by Dr Indira Hinduja at KEM Hospital Bombay. Surrogate Mothers

In case the female genital system of a woman is abnormal and incapable of developing the embryo the ovum of that woman is removed and fertilized in a test tube. The fertilized ovum is then placed in the womb of a normal woman in whose womb the embryo develops. Such women are referred to as surrogate mothers.

Glossary of Medical Terms Adam’s Apple The bulge at the front of the neck formed by thyroid cartilage which is attached to the front of the larynx (voice box). It is more prominent and larger in males than in females because the male larynx is larger in order to accommodate the larger vocal cords. Adennoids Enlarged mass of lymphoid tissue in the naso-pharynx which can obstruct breathing and even interfere with hearing. Along with tonsils they leave the purpose of preventing the entry of germs into the respiratory system. Adrenals

Ductless glands situated above the kidney.

Alimentary Canal

The digestive tract from mouth to anus.

Aorta The largest artery in the human body, that emerges from the left ventricle of the heart and distributes purified blood throughout the body. Aqueous Humour of the eye.

The transparent fluid of the anterior chamber

Auricle The two upper chambers of the heart that receive impure blood through the veins. Also pinna of the external ear. Bile Juice The alkaline juice secreted by the liver into the duodenum. Blood Count The calculation of the number of red and white cells (RBC and WBC) in a cubic millimeter of blood. A hemocytometer is used to check blood count. Blood Bank A storage of whole blood or plasma which is preserved under refrigeration for transfusion in an emergency (Refer to ‘Blood transfusion’). Blood Pressure The pressure exerted by blood on the walls of blood vessels; measured in millimeters of mercury on a sphygmomanometer. Blood Serum The fluid seen when blood clots, it consists of plasma minus the clotting agent. Blood Sugar

The amount of glucose in the blood.

Blood Urea The amount of urea (end product of protein metabolism) in the blood within a normal range. Cardiac

Pertaining to the heart.

Carpopedal

Pertaining to the hands and feet.

Cerebellum A part of the brain located below and behind the cerebrum which controls muscular activities and equilibrium of the body. Cerebrum The largest part of the brain which is the centre of intelligence, co-ordination memory, will, etc. Coagulase An enzyme which clots plasma, for example, rennin thrombin. Coagulation the blood.

The curdling of blood or the formation of a clot in

Corpuscle A microscopic mass of protoplasm, for example. Erythrocytes and Leukocytes (RBC/WBC) (Also see ‘Blood’). Dermis

The true skin, the layer below the epidermis.

Diastole The relaxation period of the cardiac cycle as opposed to Systole or expansion of the heart. Dorsal Pertaining to the back or posterior part of the body or organ. Duodenum The first part of the small intestine where digestive juices help in digestion. Ejaculatory Tubes The two fine tubes, one on either side commencing at the union of the seminal vesicle.

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Erythrocytes

Red blood cells (Refer to ‘Human Blood’).

Fallopian Tubes The two tubes opening out of the upper part of the uterus that connect the vagina to the ovaries. Each measures 10–16 cm. They carry the ova into the uterus. Gall Bladder A pouch situated at the lower border of the liver for storage of bile juice. Haemoglobin The red pigment of the blood that contains oxygen (Refer to ‘Human Blood’). Hormones Secretion of the ductless glands (Refer to ‘Glands of Human body’). Hypertension Abnormally high tension, alluding to blood pressure and involving systolic and diastolic levels. A resisting systolic pressure of 160 mm mercury and a resisting diastolic pressure of 100 mm mercury indicates hypertension. Hypotension Low blood pressure (systolic below 110 mm mercury and diastolic below 70 mm mercury). Medulla Oblongata The upper part of the spinal cord between the foramen magnum of the occipital bone and the pons cerebri—the lower most part of the brain which connects the spinal cord to the brain. Ligaments A band or sheet of fibrous tissue that connects two or more bones. Table:

Pulmonary Artery The artery carrying impure blood from the heart to the lungs. The only artery carrying impure blood as all other arteries carry pure blood. Reflex Action An involuntary motor or secretory response by tissue to a sensory stimulus, for example, sneezing, blinking, coughing, etc. Spinal Cord The continuation of the medulla oblongata; enclosed in the backbone the spinal cord controls reflex actions and communicates messages between the brain and the body. Systole The contraction of the cardiac cycle or contraction of the heart. Tendon A fibrous cord or band connecting muscles to a bone. Ventricle The two lower chambers of the heart from which purified blood flows out through arteries. Ventral Chamber The belly of the body composed of two main cavities, the thorax (which contains heart, lungs) and the abdomen (which contains digestive, excretory and reproduction organs), separated by the diaphragm. Vitreous humour The transparent fluid in the posterior chamber of the eye. (Refer to ‘Human Organs’).

Medical Discoveries

Invention/Discovery

Made by

Year

Antibiotic (Penicillin)

Alexander Fleming (London)

1928

Antiseptic

Joseph Lister (Glasgow)

1865

Aspirin

Dr Felix Hoffmann (Germany)

1897

Blood Circulation

William Harvey (Britain)

1683

Blood Group

K. Landsteiner

1900–1902

Blood Transfusion

Iean Baposte Denys (france)

1625

Cholera/TB Germs

Robert Koch (Germany)

1877

Diphtheria germs

Klebs & Loffler (Germany)

1883–84

Electro-Cardiogram (ECG)

William Einthoven (Dutch)

1903

Heart-Lung Machine

John Heynsham Gibbon (US)

1953

Hypodemic Syringe

Alexander wood (Britain)

1853

Heart Transplant

Christian Barnard (South Africa)

1967

Insulin

Federick Banting and Charles Best (Canada)

1921

Kidney Machine

W. J. Kolff (The Netherlands)

1943

Malaria Germs

Alphonse Laveran (France)

1880

Organ Transplant

John P. Merril (US)

1953

Scan (CAT)

Godfrey Hounsfield (England)

1973

Spectacles

Roger Bacon (Italy)

13th century

Sphygomomanometer

Scipione Riva-Rocci (Italy)

1896

Stethoscope

Rene Laennec (France)

1815 continued

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Invention/Discovery

Made by

Year

Tetanus

Stepice and Edward (Britain)

1978

Thermometer (Clinical)

Sir Thomas Allbutt (London)

1867

Ultrasound

Ian Donald (Ireland)

1950

Vaccine

Edward Jenner (England)

1796

X-ray

Wilhelm Roentgen (Germany)

1895

Note: Laveran was awarded Noble Prize for isolating malaria parasite in 1907.

ECOSYSTEM AND BIOSHPERE COMMUNITY AND ECOSYSTEM DYNAMICS A community is the set of all populations that inhabit a certain area. Communities can have different sizes and boundaries. These are often identified with some difficulty. An ecosystem is a higher level of organization—the community plus its physical environment. Ecosystems include both the biological and physical components affecting the community. Ecosystems can be studied from a structural view of population distribution or from a functional view of energy flow and other processes.

Community Structure Ecologists find that within a community many populations are not randomly distributed. This recognition that there was a pattern and process of spatial distribution of species was a major accomplishment of ecology. Two of the most important patterns are open community stucture and the relative rarity of species within a community; and if within a community similar geographic range and density peaks are observed among native species, the community is said to be a closed community, a discrete unit with sharp boundaries known as ecotones. An open community, however, has a population without ecotones and is distributed more or less randomly.

Classification of Communities There are two basic categories of communities: terrestrial (land) and aquatic (water). These two basic types of community contain eight smaller units known as biomes. A biome is a large-scale category containing many communities of a similar nature, whose distribution is largely controlled by climate. Terrestrial Biomes tundra, grassland, desert, taiga, temperate forest, tropical forest; Aquatic Biomes marine, freshwater

Change in Communities over Time Biological communities, like the organisms that comprise them, can and do change over time. Ecological time focuses on community events that occur over decades or centuries. Geological time focuses on events lasting thousands of year or more.

Community succession is the sequential replacement of species by the immigration of new species and local extinction of older ones following a disturbance that creates unoccupied habitats for colonisation. The initial rapid coloniser species are the pioneer community. Eventually, a climax community of more or less stable but slower growing species develops.

Disturbance of a Community The basic effect of human activity on communities is community simplification, an overall reduction of species diversity. Agriculture is a purposeful human intervention in which a monoculture of a single favoured (crop) species such as corn is created. Most agricultural species are derived from pioneering communities. Inadvertent human intervention can simplify communities and produce stressed communities that have fewer species as well as a superabundance of some species. Disturbances favour early successional (pioneer) species that can grow and reproduce rapidly.

Ecosystems and Communities Ecosystems include both living and non-living components. The living, or biotic, components include habitats and niches occupied by organisms. Non-living, or abiotic, components include soil, water, light, inorganic nutrients, and weather. An organism’s place of residence, where it can be found, is its habitat. A niche is often viewed as the role of that organism in the community, factors limiting its life, and how it acquires food. Producers, a major niche in all ecosystems, are autotrophic, usually photosynthetic organisms. In terrestrial ecosystems, the producers are usually green plants. Freshwater and marine ecosystems frequently have algae as the dominant producers. Consumers are heterotrophic organisms that eat food produced by other organisms. Herbivores are a type of consumers that feed directly on green plants (or another type of autotroph). Since herbivores take their food directly from the producer level, they are called primary consumers. Carnivores feed on other animals (or another type of consumer) and are secondary or tertiary consumers. Omnivores, the feeding method used by humans, feed on both plants and animals. Decomposers are organisms, mostly bacteria and fungi, that recycle nutrients from decaying organic

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materials. Decomposers break down detritus, non-living organic matter, into inorganic matter. Small soil organisms are critical in helping bacteria and fungi shred leaf litter and form rich soil. Even if communities do differ in structure, they have some common uniting processes such as energy flow and matter cycling. Energy flows move through feeding relationships. The term ‘ecological niche’ refers to how an organism functions in an ecosystem. Food webs, food chains, and food pyramids are three ways of representing energy flow. Food Web

It is a set of interconnected food chains by which energy and materials circulate within an ecosystem. The food web is divided into two broad categories: the grazing web, which typically begins with green plants, algae, or photosynthesising plankton, and the detrital web, which begins with organic debris. These webs are made up of individual food chains. In a grazing web, materials typically pass from plants to plant eaters (herbivores) to flesh eaters (carnivores). In a detrital web, materials pass from plant and animal matter to bacteria and fungi (decomposers), then to detrital feeders (detritivores), and then to their predators (carnivores). Generally, many interconnections exist within food webs. For example, the fungi that decompose matter in a detrital web may sprout mushrooms that are consumed by squirrels, mice, and deer in a grazing web. Robins are omnivores, that is, consumers of both plants and animals, and thus are in both detrital and grazing webs. Robins typically feed on earthworms, which are detritivores that feed upon decaying leaves. Producers absorb solar energy and convert it into chemical bonds from inorganic nutrients taken from environment. The energy content of organic food passes up the food chain; eventually all energy is lost as heat, therefore requiring continual input. Original inorganic elements are mostly returned to the soil and producers; they can be used again by the producers and no new input is required. Energy flow in ecosystems, as with all other energy, must follow the two laws of thermodynamics. The first law states that energy is neither created nor destroyed, instead it changes from one form to another (potential to kinetic). The second law mandates that when energy is transformed from one form to another, some usable energy is lost as heat. Thus, in any food chain, some energy must be lost as we move up the chain.

FIG. 18 The Flow of Energy through an four Ecosystem

The ultimate source of energy for nearly all life is the sun. Recently, scientists discovered an exception to this once unchallenged truism: communities of organisms around ocean vents where the food chain begins with chemosynthetic bacteria that oxidise hydrogen sulphide generated by inorganic chemical reactions inside the earth’s crust. In this special case, the source of energy is the internal heat engine of the earth. Energy Flow in a Biosphere Through the series of steps of eating and being eaten, energy flows from one trophic level to another. Green plants or other photosynthesising organisms use light energy from the sun to manufacture carbohydrates for their own needs. Most of this chemical energy is processed in metabolism and dissipated as heat in respiration. Plants convert the remaining energy to biomass, both above ground as woody and herbaceous tissue and below ground as roots. Ultimately, this material, which is stored energy, is transferred to the second trophic level, which comprises grazing herbivores, decomposers, and detrital feeders. Most of the energy assimilated at the second trophic level is again lost as heat in respiration; a fraction becomes new biomass. Organisms in each trophic level pass on as biomass much less energy than they receive. Thus, the more steps between producer and final consumer, the less energy remains available. Seldom are there more than four links, or five levels, in a food web. Eventually, all energy flowing through the trophic levels is dissipated as heat. The process whereby energy loses its capacity to do work is called entropy.

The Food Chain Food chains indicate who eats whom in an ecosystem; they represent one path of energy flow through an ecosystem. Natural ecosystems have numerous interconnected food chains. Each level of producer and consumers is a trophic level. Some primary consumers feed on plants and make grazing food chains; others feed on detritus. The population size in an undisturbed ecosystem is limited by the food supply, competition, predation, and parasitism. Food webs help determine consequences of peturbations: if titmice and vireos fed on beetles and earthworms, insecticides that killed beetles would increase competition among birds and probably increase predation of earthworms, and so on. The trophic structure of an ecosystem forms an ecological pyramid. The base of this pyramid represents the producer trophic level. At the apex is the highest level consumer, the top predator. Other pyramids can be recognised in an ecosystem. A pyramid of numbers is based on how many organisms occupy each trophic level. The pyramid of biomass is calculated by multiplying the average weight for organisms times the number of organisms at each trophic level. An energy pyramid illustrates the amounts of energy available at each successive trophic level. The energy pyramid always shows a decrease while moving up trophic levels because of the following: Only a certain amount of food is captured and eaten by organisms at the next trophic level; some of the food that is eaten but cannot be digested, exists the digestive tract as undigested waste; only a portion of the digested food becomes part of the organism’s body; the rest is used as a source of energy; a substantial portion of food energy goes to build up temporary ATP in mitochondria. This is then used to synthesise proteins, lipids,

General Science ▌ 2.363 carbohydrates, fuel contraction of muscles, nerve conduction, and other functions; only about 10 per cent of the energy available at a particular trophic level is incorporated into tissues at the next level. Thus, a larger population can be sustained by eating grain than by eating grain-fed animals since 100 kg of grain would result in 10 human kg but if fed to cattle, the result, by the time that reaches the human, is a paltry 1 human kg.

FIG. 19 Energy Flow and the Relative Proportions of Various Levels in the Food Chain A food chain is a series of organisms, each feeding on the one preceding it. There are two types of food chains: decomposer and grazer. Grazer food chains begin with algae and plants and end in a carnivore. Decomposer chains are composed of waste and decomposing organisms such as fungi and bacteria. Food chains are simplifications of complex relationships. A food web is a more realistic and accurate depiction of energy flow. Food webs are networks of feeding interactions among species. The food pyramid provides a detailed view of energy flow in an ecosystem. The first level consists of the producers (usually

plants). All higher levels are consumers. The shorter the food chain, the more energy organisms derive. Trophic Levels of Food Web The food web can be viewed not only as a network of chains but also as a series of trophic (nutritional) levels. Green plants, the primary producers of food in most terrestrial food webs, belong to the first trophic level. Herbivores, consumers of green plants, belong to the second trophic level. Carnivores, predators feeding upon the herbivores, belong to the third. Omnivores, consumers of both plants and animals, belong to the second and third. Secondary carnivores, which are predators that feed on predators, belong to the fourth trophic level. As the trophic levels rise, the predators become fewer, larger, fiercer, and more agile. At the second and higher levels, decomposers of the available materials function as herbivores or carnivores depending on whether their food is plant or animal material. Through these series of steps of eating and being eaten, energy flows from one trophic level to another. Most humans occupy a top carnivore role, about 2 per cent of all calories available from producers never reach the tissues of top carnivores. Leakage of energy occurs in each feeding level. Most natural ecosystems therefore do not have more than five levels in their food pyramids. Large carnivores are rare because there is so little energy available to them atop the pyramid. Food generation by producers varies greatly between ecosystems. Net primary productivity (NPP) is the rate at which producer biomass is formed. Tropical forests and swamps are the most productive terrestrial ecosystems. Reefs and estuaries are the most productive aquatic ecosystems. All of these productive areas are in danger due to human activity. Humans redirect nearly 40 per cent of the net primary productivity and directly or indirectly use nearly 40 per cent of the land food pyramid. This energy is not available to natural populations.

THE BIOSPHERE The biosphere is the sum of all living matter on earth. Highly specialized organisms have adapted to the extreme boundaries of the uppermost atmosphere and lowermost ocean depths. The biosphere is interconnected with three other spheres of the physical environment: the lithosphere, hydrosphere, and atmosphere. The hydrosphere includes all water at or near the earth’s surface. Water is very important to a number of biological and geological processes. The hydrologic cycle is the continuous recycling of water between the oceans and atmosphere. Evaporation is the movement of water from an ocean or a lake to the atmosphere. Transpiration by plants contributes to atmospheric water content. Precipitation (rainfall or snow) is the movement of water from the atmosphere to land or ocean. Water on land can either run off the surface or filter through soil to become groundwater. The atmosphere is the envelope of gas that surrounds the earth. Nitrogen and oxygen compose 99 per cent of the atmosphere today. The atmosphere becomes progressively thinner with increasing altitude.

Acid Rain Acid rain is a term for rain, snow, sleet, or other wet precipitation that is polluted by such acids as sulfuric acid and nitric acid. Acid rain harms thousands of lakes, rivers, and streams worldwide, killing fish and other wildlife. It also damages buildings, bridges, and statues. High concentrations of acid rain can harm forests and soil. Acid rain forms when water vapor in the air reacts with certain chemical compounds. These compounds, including sulfur dioxide and nitrogen oxides, come largely from the burning of coal, gasoline, and oil. Most automobiles, factories, and power plants burn such fuels for energy. Regions affected by acid rain include large parts of eastern North America, Scandinavia and central Europe, and parts of Asia. Since about the 1950s, the problem has increased in rural areas. This has occurred because the use of taller smokestacks in urban areas has enabled the winds to transport pollutants farther from their sources. Scientists and

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engineers have developed ways to reduce the acidity of rain. For example, several kinds of devices remove sulfur and nitrogen compounds from fuels or industrial emissions before they reach the atmosphere. Adding lime to lakes and rivers and their drainage areas temporarily neutralises their acidity. But the neutralization may have harmful side effects.

Biogeochemical Cycles More than 30 chemical elements are cycled through the environment by biogeochemical cycles. There are six important biogeochemical cycles that transport carbon, hydrogen, oxygen, nitrogen, sulphur, and phosphorus. These six elements comprise the bulk of atoms in living things. Carbon, the most abundant element in the human body, is not the most common element in the crust, silicon is. The Hydrologic (Water) Cycle Saltwater evaporates due to the sun’s energy, producing fresh water in clouds and leaving salts in the ocean. Water vapour cools and condenses to precipitation over oceans and land. Runoff forms freshwater lakes, streams, ponds, ground-water, and is held in plants and transpired. Some water infiltrates the ground, becoming part of the groundwater, returning very slowly to the oceans. Although the water cycle shows water to be a renewable resource, only about 3 per cent of that water is fresh and suitable for human use. The Phosphorus Cycle Weathering of rocks makes phosphate ions (PO4– and HPO4–) available to plants through uptake from the soil. The mineral apatite contains a small amount of phosphorus, sufficient for all living things to utilise. Runoff returns phosphates to aquatic systems as sediments. Organisms use phosphate in phospholipids, ATP, teeth, bones, and shells. Phosphate is a limiting nutrient because most of it is being currently used in

organisms. Humans mine phosphate ores for use in fertillizers, as an animal feed supplement, and for detergents. The Carbon Cycle There is a relationship between photosynthesis and cellular respiration. The latter releases carbon dioxide, which is used in photosynthesis, which in turn releases oxygen, used in respiration. When organisms respire, some of this carbon is returned to the atmosphere in the molecules of carbon dioxide. In aquatic ecosystems, carbon dioxide from air combines with water to give carbonic acid, which breaks down to bicarbonate ions. The Nitrogen Cycle Atmospheric nitrogen gas (N2) is unavailable to plants. Plants, therefore, depend on various types of nitrogenfixing bacteria to take up nitrogen gas and make it available to them as some form of organic nitrogen. Nitrogen fixation occurs when nitrogen gas is chemically reduced and nitrogen gas is chemically reduced and nitrogen is added to organic compounds. Atmospheric nitrogen is converted to ammonium (NH 4+ ) by some cyanobacteria in aquatic ecosystems and by nitrogen-fixing bacteria in the nodules on roots of legume (beans, peas, clover, and so on) plants in terrestrial ecosystems. Nitrification

It is the inorganic production of nitrates. Nitrogen gas (N2) is converted to nitrate (NO 3– ) by cosmic radiation, meteor trails, and lightning in the atmosphere. Human technology can now manufacture nitrates for use in fertilisers. In soil, bacteria convert ammonium (NH 4+ ) to nitrate (NO 3– ). Denitrification

It is conversion of nitrate to nitrous oxide and nitrogen gas back to atmosphere. This is done by denitrifying bacteria in both aquatic and terrestrial ecosystems. The process of denitrification almost, but not completely, counterbalances nitrogen fixation.

IMPORTANT PHENOMENA IN BIOSPHERE Disruption of the Biosphere Human technology and population growth can directly and indirectly disturb the biosphere. The key question now is: can humans cause global climate change? The human population has experienced phenomenal exponential growth since the Industrial Revolution. Modern agriculture and medicine have increased the growth rates of our population, resulting in over 90 million people being added each year. New technologies: for hunting and farming have enabled this expansion. It took 1800 years to reach a total population of one billion, but only 130 years to reach two billion, and a mere 45 years to reach four billion. Atmospheric Pollution The 1 per cent of the atmosphere (trace gases) that is neither nitrogen nor oxygen, plays an important role in global climate and in shielding the earth’s surface from solar radiation. Agricultural and industrial gases may affect the atmosphere’s ability to protect as well as alter the world’s climate. Carbon dioxide has many sources (cellular respiration and the burning of wood or fossil fuels such as coal or petroleum). There are two main sinks for carbon dioxide: plants and the oceans. Plants

convert carbon dioxide into organic molecules by photosynthesis. Oceans form calcium carbonate and, over long periods of time, store it as limestone. Temperatures, Greenhouses, and Climate Changes The average temperature of the earth has risen by 0.50°C over the past 100 years. Although a long-ternl rise of 2°C would seem minor, this is thought to be sufficient to completely melt the glacial ice caps in Antarctica and Greenland, causing sea levels to rise by 100 metres. This can alter climate patterns such as rainfall. Climate changes can have biological (such as migrations) as well as geopolitical and economic consequences. The greenhouse effect is a warming of the lower atmosphere caused by accumulation of certain greenhouse gases (notably carbon dioxide and methane) that allow the rays of the sun to pass through, but then reflect or reradiate the heat to the earth. In this way heat is trapped on earth in much the same way as heat is trapped behind the glass panels of a greenhouse. Greenhouse gases include a carbon dioxide nitrous oxide (NO2), and chlorofluorocarbons (CFCs).

General Science ▌ 2.365 The Ozone Layer Earth’s atmosphere consists of a number of different layers. The troposphere is the lower atmospheric layer. It surrounds us at ground level. The stratosphere, often referred to as the upper atmosphere, contains the ozone shield, a layer of ozone (O3) in the stratosphere, 50 km above the ground. Ozone is produced in the upper atmosphere when sunlight strikes oxygen atoms and causes them to temporarily combine. Stratospheric ozone helps filter most of the high-energy ultraviolet radiation that causes cancer and mutations. The development of the atmospheric ozone shield is one of the crucial events in the history of life, which permitted life to exist on land. Known hazards of UV radiation include increased mutation rate, which can lead to skin cancer and cataracts, depression of the immune system, impaired crop and tree growth, and the death of plankton. Ozone is being destroyed by the release of gases, such as chlorofluorocarbons (CFCs), containing chlorine (Cl –) atoms into the stratosphere. CFCs are used in refrigerators, airconditioners, and solvents. Acid Deposition Humans also alter their local atmosphere by pollution and acid rain. Burning of fossil fuels releases carbon dioxide and nitrogen and sulphur oxides. Sulphur combines with atmospheric water vapour to form sulphuric acid. Forests and lakes suffer from the pH and soil acidity changes resulting from acid rain. Coal and oil routinely burned by power plants emit sulphur dioxide (SO2) into the air. Oil from Kuwait has a naturally high sulphur content. Oil well fires, some set on purpose during the Gulf War of the early 1990s, released a lot of sulphur dioxide into the atmosphere. Origin of Life The sun and its planets formed between 5 and 4.6 billion years ago as matter in our solar system began to coalesce because of gravity. By about 3.9 billion years ago, the Earth had an atmosphere that contained the right mix of hydrogen, oxygen, carbon, and nitrogen to allow for the creation of life. Scientists believe that the energy from heat, lightning, or radioactive elements caused the formation of complex proteins and nucleic acids into strands of replicating genetic code. These molecules then organized and evolved to form the first simple forms of life. At 3.8 billion years ago, conditions became right for the fossilisation of the Earth’s early cellular life forms. These fossilised cells resemble present day cyanobacteria. Such cells are known as prokaryotes. Prokaryote cells are very simple, containing few specialised cellular structures and their DNA is not surrounded by a membranous envelope. The more complex cells of animals and plants, known as eu. karyotes, first showed up about 2.1 billion year ago. By 680 million years ago, eukaryotic cells were beginning to organize themselves into multicellular organisms. Starting at about 570 million years ago an enormous diversification of multicellular life occurred known as the Cambria explosion. During this period all but one modern phylum of animal life made its first appearance on the Earth.

Smog Smog is an urban problem caused by combustion of fuels. Pollutants react with sunlight to cause more than 100 secondary pollutants that can cause respiratory problems like asthma in humans. Most problems with acid rain and smog are caused by the use of fossil fuels. Photochemical smog is air pollution that contains nitrogen oxides (NOx) [where x is 2 or 3] and hydrocarbons (HC), that react together in the presence of sunlight to produce ozone (O3) and peroxylacetyl nitrate (PAN). Both NOx and hydrocarbons

result from the burning (or combustion) of fossil fuels. Acid deposition is responsible for: sterilization of North American and European lakes and forests; reducing the populations of small invertebrates and decomposers, threatening the ecosystem; reducing agricultural yields; causing extensive structural damage by corroding marble, metal, and stonework degrading water supplies by leaching heavy metals from the soil into drinking water supplies; and it is statistically implicated in increases in lung cancer and colon cancer Thermal Inversions Thermal inversions are local occurrences of polluted air being trapped close to the surface of the earth. A thermal inversion occurs when a layer of dense cold air is trapped under a layer of warm air. Areas around hills are susceptible because air stagnates, with little mixing between the two layers. Water Pollution is Altering the Hydrosphere Air pollutants will eventually precipitate into the hydrosphere. During cooler periods, ice acts as a water reservoir, forming glaciers that lower the sea level and affect climate. Global warming will melt this reservoir, raising the sea level (or sinking coastlines). There are three basic sources of water pollutants: municipal sewage, industrial discharges, and agriculture/mining/logging discharges of sediment. The oceans are the final recipients of wastes deposited in rivers and along the coasts. Waste dumping occurs at sea, and ocean currents sometimes transport both trash and pollutants back to the shore. Offshore mining and shipping add pollutants to the oceans. Desertification and Deforestation Deforestation is cutting the forest cover of land. Over the years, man has cut forests for agricultural purposes. Building new housing around new (or enlarging) cities also removes land from agricultural uses. Change of land use can alter heat distribution patterns and surface water runoff. Expanded urbanization also degrades the environment. In agricultural areas, wind and rain carry away about 25 billion tons of top soil yearly, worldwide. At such a rate, it is estimated that practically all top soil will be lost by the middle of the next century. Soil erosion causes a loss of productivity; it is compensated for by fertilizers, pesticides, and fossil fuel energy. One solution is to employ strip-cropping and contour farming to control soil erosion. Desertification is the transformation of marginal lands to desert conditions due to overgrazing and overfarming. Mass Extinctions Extinctions occur when environments change too fast. Local or mass extinctions can occur. They differ in scale, scope, and numbers of species involved. Five environmental changes (mostly cooling) of global proportions resulted in the five mass extinctions in the earth’s history. Recovery from these extinctions took millions of years. There have been several natural mass extinctions in the history of the earth, followed by recovery. By 2010, very little undisturbed rain forest will exist outside nationl parks. Conservation biology is a discipline that brings together many fields to attempt to solve biodiversity problems. It attempts to develop practical approaches to preventing extinction of species and destruction of ecosystems. Most conservation biologists believe biological diversity is good and each species has a value of its own.

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General Knowledge

ECOSYSTEMS AS ENVIRONMENTAL SUPPORT SYSTEMS Diversity of species is important to preserve because ecosystems are composed of species and provide us with so many of life’s essentials; dismantle them and we have a really big problem. Oxygen to breathe (for use in aerobic respiration); Medicines (to augment the body’s own immune defences); Food (which we as heterotrophic consumers in the food pyramid need); Drinkable water (purified by microbial activity) Evolution, in biology, is a complex process by which the characteristics of living organisms change over many generations

as traits are passed from one generation to the next. The science of evolution seeks to understand the biological forces that caused ancient organisms to develop into the tremendous and ever-changing variety of life seen on earth today. The study of evolution provides a focus for investigations into the nature of life itself, the origins of life, the great diversity of living things and their underlying similarities in structure and function.

PRACTICE EXERCISE 1. A plant with green leaves placed in red light will appear (a) black (b) green (c) red (d) violet 2. Electromagnetic radiation is emitted by (a) X-rays (b) Electrons (c) Ultrasonic (d) Protons 3. The common refrigerant used in the domestic refrigerator is (a) neon (b) freon/ammonia (c) spirit (d) None of these 4. A fuse wire is characterized by (a) high resistance and high melting point (b) low resistance and high melting point (c) high resistance and low melting point (d) low resistance and low melting point 5. Who said, ‘A body will remain at rest unless an external force acts on the body’? (a) Newton (b) Einstein (c) Archimedes (d) Galileo 6. Which metal is commonly used as an electromagnet? (a) Copper (b) Iron (c) Nickel (d) Cobalt 7. When ice melts in a beaker of water, the level of water in the beaker will (a) increase (b) decrease (c) remain the same (d) first increase and then decrease 8. Which of the following is not a chemical action? (a) Burning of coal (b) Conversion of water into stream (c) Digestion of food (d) Burning of paper

9. Which of the following is a physical change? (a) Burning of cooking gas (b) Souring of milk (c) Digestion of food (d) Dissolution of sugar into water 10. Conversion of a substance directly from solid to vapour state is known as (a) Vaporization (b) Sublimation (c) Decomposition (d) Ionization 11. The chemical name of table salt is (a) potassium chloride (b) sodium chloride (c) calcium chloride (d) sodium hyposulphate 12. The chemical name of Vitamin C is (a) citric acid (b) ascorbic acid (c) oxalic acid (d) nitric acid 13. Match the following: Column I

Column II

(A)

Sodium bicarbonate

1.

Bleaching powder

(B)

Calcium carbonate

2.

Baking soda

(C)

Potash

3.

Chalk

(D)

Calcium

4.

Alum hypochlorite

A B

C D

(a)

1

2

4

3

(b)

2

3

4

1

(c)

4

2

3

1

(d)

3

1

4

2

14. The chemical name of bauxite is (a) aluminium oxide (b) aluminium chloride (c) aluminium sulphite (d) anhydrated aluminium oxide

General Science ▌ 2.367 15. Crows and pigeons are _____ animals. (a) warm-blooded (b) cold blooded (c) neither (a) or (b) (d) both (a) and (b)

29. The M.K.S. unit of electric charge is (a) Volt (b) Ampere (c) Watt (d) Coulomb

16. Snakes and lizards are _____ animals. (a) warm-blooded (b) cold-blooded (c) Both (a) and (b) (d) None of these

30. The oil in the wick of an oil lamp rises up due to (a) pressure difference (b) capillary action (c) low viscosity of oil (d) gravitational force

17. Fish respire through the (a) nose (c) gills

31. The Function of the choke is to (a) give relatively large impedance of alternating current (b) give relatively large impedance of direct current (c) regulate the voltage (d) increase the voltage

(b) lungs (d) fins

18. Process of cell division can take place by (a) Heterosis (b) Fusion (c) Mitosis (d) None of these 19. Which of the following is an amphibian? (a) Whale (b) Turtle (c) Frog (d) Cow

32. If the length of a heater coil is reduced by 10% of its original length, then power consumed by the heater will (a) increase over 10% (b) decrease by 10% (c) increase by 0.5% (d) decrease by 0.5%

20. Which of the following creatures has no blood but respires? (a) Cockroach (b) Earthworm (c) Fish (d) Hydra

33. The chief constituent of gobar gas is (a) methane (b) ethane (c) propane (d) chlorine

21. Which of the following is the basic characteristic of a living organism? (a) Ability to move (b) Ability to reproduce (c) Ability to eat (d) Ability to breath 22. What is the main purpose of white blood corpuscles? (a) To carry nutrients (b) To combat infection (c) To carry oxygen (d) To give strength

34. One should not sleep in a closed room where a charcoal fire is burning because (a) charcoal produces carbon monoxide which decreases oxygen content in air (b) charcoal produces carbon dioxide which is injurious to health (c) charcoal increases temperature inside the room (d) None of these

23. Which of the following helps in clotting of blood? (a) Vitamin B1 (b) Vitamin B2 (c) Vitamin D (d) Vitamin K

35. Gelatin is added to ice cream to (a) hasten freezing (b) avoid crystal formation (d) give flavour and taste (e) None of these

24. Total volume of blood in a normal human being is (a) 5–6 litres (b) 3–4 litres (c) 8–10 litres (d) 10–12 litres 25. What shapes the fundamental form of Earth’s surface? (a) Geology (b) Anthropology (c) Politics (d) Economics 26. What is the theory of plate tectonics? (a) Earth was created 6,000 years ago. (b) Earth is made up of a large number of geological plates that move slowly across its surface. (c) Human activities are changing Earth’s surface. (d) The number of tectonic plates equals the number of continents on Earth.

36. Permanent hardness of water is due to the presence of (a) calcium bicarbonate (b) magnesium bicarbonate (c) calcium sulphate (d) sodium bicarbonate 37. Which matter is used in storage batteries? (a) Copper (b) Lead (c) Aluminium (d) Zinc 38. Which of the following constituents of pollutant exhaust is hazardous to man? (a) Carbon monoxide (b) Carbon dioxide (c) Lead particles (d) Carbon particles

27. Which of the following does not admit any division? (a) Atom (b) Molecules (c) Compound (d) None of these

39. Mammals are (a) warm-blooded (b) cold-blooded (c) Both (a) and (b) (d) None of these

28. The dynamo converts (a) mechanical energy into electrical energy (b) electrical energy into mechanical energy (c) mechanical energy into magnetic energy (d) none of these

40. A cockroach has (a) Three pairs of walking legs (b) Two pairs of walking legs (c) Four pairs of walking legs (d) One pair of walking legs

2.368

General Knowledge

41. Match the following: Column I

Column II

A. Bat

1.

Reptile

B. Amoeba

2.

Mammalia

C. Lizard

3.

Arthropod

D. Insect

4.

Protozoan

A

B C

D

(a)

2

3

4

1

(b)

2

4

1

3

(c)

4

2

3

1

(d)

2

4

3

1

42. In mammals, the embryo develops in the (a) ovary (b) uterus (c) urethra (d) placenta 43. All the following are invertebrates, except: (a) Snail (b) Crab (c) Prawn (d) Fish 44. The exchange of gases in mammals takes place in the (a) larynx (b) trachea (c) alveoli (d) bronchi 45. Red blood corpuscles are formed in the (a) liver (b) bone marrow (c) kidneys (d) heart 46. Blood does not coagulate inside the body due to the presence of (a) hemoglobin (b) heparin (c) fibrin (d) plasma

52. Which of the following is an example the possible effects of global warming? (a) The Indian wheat belt may receive more rainfall. (b) Grain production is likely to increase. (c) Canada and Russia may experience shorter growing seasons. (d) Sea level will rise. 53. A red light is used in a traffic signal because (a) it has the longest wavelength and can be easily noticed from a long distance (b) it is beautiful (c) it is visible to people even with bad eyesight (d) none of these 54. The most convenient colour combination that can be used for vehicles running day and night is (a) Orange and blue (b) White and black (c) Red and green (d) Yellow and blue 55. Metals are good conductors of electricity because (a) they contain free electrons (b) the atoms are lightly packed (c) they have high melting point (d) All of the above 56. When iron and wood are exposed to sunlight, the iron rod becomes hot quickly because (a) greater thermal conductivity of iron (b) lower thermal conductivity of iron (c) more density of iron (d) less density of iron

47. The number of bones in the human body of an adult is approximately (a) 210 (b) 250 (c) 206 (d) not fixed

57. Water cannot be used to extinguish fire caused by an electric current because (a) it may cause another short circuit (b) it may cause hydrolysis (c) it may cause electrocution (d) it will spoil the wiring

48. Lungs are situated in the (a) abdominal cavity (c) buccal cavity

58. In diesel engines, ignition takes place by (a) compression (b) electrical spark (c) dynamo (d) battery

(b) pericardial cavity (d) thoracic cavity

49. Anthropogenic emissions are caused by (a) people (b) animals (c) climate (d) rocks

59. Properties of element are determined by (a) Atomic number (b) Atomic weight (c) Neutrons (d) Protons

50. Which of the following is not one of Earth’s climate controls? (a) Incoming solar energy (b) Anthropogenic systems (c) Latitude (d) Interaction between land and water

60. During a chloride test white fumes are produced due to formation of (a) ammonium chloride (b) ammonium nitrate (c) silver nitrate (d) None of these

51. What substance accounts for more than half of the humangenerated greenhouse gases? (a) chlorofluorocarbons (b) carbon dioxide (c) methane (d) nitrous oxide

61. Milk is pasteurized in order to (a) destroy microorganisms (b) enhance taste and flavour (c) increase its food value (d) detect adulteration

General Science ▌ 2.369 62. Temporary hardness of water is due to the presence of (a) calcium sulphate (b) calcium bicarbonate (c) magnesium sulphate (d) calcium chloride 63. The chemical composition of four well known items which are used in large quantities are given below. Which one is incorrect? (a) Washing soda: sodium, hydrogen and aluminium (b) Caustic soda: sodium, hydrogen and oxygen (c) Baking soda: sodium, hydrogen, oxygen and carbon (d) Common salt: sodium and chlorine 64. Which of the following is a nonferrous metal? (a) Cobalt (b) Aluminium (c) Nickel (c) Iron 65. Which of the following is not a mammal? (a) Fish (b) Cow (c) Goat (d) Whale 66. In mammals, fertilization takes place in the (a) ovary (b) uterus (c) fallopian tubes (d) ureter 67. Fern belongs to (a) Bryophyta (c) Anthophyta

(b) Pterophyta (d) None of these

68. Aves are (a) cold-blooded (c) both

(b) warm-blooded (d) None of these

69. To which animal phyla do the organisms with round bodies marked externally into rings or segments belong? (a) Porifera (b) Annelida (c) Coelenterata (d) None of these 70. The site of photosynthesis is (a) vacuoles (b) mitochondria (c) chloroplast (d) None of these 71. The pancreas secretes (a) insulin (c) bile juice

(b) vitamin A (d) None of these

72. The seat of memory in the human brain is located in the (a) medulla oblongata (b) cerebrum (c) cortex (d) cerebellum 73. The organs of the circulatory system consist of (a) arteries (b) veins (c) capillaries (d) All of them 74. Tibia is a bone found in the (a) skull (c) leg

(b) arxm (d) face

75. What pair of countries contributes the most greenhouse emissions to the atmosphere? (a) England and Ireland (b) Germany and France (c) United States and China (d) Japan and South Korea

76. An assemblage of local plants, animals, and insects covering a large area such as a tropical rainforest or a grassland is called a ________. (a) bioregion (b) bionicle (c) biosphere (d) biogeography 77. Which of the following is a cause of tropical deforestation? (a) globalization of international wood products commerce (b) the world’s appetite for wood products (c) new settlement (d) all of the above 78. In 2001, American President George W. Bush went on record opposing the Kyoto Protocol primarily because (a) the US is and has not been a significant contributor to greenhouse gas emissions. (b) while the US has contributed greatly to greenhouse gas emissions in the past, in recent years it has made substantial efforts in reducing these emissions. (c) there is no scientific evidence whatsoever supporting a correlation between greenhouse emissions and global warming. (d) atmospheric emission reductions could potentially harm the US economy. 79. A sprayer works on the principle of (a) Newton (b) Archimedes (c) Boyle (d) Pascal 80. Cloudy nights are warmer because (a) clouds prevent radiation of heat from ground to air (b) of low atmospheric pressure (c) of the compact density of air (d) more dust particles gather in the air 81. If we go to the Himalayas, we feel breathless because (a) it is very cold there (b) the density of air is much less on the hills so oxygen content is reduced (c) the density of air is high on the hills reducing oxygen content (d) None of these 82. What is the approximate velocity of sound in air? (a) 3 m/s (b) 30 m/s (c) 300 m/s (d) 3000 m/s 83. In deserts, clouds do not precipitate because of (a) low pressure (b) Low humidity (c) high wind velocity (d) low temperature 84. An echo is produced by the _______ of sound waves. (a) refraction (b) absorption (c) reflection (d) diffraction 85. Which of the following metals is in liquid state at normal room temperature? (a) Sodium (b) Radium (c) Mercury (d) Silicon

2.370

General Knowledge

86. Match the following: i. Carbon dioxide ii. Chlorine iii. Hydrogen iv. Nitrous oxide (a) (b) (c) (d)

A 1 4 4 2

B 2 3 3 1

C 3 2 1 3

A B C D

Reducing agent Anesthetic in dentistry Bleaching powder Refrigerant

D 4 1 2 4

87. Which of the following elements behave chemically both as metal and non-metal? (a) Argon (b) Carbon (c) Xenon (d) Boron 88. Bronze is an alloy of (a) copper and zinc (c) iron and copper

(b) copper and tin (d) zinc and tin

89. Which of the following is used for preservation of grains? (a) Potassium permanganate (b) Boric acid (c) Sodium chloride (d) Sodium benzoate 90. Atomic weight of uranium is (a) 237 (b) 238 (c) 226 (d) 242

98. The largest part of the human brain is the (a) medulla oblongata (b) cerebellum (c) cerebrum (d) midbrain 99. The highest point in the blood pressure or upper reading of blood pressure is called (a) systolic pressure (b) diastolic pressure (c) hypotension (d) hypertension 100. What is the main component of bones and teeth? (a) Calcium carbonate (b) Calcium phosphate (c) Calcium sulphate (d) Calcium nitrate 101. Tropical rainforests typically (a) have rich soils. (b) have three major levels of vegetation. (c) are located 30 degrees or more from the equator. (d) have at least one dry season. 102. Which of the following was not a significant part of the Green Revolution? (a) increased use of fertilizers (b) increased use of pesticides (c) increased use of labour (d) decreased fallow (resting) time for fields 103. Which of the following is a fossil fuel? (a) petroleum (b) wood (c) hydrogen/fuel cell (d) ethanol

91. Which tissue is responsible for the passage of water in plants? (a) Sclerenchyma (b) Xylem (c) Phloem (d) Collenchynatous cells

104. What is the major source of chlorofluorocarbons (CFCs)? (a) burning fossil fuels (b) aerosol sprays and refrigerants (c) by-product of cattle and sheep digestion (d) widespread use of chemical fertilizers in agriculture

92. The vegetation of the Thar desert consists of (a) Xerophytes (b) Hydrophytes (c) Mesophytes (d) Thallophytes

105. Coating of iron with zinc is known as (a) Galvanization (b) Electroplating (c) Ionization (d) None of these

93. In which section of the plant does photosynthesis take place? (a) Roots (b) Stem (c) Leaves (d) Flowers

106. Which of the following is not a primary colour? (a) Blue (b) Green (c) Red (d) Black

94. Photosynthesis takes place faster in (a) yellow light (b) white light (c) red light (d) darkness

107. The speed of light in water is ¾ of that in air. What is the refractive index of water? (a) 3/4 (b) 4/3 (c) 1/4 (d) 0

95. Age of a tree can be determined by (a) counting the number of rings in the stem (b) counting the number of leaves (c) counting the number of branches (d) measuring the size of the tree 96. Which of the following gases are given during photosynthesis? (a) Oxygen (b) Carbon dioxide (c) Nitrogen (d) Hydrogen

108. In the case of a concave mirror, the image of an object is (a) real, erect and diminished when the object lies beyond the centre of curvature (b) real, inverted and of the same size as the object when it lies between the centre of curvature and focus (c) real, inverted and diminished when the object lies beyond the centre of curvature (d) not real, large and unproportional

97. What is the temperature of a normal man? (a) 81.1°C (b) 36.9°F (c) 21.7°C (d) 90°C

109. Sound travels fastest through (a) Vacuum (b) Steel (c) Water (d) Air

General Science ▌ 2.371 110. The process of transfer of heat by matter but without actual movement of the particles themselves is called (a) Conduction (b) Convection (c) Radiation (d) None of the above 111. A body is floating in water at 4°C. If temperature reaches 100°C, then the body will (a) sink fully (b) float freely (b) have some more part of itself submerged under water (d) have no change in its position 112. The major constituent of air is (a) Nitrogen (b) Carbon dioxide (c) Oxygen (d) Hydrogen 113. Which among the following is in liquid state at normal room temperature? (a) Mercury (b) Sodium (c) Phosphorus (d) None of these 114. Which of the following gases is used in cigarette lighters? (a) Butane (b) Methane (c) Propane (d) Radon 115. Liquefied petroleum gas (LPG) consists of (a) butane and propane (b) ethane and hexane (c) ethane and nonane (d) None of these 116. Which of the following is a chemical change? (a) Evaporation of water (b) Burning of candle (c) Glowing of an electric bulb (d) Liquefaction of air 117. Match the following columns: Column A

Column B

(A)

Mercury

1.

Element

(B)

Oxygen

2.

Compound

(C)

Water

3.

Mixture

(D)

Air

4.

Metal

A

B

C

D

(a) 1

2

3

4

(b) 4

3

2

1

(c) 4

1

2

3

(d) 4

2

3

1

120. During the day time, plants (a) take in oxygen and give out carbon dioxide (b) take in carbon dioxide and give out oxygen (c) take in nitrogen and give out oxygen (d) take in carbon dioxide and give out nitrogen 121. Plants receive their nutrients mainly from (a) chlorophyll (b) atmosphere (c) light (d) soil 122. If all plants vanish from the earth’s surface which of the following gases will also disappear? (a) Oxygen (a) Carbon dioxide (a) Hydrogen (a) Nitrogen 123. Plants that grow in saline water are called (a) Halophytes (b) Hydrophytes (c) Mesophytes (d) Thallophytes 124. The onion is a modified form of (a) stem (b) root (c) leaves (d) fruit 125. The cactus is a modified form of (a) stem (b) leaves (c) root (d) flower 126. The main constituent of hemoglobin is (a) chlorine (b) iron (c) calcium (d) None of these 127. The human cell contains (a) 44 chromosomes (c) 46 chromosomes

(b) 48 chromosomes (d) 23 chromosomes

128. The largest muscle in the human body is found in (a) hands (b) buttocks (c) neck (d) leg 129. What is the major source of methane? (a) aerosol sprays and refrigerants (b) burning fossil fuels (c) widespread use of chemical fertilizers in agriculture (d) burning associated with rainforest clearing 130. What is the major source of nitrous oxide? (a) aerosol sprays and refrigerants (b) burning fossil fuels (c) by-product of cattle and sheep digestion (d) widespread use of chemical fertilizers in agriculture

118. Which of the following is incorrect? (a) Mercury: Hg (b) Silver: Ag (c) Sodium: Na (d) Potassium: Ka

131. Which of the following criteria is used for any metal to be used as a cooking utensil? (a) Less thermal conductivity (b) More thermal conductivity (c) Less electrical conductivity (d) More density

119. The green colour of leaves is due to the presence of (a) metallic ions (b) hemoglobin (c) Iron (d) chlorophyll

132. Banking of curves is done to provide (a) centripetal force (b) centrifugal force (c) centrifugal acceleration (d) angular velocity

2.372

General Knowledge

133. What are ultrasonic waves? (a) Sound waves having frequency shorter than audio frequency range (b) Sound waves produced in a vacuum (c) Sound waves having frequency higher than audiofrequency range (d) None of these 134. In the process of galvanization, an iron sheet is coated with (a) Aluminium (b) Zinc (c) Tin (d) Galena 135. The source of solar energy is (a) Nuclear fission (b) Nuclear fusion (c) Sinking of the sun (d) None of these 136. A radar which detects the presence of an enemy aircraft uses (a) Sound waves (b) Radio waves (c) Electric waves (d) Ultrasonic waves 137. The Raman Effect is used in the study of (a) X-rays (a) Cells (a) Chromosomes (a) Molecular energy 138. Which among the following is not a natural occurring element? (a) Uranium (b) Thorium (c) Californium (d) Neptunium 139. Match the following columns: Column I

Column II

(a) Sodium hydroxide

NaCl

(b) Sodium bicarbonate

Na2CO3.10H2O

(c) Sodium chloride

NaHCO3

(d) Sodium carbonate

NaOH

(a)

A

B

C

D

1

2

3

4

(a)

4

2

3

1

(a)

4

3

1

2

(a)

4

2

1

3

140. Which of the following does not belong to the halogen family? (a) Chlorine (b) Flourine (c) Bromine (d) Morphine 141. Which of the following is present in hard water? (a) Calcium (b) Aluminium (c) Sodium (d) Chlorine 142. Which among the following gases is used to manufacture chloroform? (a) Propane (b) Ether (c) Radon (d) Methane

143. Which of the following is a parasitic plant? (a) Marchantia (b) Kelp (c) Mushroom (d) Pteris 144. Which of the following organisms does not contain chlorophyll? (a) Ferns (b) Mosses (c) Fungi (d) Algae 145. In plants organic substances are transported through (a) xylem (b) phloem (c) fibres (d) wood 146. Bacteria belongs to which of the following kingdom? (a) Protista (b) Animalae (c) Plantae (d) Monera 147. The reproductive organs in anthrophyta are in the (a) leaves (b) flowers (c) stem (d) petals 148. Epiphytes plants are those (a) which cling to other plants and take nutrition from them (b) which cling to other plants but do not take nutrition from them (c) which are adapted to grow on dry places (d) None of these 149. ‘Duodenum’ is a part of (a) brain (c) liver

(b) intestines (d) lungs

150. Chromosomes are (a) parts of the human brain (b) thread-like bodies that occur in the nuclei which carry the genetic code (c) constituent of blood which produces heat in the body (d) None of the above 151. The main function of the kidney is (a) to control blood pressure (b) to control body temperature (c) to remove waste products from the body (d) to help in digestion of food 152. A gene is a (a) sleep inducing drug (b) unit of heredity (c) a kind of vitamin (d) a type of body cell 153. The increase in greenhouse gases during the last 150 years is primarily a result of (a) the excessive burning of fossil fuels by humans. (b) the escape of these gases from plate boundaries and volcanoes. (c) increased sunspot activity that accelerate plant photosynthesis rates. (d) the release of excessive amounts of carbon from the oceans.

General Science ▌ 2.373 154. Global warming could have which of the following potential dramatic effects? (a) a regional shift in agriculture production northward (b) a decrease in worldwide grain production (c) rising sea levels as a result of the melting of the polar ice caps (d) all of the above

163. A solid needle placed horizontally on the surface of the water floats due to (a) surface tension (b) capillary action (c) cohesion (d) adhesion

155. What is the only industrialized country in the world that has not ratified the Kyoto Protocol? (a) Japan (b) Russia (c) United States (d) France

165. Which among the following is not a noble gas? (a) Argon (b) Radon (c) Neon (d) Bromine

156. What is a bioregion? (a) regions that are classified by their predominant patterns of temperature and precipitation (b) regions that are classified in regard to the similarities in their plant communities (c) regions that are classified in regard to the similarities in their animal communities (d) regions that are classified in regard to the similarities in both their plant and animal communities

167. Which of the following is the hardest element? (a) Copper (b) Diamond (c) Iron (d) Silicon

157. Fish plates are used in railway tracks (a) To avoid tracks being distorted due to temperature fluctuation (b) To control speed of the train (c) To connect two rails (d) To keep the rails equidistant 158. Food is cooked quickly in a pressure cooker because (a) Boiling point of water is decreased (b) Boiling point of water is increased (c) It absorbs heat quickly (d) It retains heat for a long duration 159. In an earthen pot, cooling of water depends upon (a) outside temperature (b) atmospheric humidity (c) wind (d) All of the above 160. A fuse wire is used to (a) to make the electrical circuit strong (b) convert AC into DC (c) convert DC into AC (d) prevent an unduly high electric current to pass through a circuit 161. A molecule is (a) a class of organic compounds (b) the smallest unit of a substance that retains the properties of the substance (c) the spectrum of a substance (d) a small mass 162. The nucleus of an atom consists of (a) Protons + Electrons (b) Only protons (c) Protons + Neutrons + Electrons (d) Protons and neutrons

164. What is the chemical name of bleaching powder? (a) Calcium carbonate (b) Calcium hypochlorite (c) Calcium phosphate (d) Calcium cyanamide

166. Which of the following is a mixture? (a) Gun powder (b) Iron sulphate (c) Brass (d) Dry ice

168. Which of the following is the source of common salt? (a) Bauxite (b) Halite (c) Pyrite (d) Calcite 169. Match the following: Column I

Column II

(a) Calomel

Copper sulphate

(b) Blue vitriol

Calcium sulphate

(c) Gypsum

Mercurous chloride

(d) Normal salt

Sodium chloride

A

B

C

D

(a)

1

2

3

4

(b)

4

3

2

1

(c)

3

2

1

4

(d)

3

1

2

4

170. Which of the following is a carnivorous plant? (a) Urn plant (b) Pitcher plant (c) Cacti (d) Ribbon plant 171. Potato is a modified form of (a) root (c) fruit

(b) stem (d) leaf

172. Rhizome is an underground modification of (a) root (b) stem (c) leaf (d) fruit 173. Man belongs to phylum (a) mollusca (c) porifera

(b) chordata (d) amphibia

174. Fertilization in frogs is (a) external (b) internal (c) both external and internal (d) aerial

2.374

General Knowledge

175. Insects belong to (a) annelida (c) crustaceans

(b) anthropods (d) coelenterata

176. Phloem is a tissue found in (a) reproductive organs of animals (b) plants (c) insects (d) mammals 177. The function of hemoglobin is (a) to transport oxygen (b) destruction of bacteria (c) prevention of anemia (d) utilization of energy 178. White blood corpuscles in the human body have a diameter of approximately (a) 0.7 mm (b) 0.07 mm (c) 0.007 mm (d) 0.0007 mm 179. In an adult human being, erythrocytes are produced in the (a) spleen and liver (b) spleen and bone marrow (c) bone marrow and liver (d) heart and lungs 180 Tropical rainforest canopies are generally (a) a two-layer canopy that is thickest in its upper layer and thin in its lower layer. (b) a two-layer canopy that is thickest in its lower layer and thin in its upper layer. (c) a three-layer canopy that is thickest at the ground level. (d) a three-layer canopy that is thickest in its middle layer. 181. The nutrients that are available for plant growth in tropical rainforests tend to be stored in (a) the uppermost layer of the soil. (b) the bottommost layer of the soil. (c) a layer that is approximately one meter below the surface. (d) the living plants. 182. Tropical forests are not well-suited for intensive agriculture because (a) the excessive amount of rainfall in the region tends to prohibit effective plant growth. (b) available nutrients are stored in tropical forest vegetation, so when the ground cover and trees are removed, the nutrients are removed, too. (c) the ground layer of vegetation in tropical forests is so thick that the land is extremely difficult to clear. (d) the temperatures in tropical forests are too high to allow feasible crop production. 183. Effect of light on photocells produces (a) energy (b) current (c) photosynthesis (d) None of these

184. Four identical kettles with the same amount of water, have bases made of different metals of the same thickness. If these are placed on identical flames water will boil first in the kettle with a base made of (a) Brass (b) Copper (c) Aluminium (d) Steel 185. A 100 watt bulb is kept on for a period of 10 hours per day in the month of September. The electrical energy consumed in that month in kilowatt hours (kwh) is (a) 30 kwh (b) 3 kwh (c) 3000 kwh (d) 1000 kwh 186. Which of the following is a good conductor of heat but bad conductor of electricity? (a) Asbestos (b) Celluloid (c) Pertex (d) Mica 187. A sudden fall in barometer reading indicates (a) clear weather (a) hailstorm (a) storm (a) heavy rainfall 188. Sound travels with a different speed in different media. In what order does the velocity of sound increase in these media? (a) Water, iron and air (b) Iron, air and water (c) Air, water and iron (d) Iron, water and air 189. Which of the following gases in air unites with many metals? (a) Hydrogen (b) Oxygen (c) Helium (d) Carbon dioxide 190. Which of the following metals is not affected by air and water under ordinary conditions? (a) Iron (b) Platinum (c) Sodium (d) None of these 191. Diamond is the _______ form of carbon. (a) crystalline (b) amorphous (c) chemical (d) alkaline 192. In a chemically pure state, diamonds are (a) monochromatic (b) polychromatic (c) colourless (d) None of these 193. Which metal has the unique tendency to readily combine with itself to form large molecules of atoms linked in long chains (rings)? (a) Sodium (b) Nitrogen (c) Carbon (d) Hydrogen

General Science ▌ 2.375 198. A plant cell is distinguished from an animal cell by the presence of (a) nucleus (b) chloroplasts (c) cell membrane (d) cell wall

194. Gun powder is a mixture of (a) sulphur, carbon and phosphorus (b) sulphur, charcoal and nitre (c) sulphur, charcoal and carbon (d) carbon, nitrogen and chlorine 195. Which of the following is fire resistant? (a) Asbestos (b) Gypsum (c) Dolomite (d) None of these

199. Clove is obtained from (a) flower (b) root (c) stem (d) leaves

196. From which part of the plant is turmeric obtained? (a) Root (b) Fruit (c) Seed (d) Stem 197. The process of the division of somatic cells or an organism is called (a) Mitosis (b) Meiosis (c) Cytokinesis (d) None of these

200. Pollen grains in plants are produced in (a) roots (b) leaves (c) flower (d) stem

ANSWER KEY 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 1 4. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.

(a) (b) (b) (c) (a) (b) (c) (b) (d) (b) (b) (b) (b) (d) (a) (b) (c) (c) (c) (d) (b) (b) (d) (a) (a)

26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50.

(b) (d) (a) (d) (b) (a) (a) (a) (a) (b) (c) (b) (a) (a) (a) (b) (b) (d) (c) (b) (b) (c) (d) (a) (b)

51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75.

(b) (d) (a) (c) (a) (a) (a) (a) (a) (a) (a) (b) (a) (b) (a) (c) (b) (b) (b) (c) (a) (c) (d) (c) (c)

76. 77. 78. 79 . 80. 81. 82. 83. 84. 85. 86. 87. 88. 89. 90. 91. 92. 93. 94. 95. 96. 97. 98. 99. 100.

(a) (d) (d) (d) (a) (b) (c) (b) (c) (c) (c) (d) (b) (d) (b) (b) (a) (c) (c) (a) (a) (b) (c) (a) (b)

101. 102. 103. 104. 105. 106. 107. 108. 109. 110. 111. 112. 113. 114. 115. 116. 117. 118. 119. 120. 121. 122. 123. 124. 125.

(b) (c) (a) (b) (a) (d) (b) (c) (b) (a) (c) (a) (a) (a) (a) (b) (c) (d) (d) (b) (d) (a) (a) (a) (a)

126. 127. 128. 129. 130. 131. 132. 133. 134. 135. 136. 137. 138. 139. 140. 141. 142. 143. 144. 145. 146. 147. 148. 149. 150.

(b) (c) (d) (d) (d) (b) (a) (c) (b) (b) (b) (d) (d) (c) (d) (a) (d) (c) (c) (b) (d) (b) (b) (b) (b)

151. 152. 153. 154. 155. 156. 157. 158. 159. 160. 161. 162. 163. 164. 165. 166. 167. 168. 169. 170. 171. 172. 173. 174. 175.

(c) (b) (a) (d) (c) (d) (b) (b) (d) (d) (b) (d) (a) (b) (d) (a) (b) (b) (d) (b) (b) (b) (b) (a) (b)

176. 177. 178. 179. 180. 181. 182. 183. 184. 185. 186. 187. 188. 189. 190. 191. 192. 193. 194. 195. 196. 197. 198. 199. 200.

(b) (a) (c) (b) (d) (d) (b) (b) (b) (a) (d) (c) (c) (b) (b) (a) (c) (c) (b) (a) (d) (a) (d) (c) (c)

2.376

General Knowledge

NDA BOX 1. If the distance between two charges is halved, then the force between them becomes (a) half (b) double (c) four times (d) one-fourth 2. To increase the magnifying power of a telescope, the focal length of (a) objective lens should be increased (b) objective lens should be decreased (c) eye-piece lens should be increased (d) eye-piece lens should be decreased 3. An object moving around in a circle is moving with (a) uniform velocity (b) uniform speed (c) variable velocity (d) variable speed 4. When water is heated from 0° C to 10° C, the volume of water (a) increases steadily (b) decreases steadily (c) first increases then decreases (d) first decreases then increases 5. Which of the following pairs is incorrect? (a) Atom: Smallest indivisible part of element (b) Molecule: Unit of a compound (c) Atom: Smallest part of a compound (d) Atom: Indivisible particle 6. Lipids are (a) hormones (c) enzyme

(b) fatty acids (d) None of these

7. Match the following: Column I

Column II

(A)

Acetic acid

1.

Milk

(B)

Tannic acid

2.

Proteins

(C)

Amino acid

3.

Tea

(D)

Lactic acid

4.

Vinegar

A

B

C

D

(a)

1

2

3

4

(b)

4

2

1

3

(c)

4

3

2

1

(d)

1

3

4

2

8. Which of the following metals is mostly used for qualitative analysis of organic compounds? (a) Cu (b) Hg (c) Na (d) Ca

9. Permanent hardness of water due to sulphates of metals can be destroyed by the use of (a) zeolites (b) sulphonides (c) lime (d) nitrates 10. Amalgams are the compounds which always contain (a) Al (b) Hg (c) Zn (d) Cu 11. The chemical name of laughing gas is (a) nitrous oxide (b) nitric oxide (c) nitrogen dioxide (d) nitrogen peroxide 12. The site of photosynthesis is (a) chloroplasts (b) mitochondria (c) vacuoles (d) endoplasmic reticulum 13. Give the chronological sequence of origin of the following plant groups. (i) angiosperms (ii) gymnosperms (iii) fungi (iv) algae (a) (b) (c) (d)

(iii), (iv), (i), (ii) (iii), (ii), (iv), (i) (iii), (iv), (ii), (i) (i), (iv), (iii), (ii)

14. Match the following: A. Yeast

1.

Protozoa

B. Amoeba

2.

Algae

C. Chlorella

3.

Bacteria

D. Salmonella

4.

Fungi

A

B

C

D

(a)

2

3

1

4

(b)

3

1

4

2

(c)

4

1

2

3

(d)

4

1

3

2

15. Which one of the following constitute the two best reasons distinguishing between the living and non-living? (a) growth and locomotion (b) respiration and excretion (c) reproduction and locomotion (d) irritability and locomotion 16. A person of which of the following blood groups is called a universal donor? (a) O (b) AB (c) A (d) B

General Science ▌ 2.377 17. Enzymes help in (a) respiration (c) immune system

(b) digestion of food (d) reproduction

18. Food is normally digested in the (a) liver (b) stomach (c) small intestines (d) large intestines 19. Ptyalin is an enzyme produced in the (a) salivary glands (b) pituitary glands (c) thyroid glands (d) pancreas 20. Some estimates suggest that logging occurs _________ times faster in South-east Asia than in the Amazon. (a) 2 (b) 3 (c) 4 (d) 5 21. What is the major cause of widespread cutting of tropical rainforests? (a) the recent globalization of commerce in international wood products (b) an expansion of farmlands in tropical regions (c) intensified efforts to protect the forests from wildfires by harvesting (d) rising worldwide demand for paper 22. Which of the following contributes significantly to the destruction of tropical rain forests? (a) the world’s growing appetite for beef (b) a tree disease that was accidentally carried to tropical forests around the world (c) intensified efforts to protect the forests from wildfires by harvesting (d) flooding 23. If current rates of deforestation continue in Southeast Asia, how many years would it take for the region to become completely stripped of its forests? (a) 5 (b) 15 (c) 25 (d) 50 24. What percentage of tropical forest timber is used in Japan? (a) 10% (b) 20% (c) 30% (d) 50% 25. In what type of bioregion is desertification most likely to occur? (a) tropical forests (b) temperate forests (c) coniferous forests (d) grasslands 26. With what bioregion is the term “steppe” associated? (a) grasslands (b) tropical forests (c) savanna (d) coniferous forests 27. In electric fittings, the wires are earthed because (a) in case of a short circuit, current passes to the earth (b) it avoids leakage of electricity (c) it completes the electric circuit (d) it reduces fluctuation

28. Which of the following metals is used in aircraft? (a) Chromium (b) Titanium (c) Palladium (d) None of these 29. The force of gravity on the moon is (a) less than on the earth as the moon has a smaller mass (b) the same as on the earth (c) more than the earth (d) there is no gravitational force at all 30. A transformer is used to (a) increase DC voltage (b) increase or decrease AC voltage (c) decrease DC voltage (d) convert DC into AC 31. When milk is churned, the cream from it is separated due to (a) gravitational force (b) centrifugal force (c) frictional force (d) heat 32. The molecular formula of phosphorus is (a) P1 (a) P2 (a) P3 (a) P4 33. The gas used for artificial ripening of green fruits is (a) carbon dioxide (b) acetylene (c) ethane (d) oxygen 34. Baking soda is also known as (a) sodium bicarbonate (b) sodium carbonate (c) calcium chloride (d) calcium carbonate 35. Which of the following is not a compound of calcium? (a) Gypsum (b) Marble (c) Chalk (d) Molybdenum 36. Which of the following gases finds its application in extinguishing fire? (a) Oxygen (b) Carbon dioxide (c) Carbon monoxide (d) All of these 37. The first metal used by man was (a) iron (b) copper (c) gold (d) bronze 38. Non-vascular plants do not contain (a) meristematic cells (b) xylem vessels (c) epidermis (d) parenchyma 39. Which of the following structures are commonly present in the animal cells and not in plant cells? 1. mitochondria 2. lysosomes 3. ribosomes 4. centrosomes (a) 2 and 3 (c) 3 and 4

(b) 1 and 3 (d) 2 and 4

40. Which of the following plants is adapted to grow in dry places? (a) Heliophytes (b) Sciophytes (c) Hydrophytes (d) Xerophytes

2.378

General Knowledge

41. Flower colours are due to (a) chlorophyll (a) phytochromes

(a) anthocyanins (a) melanin

42. The bitter-sharp taste of unripe fruits is due to high concentration of (a) phenoic compounds (b) volatile compounds (c) organic acids (d) starch 43. During day time plants produce (a) carbon dioxide (b) oxygen (c) nitrogen (d) none 44. Heightened emotion is caused by (a) pituitary glands (b) thyroid glands (c) adrenal glands (d) salivary glands 45. The diaphragm assists in (a) sigestion (c) circulation of blood

(b) respiration (d) excretion of waste

46. The smallest muscle in the human body is (a) vertebrae (b) stapes (c) phalanges (d) metacarpals 47. Which part of the human body coordinates function of various organs? (a) Heart (a) Liver (a) Brain (a) Kidneys 48. Barium is used for (a) checking blood groups (c) x-ray of brain

(b) x-ray of alimentary canal (d) none of these

49. According to a United Nations estimate, about how much of the world’s rangelands are threatened by desertification? (a) 15% (b) 20% (c) 30% (d) 60% 50. Between 1950 and 1980, an area the size of Denmark has been desertified in which country? (a) China (b) Norway (c) Brazil (d) Bangladesh 51. A bee sting leaves an acid which causes pain and irritation. The injected acid is (a) Acetic acid (b) Sulphuric acid (c) citric acid (d) Methanoic acid 52. Which one among the following is the main ingredient in cement? (a) Gypsum (b) Limestone (c) Clay (d) Ash 53 . A bar magnet is placed inside a uniform magnetic field. What does it experience? (a) A force (b) A torque (c) Both a force and a torque (d) Neither a force nor a torque

54. A mixture containing SiO2, NaCl and NH4Cl is taken for separating the constituents. The suitable steps required for this are (a) Sublimation- dissolution-filtration-crystallization (b) Dissolution-filtration-crystallization-distillation (c) Sublimation-evaporation-dissolution-decomposition (d) Dissolution-distillation-decomposition-evaporation 55. A typical human rib cage consists of how many ribs? (a) 12 (b) 14 (c) 16 (d) 24 56. A body initially at rest is acted upon by a constant force. The rate of change of its kinetic energy varies (a) Linearly with square root of time (b) Linearly with time (c) Linearly with Square of time (d) Inversely with of time 57. All enzymes are (a) Alkaloids (b) Proteins (c) Carbohydrates (d) Lipids 58. Which one of the following chemicals is used in beauty parlours for hair-settting? (a) Sulphur based (b) Phosphorus based (c) Silicon based (d) Iron based 59. A body is at rest on the surface of the earth. Which one among the following statements is correct regarding this? (a) No force is acting on the body (b) Only weight of the body acts on it. (c) Net downward force is equal to the net upward force (d) None of the above 60. Bats can ascertain distance, directions, nature and size of the obstacles at night. This is possible by reflection the emitted (a) Ultrasonic waves from the bat (b) Ultrasonic waves from the distant objects (c) Supersonic waves from the bat (d) Supersonic waves from the distance objects 61. A body is thrown upward against the gravity g with initial velocity u. Which one among the following is the correct expression for its final velocity when it attains the maximum height? u2 2g u2 g (c) 2 (a)

(b)

2g u2

(d) None of these

62. Consider the following Plants: I. Jasmine II. Sugarcane III. Turmeric

General Science ▌ 2.379 Which of the above is/are propagated by vegetative method? (a) I only (b) I and II (c) II and III (d) I, II and III 63. Cure to spinal injury is likely to emerge from (a) Gene therapy (b) Stem cell therapy (c) Xenograft (d) Tranfusion 64. From the evolutionary point of view, which one among the following is the most primitive animal? (a) Dolphin (b) Otter (c) Turtle (d) Walrus 65. The best and the poorest conductors of heat are respectively (a) Silver (Ag) and lead (pb) (b) Cooper (Cu) and aluminium (Al) (c) Silver (Ag) and gold (Au) (d) Copper (Cu) and gold (Au) 66. The reaction that takes place in soap making is called saponification. Basically, soap is sodium or potassium salts of (a) Long chain monocarboylic acids (b) Glycerol (c) Long chain dicarboxylic acids (d) Long Chain tricarboxylic acids 67. What is the correct order of oxidation states of sulphur in anions SO32-, S2 O42- and S2 O62-? (a) S2O42  SO32  S2O62 (b) S2O62  S2O42  SO32 (c) SO32  S2 O 42  S2 O62 (d) S2 O 24  S2O62  SO32 68. In a living cell, what is the site of ribosome formation? (a) Golgi bodies (b) Endoplasmic reticulum (c) Plasma membrane (d) Nucleolus 69. A body is thrown vertically upwards and then falls back on the ground. Its potential energy is maximum (a) On the ground (b) At the maximum height (c) During the return journey (d) Both on the ground and at the maximum height 70. A student heated some sulphur in spatula and collected the gas ‘X’. Which one among the following is correct about ‘X’? (a) X is SO2 and it turns moist litmus to blue (b) X is SO2 and it turns moist litmus to blue (c) X is SO2 and it turns moist litmus to red (d) X is SO2 and it turns dry litmus to blue 71. In the human bid, which of the following store/stores extra blood for release when shortages occur? (a) Liver (b) Pancreas (c) Spleen (d) Lymph nodes

72. Which one of the following elements shows variable equivalent mass? (a) Zinc (b) Silver (c) Calcium (d) Iron 73. When aqueous solutions of two salts are mixed, the third salt formed may appear as a solid precipitate or a clear solution depending upon the solubility of its ions. It is observed that all salts of Na, K, NH4 are soluble. All nitrates and bicarbonates are also soluble. All halides (Chlorides, bromides, iodides) are soluble except those of Ag, Hg (I) and Pb. All sulphates are soluble except those of Ag, Ca, Ba and Pb. Which one among the following combinations of solutions will produce a solid precipitate? (a) Sodium sulphate and barium chloride (b) Magnesium sulphate and barium bicarbonate (c) Lithium iodide and barium chloride (d) Ammnioum sulphate and potassium bromide 74. A man with a dark skin, in comparison with a man with a white skin, experiences (a) Less heat and less cold (b) Less heat and more cold (c) More heat and less cold (d) More heat and more cold 75. Jaundice in human beings is the result of (a) Incomplete metabolism of biliverdin (b) Complete metabolism of biliverdin (c) Complete metabolism of bilirubin (d) Complete metabolism of bilirubin 76. A neutral (uncharged) metal ball is suspended using a nonmagnetic string. A positively charged insulating rod is placed near the ball which is observed to be attracted to the rod. This is because (a) The ball becomes positively charged by induction (b) The ball becomes negatively charged by induction (c) There is a rearrangement of the electrons in the ball (d) The number of electrons in the ball is more than the number of electrons on the rod 77. A one-rupee coin is placed at the bottom of a vessel. Water is then poured onto the vessel such that the depth of water becomes 20 cm. If water has refractive index 4/3, the coin would be seen at a depth of (a) 20 cm (b) About 26 cm (c) 15 cm (d) 25 cm 78. A particle oscillates in one dimension about the equilibrium position subject to a force Fx (x) that has an associated potential energy U (x). if k is the force constant, which one of the following relations is true? (a) Fx ( x )  kx 2 1 (c) U ( x )  kz 2

(b) Fx ( x )  kx 1 (d) U ( x )  k 2 x 2

2.380

General Knowledge

79. Pitch is a characteristic of sound that depends upon its. (a) Intensity (b) Fequency (c) Quality (d) None of these 80. A pendulum beats faster than a standard pendulum. In order to bring it to the standard beat, the length of the pendulum is to be (a) Reduced (b) Increased (c) Reduced and the mass of the bob increased (d) Reduced and also the mass of the bob reduced 81. According to which one of the following laws it is indicated that when two or more gases react with one another, their volumes bear a simple ratio? (a) Law of mass action (b) Law of multiple proportions (c) Law of multiple proportions (d) Law of reciprocal volumes 82. Analysis shows that a binary compound of X (atomic mass = 10) and Y (atomic mass = 20) contains 50% X. What is the formula of the compound? (a) XY (b) X2 Y (c) XY2 (d) X2 Y3 83. A ray of light falls on a transparent glass plate. A part of it is reflected and a part is refracted. The reflected and refracted rays can be perpendicular to each other for (a) Angle of incidence equal to 90º (b) Angle of incidence equal to zero (c) Only one angle of incidence (d) More than one angle of incidence 84. A ray of light is incident normally on one of the faces as shown above. It undergoes total internal reflection from hypotenuse. Which one of the following is the minimum refractive index of the material of the prism?

(a) 1.0 (c) 1.414

(b) 1.33 (d) 1.6

85. Sandal wood tree is considered a (a) Total root parasite (b) Total stem parasite (c) Stem parasite (d) Partial root parasite 86. An electric lamp of 100 W is used for 10 h per day. The units of energy consumed in one day by the lamp is (a) 1 unit (b) 0.1 unit (c) 10 units (d) 100 units

87. An object is kept 5 cm in front of a concave mirror of focal length 15 cm. What will be the nature of the image? (a) Virtual, not magnified (b) Virtual, magnified (c) Real, not magnified (d) Real, magnified 88. The common edible mushroom is a (a) Mass of fungal spores (b) Type of hypha (c) Tightly packed mycelium (d) Structure used for producing asexual spores 1 of its volume 4 with water and immersed in a glass of water. The upper portion of the test tube looks bright. Reason (R): This is due to dispersion of light. (a) Both A and R are individually true and R is the correct explanation of A. (b) Both A and R are individually true but R is not the correct explanation of A. (c) A is true but R is false. (d) A is false, but R is true.

89. Assertion (A): A test tube is filled up with

90. Assertion (A): On a wet floor of a bathroom, a small circular area around a pinch of foam appears to be dry. Reason (R): Soap molecules reduce surface tension and increase surface area of the thin layer of water on the wet floor. (a) Both Assertion and Reason are true and Assertion is the correct explanation of the Reason. (b) Both Assertions and Reason are true and Assertion is not the correct explanation of the Reason. (c) Assertion is true but Reason is false. (d) Assertion is false but Reason is true. 91. The main function of the inner bark of a woody plant is to (a) Transport minerals and water from the roots to the leaves (b) Act as a membrane impermeable to water and gas (c) Transport food from the leaves to the other parts of the plant (d) Protect the plant from herbivorous animals 92. By which one among the following mechanisms, soap removes dirt (soil) from cloth? (a) Soap dissolves the soil as such (b) Soap reacts with soil and converts them into soluble silicates (c) Soap takes away the oily part of the soil and thus separates the soil from the cloth (d) The soap molecules bind with the soil, lift the soil and keep it suspended which can then be rinsed away 93. In which one of the following situations, a chemical reaction does not occur? (a) Common salt is exposed to air (b) Coal is burnt in air (c) Sodium is placed in water (d) Iron is kept in moist air

General Science ▌ 2.381 94. Neutral water with pH about 7 becomes slightly acidic when aerated. This is because (a) Oxygen form air is dissolved in the water which makes the water acidic (b) Dirt, which get contaminated with the water during aeration makes the water acidic

(c) Ultraviolet radiation dissociates water molecules and makes acidic (d) Carbondioxide from air is dissolved 95. The natural dye present in turmeric is known as (a) Cinnamin (b) Phenolphthalein (c) Methyl Orange (d) Curcumin

ANSWER KEY 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

(c) (d) (c) (d) (a) (b) (c) (c) (a) (b) (a) (a)

13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.

(c) (c) (c) (a) (b) (c) (a) (b) (a) (a) (b) (d)

25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36.

(d) (a) (a) (b) (a) (b) (b) (d) (b) (a) (d) (c)

37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48.

(b) (b) (d) (d) (c) (c) (b) (c) (b) (b) (c) (b)

49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60.

(d) () (d) (b) (b) (a) (d) (b) (b) (c) (c) (a)

61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72.

(d) (c) (b) (c) (a) (a) (a) (d) (b) (c) (c) (d)

73. 74. 75. 76. 77. 78. 79. 80. 81. 82. 83. 84.

(a, b) (c) (c) (c) (c) (b) (b) (a) (d) (c) (c) (c)

85. 86. 87. 88. 89. 90. 91. 92. 93. 94. 95.

(d) (b) (b) (c) (c) (a) (c) (d) (a) (d) (d)

This page is intentionally left blank.

C

H

6

A

P

T

E

R

Basic General Knowledge Basic General KnowledGe 1. Who among the following wrote a treatise on medicine in ancient India? (a) Aryabhatta (b) Charaka (c) Nagarjuna (d) Varahamihira 2. The author of ‘Hind Swaraj’ was (a) V. D. Savarkar (b) Mahatma Gandhi (c) B. G. Tilak (d) None of these 3. Which of the following books was written by Gandhiji? (a) Discovery of India (b) My Experiments with Truth (c) India Wins Freedom (d) Freedom at Midnight 4. Which of the following books was banned by all Muslim countries, including India? (a) Shame (b) Freedom at Midnight (c) Satanic Verses (d) None of these 5. Humayun Nama was written by (a) Babur (b) Humayun (c) Gulbadan Begum (d) Khanzada Begum 6. The oldest Hindu epic is (a) Mahabhashya (c) Mahabharata

(b) Ramayana (d) Ashtadhyayi

7. Match the following: Column I (Author)

Column II (Book)

A.

Mahatma Gandhi

1.

India Divided

B.

Ram Manohar Lohia

2.

India Wins Freedom

C.

Dr Rajendra Prasad

3.

Hind Swaraj

D.

Maulana Abul Kalam 4. Azad

The Wheel of History

A

B

C

D

(a)

1

2

4

3

(b)

2

3

1

4

(c)

3

1

2

4

(d)

3

4

1

2

8. Which of the following Pairs is correctly matched? (a) Abul Fazal—Ain-i-Akbari (b) Badauni—Akbar Nama (c) Abdul Hamid Lahori—Tabkat-i- Akbari (d) Nizamuddin Ahmed—Badshah Nama 9. Charaka Samhita is a work on (a) Politics (b) Mathematics (c) Economics (d) Medicine 10. Who was the author of the book which had been banned by all Muslim countries, including India? (a) Salman Rushdie (b) George Orwell (c) K. Asaf Ali (d) None of these 11. Match the following: A. Rigveda B. Yajur Veda C. Sama Veda D. Atharvaveda A

B

C D

(a)

2

3

4

1

(b)

3

4

2

1

(c)

4

2

3

1

(d)

4

2

1

3

1. Musical hymns 2. Sacrifices and rituals 3. Medicine 4. Hymns and prayers

2.384

▌ General Knowledge

12. Who wrote the book titled Runs and Ruins? (a) Kapil Dev (b) Imran Khan (c) Sunil Gavaskar (d) None of these

24. Who is the author of Swami and Friends? (a) R. K. Laxman (b) R. K. Narayan (c) Mulk Raj Anand (d) None of these

13. Who wrote As You Like It? (a) Bernard Shaw (c) Leo Tolstoy

(b) William Shakespeare (d) Mulk Raj Anand

25. Post Office is written by (a) R. K. Narayan (c) R. K. Laxman

14. Das Kapital is written by (a) Lenin (c) Karl Marx

(b) Mao-Tse-tung (d) None of these

15. Select the correct pair. (a) Patanjali—Mahabhashya (b) Kaildasa—Setubandha (c) Ashvaghosa—Malavikagnimitra (d) Pravarasena—Buddha Charita

(b) Mulk Raj Anand (d) Rabindranath Tagore

26. Who is the author of Anand Math? (a) Iqbal (b) Bankimchandra Chatterjee (c) Tara Shankar Bandyopadhyay (d) None of these 27. Panchatantra was written by (a) Bhasa (b) Jaya Deva (c) Vishnu Sharma (d) Banabhatta

16. Who is the author of The Great Depression of 1990? (a) Ravi Batra (b) R. K. Narayan (c) M. V. Kamath (d) Khushwant Singh

28. Who is the author of Nineteen Eighty Four? (a) J. M. Barrie (b) Walter Scott (c) George Orwell (d) Thomas Hardy

17. Ain-i-Akbari is written by (a) Firdausi (c) Abul Fazal

(b) Ghalib (d) None of these

18. Match the following: Author A. Kautilya B. Panini C. Maharishi Ved Vyas D. Sarojini Naidu

Work 1. Broken Wing 2. Arthashastra 3. Ashtadhyayi 4. Bhagawad Gita

29. The oldest book composed was (a) Mahabharata (b) Rig Veda (c) Atharvaveda (d) Upanishads

A

B

C

D

(a)

1

4

2

3

(b)

4

1

3

2

(c)

2

3

4

1

(d)

2

3

4

1

19. Who wrote A Passage to India? (a) E. M. Forster (b) Jawaharlal Nehru (c) Minoo Masani (d) None of them 20. Which among the following books was written by Jawaharlal Nehru? (a) A Passage to India (b) My Experiments with Truth (c) India Wins Freedom (d) The Discovery of India 21. Who is the author of the famous play Halla Bol? (a) Uma Shankar (b) Vatsyayan (c) Safdar Hashmi (d) Khushwant Singh

30. The authors of Shahnama and Akbarnama are respectively (a) Firdausi and Abul Fazal (b) Abul Fazal and Firdausi (c) Abul Fazal and Sheikh Sayed (d) None of these 31. Rig Veda chiefly contains (a) early Aryan culture (b) methods of Vedic yagnas (c) collection of hymns to God (d) origin of Hindu gods 32. Match the following: Column I

Column II

A. Seymour Hersh

1.

Runs and Ruins

B.

Salman Rushdie

2.

Shame

C.

Sunil Gavaskar

3.

Price of Power

A

B

C

(a) 1

3

2

(b) 2

1

3

(c) 3

1

2

(d) 3

2

1

22. Who wrote the Naked Triangle? (a) Balwant Gargi (b) Khushwant Singh (c) Amrita Pritam (d) None of these

33. Author of Arthashastra was (a) Kalidasa (b) Kautilya (c) Bhavabhuti (d) None of these

23. Kautilya’s Arthashastra is a (a) Law Book (b) Treatise (c) Hymns (d) Gazetteer

34. The author of Malavikagnimitra was (a) Harishena (b) Kalidasa (c) Vatsabhatti (d) Bhavabhuti

Basic General Knowledge ▌ 2.385 35. One of the great epics of Aryans, the Mahabharata consists of (a) 18 books and 10,000 couplets (b) 18 books and 5,000 couplets (c) 15 books and 10,000 couplets (d) 13 books and 10,000 couplets 36. The number of the Upanishads are (a) Three hundred (b) Two hundred (c) Four hundred (d) Five hundred 37. Arrange the following in chronological order: (1) Bhavabhuti (2) Asvaghosa (3) Kalidasa (a) 2 1 3 (c) 2 3 l

(b) 3 2 1 (d) None of these

38. Who recast the original single Veda into four Vedas? (a) Vyasa (b) Sri Krishna (c) Parasurama (d) Patanjali 39. The author of Geet Govinda is (a) M. K. Gandhi (b) Jaya Deva (c) Mulk Raj Anand (d) Prem Chand 40. Who wrote the book Life Divine? (a) Mulk Raj Anand (b) Abul Kalam Azad (c) Aurobindo Ghosh (d) Jawaharlal Nehru 41. India Wins Freedom is the autobiography of (a) Jawaharlal Nehru (b) Mahatma Gandhi (c) Maulana Abu Kalam Azad (d) None of these 42. Meghdootam was written by (a) Bhavabhuti (b) Kalidasa (c) Maharishi Ved Vyas (d) Maharishi Valmiki 43. Rajatarangini was written by (a) Jaya Deva (b) Kalhana (c) Somadeva (d) Banabhatta 44. Who rendered the English translation of India Wins Freedom? (a) Humayun Kabir (b) Maulana Abul Kalam Azad (c) Jawaharlal Nehru (d) None of them 45. Which is the controversial book, some pages of which were kept sealed in the Calcutta Museum to be released after 40 years of independence in 1988? (a) Freedom at Midnight (b) India Wins Freedom (c) India Divided (d) None of these 46. Who is the author of India Divided? (a) Jawaharlal Nehru (b) Rajendra Prasad (c) Maulana Abul Kalam Azad (d) Humayun Kabir

47. Agnivesa Samhita is an ancient book which deals with codification of (a) Medicine (b) Scientific rules (c) Mathematics (d) Grammar 48. Mitakshara the ancient book was written by (a) Rajashekhara (b) Vijneswara (c) Somadeva (d) None of these 49. Who was the first human space traveller? (a) Edwin Aldrin (US) (b) Frank Boreman (US) (c) Yuri Gagarin (Russia) (d) Gherman S. Titov (Russia) 50. Who among the following is associated with freedom struggle of Zambia? (a) Patrice Lumumba (b) Kenneth Kaunda (c) Jomo Kenyatta (d) None of them 51. Which of the following Acts enabled the Government of India to put a ban on The Satanic Verses by Salman Rushdie? (a) National Security Act (b) Customs Act (c) Immoral Trafficking Act (d) None of these 52. The most important divinity in the Rig Veda is (a) Agni (b) Maruti (c) Varuna (d) Indra 53. Who is the author of A Suitable Boy? (a) R. K. Narayana (b) Nirad C. Chaudhary (c) Amitava Ghosh (d) Vikram Seth 54. Who is the author of controversial book Gatha Adi Shree Guru Granth Sahib? (a) Akhilesh Jaiswal (b) Dr Piar Singh (c) Dr Rita Joshi (d) Dr Piara Singh 55. Mira Behn was a follower of (a) Vinoba Bhave (b) Mahatma Gandhi (c) Jayaprakash Narayan (d) None of these 56. Who is known as the ‘Father of Geometry’? (a) Kepler (b) Euclid (c) Pythagoras (d) Newton 57. Aryabhatta was India’s renowned (a) poet (b) physicist (c) mathematician (d) medical practitioner 58. In which of the following fields did Tenzing Norgay distinguish himself? (a) Boxing (b) Cycling (c) Mountaineering (d) Hockey 59. Who advocated the theory of laissez faire? (a) Marshall (b) Malthus (c) Adam Smith (d) None of these 60. Who among the following is known as ‘Fuehrer’? (a) Stalin (b) Lenin (c) Hitler (d) Bismarck

2.386

▌ General Knowledge

61. Who discovered the sea route to India? (a) Vasco da Gama (b) Columbus (c) Magellan (d) Hopkins 62. Who is known as the ‘Iron Man of India’? (a) Jawaharlal Nehru (b) Bal Gangadhar Tilak (c) Sardar Vallabhbhai Patel (d) Mahatma Gandhi 63. Otto von Bismarck is called (a) Grand Commoner (c) Grand Old Man

(b) Man of Blood and Iron (d) Man of Destiny

64. The Ramakrishna Mission was started by (a) Swami Vivekananda (b) Ramakrishna Paramhansa (c) Swami Dayanand Saraswati (d) None of these 65. Who among the following is not associated with the Swaraj Party? (a) C. R. Dass (b) M. L. Kelkar (c) Motilal Nehru (d) Mahatma Gandhi 66. Maria Montessori’s name is associated with (a) Christian mission (b) Child education (c) Women’s rights (d) Mission hospitals 67. Who is known as ‘Deshbandhu’? (a) Chandra Shekhar (b) C. R. Dass (c) A. O. Hume (d) Annie Besant 68. Marshal Tito was (a) President of Egypt (b) A hero of the World War II (c) President of Yugoslavia (d) None of these 69. Where was Albert Einstein born? (a) UK (b) USA (c) Germany (d) Albania

74. Who was the first woman President of the Indian National Congress? (a) Annie Besant (b) Sarojini Naidu (c) Vijayalaxmi Pandit (d) None of these 75. The first navigator to sail round the world was (a) Marco Polo (b) Columbus (c) Amundsen (d) Magellan 76. Who among the following is known as ‘Kipper’? (a) Napolean Bonaparte (b) Field Marshal K. M. Cariappa (c) Gen Erwin Rommel (d) Younger Pitt 77. John Maynard Keynes was a/an (a) Poet (b) Scientist (c) Mathematician (d) Economist 78. Who among the following was associated with the freedom struggle of Tanzania? (a) Patrice Lumumba (b) Kenneth Kaunda (c) Kemal Ataturk (d) Dr Julius Nyerere 79. Match the following: Column I

Column II

A.

Charles de Gaulle

1.

Cuba

B.

Fidel Castro

2.

Russia

C.

Vladimir Lenin

3.

France

D.

Demal Ataturk

4.

Turkey

(a) 3

1

4

2

(b) 3

2

1

4

(c) 3

1

2

4

(d) 4

2

1

3

70. Who was known as ‘Nightingale of India’? (a) Vijaylaxmi Pandit (b) Sarojini Naidu (c) Suraiya (d) None of these

80. Vasco de Gama was a (a) Australian navigator (c) British explorer

71. Kalidas was (a) A poet during the Gupta period (b) A dramatist during Harshvardhana’s reign (c) An astronomer during the Gupta period (d) None

81. Who was the first woman to reach the North Pole? (a) Mrs Fran Phipps (b) Ms Valentina Tereshkova (c) Mrs Karoline Mikkelsen (d) None of these

72. Gopal Krishan Gokhale’s name is associated with (a) Hindu Mahasabha (b) Servants of India Society (c) Brahmo Samaj (d) Shiv Sena

82. Fahien was a (a) Chinese traveller (c) Norwegian traveller

73. Who was the first lady Prime Minister of the world? (a) Sirimavo Bandaranaike (b) Margaret Thatcher (c) Indira Gandhi (d) Golda Meir

83. Who is known as ‘Little Corporal’? (a) Adolf Hitler (b) Napolean Bonaparte (c) William E. Gladstone (d) None of these

(b) Portuguese explorer (d) French navigator

(b) Portuguese sailor (d) Japanese navigator

Basic General Knowledge ▌ 2.387 84. Valentina Tereshkova was the first woman to (a) cross the English channel (b) reach North Pole (c) reach the South Pole (d) None of these 85. Who was the first Indian woman to scale Mt Everest? (a) Phu Dorjee (b) Ang Dorjee (c) Bachendri Pal (d) None of these 86. Dhanvantari’s name is associated with (a) Indian medicine (b) Sanskrit poetry (c) Sanskrit grammar (d) Classical music 87. Which of the following pairs is incorrect? (a) Alberuni: Arab historian (b) Aesop: Italian author (c) Dhyan Chand: Hockey (d) Epicurus: Greek philosopher 88. Leonardo da Vinci, who is famous for his masterpiece painting Monalisa was a/an (a) French artist (b) Italian artist (c) German painter (d) None of these 89. Shridath Ramphal’s name is associated with (a) UNO (b) Commonwealth (c) NAM (d) SAARC 90. Mother Teresa was born in (a) Switzerland (c) Germany

(b) India (d) Albania

91. The first woman film star nominated or elected to the Rajya Sabha was (a) Vaijaynthimala (b) Nargis Dutt (c) Jayalalitha (d) None of these 92. Mrs Vijayalakshmi Pandit was the first woman to become (a) Chief Minister of a state (b) Minister of a state (c) Member of the Lok Sabha (d) Member of the Rajya Sabha 93. J.B. Priestly was a well known (a) French scientist (b) German scientist (c) British author (d) Italian author 94. Sun Yat-sen was the (a) first President of the Chinese republic (b) first Chairman of Communist Party of China (c) first Premier of China (d) None of these 95. Who among the following is the inventor of a system of writing and printing for the blind? (a) Robert Wilhelm Bunsen (b) Louis Braille (c) Sir Humphrey Davy (d) Christian Huygens

96. Freud is associated with (a) detective work (c) psychology

(b) physiology and medicine (d) leprosy control

97. The background colour of Olympic Flag is (a) Yellow (b) White (c) Light Blue (d) Silver 98. The XXV Summer Olympic Games were held at (a) Amsterdam (b) Sydney (c) Barcelona (d) Los Angeles 99. Which Indian personality has been awarded ‘The Olympic Order’ in 1983? (a) Jawaharlal Nehru (b) Indira Gandhi (c) Dhyan Chand (d) Milkha Singh 100. The 2006 Winter Olympics were scheduled at (a) Italy (b) France (c) Japan (d) USA 101. What was the original name of Mira Behn, the British disciple of Mahatma Gandhi? (a) Katherine May Heilman (b) Madeleine Slade (c) Katherine Slade (d) None of these 102. Which of the following statements about Abraham Lincoln is incorrect? (a) He abolished slavery in US (b) He was the 16th President of the US (c) He was assassinated in 1865 (d) He was the first President of the US 103. Martin Luther King was a (a) Social reformer (b) Secretary General of the UN (c) Black US civil rights leader (d) None of these 104. Who pioneered the modern nursing system? (a) Florence Nightingale (b) Mother Teresa (c) Greta Garbo (d) Preity Zinta 105. Which is the world’s oldest religion? (a) Judaism (b) Islam (c) Christianity (d) Hinduism 106. Which is the largest religion in the world? (a) Islam (b) Hinduism (c) Christianity (d) Judaism 107. When was Buddhism founded? (a) 500 BC (b) 525 BC (c) 400 BC (d) 300 BC 108. The Tripitaka is the sacred text of (a) Shintoism (b) Confucianism (c) Buddhism (d) None of these

2.388

▌ General Knowledge

109. Match the following: Column I

Column II

1.

Torah

A.

Shintoism

2.

Tao-te-Ching

B.

Confucianism

3.

The Analects

C.

Taoism

4.

Non text

D.

Judaism

(a) (b) (c) (d)

1B, 2D, 3C, 4A 1D, 2C, 3B, 4A 4A, 2B, 3C, 4D 2A, 3B, 4C, 1D

110. Sikhism was found in (a) ad 1000 (c) 300 bc

121. Where was Jesus Christ born? (a) Iraq (b) Jerusalem (c) Mecca (d) Madina 122. Christianity is nearly _______ old. (a) 1500 years (b) 2000 years (c) 2500 years (d) over 2500 years

(b) (d)

ad ad

1500 1400

111. Which religion is said to have no sacred text? (a) Taoism (b) Shintoism (c) Confucianism (d) Judaism 112. Who expounded the theory of ‘Advaita’? (a) Ramanuja (b) Sri Shankaracharya (c) Madhavacharya (d) Swami Vivekananda 113. Who was the founder of the Arya Samaj? (a) Swami Vivekananda (b) Swami Dayananda (c) Sri Shankaracharya (d) None of these 114. Who was the founder of the Ramakrishna Mission? (a) Swami Vivekananda (b) Ramkrishna Paramahansa (c) Swami Dayananda (d) Sri Shankaracharya 115. Who started the Sudhi (Purification) movement in modern Hinduism? (a) Swami Vivekananda (b) Sri Shankaracharya (c) Swami Dayananda (d) Ramakrishna Paramahansa 116. What was the role played by Swami Vivekananda in modern Hinduism? (a) He founded the Arya Samaj (b) He was an exponent of the Advaita movement (c) He carried the message of Hiduism to far off western countries (d) None of these 117. Lumbini is a sacred place of the (a) Muslims (b) Buddhists (c) Christians (d) Sikhs 118. Where is Lumbini located? (a) West Bengal (c) Nepal

120. Which among the following is the sacred text of the Confucianists? (a) The Tripitaka (b) The Analects (c) Torach (d) None of these

(b) Bihar (d) Uttar Pradesh

119. Confucianism is the main religion of (a) Arabia and Iraq (b) China and Taiwan (c) Japan and Sri Lanka (d) Nepal and Afghanistan

123. The sacred place of Christians is (a) Nazareth (b) Jerusalem (c) Mecca (d) None of these 124. Who founded Judaism? (a) Prophet Mohammed (c) Moses

(b) Jesus Christ (d) None of these

125. The first recorded ancient Olympics was held in (a) 700 BC (b) 776 BC (c) 310 BC (d) 450 BC 126. Who started the ancient Olympic Games? (a) The Romans (b) The Greeks (c) The Egyptians (d) None of these 127. The ancient Olympics continued till about (a) 200 BC (b) 394 BC (c) 300 BC (d) 370 BC 128. The modern Olympic Games were revived after a lapse of nearly (a) 1000 years (b) 1200 years (c) 1500 years (d) 2000 years 129. Modern Olympics was revived by (a) A Dutch nobleman (b) A Greek nobleman (c) A French nobleman (d) A German philosopher 130. Who is said to be the father of Modern Olympics? (a) Robert Dover (b) Pierre de Coubertin (c) Theodosius I (d) Ernest Curtis 131. In which year did the first modern Olympic Games take place? (a) 1908 (b) 1904 (c) 1896 (d) 1895 132. In which year was the International Olympic Committee (IOC) formed? (a) 1845 (b) 1875 (c) 1894 (d) 1895 133. The Headquarters of the IOC is located at (a) Olympia (b) Lausanne (c) Paris (d) Athens

Basic General Knowledge ▌ 2.389 134. The symbol of the summer Olympic Games comprises (a) Five rings of the same colour (b) Five rings of the different colours (c) Four blue rings (d) Four rings of the different colours 135. The five intertwined Olympic rings that form the Summer Olympic symbol are of five different colours. Which of the following colours is not one of the five? (a) Yellow (b) Black (c) Violet (d) Blue 136. Where were the first modern Olympic Games held? (a) Olympia (b) Athens (c) Seoul (d) Los Angeles 137. The Summer Olympic Games are normally held at intervals of (a) 2 years (b) 3 years (c) 4 years (d) 5 years 138. The Winter Olympic Games came into being in the year (a) 1908 (b) 1912 (c) 1916 (d) 1924 139. Since the revival of the modern Olympic Games, how many times have the games been cancelled? (a) Once in 1916 (b) Twice: 1940 and 1941 (c) Thrice: 1916, 1940 and 1944 (d) Never 140. Since 1896 the Olympic Games have been held twice at each of the following countries, except (a) London (b) Los Angeles (c) Rome (d) Paris 141. Which of the following countries had hosted the Olympic Games three times till the Barcelona Olympics of 1992? (a) Former USSR (b) France (c) Canada (d) Australia 142. The XXIX Olympic Games were held at (a) Korea (b) China (c) India (d) France 143. Which Olympiad was held in Greece in 2004? (a) XXVI (b) XXVII (c) XXVIII (d) XXIX 144. Which Olympic Games had to be cancelled due to World War I? (a) 1908 (b) 1912 (c) 1916 (d) None of these 145. Among the following terms, pick out the one associated with hockey. (a) Short Corner (b) Dribble (c) Bully (d) Hat Trick 146. Grand Prix is associated with (a) Lawn Tennis (b) Boxing (c) Volleyball (d) Golf

147. Wimbledon is associated with which of the following games? (a) Golf (b) Lawn Tennis (c) Hockey (d) Badminton 148. The term ‘Butter fly’ in sports is associated with (a) Basketball (b) Swimming (c) Rowing (d) Golf 149. The term ‘Dribbling’ is not associated with (a) Hockey (b) Football (c) Basketball (d) Baseball 150. The term ‘smash’ in sports is associated with (a) Lawn Tennis (b) Badminton (c) Volleyball (d) Hockey 151. ‘Tokyo Trophy’ instituted in the 1964 Tokyo Olympics is awarded to (a) An athlete for bagging maximum number of gold medals in Olympics (b) An athlete for demonstrating exemplary conduct during the Games (c) IOC President who chairs the Games (d) National Olympic Committee of the next edition of Olympics Games 152. Olympiads of which year are not recognized by the IOC? (a) 1940, Tokyo Olympics (b) 1944, London Olympics (c) 1896, Athens Olympics (d) 1906, Athens (Greece) 153. In which year Olympics were not held? (a) 1920 (b) 1916 (c) 1936 (d) 1948 154. The Olympics Motto ‘Citius, Altius, Fortius’ have been taken from which language? (a) Latin (b) Spanish (c) Greek (d) French 155. Five coloured rings in the Olympics Emblem represents (a) Five sports which were competed in the first Olympics (b) Group of five nations which joined hands to revive the modern Olympics in 1894 (c) Five continents participating in the Games (d) Five Gods in whose honour the Greek instituted these Games 156. The Olympics Flame symbolizes (a) Continuity of games between the ancient and modern Olympics (b) Zeal of participating athletes to win the events (c) Burning of hatred and sending message of brotherhood across the continents (d) None of the above 157. In which Games for the first time was Olympic Mascot introduced? (a) 1952 Helsinki, Finland (b) 1960 Rome, Italy (c) 1972 Munich, Germany (d) 2000 Atlanta, USA

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158. Which coloured ring represents the Asian Continent in the Olympics Emblem? (a) Blue (b) Yellow (c) Red (d) Green 159. Who won the Hockey Gold (Men) in the 1992 Barcelona Games? (a) Pakistan (b) Britain (c) Germany (d) Holland 160. Which country led a boycott of the 1980 Olympic games held in Moscow? (a) Great Britain (b) China (c) United States (d) South Korea 161. The Commonwealth Games were earlier called (a) British Colonies Sports Festival (b) British Empire Sports Festival (c) Royal Colonial Games (d) 162. When did the first Commonwealth Games take place? (a) 1948 (b) 1930 (c) 1934 (d) 1945 163. Where were the first Commonwealth Games held? (a) New Zealand (b) Scotland (c) Kenya (d) Canada 164. Which country hosted the 2002 Commonwealth Games? (a) India (b) UK (c) Australia (d) Canada 165. The Commonwealth Games were not held in the years (a) 1942 and 1950 (b) 1946 and 1950 (c) 1942 and 1946 (d) 1986 and 1990 166. The Commonwealth Games normally take place at an interval of ______ years. (a) three (b) four (c) five (d) Not defined 167. The Commonwealth Games had to be cancelled during (a) 1942 (b) 1946 (c) 1942 and 1946 (d) 1948 168. In which year were the first Asian Games held? (a) 1947 (b) 1951 (c) 1995 (d) 1958 169. Which among the following countries hosted the First Asian Games? (a) Indonesia (b) Former USSR (c) India (d) China 170. Which country hosted the XI Asiad? (a) Former W. Germany (b) South Korea (c) China (d) USSR

171. Asian Games have been organized in India (a) once (b) thrice (c) five times (d) twice 172. How many medals did P.T. Usha win at the Xth Asiad? (a) 3 (b) 2 (c) 4 (d) 5 173. How many medals did P.T. Usha win individually at the Xth Asiad? (a) 3 (b) 2 (c) 4 (d) 5 174. Which of the following did not participate in the first Asian Games (1951) organized at New Delhi, India? (a) Afghanistan (b) Pakistan (c) Iran (d) Japan 175. The 2002 Asian Games were held at (a) Busan, South Korea (b) Bangkok, Thailand (c) Hiroshima, Japan (d) Beijing, China 176. Which country holds the record of being at the top of the medals tally in six editions in a row in Asian Games? (a) Japan (b) South Korea (c) China (d) None of these 177. Which of the following games is Davis Cup associated with? (a) Volleyball (b) Badminton (c) Tennis (d) Handball 178. Ranji Trophy and Aga Khan Cup are associated with (a) Cricket and Football (b) Cricket and Volleyball (c) Cricket and Hockey (d) Badminton and Hockey 179. Which of the trophies below is not associated with cricket? (a) Duleep Trophy (b) Charminar Challenge Cup (c) Champions Trophy (d) Rohington Trophy 180. Which of the following is associated with Badminton? (a) William Todd Memorial Trophy (b) Uber Cup (c) Holkar Trophy (d) None of these 181. Which game is associated with the Durand Cup? (a) Hockey (b) Football (c) Badminton (d) Golf 182. Match the following columns: Game

Trophy/Cup

A. Hockey

1.

Davis Cup

B. Football

2.

Jules Rimet Trophy

C. Badminton 3.

Beighton Cup

D. Tennis

Thomas Cup

4.

Basic General Knowledge ▌ 2.391

A

B

C

D

(a) 2

3

4

1

(b) 3

2

4

1

(c) 3

4

2

1

(d) 1

3

2

4

183. Brooklyn (USA) is associated with (a) Golf (b) Snooker (c) Baseball (d) Tennis 184. With which game is Brookland (England) associated? (a) Hockey (b) Golf (c) Football (d) Tennis 185. The Dronacharya Award is associated with (a) Hockey captains for the World Cup (b) Sports coaches (c) Boxing Championship (d) None of these 186. The Dronacharya Award for Sports coaches was instituted in the year (a) 1984 (b) 1985 (c) 1987 (d) 1988 187. Merdeka Cup is associated with (a) Badminton (b) Volleyball (c) Hockey (d) Football 188. Reliance Cup is associated with which of the following games? (a) Hockey (b) Cricket (c) Football (d) Tennis 189. What was the former name of Reliance Cup? (a) Uber Cup (b) Thomas Cup (c) Rothman Cup (d) Prudential Cup 190. Which of the following Cups is not associated with Badminton? (a) Uber Cup (b) Thomas Cup (c) Yonex Cup (d) Beighton Cup

193. Dr Hargobind Khorana received the Nobel Prize for (a) Interpretation of genetic code (b) Grouping of blood (c) Discovery of DNA (d) Artificial genes 194. The only Muslim professor to get the Nobel Prize is (a) Abdus Salam (b) Nurul Hassan (c) Iqbal Ahmed (d) Abdul Majid 195. The first Indian to receive the U Thant award for cultural understanding was (a) Jawaharlal Nehru (b) Indira Gandhi (c) Acharya Vinoba Bhave (d) Mother Teresa 196. India’s first Nobel Prize was for (a) Physics (b) Literature (c) Medicine (d) Chemistry 197. Dr Rabindranath Tagore received the Nobel Prize for his work (a) Evening Songs (b) Morning Songs (c) Gitanjali (d) Gora 198. The value of the Nobel Prize was enhanced in the year (a) 1988 (b) 1989 (c) 1990 (d) 1987 199. The Nobel Prize for Economics was introduced in (a) 1969 (b) 1986 (c) 1976 (d) 1956 200. Match the following columns: Column I

Column II

A.

Oscar awards

1.

Progress in Religion

B.

Templeton awards

2.

Literature

C.

Pulitzer Prize

3.

Films

191. The term ‘Knock Out’ is associated with which of the following games? (a) Football (b) Basketball (c) Cricket (d) Boxing

D.

Booker Prize

4.

Journalism

A

B

C

D

192. In cricket Long Top is (a) the position before the bowler (b) the position behind the batsman (c) the position on the left side of the batsman (d) None of these

(a) 2

3

1

4

(b) 1

3

2

4

(c) 3

1

2

4

(d) 3

1

4

2

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answer KeY 1. 2. 3. 4. 5. 6. 7. 8. 9.

(b)

10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.

(a)

(b) (b) (c) (c) (b) (d) (a) (d) (d) (c) (b) (c) (a) (a) (c) (d) (a) (d) (c) (a) (b) (b) (d)

26. 27. 28. 29. 30. 31. 32. 33. 34.

(b)

35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50.

(a)

(c) (c) (b) (a) (c) (d) (b) (b) (a) (c) (a) (b) (c) (c) (b) (b) (a) (b) (b) (a) (b) (c) (b)

51. 52. 53. 54. 55. 56. 57. 58. 59.

(b)

60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75.

(c)

(d) (d) (b) (b) (b) (c) (c) (c) (a) (c) (b) (a) (d) (b) (b) (c) (c) (b) (a) (b) (a) (a) (d)

76. 77. 78. 79. 80. 81. 82. 83. 84.

(b)

85. 86. 87. 88. 89. 90. 91. 92. 93. 94. 95. 96. 97. 98. 99. 100.

(c)

(d) (d) (c) (b) (c) (a) (b) (d) (a) (b) (b) (b) (d) (b) (b) (c) (a) (b) (c) (b) (c) (b) (a)

101. 102. 103. 104. 105. 106. 107. 108. 109.

(b)

110. 111. 112. 113. 114. 115. 116. 117. 118. 119. 120. 121. 122. 123. 124. 125.

(b)

(d) (c) (a) (d) (c) (b) (c) (b) (b) (b) (b) (a) (c) (c) (b) (c) (b) (b) (b) (b) (b) (c) (b)

126. 127. 128. 129. 130. 131. 132. 133. 134.

(b)

135. 136. 137. 138. 139. 140. 141. 142. 143. 144. 145. 146. 147. 148. 149. 150.

(c)

(b) (c) (c) (b) (c) (c) (b) (b) (b) (c) (d) (c) (c) (b) (b) (c) (c) (c) (a) (b) (b) (d) (b)

151. 152. 153. 154. 155. 156. 157. 158. 159.

(b)

160. 161. 162. 163. 164. 165. 166. 167. 168. 169. 170. 171. 172. 173. 174. 175.

(c)

(d) (b) (a) (c) (a) (c) (b) (c) (b) (b) (d) (b) (c) (b) (c) (b) (c) (c) (d) (c) (a) (b) (a)

176. 177. 178. 179. 180. 181. 182. 183. 184.

(c)

185. 186. 187. 188. 189. 190. 191. 192. 193. 194. 195. 196. 197. 198. 199. 200.

(b)

(c) (c) (c) (b) (b) (b) (c) (c) (b) (d) (b) (d) (d) (d) (a) (a) (a) (c) (b) (c) (b) (a) (d)

nda BoX 1. Pivot is associated with which of the following games? (a) Hockey (b) Basketball (c) Golf (d) Tennis

4. The term LBW is associated with (a) Tennis (b) Badminton (c) Hockey (d) Cricket

2. The term ‘deuce’ is common in which of the following two games? (a) Basketball and Badminton (b) Badminton and Tennis (c) Volleyball and Tennis (d) None of these

5. Which of the following pairs is not correct? (a) Deodhar Trophy : Hockey (b) Jules Rimet Cup : Football (c) Corbillon Cup : Table Tennis (d) Irani Cup : Cricket

3. The term Bishop is associated with which game? (a) Golf (b) Chess (c) Bridge (d) Billiards

6. How many players take part in each team in a cricket match played under international rules? (a) 7 (b) 11 (c) 6 (d) 8

Basic General Knowledge ▌ 2.393 7. How many players are there on each side in a Basketball match? (a) 11 (b) 7 (c) 5 (d) 4 8. The duration of a normal one-half of a Hockey match is (a) 45 min (b) 40 min (c) 35 min (d) 30 min 9. What is the duration of each period before and after the tenminute break in a basketball game? (a) 30 min (b) 20 min (c) 35 min (d) 45 min 10. When and where was the game of Volleyball invented? (a) England, 1904 (b) USA, 1894 (c) Canada, 1894 (d) Australia, 1890 11. The normal length of a Football ground must be (a) 110–120 m (b) 100–110 m (c) 90–100 m (d) 120–130 m 12. How many players are there in a Football team? (a) 10 (b) 11 (c) 12 (d) 6 13. How many players are there on each side in a women’s Basketball game? (a) 5 (b) 6 (c) 7 (d) 10 14. Swaythling Cup is associated with (a) Lawn Tennis (b) Table Tennis (c) Hockey (d) Golf 15. Who among the following was associated with Badminton? (a) Dhyan Chand (b) Syed Modi (c) Bishan Singh Bedi (d) None of these 16. Dhyan Chand’s name is associated with which game? (a) Football (b) Hockey (c) Badminton (d) Tennis 17. ‘Play the game in the spirit of game’ was said by (a) G. S. Sondhi (b) Coubertin (c) Pt Nehru (d) None of these 18. Who among the following has been named as ‘Wisden’s Cricketer’? (a) Vishwanath (b) Gavaskar (c) Kapil Dev (d) Vijay Manjrekar 19. Sports goods are mainly manufactured in (a) Mumbai (b) Lucknow (c) Jalandhar (d) Moradabad 20. Which cricketer was fined Rs 20,000 by the Board of Control for Cricket in India for calling national selectors a ‘bunch of jokers’? (a) B. S. Bedi (b) Kapil Dev (c) Mohinder Amarnath (d) Sunil Gavaskar

21. Who among the following players was the first to win Wimbledon Singles title six times in a row? (a) Ramesh Krishnan (b) John McEnroe (c) Irvin Mathews (d) Martina Navratilova 22. Football (soccer) is said to have originated in (a) Former USSR (b) England (c) China (d) Australia 23. Lawn tennis is said to have originated in (a) England (b) France (c) Australia (d) USA 24. The Olympic motto was first displayed on the Olympics Flag in the Games held in (a) 1894 (b) 1920 (c) 1972 (d) 2000 25. With which sport is the term Chinaman associated? (a) Boxing (b) Yachting (c) Judo (d) Cricket 26. The recent edition of SAF Games was held in (a) 1999, India (b) 1999, Nepal (c) 2003, Pakistan (d) 2006, Sri Lanka 27. ‘The Ashes’ is associated with cricket matches played between? (a) Australia and West Indies (b) Australia and England (c) England and West Indies (d) New Zealand and West Indies 28. ‘Bully’ is the term associated with which one of the following sports? (a) Hockey (b) Football (c) Boxing (d) Golf 29. ‘A Half Nelson’ is the term associated with which of the following sports? (a) Bridge (b) Wrestling (c) Golf (d) Polo 30. ‘Aga Khan Cup’ is associated with which of the following sports? (a) Basketball (b) Hockey (c) Table Tennis (d) Football 31. EZAR Cup is associated with which of the following sports? (a) Hockey (b) Football (c) Polo (d) Tennis 32. ‘Beighton Cup’ is associated with which of the following sports? (a) Football (b) Hockey (c) Boxing (d) Badminton 33. ‘Santosh Trophy’ is associated with which of the following sports? (a) Cricket (b) Hockey (c) Football (d) Table Tennis

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34. Which is India’s largest Stadium? (a) National Stadium, Delhi (b) Yuba Bharati Stadium, Kolkata (c) Yadvindra Stadium, Patiala (d) Vallabhbhai Patel Stadium, Mumbai

45. The first winners of the UN Population Award were (a) Ronald Reagan and Indira Gandhi (b) Indira Gandhi and Qian Xangzhong (c) Qian Xangzhong and Ronald Reagan (d) Yuri Androopv and Qian Xangzhong

35. Which is India’s largest Indoor Stadium? (a) Netaji Stadium, Kolkata (b) Indira Gandhi Indoor Stadium, New Delhi (c) Bharati Stadium Cuttack (d) None of these

46. The Nobel Prize was given for the first time in the year (a) 1901 (b) 1910 (c) 1911 (d) 1912

36. The stadium ‘Eden Gardens’ is situated at (a) Kolkota (b) Mumbai (c) Chennai (d) Banglore 37. ‘Bishop’ is the term associated with which sport (a) Cricket (b) Polo (c) Chess (d) Golf 38. The only cricket team to have played in the finals of the last three World Cups beore 2011 is (a) Pakistan (b) Australia (c) India (d) None of these 39. Which team has won the Football World Cup most of the times? (a) Brazil, 4 times (b) Brazil, 5 times (c) Germany and Brazil, 4 times each (d) Brazil and Germany, 3 times each 40. Who among the following shared the first UN Population Award with a Chinese? (a) Ronald Reagan (b) Indira Gandhi (c) Yuri Andropov (d) Yehudi Menuhin 41. The first recipient of the Jawaharlal Nehru Award for International Understanding was (a) Khan Abdul Gaffar Khan (b) Yehudi Menuhin (c) Martin Luther King (d) U. Thant 42. Ramon Magsaysay award is named after the former President of (a) Philippines (b) Thailand (c) Indonesia (d) None of these 43. Who among the following was the first to receive the Nobel Prize? (a) C. V. Raman (b) Hargobind Khorana (c) Rabindranath Tagore (d) None of these 44. Who in India was conferred the highest British Award the ‘Order of Merit’? (a) Indira Gandhi (b) Rajiv Gandhi (c) Mother Teresa (d) Jawaharlal Nehru

47. S. Chandrashekhar got the Nobel Prize for (a) Chemistry (b) Thermodynamics (c) Electronics (d) Physics 48. The first Indian to share the Oscar Award was (a) Hargobind Khorana (b) Rabindranath Tagore (c) Bhanu Athaiya (d) None of these 49. Professor Abdus Salam of Pakistan was the Nobel Prize winner in (a) Physics (b) Medicine (c) Chemistry (d) Zoology 50. Name the Indian-born scientist who won the Nobel Prize in the field of Medicine. (a) Hargobind Khorana (b) J. V. Narlikar (c) C. V. Raman (d) J. C. Bose.

General Knowledge and Miscellaneous 51. According to International Astronomial Union, which one of the following has lost its status as a planet? (a) Jupiter (b) Uranus (c) Neptune (d) Pluto 52. According to the International Food Policy Research Institute Report 2007, India ranks which one of the following positions on the global hunger index? (a) 94 (b) 95 (c) 96 (d) 97 53. Arrange the following Military Training Institutes in India in terms of their geographical location form North to South. I. Indian Military Academy II. Officers Training Academy III. National Defence Academy IV. National Defence College V. College of Defence Management Select the correct answer using the code given below (a) I, III, V, IV, II (b) I, IV, III, V, II (c) IV, V, III, II, I (d) III, IV, I, II, V 54. Car manufacture Maruti has stopped selling its popular brand Maruti 800 in some major cities of India from 1st of April 2010. Which among the following statements is /are correct in this regard? I. The engine of the car is unable to meet the stricter Bharat Stage IV norms that become mandatory in some major cities of Indian from that date.

Basic General Knowledge ▌ 2.395 II. The Company has decided to promote another brand of car in the small car segment in some select cities. III. The company wants to shorten its product line. Select the correct answer using the code given below: (a) I, II and III (b) I only (c) II and III only (d) II only 55. Consider the following pairs: 1. Pinaka : Multi Barrel Rocket Launcher System 2. Nag : Anti Tank Missile System 3. Lakshya : Unmanned Aerial Vehicle Which of the pairs given above is/are correctly matched? (a) 1 only (b) 1 and 2 only (c) 2 and 3 only (d) 1, 2 and 3 56. Consider the following paragraph about one of the prominent personalities of India and identify him/her using the code given below. One of the founder members of the Progressive Artists Group, he/she painted several important series of paintings like the Mahabharata and Ramayana, Christ and the Lamb, Peace Doves, Women with Roosters and Mother Teresa. He/ She made a symbolic film, through the Eyes of a Pinter, which won him/her the Golden Bear. (a) Dhiraj Choudhary (b) Satish Gujral (c) MF Hussain (d) Amrita Shergil 57. Consider the following paragraph and identify with the help of the code given below the writer referred to therein. She is the winner of the Sahitya Akademi Award, the Jnanpith Award and is also India’s first Principal Prince Claus laureate. One of the most celebrated writers of contemporary Indian literature, she is noted for her novels which include The Mothe-Easten Howdah of the Tusker, Pages Stained with Blood and The Man from Chinnamasta. (a) Anita Desai (b) Indira Goswami (c) Mashasweta Devi (d) Chitra Mudgal 58. Consider the following paragraph and identify with the help of the code given below the Church referred to there in. Considered as one of the best examples of baroque architecute in India, this world Heritage Monument is dedicated to Infant Jesus. It houses the sacred relics of St Francis Xavier, the patron saint, who died in 1552. (a) Santa Cruz Basilica (b) St Francis Church (c) Church of the Holy Cross (d) Basilica of Bom Jesus 59. Consider the following passage and identify the form of dance theatre referred thereto using the code given below: This is uniquely traditional form of dance theatre of India with a formidable classical background. The main essence of this form of dance drama is its attachment with religion, which provides the most common theme for its plays. The performance begins with a prayer offering the Lord Ganesha, Which is followed by a comedy act, accompanied by the background music of the Chenda

and the Maddale and a tala (cymbals) played by a team of three. The narrator, who is also a part of the team is called Bhagavata and is the producer, the director, and the head of the ceremony. His primary task involves the narration of the story through songs, introduction of the characters and occasionlly conversing with them. Another unique feature of it is the totally unrehearsed and unwritten use of dialogues, which makes it so special. (a) Cheraw (b) Yakshagana (c) Devarattam (d) Jatra 60. Consider the following statements. 1. The G20 was established in 2008 in the wake of the global financial crisis led by the USA. 2. It brings together the major advanced and emerging economies to stabilize the global financial market. 3. India’s stand on Mutual Assesment Process (MAP) for measuring imbalances between surplus and deficit economies in the Seoul Conference G 20 in the year 2010 was firmly endorsed by all the nations. Which of the statements given above is/are correct? (a) 1, 2 and 3 (b) 1 and 2 (c) 2 and 3 (d) 3 only 61. Consider the following statements: I can only thank a genius director like Guru Dutt for giving me an opportunity to work in his films and translate his conceptions on screen. I was able to involve (myself) in many innovative experiments he had tried. The best example is the making of ‘Kagaz ke Phool’ (1959), which is the first cinemascope venture in the Indian film industry. On receipt of a Prestigious Award of India, the above statement was made recently by (a) Santosh Sivan (b) Ravi K Chandram (c) Ramachandra Babu (d) VK Murthy 62. Consider the following statements: 1. The Green Governance Awards have been instituted by Bombay Natural History Society (BNHS). 2. Tata Chemicals has won the Green Governance Award 2008; for its project on conservation of the whale shark. Which of the statements given above is/are correct? (a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2 63. Consider the following statements: 1. Hariyali is a watershed development project sponsored by the Central Government. 2. Hariyali also aims at enabling the urban population to conserve rain water. Which of the statements given above is/are correct? (a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1nor 2 64. Consider the following statements about Astra: 1. This anti-aircraft missile is procured by India from Russia. 2. It is a single-stage, smokeless, solid-propelled missile.

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▌ General Knowledge

3. It may be launched from different altitudes. Which of the statements given above are correct? (a) 1 and 2 (b) 2 and 3 (c) 1, 2 and 3 (d) 1 and 3 65. Consider the following statements about General VK Singh. I. He is the first trained commando to take over as the Chief of the Army Staff of Indian Army. II. He belongs to the Rajputan Rifles regiment of Indian Army. III. General Singh succeeded General Deepak Kapoor. Which among the statements given above is/are correct? (a) I, II and III (b) I and III only (c) III only (d) I and II only 66. Consider the following statements about the accident of AF 447, which crashed over the Atlantic Ocean on 1 June, 2009. 1. The Airbus A 330-200 was on its way to Rio de janeiro from paris. 2. The scheduled flight usually crosses the Tropic of Cancer on its way to its destination. Which the statement given above is/are correct? (a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2 67. Consider the following statements about the visit of Indian President Smt Pratibha Patil to Mongolia in July 2011. 1. She became the first Indian President to make an official visit to the country. 2. In the area of defence, India and Mogolia agreed to expand collaboration through increased level of joint exercises and training. Which of the statements given above is/are correct? (a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2 68. Consider the following statements with regard to the Global Hunger Index. 1. India ranks within the top ten among the developing countries in the Global Hunger Index 2010. 2. According to the Global Hunger Index 2010, the proportion of undernourished in India is decreasing. 3. Normally, economic Progress of a country of a country measured by its gross national income and the hunger level are inversely correlate. Which of the statements given above is/are correct? (a) 1, 2 and 3 (b) 2 and 3 only (c) 1 and 2 only (d) 3 only 69. Consider the following statements with regard to the Renewable Energy Cetificate (REC) mechanism launched in November 2010 by the Government of India. 1. It enables the obligated entities to meet their renewable purchase obligation. 2. It is one of the pioneering efforts in any developing country for mainstreaming the renewable energy generation through market mechanism.

Which of the statements given above is/are correct? (a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2 70. Consider the following statements with respect to the Radar Imaging Satellite (RISAT 2). 1. This is the first Indian satellite with microwave imaging configuration of board. 2. The satellite will boost India’s defence capabilities. 3. It can take images of the Earth day and night, even under rainy and cloudy conditions. Which of the statement given above are correct? (a) 1 only (b) 2 and 3 (c) 1, 2 and 3 (d) 1 and 2 71. Government of India in July 2011 gave its nod for accepting international regulation for the prevention of air pollution emanating from ships. In this regard, which of the following statements is/are correct? 1. According to the regulations, limits have been set on sulphur oxide and nitrogen oxide emissions from ship exhausts. 2. India is the first country to ratify to the convention for the prevention of pollution from ships of the International Maritime Organization. Select the correct answer using the code given below. (a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2 72. Historical materialism is a tenet of which one of the following political theories? (a) Capitalism (b) Liberalism (c) Fascism (d) Marxism 73. How is Asafa Powell well-known as? (a) Swimmer (b) Football Player (c) Boxer (d) 100 m Sprinter 74. How is Ban-Ki-Moon known as? (a) President of South korea (b) Head of World Bank (c) A famous scientist of Genetic Engineering (d) Secretary General of UN 75. How is Leila Seth known as? (a) First Woman Chairperson of the Union Public Service Commission (b) First Woman Election Commissioner of India (c) First Women Chief Justice of a High Court in India (d) First Women Governor of a State in India 76. In a major eco-friendly step towards renewable energy generation in India, in June 2010 seaweed sourced ethanol (ethyl alcohol), a befoul additive in transport fuel has been used to run a car for the first time. With regard to this, which of the following statements is/correct? I. Experiment in this direction was carried out by the scientists of Central Salt and Marine Chemicals Research Institute (CSMCRI), Bhavnagar.

Basic General Knowledge ▌ 2.397 II. I ndia has become the first country in the world to produce and use green fuel to run cars. Select the correct answer using the code given below: (a) I only (b) II only (c) Both I and II (d) Neither I nor 77. In a rare case, a record number of about 20000 employees of the Government of Kerala retired on a Single day (on 31 March, 2010). Which among the following statements is/ are correct in this respect? I. The mass retirement follows the Kerala Government’s move to allow employees retiring and day during the year to work till March 31. II. In a recent bid to reduce burden on the state exchequer, the Government of Kerala reduced the retirement age of government employees from 59 to 58 years. III. More than 50% of those retired were from the Education department. Select the correct answer using the code given below (a) I, II and III (b) II and III only (c) II and III only (d) I only 78. In the year 2007, in which one of the following places was the ‘UN Framework convention on Climate change’ held? (a) Bali (b) Guam (c) Hawaii (d) Seychelles 79. In which city is the Flushing Meadow, Corona Park—the venue of US open Tennis Tournament located? (a) Chicago (b) New York (c) Washington DC (d) Los Angeles 80. In which country was the recent NAM summit held? (a) Philippines (b) Indonesia (c) Cuba (d) Brazil 81. In which game does Suman Bala represent India? (a) Chess (b) Women’s Hockey (c) Shot-put (d) Women’s Cricket 82. In which one of the following countries is the infant mortality rate (per 100 live births) highest? (a) Sri Lanka (b) China (c) India (d) Vietnam 83. In which one of the following One-Day Cricket World Cups, Did India lose in the Finals? (a) World Cup, 1992 (b) World Cup, 1996 (c) World Cup, 1999 (d) World Cup, 2003 84. India by becoming a full-fledged member of the Financial Action Task forces (a) Will be able to access information on suspect accounts in nations such as Switzerland and UK (b) Will play an important role in law enforcement matters, investigations or prosecutions of terrorist activates at an international level

(c) Has also become a member of Organization for Economic Cooperation and Development (d) All of the above 85. India is not a member of which one of the following? (a) Commonweatlth of Independent States (b) South Asian Association for Regional Cooperation (c) South Asian Free Trade Agreement (d) World Trade Organization 86. Indian Army’s Counter-insurgency school is situated in (a) Kanker (b) Srinagar (c) Tezpur (d) Vairengte 87. Indian Military Training Team (IMTRAT) imparts training to army officers of (a) Nepal (b) Bangladesh (c) Maldives (d) Bhutan 88. Match List I with List II and select the correct answer using the codes given below the lists.

A. B. C.

List (Joint Exercise) Indra 1. Cope India 2. Ind-Indo Corpat 3.

D.

Yuddhabhyas

4.

List (Type) Joint anti-terrorism exercise Joint air force exercise Coordinated maritime patrolling Joint army exercise

Codes: A 1 1 4 4

(a) (b) (c) (d)

B 2 3 3 2

C 3 2 2 3

D 4 4 1 1

89. Match List I with List II and select the correct answer using the code given below the lists. A. B. C.

List (Ship) INS Arihant INS Vikrant INS Shiwalik

1. 2. 3.

List (Type) Aircraft Carrier Landing Platform Dock Nuclear Submarine

D.

INS Airavat

4.

Stealth Frigate

Codes: (a) (b) (c) (d)

A

B

C

D

3 3 2 2

4 1 1 4

1 4 4 1

2 2 3 3

2.398

▌ General Knowledge

90. Match List I with List II and select the correct answer using the code given below the lists. List I (Winner of Booker Prize)

Codes: A

B

C

D

List II (Novel)

(a)

3

1

4

2

A.

Arundhati Roy

1.

In a free State

(b)

2

1

4

3

B

Kiran Desai

2.

The God of small Thing

(c)

2

4

1

3

C.

Salman Rshdie

3.

(d)

3

4

1

2

D.

VS Naipaul

4.

93. Match List I With List II and select the correct answer using the code given below the lists:

Codes: A

B

C

D

(a)

1

5

3

2

(b)

1

3

4

2

(c)

2

5

3

1

(d)

2

4

5

1

List I (Institue)

91. Match List I with List II and select the correct answer using the code given below the lists. List I (Exponet)

List II (Dance form )

A.

Surendranath Jena

1.

Kathakli

B.

Pushpa Bhuyan

2.

Kathak

C.

Birju Mharaj

3.

Bharat Natyam

D.

Sankaran Embranthiri

4.

Odissi

Codes:

List II (Location )

A.

National Institute of Immunology

1.

New Delhi

B.

National Central for Cell Science

2.

Pune

C.

Centre for DNA Fingerprinting and Diagnostics

3.

Hyderabad

D.

National Brain Research Centre

4.

Manesar

Codes: A

B

C

D

(a)

2

3

1

4

(b)

4

1

3

2

(c)

4

3

1

2

(d)

2

1

3

4

94. Match List I with List II and select the correct answer using the code given below the lists:

A

B

C

D

(a)

1

2

3

4

(b)

4

3

2

4

(c)

1

3

2

4

A.

24th October

1.

(d)

4

2

3

1

B.

10th December

2.

C.

24th May

3.

m

21st September

4.

List I (Date)

92. Match List I with List II and select the correct answer using the codes given below the lists: List I (Person)

List ii (Industrial Unit)

A.

Indira Nooyi

1.

Pepsi

B.

Mukesh Ambani

2.

Wipro

C.

Sunil Mittal

3.

Airtel

D.

Azim Premji

4

Reliance

List II (Event )

Codes: A

B

C

D

(a)

4

1

2

3

(b)

3

1

2

4

(c)

3

2

1

4

(d)

4

2

1

3

Basic General Knowledge ▌ 2.399 95. Match List I with List II and select the correct answer using the code given below the lists:

A. B. C. D.

List I (Specialized Agency of UN) International Bank for Reconstruction and Development International Civil Aviation Organization International Maritime Organization International Labour Organization

1.

List II (HQ) Montreal

2.

London

3.

Rome

4.

Washington DC

Codes: A

B

C

D

(a)

3

1

2

4

(b)

3

2

1

4

(c)

4

2

1

3

(d)

4

1

2

3

C.

List I (UN Bodies) International Court of Justice Food and Agriculture Organization UNESCO

D.

WHO

B.

1.

List II (Headquarters) Geneva

2.

Rome

3.

Washington

4. 5.

The Hague Pairs

Codes: A

B

C

D

(a)

2

3

5

4

(b)

1

2

3

5

(c)

3

4

1

5

(d)

4

2

5

1

97. Match List –I with List-II and select the correct answer using the code given below the list. List-I (Jnanpith Laureate) A.

Birendar kumar Bhattacharya

A

B

C

D

(a)

4

3

2

1

(b)

1

2

3

4

(c)

1

3

2

4

(d)

4

2

3

1

98. Milan was 1. an annual gathering of Navies of Indian Ocean countries 2. Conducted at Chennai 3. A joint terrorism exercise 4. Aimed at enhancing inter-operability among navies Select the correct answer using the codes given below (a) 1 only (b) 2 and 3 (c) 1 and 4 (d) 3 and 4 99. Ms Reena Kaushal Dharmshaktu is the first Indian woman (a) To reach Antarctica (b) Pilot inducted in Indian Air Force (c) To Ski to the South Pole (d) To win the gold medal in Asian Boxing Championship

96. Match List I with List II and select the correct answer using the code given below the lists. A.

Codes:

List- II (Language) 1.

Bengali

B.

Rahman Rahi

2.

Marathi

C.

Vinda karandikar

3.

Kashmiri

D.

Mahasweta Devi

4.

Assomese

100. On one Sunday in December 2006, what was the reason for hundreds of thousands of agitators to gather in downtown Beirut? (a) To protest against US support to Israel in Lebanon’s Conflict with Israel (b) To protest against the US interference in Iraq and Afghanistan (c) To demand their Prime Minister to step down or cede power to the opposition (d) To show their solidarity with the Palestinians in their conflict with Israel 101. Operation Vajraprahar is the joint counter terrorism exercise between Indian Army and (a) US Army (b) Chinese Army (c) Russian Army (d) Nepalese Army 102. Recently, in which one of the following countries did a large number of monks make protest marches against the government ? (a) Cambodia (b) Japan (c) Myanmar (d) Vietnam 103. Recently, the Inter-Government Panel on Climate Change shared the Nobel Prize with who of the following? (a) Al Gore (b) Ban-Ki Moon (c) Bill Clinton (d) John Howard 104. START (Strategic Arms Reduction Treaty) is a treaty between/ among (a) China and Japan (b) USA and Russia (c) USA and European Union (d) China, India, USA and Russia

2.400

▌ General Knowledge

105. The 1770 km long Baku Tbilisi-Ceyhan (BTC) oil pipeline connects which of the following seas? (a) Aral Sea and Mediterranean Sea (b) Caspian Sea and Mediterranean sea (c) Black Sea and Baltic Sea (d) Caspian Sea and Arabian Sea 106. The Clean Development Mechanism (CDM), a mechanism to reduce greenhouse gas emission as per Kyoto implies that (a) Industrial counties receive carbon credits by funding carbon saving projects in another relatively affluent nation (b) Industrial countries reduce their carbon emission by using environment friendly technology in production (c) Developed countries invest in carbon reduction in developing countries and receive carbon credit in return (d) Developed nations purchase carbon credit from other nations 107. The Copenhagen Summit for climate change, 2009 saw (a) An agreement among the participants over CO 2 emission (b) Unanimity of views among the developed and BASIC bloc of counties (c) Rise of African countries as a bloc (d) Acceptance of the US as a leader to attain the objective of the Copenhagen Summit 108. The famous artist Uma Sharma is an exponent of which one of the following forms of dance? (a) Bharatnatyam (b) Kathak (c) Kuchipudi (d) Manipuri 109. The Film Director of which country owns the Best Film Award and Golden Peacock at the 37th International Film Festival of India held recently in Goa? (a) China (b) Iran (c) Bangladesh (d) Russia 110. The following statements about a report on ‘Status of Tigers, Co-predators and Prey in India, 2010’ released by the Ministry of Environment and Forests in July 2011. 1. India registered a 20 per cent decline in tiger population over the last survey in the year 2006. 2. The report warns that tigers are in danger due to an overall more than 12 per cent loss of habitat. Which of the statements given above is/are correct? (a) 1only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2 111. The Governments of which one of the following pairs of countries are planning to build a rail tunnel link between Eurpoe and Africa? (a) Algeria and France (b) Egypt and Greece (c) Morocco and Spain (d) Libya and Italy 112. The immediate cause of the revival of the revival of human rights in post–Second World War period was (a) Massive loss of lives in the War (b) Nuclear bombing of Hiroshima and Nagasaki by the Americans

(c) Growing knowledge of brutal atrocities of Nazis over the jews (d) Emergence of Soviet Union as a Superpower 113. The Parliament of which one of the following proclaimed its country as an Independent State in the year 2006 after taking the referendum of its people? (a) Macedonia (b) Montenegro (c) Belarus (d) Democratic Republic of Congo 114. The Supreme Court of India on 29 July 2011 directed the suspension of all mining operations and transportation in Bellary district of Karnataka till further orders. In this respect, which of the following statement is/are correct? 1. The court was convinced that over-exploration of the area has caused large-scale environmental degradation. 2. The court felt that Article 21 of the Constitution (Right to life) was violated by such exploration. 3. The court directed the Central Government to prepare rehabilitation scheme for restoration of the forest wealth. Select the correct answer using the code given below. (a) 1, 2 and 3 (b) 1 and 2 only (c) 2 and 3 only (d) 1 only 115. The tourism phrase ‘Paradise Unexplored’ refers to (a) Jammu and Kashmir (b) North-East India (c) Chhattisgarh (d) Uttarakhand 116. Venkataraman Ramkrishnan was jointly awarded Noble Prize in Chemistry in the year 2009 for the (a) Theory of electron transfer (b) Studies of the structure and function of the ribosome (c) Palladium catalysed cross counlings in organic synthesis (d) Work in the area of olefin metathesis 117. What is ‘Look East policy’ often in the news? (a) Government of India’s initiative for the infrastructural development in the North-Eastern States (b) India’s search for oil and gas in its Eastern shoreline (c) India’s collaboration with some East Asian countries in the exploration of oil and gas (d) India’s continuing pursuit of close relations with SouthEast Asian countries 118. What is the number of the permanent members of the UN Security Council? (a) Five (b) Six (c) Ten (d) Fifteen 119. Where is the Counterinsurgency and Jungle Warfare School (CIJWS), a training institution of the Indian Army specializing in unconventional warfare, specially guerilla warfare, located? (a) Vairengte, Mizoram (b) Kirkee, Pune (c) Raiwala, Dehradun (d) Dighi Hills, Pune

Basic General Knowledge ▌ 2.401 120. Where was the last meeting of Asia-Pacific Economic Cooperation (APEC) held? (a) kuala Lumpur (b) Manila (c) Hanoi (d) Jakarta 121. Where were the Olympic Games, 2004 held? (a) Madrid (b) Buenos Aires (c) Athens (d) Rome 122. Which among the following statements about Nirupam Sen is/ are correct? I. In the year 2009 he was appointed as Special Senior Adviser to the President of the UN General Assembly on the world financial and economic crisis. II. He was formerly the Indian Ambassador to the UK. Select the correct answer using the code given below. (a) I only (b) II only (c) both I and II (d) Neither I nor II 123. Which among the following statements about the ZaranjDelaram road is/are correct? I. It was constructed by Border Road Organization in Afghanistan. II. It links India with Afghanistan via Pakistan III. The road will open up market for Indian goods in Afghanistan via Iran. Select the correct answer using the code given below. (a) III only (b) I and III only (c) I and II (d) II only 124. Which amongst the following statements is /are correct about the INS Arihant? I. It is India’s first nuclear submarine. II. It is India’s first aircraft carrier. III. It has been indigenously built in India. (a) I only (b) II and III only (c) I and III only (d) II only 125. Which country won the 2006 FIFA Cup (World Football Cup)? (a) France (b) Brazil (c) Italy (d) Germany 126. Which of the following aspects with regard to the T-20 World Cup (Men’s) Cricket and T-20 World Cup (Women’s) Cricket tournaments held in the year 2010 was/were common? I. Both the tournaments were won by England. II. The tournaments were held in West Indies. III. There was a tie match each in both the tournaments. Select the correct answer using the code given below. (a) I only (b) II only (c) I, II and III (d) II and III only 127. Which of the following pairs are correctly matched? 1

Arundhati Roy

The god of Samall Things

2

Kiran Desia

The Inheritance of Losss

3

Jhumpa Lahiri

The Namesake

Select the correct answer using the codes given below: (a) 1, 2, and 3 (b) 2 and 3 (c) 1 and 3 (d) 1 and 2 128. Which one among the following films won the Nargis Dutt Award for best feature film on national integration in the57th National Film Award given in the year 2010? (a) Well Done Abba (b) Lahore (c) Delhi-6 (d) Paa 129. Which one among the following is the latest milestone of Doordarshan, the public broadcaster of India? (a) Lauch of enrichment/Cultural Channel −DD Bharati (b) Lauch of 24 hours news channel − DD News (c) Launch of sports channel−DD Sports (d) Formation of Prasar Bharati ( Broadcasting Corporation of India) 130. Which one among the following is the unit raised to protect the naval assets? (a) Sagar Rakshak Bal (b) Sagar Suraksha Bal (c) Sagar Prahari Bal (d) Sagar Nigrani Bal 131. Which one among the following led to the Greece economic Crisis 20120 ? (a) Excessive borrowing from IMF (b) Sudden depreciation of Euro (c) Unrestricted spending and cheap loans (d) Outflow of foreign capital due to political unrest 132. Which One among the following statements regarding South Sudan, who officially joined the international community as an independent nation on July 9, 2011, is not correct? (a) The capital of the new nation is Juba (b) It is Africa’s first Post-Colonial nation (c) India has stationed peacekeepers in South Sudan (d) The Nile flows through Sudan before entering South Sudan and then flowing into Egypt 133. Which one of the following has recently made it to the Guinness Book of World Records for being the world’s largest Hindu temple comlex? (a) Sri Venkateshvara Temple Complex, Tirupati (b) The ISKCON Temple Comlex, New Delhi (c) Swaminarayan Akshardham Temple Complex, New Delhi (d) Lingaraja Temple Complex Bhubaneshwar 134. Which one of the following is mixed with petrol by the Government of India in its nationwide programme of blended petrol? (a) Butanol (b) Ethanol (c) Methanol (d) Propanol 135. Which one of the following is not a component of the scheme Innovation in Science Pursuit for Inspired Research (INSPIRE) launched by the Union of India? (a) Scheme for early attraction of talents for science (b) Scholarship for higher education

2.402

▌ General Knowledge

(c) Assured opportunity for research careers (d) Vaocational Training after higher education

(a) Roger Federer (c) Novak Djokovic

(b) Rafael Ndal (d) Andy Roddick

136. Which one of the following is the international aircraft identification mark (registration prefix) for India? (a) AT (b) VT (c) IX (d) VX

145. Who among the following sports personalities is known as the Payyoli Express? (a) Shiny Abraham (b) PT USha (c) Jyotimoyee Sikdar (d) KM Beenamol

137. Which one of the following political theories advocates the ‘Dictatorship of Proletatiat’? (a) Democratic Socialism (b) Fabian Socialism (c) Liberalism (d) Marxism

146 Who among the following was awarded to Ramon Magsaysay award in category of Journalism, Literature and Creative Communication in 2007? (a) a Kejriwal (b) P Sainath (c) L Ramdas (d) V Shanta

138. Which one of the following Statements about salwa Judum, the Special Force created to counter Maoist insurgency, is not correct? (a) It was created by the Chhatisgarh Government in 2005, comprising of about 6500 Special Police Officers or Koya Commandos (b) Salwa Judum was initially known as Jan Jagran Abhiyan (c) The force was fully funded by the Chhattisgarh Government (d) The Supreme Court of India in July 2011 ordered the Chhattisgarh Government to disband the Special Force and desist from using them to counter Maoist activites 139. Who among the following has been declared as best footballer for the year 2007 by FIFA? (a) Ronaldo (b) Kaka (c) Ronldino (d) Boberto Carlos 140. Who among the following has been presented the Leontief Award, 2009 for his/her outstanding contributions to the field of economic theory? (a) Montek Singh Ahluwalia (b) Prof Bina Agarwal (c) prof Amartya Sen (d) Jagdish Bhagwati 141. Who among the following have scored more than 3000 runs in cricket test matches and taken more than 300 test wicket? 1. Kapil Dev 2. Shane Warne 3. Imran Khan 4. Shaun Pollock Select the correct answer using the code given below (a) 1, 2 and 3 only (b) 3 and 4 only (c) 1 and 4 only (d) All Four 142. Who among the following Indian origin immigrants has become the Governor of Lousiana province of the USA in 2007? (a) Swaraj Paul (b) Laxmi Mittal (c) Vikram Pnadit (d) Bobby Jindal 143. Who among the following is not a Dada Sahib Phalke awardee? (a) Shyam Benegal (b) Adoor Goplakrishnan (c) Mrinal Sen (d) JP Dutta 144. Who among the following is the winner of the Wimbledon Men singles title 2008?

147. Who among the following was recently presented with the Indira Gandhi Prize for Peace, Disarmament and Development for the year 2005? (a) Hamid Karzai (b) Hu Jintao (c) Mahinda Rajapakse (d) Gloria Macapagal Arroyo 148. Who among the following was the Director of the film Kaal Purush, which was adjudged as the best feature film in the 53rd National Film Awards (2007)? (a) Ritwik Ghatak (b) Buddhadev Dasgupta (c) JP Dutta (d) Asit Sen 149. Who of the following invented the cotton gin that separates the seeds from cotton three hundred times faster than by hand? (a) Eli Whitney (b) George Stephenson (c) McAdam (d) James Watt 150. Who of the following is an accomplished vocalist? (a) Monu Mazumdar (b) Shiv kumar Sharma (c) Satish Gujral (d) Dinanath Mishra 151. Who of the following published a famous pamphlet known as ‘Right of Mass’ and urged the people in England, America and France to Fight for their liberty? (a) Thomas Jefferson (b) Thomas Paine (c) John Locke (d) Jean Jacques Rousseau 152. Who was the first Foreign Minister of free India? (a) Jawahar Lal Nehru (b) Gulzari Lal Nanda (c) Lal bhadur Shastri (d) John Mathai 153. Who won the Frech open Tennis tournament, 2007 (Men’s Singles)? (a) Carlos moya (b) Andy Roddick (c) Rafael Nadal (d) Roger Federer 154. Who, among the following is the latest Indian to win the Pulitzer Prize for literature? (a) Jhumpa Lahiri (b) Geeta Anand (c) Gobind Behari Lal (d) Siddhartha Mukherjee 155. Why was Sucheta Kadethankar in the news recently? (a) She secured highest rank among women in Civil Services Examination 2010 (b) She is the first Indian to cross the Gobi Desert (c) She is the first Indian to cross the English Channel (d) She is the first Indian to climb Mount Everest twice

Basic General Knowledge ▌ 2.403 156. With which one of the following countries is ‘Orange revolution’ associated? (a) Brazil (b) Sudan (c) Turkey (d) Ukraine

157. With Which one of the following games is the Hopman Cup associated? (a) Badminton (b) Lawn Tennis (c) Hockey (d) Football

answer KeY 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

(b) (b) (b) (d) (a) (b) (c) (c) (b) (b) (b) (b) (b) (b) (b) (b) (c) (b) (c) (c)

21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40.

(d) (c) (b) (b) (d) (d) (b) (a) (b) (b) (c) (b) (c) (b) (b) (a) (c) (b) (b) (b)

41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60.

(d) (a) (c) (c) (b) (a) (d) (c) (a) (a) (d) (a) (b) (b) (d) (c) (b) (d) (b) (c)

61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80.

(d) (a) (c) (c) (a) (a) (c) (b) (c) (c) (a) (d) (d) (d) (c) (c) (b) (a) (b) (c)

81. 82. 83. 84. 85. 86. 87. 88. 89. 90. 91. 92. 93. 94. 95. 96. 97. 98. 99. 100.

(b) (c) (d) (b) (a) (d) (d) (a) (b) (d) (b) (d) (b) (b) (c) (d) (a) (d) (c) (c)

101. 102. 103. 104. 105. 106. 107. 108. 109. 110. 111. 112. 113. 114. 115. 116. 117. 118. 119. 120.

(a) (c) (a) (b) (b) (c) (a) (b) (a) (b) (c) (c) (b) (a) (b) (b) (d) (a) (a) (a)

121. 122. 123. 124. 125. 126. 127. 128. 129. 130. 131. 132. 133. 134. 135. 136. 137. 138. 139. 140.

(c) (c) (b) (c) (c) (b) (a) (c) (c) (c) (c) (b) (c) (b) (d) (b) (d) (b) (b) (b)

141. 142. 143. 144. 145. 146. 147. 148. 149. 150. 151. 152. 153. 154. 155. 156. 157.

(d) (d) (d) (c) (b) (a) (b) (b) (a) (a) (b) (a) (c) (d) (b) (d) (b)

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MATHEMATICS CHAPTER 1. ALGEBRA CHAPTER 2. MATRICES

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DETERMINANTS

CHAPTER 3. TRIGONOMETRY CHAPTER 4. ANALYTICAL GEOMETRY

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CHAPTER 5. DIFFERENTIAL CALCULUS CHAPTER 6. INTEGRAL CALCULUS

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DIFFERENTIAL EQUATIONS

CHAPTER 7. VECTOR ALGEBRA CHAPTER 8. STATISTICS

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PROBABILITY

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Algebra SE TS AND SUBSE TS The concept of set is fundamental to modern mathematics. A welldefined collection of distinct objects is called a set. A set can be written in two forms: (i) Tabular form (roster form): Here, the numbers of the set are listed, separated by comma and enclosed within curly brackets {}. (ii) Set-builder form: In this form, any typical element of the set is represented by a variable, followed by properties that must be satisfied by elements of the set.

General definitions (i) Finite and infinite sets: A set containing a finite number of elements is called a finite set and one containing an infinite number of element is called an infinite set. (ii) Equal and equivalent sets: If two sets have same number of elements and also the same elements, then two sets are equal sets, but if only the number of elements are same, they are equivalent set. (iii) Unit or singleton set and empty or null or void set: If a set contains only one element, it is called unit (Singleton) set, and if there is no element in the set, it is called empty, (Null or Void) set. (iv) Disjoint sets: Two sets are disjoint sets if they do not have any common element. (v) Subset and superset: If every element of set A is also present in set B, then A is called Subset of B denoted as A Ì B and read on A is contained in B or A is subset of B. Conversely, if A Ì B their B É A, which is read as B contains A or B is superset of A.

(vi) Universal set (U): When we consider problems involving sets which are all subsets of a given set, then set referred to is called universal set. (vii) Power set: P(A) is the power set of set A, if P(A) contains all subsets of set A. (viii) Proper set: if set A is subset of set B and has at least one element less than set B, then set A is called proper set of B. (ix) Complement of a set: If a is a set of elements which belong to Universal set U but do not belong of A is called complement of A and denoted by A k or A¢.

Important Statements (i) f is subset of every set. (ii) Every set is subset of itself. (iii) Equal sets are always equivalent sets, but equivalent sets, may not be equal sets. (iv) if A contains n elements, then the number of all subsets of A is 2n. (v) If A Ì B and B Ì A that A 5 B. (vi) The set N 5 {1, 2, 3.......} of all natural numbers. (vii) The I 5 {....23, 22, 21, 0, 1, 2, 3......} of integer (viii) Io is set of all non-zero integers as elements of the set.   p (ix) Q 5  x : x 5 ,where p and q are integers and q ≠ 0  q   of all rational numbers. (x) Qo is set of all non-zero rational numbers. (xi) Q1 is set of all positive rational numbers.

3.4

▌ Mathematics

(xii) R is set of all real numbers, Ro set of all non-zero real numbers and R1 is set of all positive real numbers. (xiii) C is set of all complex numbers, Co is set of all non-zero complex numbers. (xiv) Nk 5 {1, 2, 3…k}, is set of all natural numbers which are less than or equal to k where k is a positive integer. (xv) Every subset of an infinite set is a finite set.

The union of sets {Al : l Î Ù} is denoted by {Al : l Î Ù} and is defined by {Al : l Î Ù} 5 {x : x Î Al for at least one l Î Ù} if A, B, C are any subsets of U, then 1) A Ì A È B and B Ì A È B [Idempoetent law] 2) A È A 5 A

[Identity law]

3) A È f 5 A

Operations on Sets (a) Logical Connectives A sentence which is either true or false but not both is called a statement. Hence, if p and q are two statements, then the logical connectives may be defined on: (i) Conjunction (p Ù q): it denotes a statements which is true if and only if both p and q are true. (ii) Disjunction (p Ú q): it denotes a statement which is true if p is true or q is true or both are true and is false if both p and q are false.

4) A È U 5 U

[Commutative law]

5) (A È B) È C 5 A È (B È C) [Associative law] 6) Ù 5 f, than U{Al : l Î Ù} 5 f (ii) Intersection of two sets: Intersection A and B is two sets of all elements which belong to both A and B and is denoted by A Ç B read as ‘A intersection B’.

(iii) Negation ~ p: ~p is any statement which is true if p is false and is false if p is true. (iv) One way implication p Þ q: it is a statement which is false only when p is true and q is false and is true in all other cases. It is read as ¢p implies q¢ or if p then q¢ or ¢q if p¢ or p¢ only if q¢. (v) Contra-positive. If p Þ q then the statement ~ q Þ ~ p is called the contra-positive of p Þ q. (vi) Two contra-positive implications. p Û q is a statement which is true if both p and q are true or both are false and is false when one of the statements is true and other is false. (b) Venn-Euler Diagrams sets by set of points.

It is a schematic representation of

(i) Starting with universal set is represented by the points within a rectangle and a subset A of the universal set U is represented by the interior of a circle. (ii) Set inclusion is depicted by taking one region lying entirely within the other. (iii) Disjoint sets are depicted by non-overlapping regions. (c) Basic Operations (i) Union of two sets: If A and B are two sets, then the set of all elements which either belong to A or to B is called union of the two sets and is denoted by A È B and as ‘A union B’ or ‘A cup B’.

A

B

Let {Al ÎÙ} be an indexed family of subsets of U. The arbitrary intersection is denoted by {Al : l ÎÙ} and defined by {Al : l ÎÙ} 5 {x : x Î A l ∀ Î Ù}. If A, B, C, are any sets, then 1) A Ç B Ì A and A Ç B Ì B

[Idempoetent law]

2) A Ç A 5 A

[Identify law]

3) A Ç f 5 f 4) A Ç U 5 A 5) A Ç B 5 B Ç A

[Commutative law]

6) (A Ç B) Ç C 5 A Ç (B Ç C)

[Associative law]

7) if Ù 5 f, than {Al : l ÎÙ} 5 È (iii) Complement of a set: The complement of a set B relative to another set A is the set of all elements which belong to A but which do not belong to B and is denoted by A ~ B or A 2 B. Thus, A 2 B 5 {x : x Î A and x Ï B}.

A∪B

A

B

AB

A

Algebra ▌ 3.5 The complement of set A relative is universal set U is denoted by Aa or A¢ and is A¢ 5 {x : x Î È and x Ï A} or simply A¢ 5 {x : x Ï A} if A, B, C are any subsets of U, that 1) A 2 A 5 f 2) A 2 f 5 A 3) (A 2 B) Ì A 4) (A 2 B) Ç B 5 f 5) (A 2 B) È B 5 A 6) (A 2 B) 2 C 5 (A 2 C) 2 (B 2 C) 7) A 2 (A 2 B) 5 A Ç B 8) A Ç (B 2 C) 5 A Ç B 2 A Ç C 5 A Ç B 2 C 9) (A 2 B) È (B 2 A) 5 (A È B) 2 (A Ç B) 10) A È A¢ 5 È 11) A È A¢ 5 f 12) È 5 f 13) f¢ 5 È 14) (A¢)¢ 5 A 15) A 2 B 5 A Ç B¢ 5 B¢ 2 A¢ 16) B 2 A Ì A¢

20) if {Al : l Î Ù} be subsets of U indexed by a set B and if B Ì U, then [Distributive law] (i) B È [Ç{Al : l Î Ù}] 5 Ç{B È Al : l Î Ù} (ii) B Ç [È{Al : l Î Ù}] 5 È{B Ç Al : l Î Ù} 21) [De-Morgan’s law] if {Al : l Î Ù} be an indexed family of U, then (i) [ ∪ {Aλ ∈∧}]' 5 ∩ {Aλ' : λ ∈∧} (ii) (ii) [  {Aλ : λ ∈∧}]' 5  {Aλ' : λ ∈∧} 22) (A È B)¢ 5 A¢ È B¢ 23) (A Ç B)¢ 5 A¢ È B¢ (iv) Cartesian product of two sets: It is the set and {(a,b): a ∈A, b ∈B} is denoted to have, denotes an ordered pair whose first number, is a and second members is b. Hence, the cartesian product of two sets A and B tri set of all ordered pairs whose first member belongs to A and second member belongs to B. Also, note (a, b) is not the same thing as the set {a, b} for we always know {a, b} 5 {b, a} If A, B, C, S, T are sets, then 1) A ´ (B Ç C) 5 (A ´ B) Ç (A ´ C) 2) A ´ (B È C) 5 (A ´ B) È (A ´ C) 3) (A ´ B) Ç (S ´ T) 5 (A Ç S) ´ (B Ç T)

17) B 2 A¢ 5 B Ç A

4) (A 2 B) ´ S 5 (A ´ S) 2 (B ´ S)

18) A Ì B iff B¢ Ì A¢

5) A Ì B and C Ì D Þ (A ´ C) Ì (B ´ D)

19) A 5 B iff A¢ 5 B¢

6) If A Ì B, then A ´ A 5 (A ´ B) Ç (B ´ A)

RELATIONS A relation between two sets, A and B is any subset of A ´ B. Symbolically, R is relation from A to B. If R Ì A ´ B and A 5 B, we say that R is a relation on A. Thus, R Ì A ´ B and R 5 {(a, b) | (a, b) Î AB, a is R 2 related to B} We write it as ‘aRb’ and read it as ‘a’ is in relation to ‘b’. The domain of a relation R in the set of first element of the ordered pair in R. Thus, domain of R 5 {a\(a, b) Î R, for some b Î B} The range of a relation R is the set of second elements of the ordered pair in R Thus, range of R 5 {b\(a, b) Î R, for some a Î A} A relation in a set A is a subset of A ´ A. Thus, R Ì A ´ A, and R 5 {(a, b)\(a, b) Î A ´ A, aRb} The inverse of a relation R is: R 21 5 {(b, a) | (a, b) Î R}. A relation R in a set A is said to be symmetric if aRa ⇒ bRa if (a,b) Î R, then (b, a) Î R, ∀a, b Î A. A relation R in a set A is said to be reflexive of aRb if (a, a) Î r, ∀ a Î A. A relation R in a set A is said to be transitive if aRb and bR Þ aR, i.e., if (a, b) Î R and (b, c) Î R then (a, b) Î R, ∀a, b, c Î A A relation R in set A is said to be an equivalance relation or an RST relation) if R is reflexive, symmetric and transitive. An equivalance relation divides the sets into pair-wise disjoint sets.

1) Let A, B, C be set and let R Ì A ´ B and S Ì B ´ C, then (SoR)21 5 R 21oS 21 2) If R, S, T be relations on a set x, then (i) (R 21)

21

5R

(ii) Ro(SoT) 5 (RoS)o T (iii) R o Ix 5 IxoR 5 R 3) A relation R on a set A is transitive iff, RoR Ì R 4) Let R be a relation on a set A, then R is said to be antisymmetric iff aRb and bRa Þ a 5 b, also R Ç R 21 Ì R whose is the identify relation on A. 5) R is said to be an equivalance relation iff. (i) R is reflexive (ii) R is symmetric (iii) R is transitive 6) If R is an equivalance relation on set A, then for any a, b, Î A, we have either [a] Ç [b] 5 f or [a] 5 [b] and further È{[a]; a Î a} 5 A. 7) An equivalence relation R on a non-empty set x determines a partition of x and conversely, a partition of x induces an equivalance relation on X.

3.6

▌ Mathematics

Functions

(iii) If f : A ® B is a bijection, that f 210 f 5 IA, f0 f 21 5 IB

Let x and y be two non-empty sets. A subset of x ´ y is called a function from x to y iff is each x Î x there exists a unique y in y such that (x, y) ∈f functions are also known as ‘mappings’, ‘tranformation’ and ‘operations’ and denoted by f:x2y it follows from the above, that a relation from x to y is a function from x to y iff. (i) To each x∈x there exists a y Î Y such that (x, y)∈f (ii) (x, y1) Îf and (x, y2) Î f Þ y1 5 y2 The condition (i) ensures that each x in y,f associates an element y in y and condition (ii) guarantees that y is unique. • y is called the image of x under f and is denoted y f(x), i.e. y 5 f(x) • x is called the pre-image of y. • x is called domain of function f and Y is called the co-domain of function f. • The sets of images of all the elements of x under f is called the range of function f and is denoted by f(x). • if F: x ® y is such that each element of B is the f-image of at least one element of x, i.e. f(x) 5 y, then f is called subjective function or ONTO function. • If f: x ® y is such that district element of x have district f-image in y, then f is called Injective Function or ONE2ONE function. • A function is called Bijective Function if it is subjective as well as injective function. • Inverse Function: Let f 5 A ® B be subjective function, that f 21 5 B ® A is called inverse of this function. (i) Domain of f 21 2 B and f21 2 A range of f21 5 A (ii) f 21 : B ® A is also ONE2ONE function (f21)

21

5f

(iii) if f : A ® B and D Ì B, then f 21(D) 5 {x/x Î A and f(x) Î D} (iv) if f : A ® B is one2one function, then f 21 is unique. (v) f : A ® B is constant function, if f(a) 5 b ∀ a Î A for a fixed b Î B Composition or product of two functions: Let f : A ® B and g : B ® C then product of f and g denoted by gof is mapping of A into C given by: gof : A ® C such that (gof )(x) 5 g( f(x)) ∀ x Î A (gof ) exists only if range of f is a subset of domain of g for f : A ® A and g : A ® A gof and fog exists, and gof : A ® A and fog : A ® A Identity function (IA) IA(a) 5 a ∀ a Î A IA is one2one onto function and is its own inverse. (i) If, f : A ® B than f0 IA 5 f, IB of 5 f (ii) If f : A ® A (self-function) that f0 IA 5 IA of 5 f

(iv) If f : A ® A is a injection then f21of 5 IA, f0f 21 5 IB (v) If f : A ® A and g : A ® A are functions such that gof 5 fog 5 IA than f and g are bijections and inverse of each other.

Examples 1. Of the players of three teams in the Punjab University, 21 are on cricket team, 26 on the football team and 29 on the swimming team. 14 are footballers and cricketers, 15 are footballers and swimmers and 12 are swimmers and cricketers. 8 are in all the three teams. How many players are there together? Solution: Let A 5 set of cricketers, B 5 set of footballers and C 5 set of swimmers. then, n(A È B È C) 5 n(A) 1n(B) 1 n(C) 2 n(A B) 2 n(B Ç C) 2 n(C Ç A) 1n(A Ç B Ç C) 5 21 1 29 1 26 2 12 2 15 2 14 1 8 5 43 2. If A 5 {a, b, c} and B 5 {x, y}. Find (i) A ´ B (ii) A ´ A (iii) n(A ´ B) and n(A ´ A) Solution: (i) A ´ B 5 {a, b, c,} ´ {x, y} 5 {(a, x), (a, y), (b, x), (b, y), (c, x), (c, y)} (ii) A ´ A 5 {a, b, c,} ´ {a, b, c} 5 {(a, a), (a, b), (a, c), (b, a), (b, b), (c, a), (c, b), (c, c)} (iii) n(A ´ B) 5 6 and n(A ´ A) 5 9 3. Some elsments of A ´ B are (x, 1), (y, 3)(z, 4)(u, 5) if B 5 {1, 2, 3, 4, 5}, find the remaining elements of A ´ B such that n(A ´ B) is the least. Solution: Since (x, 1), (y, 3), (z, 4), (u, 5) are some of the elements and n(A ´ B) is to be the last therefore, A 5 {x, y, z, u), then A ´ B 5 {x, y, z, u} {1, 2, 3, 4, 5} Hence, the required remaining elements will be (x, 2), (x, 3), (x, 4), (x, 5), (y, 1), (y, 2), (y, 4), (y, 5), (z, 1) (z, 2), (z, 3), (z, 5), (u, 1), (u, 2), (u, 3), (u, 4). 4. Determine the domain and range of the following relations: (i) (3, 1)(6, 2)(15, 5) (ii) R 5 {(x, y) | x2 1 y2 5 25, x and y are integers}. Solution: (i) The relation can be described and is three times and Domain of relation 5 {3, 6, 15} Range of relation5 {1, 2, 5}

Algebra ▌ 3.7 (ii) Domain 5 {0, 3, 23, 4, 24,5, 25} Range 5 {0, 3, 23, 4, 24, 5, 25} 5. Determine if R 5 {(1, 1), (2, 2), (3, 3), (4, 4), (1, 3), (3, 1), (1, 4), (4, 1)} is equivalence relation in {1, 2, 3, 4} Solution: (i) Because (1, 1)∈R, (2, 2)∈R, (3, 3)∈R, (4, 4)∈R therefore, R is reflexive. (ii) R is symmetric because (1, 3) and (3, 1) belongs to R and (1, 4) and (4, 1) belongs to R. (iii) R is transitive, because (1, 3),(3, 1)and (1, 1) belong to R and (1, 4), (4, 1) and (1, 1) belong to R. Now, since R is reflexive, symmetric and transitive therefore R is equivalance relation. 6. If A 5 {a, b, c, d} and F corresponds, to line Cartesian product {(a, b), (b, d), (c, a), (d, c)}, show that f is one2one from A onto A. Find f21. Solution: Because, f(a) 5 b, f(b) 5 d, f(c) 5 a, f(d) 5 c \ District elements of A have district f images of A. \ f is one2to2one function, because f(A){b, d, a, c} 5 A \

f is onto function. Hence, F is one2to2one from A onto A.

\

f 21 5 {(b, a), (d, b), (a, c), (c, d)}

7. Let A 5 R 2 {3}, B 5 R 2 {1}, R being the set of real numbers, let the function f : A ® B be defined by (x  2) f(x)  . Is this function bijective and how? (x  3) Solution: ( x  2) Let y  , ( x  3)

Since, f(x) 

Solution: (i) f1(21) 5 f1(1) 5 1, therefore, f1 is not one2to2one function and have f121 does not exist. (ii) f2(1) 5 sin1 which does not belong to co-domain of A. Hence, f is not a function at all, so no f221 exit. 9. Find a function of : R ® R such that gof 5 fog 5 IB, where R is set of all real numbers and f  R → R, f ( x ) 

x 1 . 4

Solution: Since f 5 R ® R, g 5 R ® R, and gof 5 fog 5 IB. Therefore, g and f are bijections and inverse of each other. \ g 5 f 21 x 1 then x 5 4y 2 1 or f 21(y) 4 5 (4y 2 1) g( y) 5 (4y 2 1) Þ g(x) 5 (4x 2 1) Now, Let y  f(x) 

10. Let f(x) 

(x 2  2x  11) if f : R 2 {3} R {x : a < x < b, (x  3)

x Î R} is subjective find value of a and b. Solution:  x 2  2 x  11  Let y    , then y(x 2 3) 5 x 2 1 2x 2 11  (x  3)  Þ x2 1 x(2 2 y) 1 3y 2 11 5 0 Since x ∈ R therefore, (2 2 y)2 2 4(3y 2 11) ³ 0

then y (x  3)  x  2 ⇒ x ( y  1)  (3 y  2) or x 

8. Let A 5 {21, 1} and let the function since f1, f2 of A into A be defined by. f1(x) 5 x2 and f2(x) 5 sin x. State whether or not f121 and f221 exists.

(3y  2) ( y  1)

( x  2) therefore, f (x1 ) ≠ f (x2 ) ( x  3)

Þ y 2 2 16y 1 48

³0

Þ ( y 2 4)(y 2 12)

³0

Þ

Either (i) (y 2 4) ³ 0 and (y 2 12) ³ 0

y ³ 4 and y ³ 4 Þ y ³ 12 or (ii) ( y 2 4) £ 0 and (y 2 12) £ 0 y £ 4 and y £ 12 y £ 4

 ( x  2) ( x2  2)  ≠ if x1 ≠ x2  1   ( x1  3) ( x2  3) 

\

3y 2 2 , therefore, every real y 21 number y other than 1, has pre-image and consequently

5 R 2 {x : 4 < x < 12, x Î R}

f is subjective. Since f is injective, as well as subjective therefore, it is bijective function.

\ Range of f 5 R 2 {x : 4 < x < 12, x Î R} 5 R 2 {x : a < x < b, x Î R} therefore a 5 4 and b 5 12.

\

f is injective. Since

Range of f 5 {y : y ³ 12 or y £ 4, y Î R}

5 R 2 {y : 4 < y < 12, y Î R} Since f is subjective and f : R 2 {3} ® R 2 {x : a < x < b, x Î R}

3.8

▌ Mathematics

QUADRATIc EQUATIONS Quadratic Equation in Standard Form A quadratic equation in standard form is an equation that can be written in the form ax2 1 bx 1 c 5 0 a ¹ 0 where, x is a variable and a, b and c are constants. Methods of solution include: 1. Factoring and using the zero property: m.n 5 0 if and only if m 5 0 n 5 0 (or both).

or,

2. Using the square root property: If A2  C , then A  C . 3. Completing the square: 2

2

b  b x2  b x      x   . 2 2 

4. Using the quadratic formula: x

b ± b2  4 ac . 2a

If the discriminant b2 2 4ac is positive, the equation has two distinct real roots; if the discriminant is 0, the equation has one real double root; and if the discriminant is negative, the equation has two imaginary roots, each the conjugate of the other.

Equations Reducible to Quadratic Form A square root radical can be eliminated from an equation by isolating the radical on one side of the equation and squaring both sides of the equation. The new equation formed by squaring both sides may have extraneous solutions. Consequently, every solution of the new equation must be checked in the original equation to eliminate extraneous solutions. If an equation contains more than one radical, then the process of isolating a radical and squaring both sides can be repeated until all radicals are eliminated. If a substitution transforms an equation into the form au2 1 bu 1 c 5 0, where u is an expression in some other variable, then the equation is a quadratic form that can be solved by quadratic methods.

MAThEMATIcAL INDUcTION Let P n be a statement associated with each positive integer n and suppose, the following conditions are satisfied: 1. P1 is true. 2. For any positive integer k, if Pk is true, then Pk 1 1 is also true.

To deal with conjectures that may be true only for n ³ m, where m is a positive integer, we use the extended principle of mathematical induction: Let m be a positive integer, let Pn be a statement associated with each integer n ³ m, and suppose the following conditions are satisfied:

Then, the statement Pn is true for all positive integers n.

1. Pm is true.

`To use mathematical induction to prove statements involving laws of exponents, it is convenient to state a recursive definition of an:

2. For any integer k ³ m, if Pk is true, then Pk 1 1 is also true.

a1 5 a and an 1 1 5 ana for any integer n > 1.

Then, the statement Pn is true for all integers n ³ m.

PROgRESSION Arithmetic Progression A sequence is called an arithmetic sequence, or arithmetic progression, if there exists a constant d, called the common difference, such that an 2 an21 5 d or an 5 an 2 1 1 d for every n > 1. The following formulas are useful when working with arithmetic sequences:

an  a1  (n  1) dnth-Term Formula n S n  [2a1  (n  1)d ] 2 Sum Formula — First Form Sn 

n (a  an ) 2 1 Sum Formula — Second Form

Algebra ▌ 3.9

Geometric Progression

an  a1r n 1 nth-Term Formula

A sequence is called a geometric sequence, or a geometric progression, if there exists a non-zero constant r, called

Sn 

an  r or an  ran  1 for every n  1. an  1 The following formulas are useful when working with geometric sequences:

a1  a1r n r ≠ 1 Sum Formula — First Form 1 r

a1  ran r ≠ 1 Sum Formula — Second Form 1 r a S x  1 r  1 Sum of an Infinite Geometric Series. 1 r Sn 

PERMUTATION AND cOMBINATIONS (a) Fundamental Principle of Counting or Multiplication Principle If an event occurs in m distinct ways following which a second event occurs in n distinct ways, then the two together will occur in mn ways in the given order. (b) Addition Principle If an operation can be performed in m different ways and if another operation can be performed in n different ways and the two operations are mutually exclusive, then either of the two operations can be performed in (m 1 n) different ways. (c) The number of ways of arranging n distinct objects in a row taking r (0 £ r £ n) of them at a time is denoted by n Pr and its value is given by n

Pr 

n! (n r )!

Note that n

P0  1, n P1  n , n Pn1  n Pn  n!

(d) The number of ways of arranging n distinct objects along a circle is (n 2 1) ! Note that even if one object is lying on the circle, then the circle has to be treated as a row so far as arranging of objects is concerned along it. (e) If out of n objects a are exactly identical and are of one kind, b are exactly identical and are of a second kind, and so on, then the number of ways of permuting the n objects is given by n! a! b!... (f) The number of ways of permuting n distinct objects taking r of them at a time when any object may be repeated any number of times is nr. (g) The number of ways of selecting, (0 £ r £ n) objects out of n distinct objects is denoted by nCr

and its value is given by n

Cr

n! r!(n 2 r )!

Note that (i) nC0 1, nCn 2 1, nC1 n. (ii) nCr nCn 2 r (0 £ r £ n). (iii) nCr 2 1 nCr 5 n 1 1Cr (1 £ r £ n) (iv) nC¥ 2 nCv implie, x y or x 1 y 2 n. (v) If n is even, then the greatest value of nCr is given by nCm where m 5 n/2. If n is odd, then the greatest value of nCr is given by nCm, where, m 5 (n 2 1) 2 or m 5 (n 1 1 )/2. (h) The number of combinations of n distinct objects taken r at a time when k(1 £ k £ r) particular objects always occur is n2kCr2k. (i) The number of combinations of n distinct objects taken r at a time when k(1 £ k £ n 1 r) never occur is n2kCr. (j) If there are a objects of one kind, b objects of a second kind, c objects of a third kind and so on, then the number of ways of choosing r object out of these objects is given by the coefficient of xr in (l 1 x 1 x2 ... |xa) (11 x ... |xc) (1 1 x2 1 ... 1 xc) ... If at least one object of each kind is to be included in the collection, then the number of ways of choosing r objects is given by the coefficient of xr in (x 1 x2 1 ... 1 xa) (x 1 x2 1 ... 1 xb) (x 1 x2 1 ... 1 xc)... (k) The following result is also useful: (a0  a1 x  a2 x 2 | ) ( b0  b1x  b2 x 2  ...) c0  c1 x  c2 x 2  ... cn x n  ... where, cn  a0 bn  a1 bn1  a2 bn  2  ......| an b0 for each n > 0.

3.10

▌ Mathematics

BINOMIAL ThEOREM For any positive integer n, (x 1 a) 5 C0 x 1 C1x a 1 C2x a 1 … 1 nCrxn2rar 1 … 1 nCn21 xan21 1 nCnan. n

n

n

n

n 21

n

n22 2

The (r 1 1)th term in the expansion of (x 1 a)n is given by Tr11 5 nCr xn2r ar

Middle Terms(s) (i) If n is even, then there is just one middle term in the expansion of (x 1 a)n. This term is (n/2 1 1)th term and is given by Tn /211 5 nC(n/2) xn/2 an/2 (ii) If n is odd, then there are two middle terms in the expan n 11   n13 sion of (x 1 a)n. These are   th terms and   2    2  th term. These terms are given by T(n 11)/2 5 nC(n 11)/2 x(n 11)/2 a(n 21)/2 and, T(n 13)/2 5 nC(n 11)/2 x(n 21)/2 a(n 11)/2

Greatest Term in the Expansion of (x 1 a)n Suppose, x and a are positive real numbers. ( n 1 1) a (i) If is an integer, say equal to k then the expansion x1a of (x 1 a)n has two greatest terms. These are kth and (k 1 1)th terms. ( n 1 1) a is not an integer, then the expansion of x1a (x 1 a)n has just one greatest term. This is, (k 1 1)th term  ( n  1) a  where k   ([x] denotes the greatest integer less  xa  than or equal to x).

(ii) If

Some Important Results (i) nC0 1 nC1 1 ... 1 nCn 5 2n (ii) nC0 1 nC2 1 ... 1 nC1 1 nC3 1 ... 5 2n21 (iii) (nC0)2 1 (nC1)2 1 ... 1 (nCn)2 5 nCn C0 1 2n11C1 1 ... 1 2n11Cn 5 2n11Cn11 1 2n12Cn12 1 ... 1 2n11C2n11 5 22n

(iv)

2n11

(v )

n

1 C0  nC2 x 2  nC4 x 4  ...  (1  x ) n  (1  x ) n  2

( vi )

n

1 C1x  nC3 x 3  ...  (1  x ) n  (1  x ) n  2

De Moivre’s Theorem (i) If n is any integer, then (cos q 1 i sin q)n 5 cos (nq) 1 i sin (nq) (ii) If n is a rational number, then one of the values of (cos q 1 i sin q)n is cos (nq) 1 i sin (nq). If n is a rational number such that n 5 p/q where p, q Î Z, q > 0 and p, q have no factor in common, then (cos q 1 i sin q)n has q values, one of which coincides with cos (nq) 1 i sin (nq). If z 5 r (cos q 1 i sin q) and n is a positive integer, then   2 kπ  θ   2 kπ  θ   z1/ n  r 1/ n cos    i sin   n n      where, k 5 0, 1, 2, …, n 2 1 We sometimes denote cos q 1 i sin q by cis q or by eiq, that is, cis q 5 cos q 1 i sin q and eiq 5 cos q 1 i sin q.

ExPONENTIALS AND LOgARIThMS Exponential Functions

2. ax 5 ay if and only if x 5 y.

The equation f(x) 5 b x, b > 0, b ¹ 1, defines an exponential function with base b. The domain of f is (2 ¥, ¥) and the range is (0, ¥). The graph of an exponential function is a continuous curve that always passes through the point (0, 1) and has the x axis as a horizontal asymptote. If b > 1, then bx increases as x increases, and if 0 < b < 1, then bx decreases as x increases. The function f is one-to-one and has an inverse. We have the following exponential function properties:

3. For x ¹ 0, then ax 5 bx if and only if a 5 b. Exponential functions are used to describe various types of growth. 1. Population growth can be modelled by using the doubling time growth model P 5 P02t /d, where P is population at time t, P0 is the population at time t 5 0, and d is the doubling time—the time it takes for the population to double. 2. Radioactive decay can be modelled by using the half-

x y x y 1. a a  a x

a a    x b b  

x

(a x ) y  a xy (ab) x  a x bx x

a  a x y by

t/h

1 t/h life decay model A 5 A0   5 A0 2 , where A is the 2   amount at time t, A0 is the amount at time t 5 0, and h

Algebra ▌ 3.11 is the half-life—the time it takes for half the material to decay. 3. The growth of money in an account paying compound interest is described by A 5 P(1 1 r/n)nt, where P is the principal, r is the annual rate, n is the number of compounding periods in one year, and A is the amount in the account after t year. We also call P the present value and A the future value of the account.

The Exponential Function with Base e As m approaches ¥, the expression (1 1 1/m)m approaches the irrational number e » 2.718 281 828 459. The function f(x) 5 ex is called the exponential function with base e. Exponential functions with base e are used to model a variety of different types of exponential growth and decay, including growth of money in accounts that pay continuous compound interest. If a principal P is invested at an annual rate r compounded continuously, then the amount A in the account after t years is given by A 5 Pert.

Logarithm Functions The logarithmic function with base b is defined to be the inverse of the exponential function with base b and is denoted by y 5 logb x. Thus, y 5 logb x if and only if x 5 by, b > 0, b ¹ 1. The domain of a logarithmic function is (0, ¥) and the range is (2 ¥, ¥). The graph of a logarithmic function is a continuous curve that always passes through the point (1, 0) and has the y-axis as a vertical asymptote. We have the following properties of logarithmic functions: 1. logb 1 5 0 2. Logb b 5 1 3. logb bx 5 x 4. blogbx 5 x, x > 0

5. logb MN 5 logb M 1 logb N 6. Log b

M  log b M  log b N N

7. logb M p 5 p logb M 8. Logb M 5 logb N if and only if M 5 N.

Common and Natural Logarithms Logarithms to the base 10 are called common logarithms and are denoted by log x. Logarithms to the base e are called natural logarithms and are denoted by ln x. Thus, log x 5 y is equivalent to x 5 10y, and ln x 5 y is equivalent to x 5 ey. The following applications involve logarithms: 1. The decibel is defined by D 5 10 log(I/I0), where D is the decibel level of the sound, I is the intensity of the sound, and I0 5 10212 watt per square metre is a standardised sound level. 2. The magnitude M of an earthquake on the Richter scale 2 is given by M 5 log (E /E0 ), where E is the energy 3 released by the earthquake and E0 5 104.40 joules is a standardised energy level. 3. The velocity v of a rocket at burnout is given by the rocket equation v 5 c In (Wt/Wb), where c is the exhaust velocity, Wt is the takeoff weight, and Wb is the burnout weight.

Exponential and Logarithmic Equations Various techniques for solving exponential equations, such as 23x 22 5 5, and logarithmic equations, such as log (x 1 3) 1 log x 5 1, are illustrated by examples. The change-of-base formula, logb N 5 (loga N) (loga b), relates logarithms to two different bases and can be used, along with a calculator, to evaluate logarithms to bases other than e or 10.

SURDS AND INDIcES 1. am ´ an 5 am 1n

9. am 5 bm Þ a 5 b

2. am ¸ an 5 am2n

10. am 5 an Þ m 5 n

3. (a ) 5 a

11. am/n 5 b Þ a 5 bn/m

m n

mn

4. a° 5 1 when a50 5. ambm 5 (ab)m 6. am  m

1 am

Cross-multiplication method for the solution of the equations p1 x  q1 y  r1  0 p2 x  q2 y  r2  0 , we get,

a a 7.   5 m b b 8.

n

m

am 5 am/ n

where a > 0 and m, n are positive integers and m > 0

x y 1   q1r2  q2 r1 r1 p2  p1r2 p1q2  p2 q1 ∴

x

q1r2  q2 r1 r p  p1r2 . ; y 1 2 q1q2  p2 q1 p1q2  p2 q2

3.12

▌ Mathematics

INEQUALITIES Inequalities are based on the order relation ‘>’, ‘5’ and ‘ 0, a 5 0, a < 0 Between any two real numbers, a and b, again, one of the following holds: (1) If a > b, then

a > b, a 5 b, a < b a 1 x > b 1 x for any x a 2 x > b 2 x for any x

ka  kb

if k  0

ka  kb

if k  0

a m  bm   (m  0) am  bm 

(2) If (a/b) > 1, then 1 < (a 1 x)/(b 1 x) < (a/b) Thus, any real number of the form a/b (a > b) decreases if the same positive x is added to both the numerator and the denominator and continues to be > 1. If (a/b) < 1, then (a/b) < (a 1 x)/ (b 1 x) < 1 Any real number of the form a/b (a < b), therefore, increases if the same positive x is added to both the numerator and the denominator, and, continues to be < 1.

(3) The Arithmetic Mean of any (n) positive quantities (not equal to one another) is always greater than their geometric mean, i.e. a1  a2  ......  an n  a1 a2 ... an n If a1 5 a2 5 ¼ an, then the two values are equal. Generally, all problems on inequalities are solved by the use of these properties. (4) If a1, a2, ... an are n positive numbers (not all equal), then a1 p  q  a2 p  q  ...  an p  q n a1 p  a2 p  ...  an q a1q  a2 q  ...  an q  n n according as p and q have the SAME or OPPOSITE signs. (5) If a1, a2, ... an are positive and a1 1 a2 1 ... 1 an 5 s, then (1 1 a1) (1 1 a2) ..... (1 1 an) > 1 1 s (1 2 a1) (1 2 a1) ..... (1 2 an) > 1 2 s (6) a2 1 b2 1 c2 2 bc 2 ca 2 ab 5 {(a 2 b)2 1 (b 2 c)2 1 (c 2 a)2}/2 > 0, (a ¹ b ¹ c)

PRAcTIcE ExERcISE SET ThEORy, RELATIONS AND FUNcTIONS 1. The smallest set A such that AÈ{1, 2} 5 {1, 2, 3, 5, 9} is (a) {2, 3, 5} (b) {3, 5, 9} (c) {1, 2, 5, 9} (d) None of these 2. A has 3 elements and B has 6 elements such that A Ì B. The number of elements in A È B is (a) 3 (b) 9 (c) 6 (d) None of these 3. Let A{1, 2, 4} and B 5 {2, 6, 10}, then n(A ´ B) is equal to (a) 6 (b) 9 (c) 3 (d) None of these 4. Let A{a, b, c}, B 5 {b, c, d}, C 5 {a, b, d, e} then A Ç (B È C) is (a) {a,b,c} (b) {b,c,d} (c) {a,b,d,e} (d) {e} 5. If A is any set that A-A is equal to (a) A (b) Ac (c) F (d) None of these 6. Let A and B be two disjoint subsets of a universal set U. Then, (AA È B) Ç B¢ (a) A (b) B (c) F (d) None of these

7. If x and y are two sets, then X Ç (X È Y ) equals (a) x (b) y (c) F (d) None of these 8. If A 5 {f{f}}, then the Power set P(A) of A is (a) A (b) {f{f}, A} (c) {f{f}, {f} A} (d) None of these 9. If A is any set, the (a) A È A¢ 5 f (c) A È A¢ 5 X

(b) A Ç A¢ 5 X (d) None of these

10. Which of the following is a Null set? (a) {0} (b) {x;x > 0 or x < 0} (c) {x:x2 5 4 or x 5 3} (d) {x:x2 1 1 5 0, x∈R} 11. If A and B are two sets, then A È B 5 A Ç B iff (a) A Ì B (b) A 5 B (c) B Ì A (d) None of these 12. If A and B are two sets, then A Ç (A È B)¢ is (a) A (b) B (c) F (d) None of these 13. A, B be any two sets, then (A Ç B)¢ is equal to (a) A¢ È B¢ (b) A Ç B (c) A¢ Ç B¢ (d) A È B

Algebra ▌ 3.13 14. If A Ç B 5 A and B Ç C 5 B, then A È C is equal to (a) A (b) C (c) B (d) B È C 15. If A, B be any two sets, then (A Ç B)¢ equals (a) A¢ Ç B¢ (b) A¢ È B¢ (c) A È B (d) A Ç B

30. Let A be the set of squares of natural numbers and let x Î A, y Î A then, (a) x  y ∈ A (b) x  y ∈ A (c)

x ∈A∈A y

(d) x y ∈ A

16. If A Ì B then A Ç B is equal to (a) A (b) B (c) AC (d) B C

31. If aN 5 {an : x∈N}, then the set 3N Ç 7N (a) 21N (b) 10 N (c) 4N (d) None of these

17. If A Ì B then A Ç B is equal to (a) A (b) B (c) AC (d) B C

32. A 5 {x:x ≠ x} represents (a) {0} (c) {1}

18. Two Sets A, B are disjoint iff (a) A È B 5 f (b) A Ç B ¹ f (c) A Ç B 5 f (d) A 2 B 5 A

  1 33. If φ 5  x : x 5 , y ∈ N  then y  

19. If A and B are any two sets, then A È (A Ç B) is equal to (a) A (b) B (c) AC (d) B C 20. If A and B are any two sets, then A Ç (A È B) is equal to (a) A (b) B (c) AC (d) B C 21. If A and B are sets, then A Ç (B 2 A) is (a) F (b) B (c) A Ç B (d) None of these 22. If A, B, C are three sets, then A Ç (B È C) is equal to (a) (A È B) Ç (A È C) (c) (A È B) È (A È C)

(b) (A Ç B) È (A Ç C) (d) None of these

(b) {} (d) {x}

(a) O ∈ Q

(b) 1 ∈φ

(c) 2 ∈φ

(d)

2 ∈φ 3

34. Which of the following in an empty set? (a) The set of prince numbers which are even. (b) The solution set of the equation 2(2x  3) 2   3  0, x ∈ R x 1 x 1 (c) (A ´ B) Ç (B ´ A) where A and B are disjoint (d) The set of reals which satisfy x2 1 ix 1 i 2 1 5 0

23. If A and B are two sets, then A ´ B 5 B ´ A iff (a) A Ì B (b) B Ì A (c) A 5 B (d) None of these

35. Which of the following statement is true? (a) 3 Í {1, 3, 5} (b) 3 Î {1, 3, 5} (c) {3} Î {1, 3, 5} (d) {3, 5} Î {1, 3, 5}

24. If A, B and C are any three sets, than A ´ (B È C) is equal to (a) (A ´ B) È (A ´ C) (b) (A ´ B) Ç (A ´ C) (c) (A È B) ´ (A È C) (d) None of these

36. Which of the following is a null set? (a) A 5 {x:x >1 and x < 1} (b) B 5 {x:x 1 3 5 3} (c) C 5 {F} (d) D 5 {x:x1 ³ and x £ 1}

25. Let A and B be the subset of a set X, then (a) A 2 B 5 A È B (b) A 2 B 5 A Ç B (c) A 2 B 5 AC Ç B (d) A 2 B 5 A Ç B c

37. Let A 5 {a, b, c}, B 5 {a, b}, C 5 {a, b, d}, D 5 {c, d} and E 5 {d}. then which statement is not correct? (a) B Ì A (b) D É E (c) D Ì B (d) B Ì C

26. If A and B are finite sets (non-empty), then number of elements in A ´ B is (a) n(A È B) (b) n(A Ç B) (c) n(A) ´ n(B) (d) None of these

38. If A Ì B, then (a) AC Ì BC (c) BC Ì AC

(b) AC 5 BC (d) BC Ë AC

27. If A and B are not disjoint, then n(A È B) is equal to (a) n(A) 1 n(B) (b) n(A) 1 n(B) 2 n(A Ç B) (c) n(A) 1 n(B) 1 n(A Ç B) (d) n(A) n(B)

39. If A, B and C are any three sets, then A ´ (B Ç C) is equal to (a) (A ´ B) È (A ´ C) (b) (A È B) ´ (A È C) (c) (A ´ B) Ç (A ´ C) (d) (A Ç B) ´ (A Ç C)

28. A 2 B is equal to (a) B 2 A (c) A 2 (A Ç B)

40. If A, B and C are any three sets, then A 2 (B È C) is equal to (a) (A 2 B) È (A 2 C) (b) (A 2 B) Ç (A 2 C) (c) (A 2 B) È C (d) (A 2 B) Ç C

(b) A È B (d) A Ç B

29. If A is finite, then the number of distinct subsets of A is (a) 2n(A) (b) 2n(A) 2 (c) (n(A)) (d) None of these

41. If A, B and C are any three sets, then A 2 (B Ç C) is equal to (a) (A 2 B) È (A 2 C) (b) (A 2 B) Ç (A 2 C) (c) (A 2 B) È C (d) (A 2 B) Ç C

3.14

▌ Mathematics

42. If A and B are disjoint, then n(A È B) is equal to (a) n(A) (b) n(B) (c) n(A) 1 n(B) (d) n(A) n(B)

51. The set { x : (x 2 2)(x 2 3) > 0} is equal to (a) {x : 2 < x < 3} (b) {x : x < 3}È{x : x < 2} (c) {x : x > 3}È{x : x < 2} (d) None of these

43. A set is (a) a collection of objects (b) a group of objects (c) a collection of objects with a fixed common property (d) a well-defined collection of objects.

52. If x and y are two sets and x1 denotes the complement of x, the x Ç (x È y)1 equals (a) x (b) y (c) F (d) None

44. If A 5 {1}, which of the following statement is correct? (a) A 5 1 (b) 1 Ì A (c) {1} Î A (d) 1 Î A 45. If A 5 {1, 2, {3, 4}}, which of the following is a correct statement? (a) 3 Î A (b) {1} Ì A (c) {2} Î A (d) 1 Í A 46. If A 5 {1, {2, 3}, 5}, which of the following statements is incorrect? (a) {2, 3} Î A (b) {{2, 3}} Í A (c) {3} Í A (d) {5} Í A 47. If A 5 {2, {3, 4}, 6}, which of the following is correct? (a) {2} Î A (b) {3} Î A (c) {4} Î A (d) {3, 4} Î A 48. If A 5 {1, {2, 3}, 4}, which of the following is a subset of A? (a) {1, 2} (b) {2, 3} (c) {3, 4} (d) {1, 4}   2 49. Let A  3, p , 2 , , 5, 3  7  the subset of A containing 7   all the elements from it which are irrational number is

{

(a) π , 2 , 3  7

}

  2 (c)  3, π , , 5   7

 2  (b)  3, π , , 5, 3  7    7 (d) {3, 5}

53. Consider the following statements: (1) A¢ È B 5 (A Ç B)¢ (2) (f¢)¢ 5 x (3) A Ç (B È C) 5 (A Ç B) È (A Ç C) (4) f¢ 5 x Of these statements. (a) 2 and 3 are correct (b) 1 and 3 are correct (c) 3 and 4 are correct (d) 2 and 4 are correct 54. If A Í B, which of the following statements is correct? (a) A È B 5 B Þ A 5 B (b) A Ç B 5 B Þ A B (c) A 2 B 5 A Þ A 5 B (d) B 2 A 5 B Þ A 5 B 55. If A and B are two sets such that n(A) 5 m, n(B) 5 n and n(A Ç B) 5 P, then n(A È B) is equal to (a) m 1 n (b) m 1 n 1 P (c) m 1 n 2 p (d) m 2 n 2 P 56. Which one of the following is an empty set? (a) {0} (b) {x Î N: 5x 5 9} (c) {x Î 1: 0 < x < 4x is even} (d) {x Î 1: (x 2 1)(x 2 2) 5 0 and x is odd} 57. Which of the following pairs of sets is disjoint? (a) {0, 1, 2} and {0, 21, 22} (b) {1, 3, 4, 5} and {3, 5, 7} (c) {1, 3, 5, 7} and {2, 4, 7, 9} (d) {1, 2, 3} and {21, 22, 23} 58. Let A 5 {1, 2, 3, 4, 5, 6}. Then, the incorrect statement is (a) f Í A (b) A Í A (c) {1, 5, 7} Í A (d) {1} Í A

50. Consider the following statements: 1. Any set A is comparable with itself 2. {} is a Singleton set. 3. {F} is an Empty set. of these statements, the correct ones are (a) 1 and 2 (b) 1 and 3 (c) 2 and 3 (d) 1, 2 and 3

59. Let A 5 {x Î R: o < x < 1} which of the following is not an element of A? 1 2 1 (d) 4

(a) 1 (c)

(b)

1 3

ANSwER KEy 1. 2. 3. 4. 5. 6. 7. 8.

(b) (c) (b) (a) (c) (c) (a) (c)

9. 10. 11. 12. 13. 14. 15. 16.

(c) (d) (a) (c) (c) (b) (b) (a)

17. 18. 19. 20. 21. 22. 23. 24.

(b) (c) (a) (a) (a) (b) (c) (a)

25. 26. 27. 28. 29. 30. 31. 32.

(c) (c) (b) (c) (b) (c) (a) (b)

33. 34. 35. 36. 37. 38. 39. 40.

(a) (c) (b) (a) (c) (c) (c) (b)

41. 42. 43. 44. 45. 46. 47. 48.

(a) (c) (a) (d) (b) (c) (d) (d)

49. 50. 51. 52. 53. 54. 55. 56.

(a) (a) (c) (c) (c) (b) (c) (b)

57. (b) 58. (c) 59. (b)

Algebra ▌ 3.15

SOLUTIONS 1. A È [1, 2] 5 {1, 2, 3, 5, 9} Þ A 5 {3, 5, 9}. 2. A Ì B Þ A È B \ Number of elements in A È B 5 6. 3. n(A ´ B) 5 n(A)n(B) 5 (3)(3) 5 9. 4. B È C 5 {a, b, c, d, e} \ A Ç (B È C) 5 {a, b, c} Ç {a, b, c, d, e} 5 {a, b, c}. 5. A 2 A 5 {x : x Î A and x Ï A} 5 f. 6. (A Ç B¢) È (B Ç B¢) 5 A È f 5 A Since A and B are disjoint. 7. If x Ì y then x È y 5 y and x Ç y 5 x If y Ì x then x È y 5 y and x Ç y 5 x. 8. 22 = 4 9. A È A¢ 5 X (by def.). 10. x 2 1 1 5 0 Þ x 5 ± i {x : x 2 1 1 5 0, x Î R} 5 f, will set.

27. n(A) 1 n(B) 2 n(A Ç B) An important formula. 28. A – (A Ç B) x Î A 2 B Þ x Î A and x Ï B Þ x Î A and x Ï (A Ç B) Þ x Î A 2 (A Ç B) [ A Ç B Ì B] \ A 2 B = A 2 (A Ç B). 29. Let A be a set consisting of n distinct elements; then the number of subsets of A is the same as the number of combination of n elements taken, one, two, all at a time. Hence the number of subsets of A. 5 nC0 1 nC1 1 nC2 1 … 1 nCn 5 2n 5 2n(A). 30. Let x 5 m2 and y 5 n2 for some m, n Î N. Then, xy 5 m2n2 5 (mn)2, where mn Î N. Note that 4 ∈ A, 16 ∈ A but

31. 3N 5 {x : xÎN, x is a multiple of 3}

11. B Ì A Let x Î A Þ x Î A È B [ A Ì A È B] Þ x Î A Ç B [ A È B = A Ç B] x Î A and x Î B x Î B Similary x Î B, Þ x Î A \ A Ì B. 12. A Ç (A È B) 5 A Ç (A¢ Ç B¢) 5 (A Ç A¢) Ç B¢ 5 f Ç B¢ 5 f. 13. (A È B)¢ 5 A¢ Ç B¢ [by De-Morgan’s law]. 15. (A Ç B)¢ 5 A¢ È B¢ [by De-Morgan’s law]. 16. A Ì B Þ A Ç B 5 A. 17. A Ì B Þ A È B 5 B. 18. A and B are disjoint if A Ç B 5 f [Def. of disjoint sets]. 19.  A Ç B Ì A \ A È (A Ç B) = A. 20.  A Ì A È B \ A Ç (A È B) = A. 21. x Î B 2 A Þ x Ï A \ x Î A Ç (B 2 A) Þ x Î A and x Ï A \ A Ç (B 2 A) 5 f. 22. A Ç (B È A) 5 (A Ç B) È (A Ç C) [by Distribution law]. 23. In gernal A ´ B ¹ B ´ A A ´ B = B ´ A is true if A = B [ A ´ B = A ´ A, B ´ A = A ´ A]. 24. (x, y) Î A(B È C) Þ x Î A and y Î B È C Þ x Î A and ( y Î B or y Î C) Û (x, y) Î (AB) È (AC). 25. x Î (A 2 B)x Î A and x Ï B Û

x Î A and x Î BC

Û

n Î A Ç BC \ A 2 B 5 A Ç BC.

26. n(A) ´ n(B): [Property of cartesian product of two sets].

4 ∉ A. 16

7N 5 {x : xÎN, x is a multiple of 7} \ 3N Ç 7N

5 {x Î N, x is a multiple of 3 and 7}

5 {xÎN, x is a multiple of 21} 5 21N. 32. Clearly, A 5 φ 5 {}. 33.

1 1 1 2 ≠ 0, ≠ 2, ≠ [  y ∈ N ] y y, y 3 ∴

1 can be 1 ⇒ 1 ∈ Q. y

34. For any and (a, b) Î A ´ B, a Î A and bÎB Now(a, b) will belong to B´ A only if a Î B and a Î A and that can happen only if A Ç B 5 f. But, in this case, A Ç B 5 f, therefore, (A ´ B) Ç (B ´ A) 5 f 35. Clearly, 3 Î {1, 3, 5}. 36. As x > 1 and x > 1 are not possible \ A 5 f. 37. Since ED is not a member of B \ D Ì B is not true. 38. (c). x Î BC Þ x Ï B [ A Ì B] Þ x Ï A Þ x Î AC Þ BC Ì AC. 39. A ´ (B Ç C) 5 (A ´ B) Ç (A ´ C) [by Distributive law]. 40. x Î A 2 (B È C) Þ x Î A and x Ï (B È C) Þ x Î A and (x Ï B and x Ï C) Þ (x Î A and x Î B) and (x Î A and x Ï C) Þ x Î (A 2 B) Ç (AC) Û x Î (A 2 B) Ç (A 2 C) \ A(B È C) 5 (A 2 B) Ç (AC).

3.16

▌ Mathematics

42. As A and B are disjoint \ A Ç B 5 f Þ n(A Ç B) 5 0 n(A È B) 5 n(A) 1 n(B) 2 n(A Ç B) 5 n(A) 1 n(B) 2 0 5 n(A) 1 n(B). 43. A set is a well-defined collection of objects. 44. Clearly, 1 Î A is true. 45. {1} Í A is true. 47. Clearly, {3, 4} Î 4. 48. Since 1 Î A and 4 Î A, So {1, 4} Í 4.

(

{

(

51. (x 2 2) (x 2 3) > 0 Þ [(x 2 2) > 0 and (x 2 3) > 0] or [(x 2 2) < 0 and (x 2 3) < 0] Þ x > 3 or x < 2 Þ {x : x > 3}  {x : x < 2}. 52. x Ç (x È y)¢ 5 x Ç (x¢ Ç y¢) 5 (x Ç x¢) Ç y ¢ 5 f Ç y¢ 5 f.

46. Since 3 A, So, {3} Í A is incorrect.

49. In the given set, π , 2 and 3 1 7

Also, {o} is clearly a Singleton set. So, statements 1 and 2 are correct.

)

are irrational So, the

)}

required subset is π , 2 3 1 7 . 50. {f} is not an empty set as it contains one element, namely, f. Two sets A and B are said to be comparable, if either A Í B or B Í A. Since A Í A, So A is comparable with A.

53. By Distributive law, 3 is correct and by definition of complement, f` 5 x. So, 3 ´ 4 are correct. 54. Given A Í B. Also, A Ç B 5 B Þ B Í A So, A 5 B. Thus, (B) is true. 55. n(A È B) 5 n(A) 1 n(B) 2 n(A Ç B). 56. Clearly, {x Î N : 5x 5 9} 5 f. 57. Clearly, no element is common to the sets given in (D). 58. Since 7 Ï A, So (C) is false.

QUADRATIc EQUATIONS 3x

1. If a 1 b 1 c 5 0 and a, b, c are rational, then the roots of the equation (b 1 c2a) x2 1 (c 1 a2b) x 1 (a 1 b2c) 5 0 are (a) rational (b) irrational (c) imaginary (d) equal 2. The number of values of the pair (α, β) for which the equation α(x 1 1)2 1 β (x223x 2) 1 x 1 1 5 0, ∀ x ∈ R is (a) 1 (b) 0 (c) infinite (d) none of these. 3. If α, β are the roots of the equation x223x 1 a 5 0, a R and α < 1 < β, then a belongs to  9 (a)  2∞, 4  (c) (2, ∞)

   

(b) ( 2∞, 2) 9  (d)  , ∞ 4 

4. In a quadratic equation with leading coefficient 1, a student reads the coefficient 16 of x wrongly as 19 and obtain the roots as 15 and 4. The correct roots are (a) 6, 10 (b) 26,210 (c) 27,29 (d) none of these 5. If the equations ax2 1 bx 1 c 5 0 and x2 1 2x 1 3 5 0 have a common root, then a : b : c 5 (a) 2 : 4 : 5 (b) 1 : 3 : 4 (c) 1 : 2 : 3 (d) none of these

6. The roots of the equation 2 x + 2 . 3 x1  9 are given by  2 (a) log 2   ,2 2 (b) 3,23  3 (c) 2 2, 1 2

log 3 log 2

(d) 1 2 log 2 3, 2

7. The sum of the real roots of the equation | x22 |2 1 | x22 |22 5 0 is (a) 2 (c) 4

(b) 6 (d) 8

8. If f (x) 5 x [x], x (≠ 0) ∈ R, where [x] is the greatest integer less than or equal to x, then the number of solutions of 1 f (x ) + f   5 1 are x (a) 0 (c) infinite

(b) 1 (d) 2

9. The number of solutions of the equation sin (ex) 5 5x 1 5 2x is (a) 0 (b) 1 (c) 2 (d) infinite 10. If x is real, the expression

x 2  2 x  11 takes all real values x 3

except those which lie between a and b, then a and b are (a) 212,24 (c) 4, 12

(b) 212, 2 (d) 24, 4

Algebra ▌ 3.17 11. The solution of the inequation 42x 1 0.527.2x < 4, x ∈ R, is (a) ( 2 2 , ∞) (b) (2 , ∞)  7 (c)  2,  (d) none of these  2 x

x

1 1 12. The real values of x for which 372      3  3 (a) x ∈ [0, 64] (c) x ∈ [0, 64)

1, are

(b) x ∈ (0, 64) (d) none of thes

13. If x is real, then the maximum value of 326x28x2 is 17 8 21 (c) 8 (a)

(b)

33 8

14. If x223x 1 2 be one of the factors of the expression x42px2 1 q, then (a) p 5 4, q 5 5 (b) p 5 5, q 5 4 (c) p 525, q 524 (d) none of these x2  2 x  4 lies x2  2 x  4

1 and 3. The values between which the expression 3 9.32 x  6.3x  4 lies are 9.32 x  6.3x  4

between

(a) 0 and 2

(b) 21 and 1 1 (c) 22 and 0 (d) and 3. 3 16. If the roots of x22bx 1 c 5 0 are two consecutive integers, then b224c is (a) 1 (b) 0 (c) 2 (d) none of these 17. If p (q2r) x2 1 q (r2p) x 1 r (p2q) 5 0 has equal roots, then 5 (a) p 1

1 r

1 1r p 1 1 (d) 1 p r

2

4 x 3

 (7  4 3 ) x

2

4 x 3

5 14, then the value

of x is given by (a) 2 , 2  2

(b) 2  3 , 3

(c) 3  2 , 2

(b) none of these

19. The number of negative integral solutions of x 2 ⋅ 2 x 1  2|x 3|  2  x 2 ⋅ 2 (|x 3|  4)  2 x 1 is (a) 4 (c) 1

x2 1 qx 1 r 5 0 be the same, then (a) r2c 5 b2q (b) r2b 5 c2q (c) rb2 5 cq2 (d) rc2 5 bq2 22. The number of solutions (s) of the equation 3x 2  6x  7  5x 2  10 x  14  4  2 x  x 2 is (b) two (d) infinite

α 23. If α, β are the roots of the equation x 2  px  q  0 then is β a root of the equation (a) px2 1 (2q2p2) x 1 p 5 0 (b) qx2 1 (p222q) x 1 q 5 0 (c) qx2 1 (2q2p2) x 1 q 5 0 (d) none of these 24. If a < 0 the positive root of the equation x222a |x2a|23a2 5 0 is (a) a( 1  6 ) (b) a( 1  6 ) (c) a(1  2 ) (d) none of these 25. If α, β are the roots of the equation 375x 2  25x  2  0 and S n  α n  β n , then Lt

n→∞

n

∑S r 1

r

is

(b)

(c) p 1 r 18. If (7  4 3) x

21. If the ratio of the roots of x2 1 bx 1 c 5 0 and

(a) one (c) four

(d) none of these

15. Given that, for all real x, the expression

20. If a, b, c, d and p are distinct real numbers such that (a2 1 b2 1 c2) p222 (ab 1 bc 1 cd) p 1 (b2 1 c2 1 d2) ≤ 0 then a, b, c and d (a) are in A.P. (b) are in G.P. (c) are in H.P. (d) satisfy ab 5 cd

(b) 2 (d) 0

(a)

7 12

(b)

1 12

(c)

35 12

(d) none of these 26. If α, β be the roots of x2 1 px2q 5 0 and γ, δ be the roots of (α  γ ) (α  δ )  x2 1 px 1 r 5 0, q 1 r ≠ 0, then (β  γ ) (β  δ ) (a) 1 (c) r

(b) q (d) q 1 r

3.18

▌ Mathematics

27. If the roots of the equation ax2 1 bx 1 c 5 0 are of the form

α α 1 and , then the value of (a 1 b 1 c)2 is α 1 α (b) b224ac (d) 2b22ac

(a) b222ac (c) 4b22ac

29. If the roots of x2 1 ax 1 b 5 0 are c and d, then roots of x2 1 (2c 1 a) x 1 c2 1 ac 1 b 5 0 are (a) 1, d2c (b) 0, d2c (c) 1, c2d (d) none of these x

 3 30. The solution set of    x  x 2  9 is 5

28. If log9 (x2 5x 1 6) > log3 (x 4), x belongs to (a) (∞, 4) (b) (4, ∞) (c) (∞, 4) ∪ (4, ∞) (d) no real value of x

(a) {0} (c) φ

(b) {1} (d) none of these

ANSwER KEy 1. 2. 3. 4.

5. 6. 7. 8.

(a) (b) (b, c) (b)

(c) (c) (c) (c)

9. 10. 11. 12.

(a) (c) (a) (c)

13. 14. 15. 16.

(b) (b) (d) (a)

17. 18. 19. 20.

(d) (a) (d) (b)

21. 22. 23. 24.

(c) (a) (c) (c)

25. 26. 27. 28.

(b) (a) (b) (b)

29. (b) 30. (c)

SOLUTIONS 1. We have, D 5 (c 1 a2b)224 (b 1 c2a) (a 1 b2c) 5 (a 1 b 1 c22b)2 24 (a 1 b 1 c22a) (a 1 b 1 c22c) 5 (22b)224 (22a) (22c) 5 4 (b224ac) 5 4 [(2a2c)224ac] 5 4 (a2c)2 5 {2 (a2c)}2 5 perfect square ∴ Roots are rational. 2. The equation (α 1 β )x2 1 (2α23β 1 1) x 1 (α 1 2β 1 1) 5 0 is an identify in x if α1β50 ...(1) 2α 23β 1 15 0 ...(2) and α 22β 1 1 5 0 ...(3) 1 1 Solving (1) and (2), we get α  and β  . 5 5 But these values do not satisfy (3). Hence the given equation cannot be an identify for any values of α and β. 3. Since 1 lies between the roots of the given equation, therefore, D > 0 and f (1) < 0 ⇒ 924a > 0 and 123 1 a < 0 ⇒ a,

9 and a , 2 4

∴a , 2

4. Since coefficient of x 5 16, ∴ sum of roots 5216 Since constant term 5 (215) (24) 5 60, ∴ correct answer is26,210. 5. For the equation x2 1 2x 1 3 5 0, Discriminant 5 (2)2 24 ⋅ 1 ⋅ 3 < 0.

∴ roots of x2 1 2x 1 3 5 0 are imaginary. Since the equations x2 1 2x 1 3 5 0 and ax2 1 bx 1 c 5 0 are given to have a common root, therefore both roots will be common. Hence both the equations are identical. a b c ∴ 5 5 i .e., a : b : c 5 1 : 2 : 3. 1 2 3 6. We have, 2x 1 2 ⋅ 33x /(x – 1) 5 9 5 32 3x ⇒ ( x  2) log 2  log 3  2 log 3 x 1 ⇒ ⇒ ⇒

 3x  ( x  2) log 2    2  log 3  0 x  1     1 ( x  2)  log 2  log 3   0 1 x   log 3 x  2 or x  1  . log 2

7. Put | x22 | 5 t. The given equation becomes t2 1 t22 5 0 or (t 1 2) (t21) 5 0 Since t 1 2 5 | x22 | 1 2 > 0 ∴ we get t21 5 0 ⇒ | x22 | 5 1 ⇒ x22 5 ± 1 ⇒ x 5 3, 1. Thus, the sum of roots is 4. 1 8. We have, f (x )  f    1 x ⇒

1 1 x  [x ]      1 x x

Algebra ▌ 3.19

⇒

1 1 x  1  [x ]    x x

⇒

x2 1 x  (integer) k (say) x

⇒ x2 2 (k 1 1) x 1 1 5 0 Since x is real, so (k 1 1)224 ≥ 0 ⇒ k2 1 2k23 ≥ 0 ⇒ (k 1 3) (k21) ≥ 0 ⇒ k ≤23 or k ≥ 1. Therefore, number of solutions is infinite. 9. Put 5x 5 y. Then the given equation becomes 2

sin (e x )  y 

1  1    2 [  5x  0]  y  y  y 

⇒ sin (ex ) ≥ 2. which is not possible for any real value of x. Hence the given equation has no real solution. 10. Let y  ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒

x 2  2 x  11 x 3

13. Let y 5 326x28x2 then 8x2 1 6x 1 y23 5 0. Since x is real, ∴ 6224 ⋅ 8 (y23) ≥ 0, or 36232y 1 96 ≥ 0 or 32y ≤ 132 132 33 ∴ y or y  . 32 8 33 Hence maximum value of y 5 8 14. Since x223x 1 2 is one of the factors of the expression x42px2 1 q, therefore, on dividing the expression by factor, remainder 5 0 i.e., on dividing x42px2 1 q by x223x 1 2, the remainder (1523p) x 1 (2p 1 q214) 5 0. On comparing both sides, we get 15 23p 5 0 or p 5 5 and 2p 1 q 214 5 0 or q 5 4. 1 x2  2x  4  3 for all x ∈ R. 15. Given  2 3 x  2x  4 ⇒

xy 23y 5 x2 1 2x211 x2 1 (22y) x 1 (3y211) 5 0 (D ≥ 0) (22y)224 (3y211) ≥ 0 4 1 y224y212y 1 44 ≥ 0 y2216y 1 48 ≥ 0 y2212y24y 1 48 ≥ 0 (y24) (y212) ≥ 0 y ≤ 4 or y ≥ 12

11. The given inequation is

1 x2  2x  4   3 for all x ∈ R. 3 x2  2x  4

Let 3x 1 1 5 y Then y ∈ R for all x ∈ R. ∴

9.32 x  6.3x  4 32 x 2  2.3x 1  4 y 2  2 y  4   9.32 x  6.3x  4 32 x 2  2.3x 1  4 y 2  2 y  4

1 y2  2 y  4 From (1),  2 3 3 y 2y 4 ∴

4x  0.5  7.2x < 4, x ∈ R

1 9.32 x  6.3x  4  3 3 9.32 x  6.3x  4

16. The roots of the equation are given by

Let

2− x  t

∴

2t  7t  4

⇒

2t 2  7t  4  0 ⇒ (2t  1) (t  4)  0

x 

⇒

1   t  4 but 2− x  0 so 0  t  4 2 0  2− x  4 ⇒  2  x  ∞ or x ∈ (2, ∞ )

If α 

⇒

2

12. The given inequation is valid only when x ≥ 0 ...(1) The given inequation can be written in the form 72 − x x

3

1

⇒

72  x  x  0

⇒

x  x  72  0

⇒

( x  9) ( x  8)  0

But ∴ ∴

...(1)

( 3 > 1)

x  9  0 for all x  0 x 8  ⇒ x  8 0  x  64 {from (1)}.

b  b2  4 c 2 b  b2  4 c 2

and β 

b  b2  4 c 2

Then, α  β  1 ⇒ b2  4c  1 ⇒ b2  4c  1. 17. Since p (q2r) 1 q (r2p) 1 r (p2q) 5 0 ∴ one root is 1 r ( p  q) ∴ other root  . p( q  r ) Since roots are equal ∴

rp  rq  1 ⇒ rp  rq  pq  pr ⇒ 2rp  q ( p  r ) pq  pr

∴

2 pr 1 1   .  pr p r q

▌ Mathematics

3.20

18. Since (7  4 3 ) (7  4 3 )  1, ∴ the given equation becomes 2 1 y   14 where y  (7  4 3 ) x 4 x 3 y ⇒ y 2  14 y  1  0 ⇒ y  (7  4 3) Now y  ⇒ 7  4 3 ⇒ x  4 x  3 1 ⇒ x  2, 2 2

∴ L.H.S. ≥ 5 R.H.S. 5 422x2x2 5 52(x 1 1)2 ≤ 5 ∴ the equation holds only when L.H.S. 5 R.H.S. 5 5 ∴ x 521. 23. Since α, β are roots of the equationx2 1 px 1 q 5 0 ∴ α 1 β 52p and αβ 5 q. 2

α  α q    ( 2q  p 2 )  q β β  1 1  2 [q(α + β )2  p 2αβ ]  2 [qp 2  p 2 q]  0. β β

Also, y  7  4 3 ⇒ x 2  4 x  3  1 ⇒ x  2  2 .

Now

19. The given equation can be written as  1 x 3  2 x 1  x 2    2 .4 [4 x 2  1] 4    16 ⋅ 2

x 3

Thus,

 2 1 x   4 

⇒ 2 x 3  2

x 3

1 [ x 2 5 does not give negative integral valuee] 4 ⇒ x23 5 ± (x23) ⇒ either x23 5 x23 or x23 52x 1 3 ⇒ 2x 5 6 or x 5 3 ∴ Given equation does not give any negative integral solution. 20. We have, (a2 1 b2 1 c2) p222 (ab 1 bc 1 cd) p 1 (b2 1 c2 1 d2) ≤ 0 ⇒ ( ap  b)2  ( dp  c )2  ( cp  d )2  0 ⇒ ( ap  b)2  ( dp  c )2  ( cp  d )2  0 (∴ a, b, c, d , p ∈ R ) ⇒ ap  b  0, bp  c  0, cp  d  0 b c d ⇒   p ⇒ a, b, c, d are in G.P. a b c 21. Let α, β be the roots of equation x2 1 bx 1 c 5 0 and α’, β’ be the roots of the x2 1 qx 1 r 5 0. Then,

α 1 β 52b ; αβ 5 c, α’ 1 β’ 52q, α’ β’ 5 r. α α' α  β α'  β ' It is given that  ⇒  β β ' α  β α'  β ' ⇒

(α  β )2 (α'  β ')  (α  β )2 (α'  β ')

⇒

b2 q2 ⇒ b2 r  q 2 c.  2 b  4c q  4r

 5x 2  10 x  14  5( x  1)2  9  3

and

α is a root of the equation β

qx2 1 (2q 2p2) x 1 q 5 0. 24. Since a < 0, in case of positive root of the equation x > a ∴ The equation is x222a (x2a)23a2 5 0 ⇒ x222ax2a2 5 0 Thus, the roots are

 a(1  2 ) or a(1  2 ) ∴ the only positive root possible is a(1 2 2 ). 25.

n

∑S r 1

r

 (α  β )  (α 2  β 2 )  ...  (α n  β n )

 (α  α 2  ...  α n )  ( β  β 2  ...  β n ) Lt

n→∞

n

∑S r 1

r

 (α  α 2  ...  ∞ )  ( β  β 2  ...  ∞ )

α β  1 α 1 β α  αβ  β  αβ 1  ( α  β)  αβ



α  β  2α β 1  ( α  β)  αβ 25 4  29 1 375 375    25 2 348 12 1  375 375 26. Here α 1 β 52p ⇒ α 1 β 5 γ 1 δ γ 1 δ 52p Now (α  γ ) (α  δ )  α 2  α (γ  δ )  γδ  α 2  α (α  β )  r

2

 αβ  r  (q )  r

22. We have, 3x 2  6 x  7  3( x  1)2  4  2

2a  4 a 2  4 a 2 2a  2 2 a  2 2

qr By symmetry (β 2γ) (β 2δ) 5 q 1 r ∴ Ratio is 1.

Algebra ▌ 3.21

27. We have,

⇒

b α α 1 α α 1   and.  ⋅  a α 1 α α 1 α

2α 2  1 b ca  and α  2 ca α α a

29. If f (x) 5 x2 1 ax 1 b f (x 1 c) 5 x2 1 (2c 1 a) x 1 c2 1 ac 1 b

⇒ ( c  a )2  4 ac  2b ( c  a )

∴ Roots of the given equation are 0 and d2c.

⇒ ( c  a )2  2b( c  a )  b2  b2  4 ac

(since roots of x2 1 ax 1 b 5 0 are c and d.)

⇒ ( a  b  c )2  b2  4 ac. 28.

Also, x225x 1 6 > 0 ⇒ x > 3 or x < 2 ...(2) and x24 > 0 ⇒ x > 4 ...(3) Common solution from (1), (2) and (3) is x > 4

30. x2x229

log e ( x  5x  6) log e ( x  4)  log e 9 log e 3

52(x22x 1 9)

2

2  1 35    x      0 for all x ∈ R 2 4   

⇒ x 2  5x  6  x 2  8 x  16 10 ⇒ 3x  10  0 ⇒ x  3

...(1)

  3 x  ∴ no solution i.e., solution set  φ     0 for all ∈ R   4   

PROgRESSSION 5. If a is the first term, d the common difference and Sk S the sum to k terms of an A.P., then for kx to be Sx independent of x (a) a 5 2d (b) a 5 d (c) 2a 5 d (d) none of these

1. The maximum sum of the series 1 2 20 1 19 1 18 1 18 1... is 3 3 (a) (b) (c) (d)

310 290 320 none of these

2. Let Sn denotes the sum of n terms of an A.P. whose first term is a. If the common difference d 5 Sn2k Sn21 1 Sn22 then k 5 (a) 1 (b) 2 (c) 3 (d) none of these 3. The sum Sn to n terms of the series 1 3 7 15 1 1 1 1.... is equal to 2 4 8 16 (a) (b) (c) (d)

2n2n21 1222n 2 n 1 n21 2n21

4. If S1 is the sum of an arithmetic series of ‘n’ odd number of terms and S2, the sum of the terms of the series in S odd places, then 1 5 S2 2n n 11 n 11 (c) 2n (a)

n n 11 n 11 (d) n (b)

6. The sum of n terms of m A.P.s are S1, S2, S3, ..., Sm. If the first term and common difference are 1, 2, 3, ..., m respectively, then S1 1 S2 1 S3 1 ... 1 Sm 5 1 mn ( m  1) ( n  1) 4 1 (c) mn ( m  1) ( n  1) 4 (a )

1 mn ( m  1) ( n  1) 2 1 (d) mn ( m  1) ( n  1) 2

( b)

7. If the sum of three numbers in G.P. is 63 and the product of 3 of the third term, then the the first and the second term is 4 numbers are (a) 3, 12, 48 (b) 4, 12, 36 (c) 2, 10, 50 (d) none of these 8. In a set of four numbers, the first three are in G.P. and the last three are in A.P. with a common difference 6. If the first number is same as the fourth, the four numbers are (a) 3, 9, 15, 21 (c) 8, 24, 2, 8

(b) 1, 7, 13, 19 (d) none of these

9. The three numbers a, b, c between 2 and 18 are such that their sum is 25; the numbers 2, a, b are consecutive terms of an A.P. and the numbers b, c, 18 are consecutive terms of a G.P. The three numbers are (a) 3, 8, 14 (c) 5, 8, 12

(b) 2, 9, 14 (d) none of these

3.22

▌ Mathematics

10. The sum to n terms of the series 2

14. If a, b, c are digits, then the rational number represented by 0 ⋅ cababab ... is

3

3 1 4 1 5 1 ⋅ 1   1   1 ... is 1.2 2 2.3  2  3.4  2  (a ) 1 

1 ( n  1)2 n

( b) 1 

1 ( n  1)2 n1

(c ) 1 

1 ( n  1)2 n1

99c 1 ab 990 99c 1 10 a 1 b (c) 990

(a)

17. The sum of the series 1 1 2⋅2 1 3⋅22 1 4⋅23 1 5⋅24 1 ... 1 100⋅299 is (a) 99⋅2100 1 1 (b) 100⋅2100 100 (c) 99⋅2 (d) 99⋅2100 1 1 18. Four different integers form an increasing A.P. If one of these numbers is equal to the sum of the squares of the other three numbers, then the numbers are (a) 22,21, 0, 1 (b) 0, 1, 2, 3 (c) 21, 0, 1, 2 (d) none of these

12. Sum to infinity of the series 2 5 2 11     ... is 3 6 3 24 4 9 2 (c ) 9

( b)

1 3

(d ) none of these log

5

 1 1 1   1 1 1...∞  4 8 16 

(b) 2

1 2

(d) 4

(c)

19. If there are (2n11) terms in A.P., then the ratio of the sum of odd terms and the sum of even terms is (a) n : (n 1 1) (b) (n 1 1) n (c) (n21) n (d) none of these 20. Number of increasing geometrical progression(s) with first term unity, such that any three consecutive terms, on doubling the middle become an A.P. is (a) 0 (b) 1 (c) 2 (d) infinity

is

(a) 1

(b) 35 ≥ 5! (d) none of these

16. The largest value of the positive integer k for which nk 1 1 divides 1 1 n 1 n2 1 ... 1 n127 is divisible by (a) 8 (b) 16 (c) 32 (d) 64

11. If 1, logy x, logz y,215logx z are in A.P., then (a) x 5 y23 (b) y 5 z22 (c) x 5 z3 (d) none of these

(a )

99c 1 10 a 1 b 99

(d) none of these

15. Let a 5 1 ⋅ 2 ⋅ 3 ⋅ 4 ⋅ 5. Then (a) 55 ≥ a (c) 55 ≥ 6a

(d ) none of these

13. The value of (0.2)

(b)

ANSwER KEy 1. (a) 2. (b) 3. (c)

4. (a) 5. (c) 6. (a)

7. (a) 8. (c) 9. (c)

10. (a) 11. (c) 12. (c)

13. (d) 14. (c) 15. (a, b)

16. (d) 17. (d) 18. (c)

19. (b) 20. (b)

SOLUTIONS 1. The given series is arithmetic whose first term 5 20, common 2 difference 5 . 3 As the common difference is negative, the terms will become negative after some stage. So the sum is maximum if only positive terms are added.  2 Now tn  20  ( n  1)     0 if 60  2 (n  1)  0;  3

or 62 ≥ 2n or 31 ≥ n. ∴ The first 31 terms are non-negative.  S31 

∴ Maximum sum 

 2  31  2  20  (31  1    3   3 

31 {40  20}  310. 2

Algebra ▌ 3.23 2. We have, an 5 Sn2Sn21

...(1)

and an21 5 Sn212Sn22

...(2)

n( n  1)  (1  2  ...  m) 2 2 n  n m( m  1)   n   2  2  m( m  1) n( n  1) 1 ⋅   mn ( m  1) ( n  1). 2 2 4  n(1  2  3 ...  m) 

∴ d 5 an2an21 5 (Sn2Sn21)2(Sn212Sn22) [From (1) and (2)] 5 Sn22Sn21 1 Sn22.

7. Let the three numbers be a, ar, ar 2 Given a 1 ar 1 ar2 5 63

3. We have,   1  1  1 1    1     1     1  3   ...   1  n  2 4 2 2        

Sn

3 3 and a ⋅ ar 5 ⋅ ar or a 5 r 4 4

1 1 1   n    2  ...  n  2 2 2   1 1 1    2 2 n  1 1 2

n

    n  1 2n

5  25  84 2 ∴ real value of r is 4. Putting this value in (2) 3 a  43 4 ∴ The three number are 3, 3 × 4, 3 × 42 i.e. 3, 12, 48 ∴r  4,

a, a 1 d, a 1 2d, a 1 3d, a 1 4d, ....., a 1 (n 21) d Then, S1 5 {2a 1 (n21) d}

8. Let the last three numbers in A.P. be b, b 1 6, b 1 12 and the first number be a.

n n 1 terms S2  a  ( a  2 d )  ( a  4 d )  ... to 2 2

Hence the four numbers are a, b, b 1 6, b 1 12 Given, a 5 b 1 12 ..(i)

  n 1  n  1    1  2d  2 a   2  2   2  

or b2 5 (b 1 12) (b 1 6) [ a 5 b 1 12]

n 1 {2 a  ( n 1) d }. 4

kx [2 a  ( kx  1) d S kx  2 5. We have, x Sx [2 a  ( x  1) d 2 

and a, b, b 1 6 are in G.P. i.e. b2 5 a (b 1 6) or 18b 5 272 ∴ b 5 24, From (i), a 5 24 1 12 5 8.

S 2n  1 . S2 n  1

∴

k [( 2 a  d )  kxd ] ( 2 a  d )  xd

For

S kx to be independent of x, 2a2d 5 0 or 2a 5 d. Sx

6. We have, S1 5 (n/2) [2 ⋅ 1 1 (n21) ⋅ 1] S2 5 (n/2) [2 ⋅ 2 1 (n21) ⋅ 2] Sm 5 (n/2) [2 ⋅ m 1 (n21) ⋅ m] ∴ S1 1 S2 1 ... 1 Sm

...(2)

3 3 3 Putting in (1), r  r ⋅ r  r ⋅ r 2  63 4 4 4 or r3 1 r2 1 r 2 84 5 0 or (r 24) 1 (r2 1 5r 1 21) 5 0

4. Let the odd number of terms of an arithmetic series be



...(i)

Hence, the four numbers are 8, 24, 2 and 8.

9. We have, a 1 b 1 c 5 25 ...(1)  2, a, b are in A.P., ∴ 2a 5 2 1 b ...(2) 2 ...(3)  b, c 18 are in G.P., ∴ c 5 18b From (1) and (2), 3b 1 2c 5 48; or 3b 5 4822c. ∴ From (3), c2 5 6 (4822c) 5 288 212c or c2 1 12c 2288 5 0; or c2 1 24c 212c2288 5 0 or (c 1 24) (c212) 5 0; ∴ c 5 12, as c ≠224. ∴ From (3), b 5 8 and from (2), a 5 5. n

10. We have, tn  

1 1   n 2

n 1



n2  1  2( n  1)  n  1  ⋅     n( n  1)  2  n( n  1)  2 

1 1   n 1  2 

n

n

3.24

▌ Mathematics

1  1  1  1 1   1  1 1 1  1 2  ∴ S n  ∑ tn                 1 2 2  2   2  2  3  2   n =1       n

⇒ (1042102) R 5 caba2ca ⇒ R

n  1  1 n1 1  1   1  ...         1 n n 2  1 2 ( n  1)2 n      

11. Let d be the common difference of the A.P. Then, logy x 5 1 1 d ⇒ x 5 y1 1 d logz y 5 1 1 2d ⇒ y 5 z1 1 2d 15 and 2 logx z 5 1 1 3d ⇒ z 5 x(1 1 3d)/15 ∴ x 5 y1 1 d 5 z(1 1 2d) (1 1 d) 5 x2(1 1 d)(1 1 2d)(1 1 3d)/15 ⇒ (1 1 d)(1 1 2d)(1 1 3d) 5215 ⇒ 6d 3 1 11d 2 1 6d 1 16 5 0 ⇒ (d 1 2)(6d22d 1 8) 5 0 ⇒ d 522 ∴ x 5 y1 1 d 5 y21, y 5 z1 1 2d 5 z23 and x 5 (z23)21 5 z3. 2 5 2 11 12. Let S      ... to ∞ ...(1) 3 6 3 24 1 Multiplying both sides by 2 , the common ratio of G.P. 2 1 2 5 8  S     ... to ∞ ...( 2) 2 6 12 24 Subtracting (2) from (1), we have 3 S 2

2   3

1 2 1 1 2    .  1 3 3 3 1    2

1 2 2 ∴S    . 3 3 9

⇒

1 1 1  log 5     ... ∞   4 8 16 

1

 (51 )1/ 2

1   5

1 4  1  21. 1 2 1 2

log

5

log5 2

2 log5 2 5 log5 ( 2 )2

5

1 2  3  4  5 5  1⋅ 2 ⋅ 3 ⋅ 4 ⋅ 5 5

⇒ 3  5 a ⇒ 35  a  5! Also, 55  1 ⋅ 2 ⋅ 3 ⋅ 4 ⋅ 5  a. 16. We have, 1  n  n2  ...  n127



n128  1 ( n64  1) ( n64  1)  n 1 n 1

 (1  n  n2  ...  n63 ) ( n64  1) ∴ k 5 64 which is divisible by 8, 16, 32 and 64. 17. Let S 5 1 1 2⋅2 1 3⋅22 1 4⋅23 1 5⋅24 1 ... 1 100⋅299 ∴ 2S 5 1⋅2 1 2⋅22 1 3⋅23 1 ... 1 99⋅299 1 100⋅2100 Substracting, we get S

 1 1.2  1.22  ...  1.299  100 ⋅ 2100  (1  2  22  ...  299 )  100 ⋅ 2100 

1(2100  1)  100 ⋅ 2100  2100  1  100 ⋅ 2100 2 1

∴ S 5 100⋅210022100 1 1 5 99⋅2100 1 1. 18. Let the numbers be a2d, a, a 1 d, a 1 2d where a, d ∈ Z and d > 0 Given : (a2d)2 1 a2 1 (a 1 d)2 5 a 1 2d ⇒ 2d 222d 1 3a2 a 5 0 1 ∴ d  1  1  2 a − 6 a 2 )   2  Since d is positive integer, ∴ 1 1 2a 6a2 > 0

1 1 1 13. We have,    ... ∞  4 8 16

∴ (0.2)

15. Since A.M. > G.M.

2 3 3 3      ... to ∞ 3 6 12 24  2 1 1 1       ... to ∞  3 2 4 8 

caba  ca 1000c  100 a  10b  a  10c  a  9900 9900 999c  10 a  b  . 990

 22  4..

14. Let R 5 0 ⋅ cababab... ⇒ 102R 5 ca ⋅ bababa ... and 104R 5 caba ⋅ baba ...

2

1

 1 7   1 7   a  ⇒   6   6      Since a is an integer, ∴ a 5 0, 1 then d  [1  1]  1 or 0. Since d  0, 2 Hence, the numbers are 21, 0, 1, 2.

∴d  1.

19. Let the A.P. containing (2n 1 1) terms be a, a 1 d, a 1 2d, a 1 3d, a 1 4d, a 1 5d, ..., a 1 2nd. The sum of odd terms of this A.P. 5 a 1 (a 1 2d) 1 (a 1 4d) 1 ... to (n 1 1) terms 

n 1 {2 a  ( n  1  1) ⋅ 2 d }  ( n  1) ( a  nd ) 2

Algebra ▌ 3.25 The sum of even terms of this A.P.  ( a  d )  ( a + 3d )  ( a  5d )  ... to n terms n  {2 ( a  d )  ( n  1) ⋅ 2 d }  n ( a  nd ) 2 Hence, the required ratio 5

n 11 . n

20. Let a, ar, ar2 be any three consecutive terms, then according to the given condition, a, 2ar, ar2 are in A.P. i.e., 4ar 5 a 1 ar2 or

r 2 3

Since, given G.P. is increasing, r  2  3. Hence, there is only one such G.P.

PERMUTATION AND cOMBINATIONS 1. A gentleman invites 13 guests to a dinner and places 8 of them at one table and remaining 5 at the other, the tables being round. The number of ways he can arrange the guests is 11! 40 12! (c ) 40 (a )

( b) 9! (d )

13! 40

2. If there are 12 persons in a party, and if each of them shakes hands with each other, then number of handshakes happen in the party is (a) 66 (b) 48 (c) 72 (d) none of these 3. Three boys and three girls are to be seated around a table, in a circle. Among them, the boy X does not want any girl neighbour and the girl Y does not want any boy neighbour. The number of such arrangements possible is (a) 4 (b) 6 (c) 8 (d) none of these 4. There are 10 lamps in a hall. Each one of them can be switched on independently. The number of ways in which the hall can be illuminated is (a) 102 (b) 18 (c) 210 (d) 1023 5. A telegraph has 5 arms and each arm is capable of 4 distinct positions, including the position of rest. The total number of signals that can be made is (a) 473 (b) 1023 (c) 1173 (d) none of these 6. The number of ways in which two 10-paise, two 20-paise, three 25-paise and one 50-paise coins can be distributed among 8 children so that each child gets only one coin, is (a) 1720 (b) 1680 (c) 1570 (d) none of these 7. There are six teachers. Out of them two are primary teachers, two are middle teachers and two secondary teachers. They are to stand in a row, so as the primary teachers, middle teachers and

secondary teachers are always in a set. The number of ways in which they can do so, is (a) 34 (b) 48 (c) 52 (d) none of these 8. A ‘Letter Lock’contains 5 rings each marked with four different letters. The number of all possible unsuccessful attempts to open the lock is (a) 625 (b) 1024 (c) 624 (c) 1023 9. If each of 10 points on a straight line be joined to each of 10 points on a parallel line then the total number of triangles that can be formed with the given points as vertices, is (a) 860 (b) 900 (c) 920 (d) none of these 10. A boat is to be manned by eight men of whom 2 can only row on bow side and 1 can only row on stroke side; the number of ways in which the crew can be arranged is (a) 4360 (b) 5760 (c) 5930 (d) none of these 11. A family consists of grandfather, 5 sons and daughters and 8 grandchildren. They are to be seated in a row for dinner. The grandchildren wish to occupy the 4 seats at each end and the grandfather refuses to have a grand child on either side of him. The number of ways in which the family can be made to sit is (a) 11360 (b) 11520 (c) 21530 (d) none of these 12. The number of ways in which a mixed doubles game in tennis can be arranged from 5 married couples, if no husband and wife play in the same game, is (a) 46 (b) 54 (c) 60 (d) none of these 13. Four boys picked up 30 mangoes. The number of ways in which they can divide them if all mangoes be identical, is (a) 5456 (b) 3456 (c) 5462 (d) none of these

3.26

▌ Mathematics

14. In a class tournament where the participants were to play one game with another, two class players fell ill, having played 3 games each. If the total number of games played is 84, the number of participants at the beginning was (a) 22 (b) 15 (c) 17 (d) none of these 15. Given 5 different green dyes, 4 different blue dyes and 3 different red dyes, the number of combinations of dyes that can be chosen by taking atleast one green and one blue dye is (a) 248 (b) 120 (c) 3720 (d) 465 16. Out of 18 points in a plane no three are in the same straight line except five points which are collinear. The number of straight lines that can be formed joining them is (a) 143 (b) 144 (c) 153 (d) none of these p ∈ (0, 1) such q that p, q ∈ {1, 2, 3, 4, 5} and n 5 number of mappings from {1, 2, 3} onto {1, 2}, then m2n is

17. If m 5 number of distinct rational numbers

(a) 1 (c) 0

(b) 21 (d) none of these

18. If eight persons are to address a meeting then the number of ways in which a specified speaker is to speak before another specified speaker, is (a) 40320 (b) 2520 (c) 20160 (d) none of these 19. The number of ways of selecting 10 balls from the unlimited number of red, green, white and yellow balls, if selection must include 2 red and 3 yellow balls, is (a) 36 (b) 56 (c) 112 (d) none of these 20. If a represents the number of permutations of (x 1 2) things taken together, b represents the number of permutations of 11 things taken together out of x things, and c represents the number of permutations of (x211) things taken together so that a 5 182bc, then x 5 (a) 15 (b) 12 (c) 10 (d) 18

ANSwER KEy 1. (d) 2. (a) 3. (a)

4. (d) 5. (b) 6. (b)

7. (b) 8. (b) 9. (b)

10. (b) 11. (b) 12. (c)

13. (a) 14. (b) 15. (a)

16. (b) 17. (d) 18. (c)

19. (b) 20. (b)

SOLUTIONS 1. The number of ways in which 13 guests may be divided into 13 groups of 8 and 5 5 C5 5

13! . 5!8!

Now, corresponding to one such group, the 8 guests may be seated at one round table in (821)! i.e., 7! ways and the five guests at the other table in (521)! i.e., 4! ways. But each way of arranging the first group of 8 persons can be associated with each way of arranging the second group of 5, therefore, the two processes can be performed together in 7! × 4! ways. Hence required number of arrangements 13! 13! 13!   7 !  4!   7 !  4!  . 5!8! 5 ⋅ 4!8 ⋅ 7 ! 40 2. Total number of handshakes 5 The number of ways of selecting 2 persons from among 12 persons 12  11  12C2   66. 2 1 3. As shown in figure, 1, 2 and X are the three boys and 3, 4 and Y are three girls, Boy X will have neighbours as boys 1 and 2

and the girl Y will have neighbors as girls 3 and 4.] 1 and 2 can be arranged in P (2, 2) ways X

1

2

3

4

Y 5 2! 5 2 × 1 5 2 ways. Also, 3 and 4 can be arranged in P (2, 2) ways 5 2! 5 2 × 1 5 2 ways. Hence required no. of permutations 5 2 × 2 5 4. 4. Each bulb has two choices, either switched on or off ∴ Required number of ways 5 21021 5 102421 5 1023

Algebra ▌ 3.27 [Since in one way when all are switched off, the hall will not be illuminated]. 5. Each arm can be set in 4 ways. ∴ Five arms can be set in 4 × 4 × 4 × 4 × 4 ways. But this includes the way when all the arms are in the position of rest, when no signal is sent. Hence, required number of signals 5 4521 5 102421 5 1023. 6. Total no. of coins 5 2 1 2 1 3 1 1 5 8. 2 coins are 10 paise, 2 are 20 paise, 3 are 25 paise and 1 is of 50 paise. ∴ Required no. of ways   8  7  6  5  1680. 7. There are 2 primary teachers. They can stand in a row in P (2, 2) 5 2! 5 2 × 1 ways 5 2 ways. There are 2 middle teachers. They can stand in a row in P (2, 2) 5 2! 5 2 × 1 ways 5 2 ways. There are 2 secondary teachers. They can stand in a row in P (2, 2) 5 2! 5 2 × 1 ways 5 2 ways. These three sets can be arranged in themselves in 5 3! 5 3 × 2 × 1 5 6 ways. Hence the required number of ways 5 2 × 2 × 2 × 6 5 48. 8. Number of options on 1st Ring 5 4 Number of options on 2nd Ring 5 4 Number of options on 3rd Ring 5 4 Number of options on 4th Ring 5 4 Number of options on 5th Ring 5 4 ∴ Total number of options/arrangements 5 4 × 4 × 4 × 4 × 4 5 1024 9. A triangle is formed for each selection of 2 points from one line and 1 point from the other line. ∴ The number of triangles  10C2  10C1  10C1  10C2 10  9 10  9   10  10   900. 2 2 10. First we have to select 2 men for bow side and 3 for stroke side. ∴ The number of selections of the crew for two sides 5 5C2 × 3C3. For each selection there are 4 persons each on both sides who can be arranged in 4! × 4! ways. ∴ Required number of arrangements 5 4  5C2  3C3  4! 4! 1  24  24  5760. 2

11. The total number of seats 5 1 grand father 1 5 sons and daughters 1 8 grand children 5 14 The grand children wish to occupy the 4 seats on either side of the table 5 4! ways 5 24 ways and grand father can occupy a seat in (521) ways 5 4 ways (Since 4 gaps between 5 sons and daughters) and the remaining seats can be occupied in 5! ways 5 120 ways (5 seats for sons and daughters) Hence, required number of ways 5 24 × 4 × 120 5 11520. 12. Let the sides of the game be A and B. Given 5 married couples, i.e., 5 husbands and 5 wives. Now, 2 husbands for two sides A and B can be selected out of 5 5 5C2 5 10 ways. After choosing the two husbands their wives are to be excluded (since no husband and wife play in the same game). So we are to choose 2 wives out of remaining 5 2 2 5 3 wives, i.e., 3C2 5 3 ways. Again two wives can interchange their sides A and B in 2! 5 2 ways. Therefore, the required number of ways 5 10 × 3 × 2 5 60. 13. Clearly, 30 mangoes can be distributed among 4 boys such that each boy can receive any number of mangoes. Hence total number of ways5 30 1 421C421 5 33C3 33.32.31 5 5456. 1.2.3

5

14. Suppose the two players did not play at all so that the remaining (n 2 2) players played n22C2 matches. Since these two players played 3 matches each, hence the total number of matches is C2 1 3 1 3 5 84 (given)

n21

or

( n  2)( n  3)  78 or n2  5n  6  156 1.2

or n225n2150

5 0 or (n215) (n 1 10) 5 0

∴ n 5 15 (n ≠210). 15. Number of ways of selecting at least one green dye out of 5 different green dyes 5

C1  5C2  5C3  5C4  5C5  25  1

Also, at least one blue dye can be selected out of 4 blue dyes in 4

C1  4C2  4C3  4C4  24  1

Again, 3 different red dyes can be selected in 3

C0  3C1  3C2  3C3  23 ways

∴ Required ways 5 (25 21) (24 21) (23) 5 3720

3.28

▌ Mathematics

16. The number of st. lines

(1 1 x 1 x2 1...) (x3 1 x4 1...)

5 18C22 (5C221) 5 144.

5 coefficient of x5 in (1 1 x 1 x2 1...)4 5 coefficient of x5 in (1 x)24

17. Now, n 5 2322 5 6  2 1 Also, m  4  3  2  1  1  9  as    4 2 ∴ m2n 5 3 18. Let A, B be the corresponding speakers. Without any restriction the eight persons can be arranged among themselves in 8! ways, but the number of ways in which A speaks before B and the number of ways in which B speaks before A make up 8!. Also number of ways in which A speaks before B is exactly same as the number of ways in which B speaks before A. ∴ 1 the required number of ways 5 (8!) 5 20160. 2 19. Number of 10 balls selections 5 coefficient of x10 in (x2 1 x3 1 ...) (1 1 x 1 x2 1...)

5 (4 1 521)C5 5 8C5 5 56 ways. 20. a  x 2 Px  2 , b  x P11 ,C  x 11 Px 11 ⇒ a  ( x  2)!, b 

x! , c  ( x  11)! ( x  11)!

Now, a 5 182 bc ⇒ ( x  2)! 182

x! ( x  11) ( x  11)!

⇒ ( x  2) ( x  1) x ! 182 x ! ⇒ ( x  2) ( x  1)  182  14  13 ⇒ x  1  13 ∴ x  12

BINOMIAL ThEOREM 1. The greatest term (numerically) in the expansion of 1 (3  5x )11 when x  is 5 (b) 46 × 39 (d) none of these

(a) 55 × 39 (c) 55 × 36

2. If (1 1 x 2 2x2)6 5 1 1 a1 x 1 a2 x2 1 ... 1 a12 x12, then a2 1 a4 1 a6 1 ... 1 a12 5 (a) 21 (b) 11 (c) 31 (d) none of these 3. If C0, C1, C2, ..., Cn are the coefficients of the expansion of (1 1 x)n, then the value of

n

Ck

∑ k 11

is

0

(a ) 0

( b)

2 n1  1 (c ) n 1

2n 1 n

(d ) none of these

∑

8. Given positive integers r > 1, n > 2 and the coefficients of (3r)th term and (r 1 2)th term in the binomial expansion of (1 1 x)2n are equal, then r 5 n n (a ) , n even ( b) 2 2 (c ) n (d ) 1 9. 79 1 97 is divisible by (a) 16 (b) 24 (c) 64 (d) 72 10. The sum of the coefficients of all the integral powers of x in the expansion of (1 1 2 x )80 is 1 80 (3  1) 2 80 (c) (3  1)

1 80 (3  1) 2 80 (d) (3  1) (b)

11. If Cr stands for nCr, then the sum of the series

( Ci  C j ) 2 

(a) (n21) ⋅ 2nCn 1 22n (c) (n 1 1) ⋅ 2nCn 1 22n

7. If A 5 2nC0 · 2nC1 1 2nC1 2n21C1 1 2nC2 2n22C1 1 ..., then A is (a) 0 (b) 2n 2n (c) n 2 (d) 1

(a)

4. If (1 1 x)n 5 C0 1 C1 x 1 C2 x2 1 ... 1 Cn xn, then 0i jn

6. In the expansion of (1 1 x 1 x3 1 x4)10, the coefficient of x4 is (a) 40C4 (b) 10C4 (c) 210 (d) 310

(b) n ⋅ 2nCn 1 22n (d) none of these

5. The number of irrational terms in the expansion of (41/5 1 71/10)45 is (a) 40 (b) 5 (c) 41 (d) none of these

n n 2   !  !  2   2  [c 2  2c 2  3c 2  0 1 2 n! n 2 ...  ( 1) ( n 1)cn ] where n is an even positive integer, is (a) 0 (b) (21)n/2 (n 1 1) (c) (21)n/2 (n 1 2) (d) (21)n n

Algebra ▌ 3.29

12. The sum

m

 10   20 

 p

∑  i   m  1 , (where  q   0 if p  q)

i 0    is maximum when m is

(a) 5 (c) 15





(b) 10 (d) 20

13. If coefficient of xn in (1 1 x)101 (12x 1 x2)100 is non-zero, then n can not be of the form (a) 3t 1 1 (b) 3t (c) 3t 1 2 (d) 4t 1 1 14. The number of integral terms in the expansion of

(2

6

51 7

(a) 105 (c) 321

)

642

is (b) 107 (d) 108

6

  1 16. If the fourth term in the expansion of  log x 1 1 1 x1/ 12  is equal x    to 200 and x > 1, then x is equal to (b) 10 (d) none of these n 3

 1  17. If there is a term containing x 2 r in  x  2  , then x   (a) n22r is a positive integral multiple of 3 (b) n22r is even (c) n22r is odd (d) none of these 18. Coefficient of t24 in (1 1 t2)12 (1 1 t12) (1 1 t24) is (a) 12C6 1 3 (b) 12C6 1 1 (c) 12 C6 (d) 12C6 1 2 19. Let n(> 1) be a positive integer. Then largest integer m such that (nm 1 1) divides 1 1 n 1 n2 1 ... 1 n255 is (a) 128 (b) 63 (c) 64 (d) 32 3 4 5 20. The value of 2( nC0 )  ( nC1 )  ( nC2 )  ( nC3 ) ... is 2 3 4 2 n (1  n)  1 2 n ( n  3)  1 (a ) ( b) n 1 n 1 2n  1 (c ) n 1

2n 1 (d ) n 1

22. If n is a rational number, which is not a whole number, then the number of terms in the expansion of (1 1 x)n where |x| < 1 is (a) n (b) n 1 1 (c) infinitely many (d) None of these 10

 1 23. The middle term in the expansion of  x 2  is x  C5

(a )

10

(c )

210

C5

C6

( b)

10

(d )

210

C6 5

15. The coefficient of xn in the polynomial (x 1 nC0) (x 1 3 nC1) (x 1 5 nC2)... (x 1 (2n 1 1) nCn) is (a) n.2n (b) n.2n11 n (c) (n 11).2 (d) n.2n21

(a) 10 2 (c) 104

21. In n is a positive integer, then the number of terms the binomial expansion of (x 1 a)n is (a) n (b) n 1 1 (c) n21 (d) None of these

 c 24. The coefficient of y in the expansion of  y 2 1  y  (a) 20 (c) 10c3

(b) 10c (d) 20c3

25. The coefficient np and nq (p,q are positive integers) in the binomial expansion of (1 1 x)p 1 q (a) equal (b) equal numerically (c) reciprocal of each other (d) unequal 5

 1 26. The 4th term in the expansion of  x 2   , ( x  0) is x  (a) (b) (c) (d)

independent of x positive negative the only middle term

27. The number of terms in the expansion of [(x 1 4y)3 (x24y)3]2 is (a) 6 (b) 7 (c) 8 (d) 32 28. The coefficient of x17 in the expansion of (x21) (x22) (x23) ….(x218) is (a) 342 (b) 2171 (c)

171 2

(d) 684

29. For a positive integer n nc1 1 nc2 1 nc3 1 ... 1 ncn is equal to (a) 2n (c) n2

(b) 2n 21 (d) n221

30. The value of nc0 2 nc1 1 nc2 2 nc3 1... 1(21) ncn (a) 1 (b) n (c) 2n (d) 0

3.30

▌ Mathematics

ANSwER KEy 1. 2. 3. 4.

5. 6. 7. 8.

(a) (c) (c) (a)

9. 10. 11. 12.

(c) (d) (c) (a)

13. 14. 15. 16.

(c) (a) (c) (c)

(c) (d) (c) (b)

17. 18. 19. 20.

21. 22. 23. 24.

(a) (d) (a) (b)

25. 26. 27. 28.

(b) (c) (c) (c)

(a) (c) (b) (b)

29. (b) 30. (d)

SOLUTIONS 4.

1. We have, 11

11

  5x  1  1 (3  5x )11  311  1    311  1    x   3 3 5      ∴ m



x ( n  1) ( x  1)  1   3  (11  1)  1    3  1

∑ ∑ (C  C ) i  0, 1, 2, ..., ( n  1) 2

0i jn

i

j

j 5 1, 2, 3, ..., n and i < j 5 n –C 02 + C 12 + .... C n2 )] 1 2 Σ Σ Ci Cj 0 ≤ i < j ≤ n 5 n ⋅ 2nCn 1 [(C0 1 C1 1 ... 1 Cn)2 –C 02 + C 12 + ....+ C n2 )] 5 n ⋅ 2nCn 1 (2n)222nCn 5 (n21) ⋅ 2nCn 1 22n. 5. Total number of terms in the expansion of (41/5 1 71/10)45 is 45 1 1 i.e., 46. The general term in the expansion is

3

The greatest terms in the expansion are T3 and T4 ∴ Greatest term (when r 5 2) 5 311 | T2 1 1 | 5 311 5 311 5 55 × 39 2

 1 11.10 1  311 11C2     311   55  39 1.2 9  3 and greatest term (when r 5 3) 5 311 | T3 1 1 | 3

 1 11.10 ⋅ 9 1  311 11C2     311  55  39 3 1 . 2 ⋅ 30 27   From above we see that the values of both greatest terms are equal. 2. Given (1 1 x22x2)6 5 1 1 a1x 1 a2x2 1 ... 1 a12x12 Putting x 5 1, we get 0 5 1 1 a1 1 a2 1 ... 1 a12

...(1)

Putting x 521, we get 64 5 12a1 1 a22 ... 1 a12

...(2)

Tr 1  45Cr ⋅ 4

45 r 5

r

7 10 .

Tr 1 1 is rational if r 5 0, 10, 20, 30, 40. ∴ Number of rational terms 5 5. ∴ Number of irrational terms 5 46 25 5 41. 6. (1 1 x 1 x3 1 x4)10 5 [(1 1 x) (1 1 x3)]10 5 (1 1 x)10 (1 1 x3)10 5 (1 1 10C1 x 1 10C2 x2 1 10C3 x3 1 10C4 x4 ...) × (1 1 10C1 x3 1 10C2 x6 ...) ∴ Coefficient of x4 5 (10C1) (10C1) 1 10C4  100 

10 ⋅ 9 ⋅ 8 ⋅ 7  100  210  310. 1⋅ 2 ⋅ 3 ⋅ 4

7. A 5 coeff. of x in [2nC0(1 1 x)2n 2n C1 (1  x )2n1  ...)]  coeff. of x in (1  (1  x ))2n

Adding (1) and (2), we get

 coeff. of x in (2  x )2n

64 5 2 (1 1 a2 1 a4 1 ...) ∴ a2 1 a4 1 a6 1 ... 1 a12 5 31.

 x  coeff. of x in 22n  1    n ⋅22 n. 2 

3. Here, tr 1 1 5 5 ⋅ nCr 5 ⋅ n 1 1Cr 1 1 Putting r 5 0, 1, 2, ... n and adding we get, 5 {n 1 1C1 1 n 1 1C2 1 n 1 1C3 1 ... 1 n 1 1Cn 1 1} 5 {2n 1 1 2 n 1 1C0} 5 .

2n

8. We have, t3r 5 2nC3r21 x3r 21 and tr 1 2 5 2nCr 1 1 xr 1 1. Given, 2nC3r21 5 2nCr 1 1 ⇒ 3r21 5 r 1 1; or (3r21) 1 (r 1 1) 5 2n ⇒ 2r 5 2 ; or 4r 5 2n

Algebra ▌ 3.31

n r 5 1 (impossible); or r 5 . 2 But r is a positive integer greater than 1. So the value of r is provided n is an even integer (> 2), otherwise r has no value. ⇒

14. (2.51/2 1 71/6)642 has a general term of the form 642 Cr (2.51/2)6422r (71/6)r 5 642Cr 26422r. 53212r/2. 7r/6 and will be rational if only r is a multiple of 2 and 6. ∴ r must be the LCM of 2 and 6 which is 6. ∴ r takes the values 0, 6, 12, 18,..., 642. There are 108 values.

9. We have, 79 1 97 5 (1 1 8)72(128)9 5 (1 1 7C1 ⋅ 81 1 7C2 ⋅ 82 1 ... 1 7C7 ⋅ 87) 2(129C1 ⋅ 81 1 9C2 ⋅ 822... 9C9 ⋅ 89) 5 16 × 8 1 64 [(7C2 1 ... 1 7C7 ⋅ 85) (9C22...29C9 ⋅ 87)] 5 64 (an integer) Hence 79 1 97 is divisible by 64. 10. The coefficients of the integral powers of x are 80 C0, 80C2 ⋅22, 80C4 ⋅24, ..., 80C80 ⋅280 Now, (1 1 2)80 5 80C0 1 80C1 ⋅2 1 80C2 ⋅22 1 ... 1 80C80 ⋅280 (1) and (122)80 5 80C0 2 80C12 1 80C2 ⋅22 2... 1 80C80 ⋅280 (2) Adding (1) and (2), we get 380 1 1 5 2(80C0 1 80C2 ⋅22 1 80C4 ⋅24 1 ... 1 80C80 ⋅280) ∴

80

 ( 1) n/ 2

 ( 1) n/ 2

n!  (1) n n  ! ! 2 2

n 1 2

n Cn 2 2

n

 n! n 1  n n  2  ! ! 2 2



n

r 0

r 0

n

 2∑ r r 0

n r

n 1

Cr − 1  2 n

n

 2 n∑ n1 Cr − 1  2 n r 0

 2 n.2 n1  2 n  ( n 1) 2 n 16. Given, T4 5 200 6

 1 C3  log x 1  x 10  3 1  2 (log10 x 1) 4

3

 1/12 3  (x )  200    

3

 200 ⇒ x  2 (log x 1)

1    4 

 10

1 1   log x 10  ⇒ log10 x 20(log10 x  1) 4 ⇒

5 Coefficient of x in (1 1 x) (1 1 x) 5 Cm m 5 15 (for maximum value) 10

20

1 1 3   where y  log10 x 2( y  1) 4 y

⇒ y 524 or y 5 1 ⇒ log10 x 524 or log10 x 5 1 ⇒ x 5 1024 or 10 ⇒ x 5 10 ( x > 1)

 m

n

Cr  ∑ 2 r n Cr  ∑ n Cr

3

 10   20  

r 0

⇒ 20.x

∑  i   m  1 i 0

n

∑ (2r  1)

⇒

n

n n 2  ! ! 2 2 ∴     [C02  2C12 + 3C22  ... n! (11) n ( n  1) Cn2 ]  (1) n/ 2 ( n  2) m

15. Given polynomial is (x 1 nC0) (x 1 3. nC1) (x 1 5. nC2) ...(x 1 (2n 1 1). nCn) 5 xn11 1 xn {nC0 1 3. nC1 1 5. nC2 1 ... 1 (2n 1 1). nCn} 1 xn21 (...) 1 ... ∴ Coefficient of xn in the expression is

1 C0  80C2 ⋅ 22  80C4 ⋅ 24  ...  80C80 ⋅ 280  (380  1) 2

11. C 02 2 2C 12 1 3C 22 2 4C 32 + .... 1 (21)n (n 1 1)C n2 1C 02 2 C 12 1 C 22 2 C 32 + .... 1 (21)n C n2 = 2 [C 12 2 2C 12 2 3C 32 2 .... 1 (21)n nC n2 ]

12.

Clearly in this expression x λ will be present if λ 5 3t, or λ 5 3t 1 1 So, λ can not be of the form 3t 1 2.

30

13. (1 1 x)101 (12 x 1 x2)100 5 (1 1 x) (1 1 x3)100 5 (1 1 x) (C0 1 C1x3 1 C2x6 1 ... 1 C100x300)

 1  17. General term in  x  2  x  

n 3

 1 k tk 1  n  3 Ck x n  3  k  2  x  ⇒ tk 1  n  3 Ck x n  3( k1)

is

▌ Mathematics

3.32

There is a term containing x2r, if n 2 3(k11) 5 2r ⇒ n 2 2r 5 3(k 1 1), k ∈ N ∴ n22r is a positive integral multiple of 3 18. Coefficient of t24 in (1 1 t2)12 (1 1 t12) (1 1 t24) 5 coefficient of t24 in (1 1 12C6 t12 1 12C12 t24) (1 1 t12 1 t24) 5 coefficient of t24 in (12C6 1 2)t24 5 12C6 1 2 19. We have, S 5 1 1 n 1 n2 1 ... 1 n255

5

 1  T6  10 c5 x 5    10 c5 1  x 2

23.  T10

r

c 24. Tr 1  5cr ( y 2 )5  r   = 5cr y103r c r ;  y ∴ 10 2 3r = 1 ⇒ 3r – 9 ⇒ r = 3; and ∴ Coefficient of 5C3c3 = 10c3 25. Tr + 1 = p + q crxr Tp + 1 = p + q cpx p ;

1( n256  1) n128  1  ( n128  1) n 1 n 1 ∴ S 5 (n128 1 1) (1 1 n 1 n2 1 ... 1 n127) Thus, the largest value of m for which 1 1 n 1 n2 1 ... 1 n255 is divisible by nm 1 1 is 128. ⇒S

Now, coefficient of x p  p qC p  Tq + 1 = p + q cqx q Coefficient of x p  p qCq 

20. (1 1 x)n 5 nC0 1 nC1x 1 nC2x2 1 nC3x3 ... nCnxn On integrating between the limits 0 and, we get n C x2 n C x3 n C x4 (1  x ) n1  1 n  C0 x  1  2  3 ... 3 4 n 1 2

Multiplying with x and differentiating, we get  ⇒

n C1x 3 n C2 x 4 n C3 x 5  d  n ...    C0  2 3 5 dx  

(1  x ) n1  x ( n 1) (1  x ) n  1 n 1

3n C1x 2 4 n C2 x 3  2 C0 x   3 2 n

pq q⋅ p

pq q⋅ p

∴ coefficient of xp 5 coefficient of xq 3  3  26. T = T  5c (x )53  1  5c (x )2  1 4 31 3 3  x   x  10   0 [∴x  0], so it is negative. x 3 3 2 2 2 6 27. [( x  4 y ) ( x  4 y ) ]  ( x  16 y ) ∴ Number of terms

5 6 1 1 5 7. 28. Coefficient of x17  (1  2  3  ...  18) 

put x  1, we get 2 n1  ( n 1) 2 n  1 3 4  2 n C0  nC1  nC2  ... 2 3 n 1  2 n ( n  3)  1 n 1

29.

18 (1  18)  9(19) 171. 2

c 0 1 n c 1 1 n c 3 1 .... 1 n c n 5 2 n (Sum of binomial coefficients)

n

⇒ nc1 1 nc2 1 nc3 1 .... 1 ncn 5 2n 2 nc0 5 2n 2 1

21. Number of terms 5 One more than the power of the binomial. 22. As n is a rational number but not a whole number.

30. ⇒ nc1 1 nc3 1 nc5 1 .... 5 nc0 1 nc2 1 nc4 1 ... ⇒ nc0 2 nc1 1 nc2 2 nc3 1 nc4 2 nc5 1.... 5 0

cOMPLEx NUMBER 1. The centre of a regular polygon of n sides is located at the point z 5 0, and one of its vertex z1 is known. If z2 be the vertex adjacent to z1, then z2 is equal to  2π 2π  (a ) z1  cos  i sin  n n  

 π π ( b) z1  cos  i sin  n n 

 π π  (c ) z1  cos  i sin  2n 2n  

(d ) none of these

2. If 1, ω, ω2 be the three cube roots of unity, then (1 1 ω) (1 1 ω2) (1 1 ω4) (1 1 ω8) ... to 2n factors 5 (a) 1 (b) 21 (c) 0 (d) none of these 6

6

 3i   i 3  3.       2   2     

Algebra ▌ 3.33 (a) 22 (c) 21

(b) 2 (d) 1

13. ( i  3 )100  ( i  3 )100  2100 

4. The complex number z satisfying the equations | z2i | 5 | z 1 1 | 5 1 is (a) 0 (b) 1 1 i (c) 1 1 i (d) 1 2i 5. The equation | z21 |2 1 | z 1 1 |2 5 4 represents on the Argand plane (a) a straight line (b) an ellipse (c) a circle with centre origin and radius 2 (d) a circle with centre origin and radius unity 6. If z 5 cos θ 1 i sin θ, then 1  2 cos nθ zn 1 (c ) z n  n  2i sin nθ z (a ) z n 

( b) z n 

1  2 n cos nθ zn

(d )  ( 2i ) n sin nθ

7. The locus of the point z satisfying the condition arg z 1 π  5 is z 1 3 (a) a straight line (c) a parabola 8. If

(b) circle (d) none of these

z 1 is purely imaginary, then z 1

(a) | z | > 1 (c) | z | 5 1

(b) | z | < 1 (d) none of these

9. If i z 1 z z 1 i 5 0, then (a) | z | < 1 (c) | z | 5 1 3

2

(b) | z | > 1 (d) | z | 5 0

10. arg bi (b > 0) is (a) π

(b)

(c) 2

π 2

π 2

(d) 0 n

 z 1  11. If ω    , integral, then ω lies on the unit circle for  1  iz  (a) only even n (c) only positive n

(b) only odd n (d) all n

12. The maximum value of | z | when z satisfies the condition z

2  2 is z

(a) 1 (c) 0

(b) 21 (d) none of these

14. If a, b, c, p, q, r are three complex numbers such that p q r a b c    1  i and    0, then the value of a b c p q r p2 q2 r 2 1 1 is a 2 b2 c 2 (b) i (d) None of these.

(a) 2i (c) 22i

15. The region in the Argand diagram difined by z 2 31z13,6 is the interior of the ellipse is in major axis along (a) real axis (b) imaginary axis (c) y 5 x (d) y = 2x 16. If z  z  z  z 8, then z lies on (a) a circle (c) a square

(b) a straight line (d) none of these

17. The cube roots of unity (a) lie on the circle | z | 5 1 (b) are collinear (c) form an equilateral triangle (d) none of these 18. If arg (z) < 0, then arg (2z)2arg (z) 5 (a ) π (c ) 2

( b) 2 π

π 2

(d )

π 2

19. The number of solutions of the equation z2 1 | z |2 5 0, where z ∈ C is (a) one (b) two (c) three (d) infinitely many  a  ib  20. tan i log  is equal to a  ib   2 ab 2 a  b2 2 ab (c) 2 a  b2 (a)

(b)

a 2  b2 2 ab

(d) ab

21. The real value of α for which the expression is purely real is

(a)

3 1

(b)

3 1

(a) (2 n 1 1)

(c)

3

(d)

2 3

(c) nπ

π 2

1  i sin α 1  2i sin α

π 2 (d) none of these

(b) ( n 1 1)

▌ Mathematics

3.34

4

2z z  z2 is equal to 22. If 1 is purely imaginary number, then 1 z1  z2 3z2

3 2 2 (c ) 3 (a )

( b) 1 (d )

4 9

23. If x6 5 (423i)5, then the product of all of its roots is (where θ 52tan21 (3/4)) (a) 55 (cos 5θ 1 i sin 5θ (b) 255 (cos 5θ 1 i sin 5θ) (c) 55 (cos 5θ2i sin 5θ) (d) 255 (cos 5θ2i sin 5θ) 24. If (1 1 i) (1 1 2i) (1 1 3i) ... (1 1 ni) 5 α 1 iβ then 2 ⋅ 5 ⋅ 10 ... (1 1 n2) 5 (a) α2iβ (b) α22β2 (c) α2 1 β2 (d) none of these 25. If

3

a  ib  x  iy , then

(a) 4 (x2 1 y2) (c) 2 (x22y2)

a b   x y (b) 4 (x22y2) (d) none of these

26. If (1 1 x 1 x2)n 5 a0 1 a1 x 1 a2 x2 1 ... 1 a2n x2n, then a0 1 a3 1 a6 1 ... 5 (a) 3n (b) 3n21 (c) 3n22 (d) none of these 27. The roots of the equation z4 1 1 5 0 are (a ) ( 1  i ) ( b) ( 2  2i ) (c )

1 2

( 1  i )

(d ) none of these

28. The locus of the complex number z in an argand plane satisfying the equation

π Arg (z + i )  Arg (z  i )   is 2 (a) boundary of a circle (b) interior of a circle (c) exterior of a circle (d) none of these

z  2 z2 29. z1 and z2 are two complex numbers such that 1 is 2  z1z2 unimodular whereas z2 is not unimodular. Then | z1 | 5 (a) 1 (b) 2 (c) 3 (d) 4 30. If all the roots of z3 1 az2 1 bz 1 c 5 0 are of unit modulus, then (a) | a | ≤ 3 (b) | b | > 3 (c) | c | ≤ 3 (d) none of these 31. If z1 ≠  z2 and z1  z2 

1 1  , then z1 z2

(a) (b) (c) (d)

at least one of z1, z2 is unimodular z1 ⋅ z2 is unimodular both z1, z2 are unimodular none of these

32. If z1, z2, z3, z4 are the four complex numbers represented by the vertices of a quadrilateral taken in order such that z  z1 π   then the quadrilateral z12 z4 5 z22 z3 and amp 4 z2  z1 2 is a (a) square (c) rectangle

(b) rhombus (d) a cyclic quadrilateral

2π , 33. Let z be a complex number with modulus 2 and argument 3 then z is equal to (a) 1  i 3

(b) 1  i 3

1 i 3 (c)   2 2

(d) none of these

34. If z1 and z2 are complex numbers, such that z1 1 z2 is a real number, then (a) z1  z2 (b) z2 5 z1 (c) z1 and z2 are any two complex numbers (d) z1 5 z1 , z2 5 z2 35. Let the affix of 22 4i be P. Then OP is rotated about O through an angle of 180º and is stretched 5/2 times. The complex number corresponding to the new position of P is (a) 5210i (b) 5 1 10i (c) 25 1 10i (d) none of these 36. If a, b, c, p, q, r are three non-zero complex numbers such p q r a b c that     1  i and   0, then value of q b c p q r p2 q2 r 2 1 1 is a 2 b2 c 2 (a) 0 (c) 2i

(b) 21 (d) 22i

37. If z1, z2 are two complex numbers such that

z1  z2  1 and z1  z2

tz1 5 kz2 where k ∈ R, then the angle between (z12z2) and (z1 1 z2) is  2k   (a) tan1  2  k  1   1 (c)  2 tan ( k )

 2k   (b) tan1   1  k 2    1 (d) 2 tan ( k )

38. If the complex numbers z1, z2, z3 are in AP, then they lie on a (a) circle (b) parabola (c) line (d) ellipse

Algebra ▌ 3.35 39. Common roots of the equations z 3 1 2z2 1 2z 1 1 5 0 and z1985 1 z100 1 1 5 0 are (a) ω, ω2 (b) 1, ω, ω2 (c) 21, ω, ω2 (d) 2ω,2ω2

40. The solutions of the equation

)

(

z z  2i  2 ( 2  i) are (a) 3 1 i, 32i (c) 1 1 3i, 12i

(b) 1 1 3i, 123i (d) 123i, 1 1 i

ANSwER KEy 1. 2. 3. 4. 5.

6. 7. 8. 9. 10.

(a) (a) (a) (a, c) (d)

(a, c) (b) (c) (c) (b)

11. 12. 13. 14. 15.

(d) (b) (c) (a) (a)

16. 17. 18. 19. 20.

(c) (a, c) (a) (d) (c)

21. 22. 23. 24. 25.

26. 27. 28. 29. 30.

(c) (b) (b) (c) (b)

31. 32. 33. 34. 35.

(b) (c) (a) (b) (a)

36. 37. 38. 39. 40.

(b) (c, d) (a) (b) (c)

(c) (c) (a, c) (a) (c)

SOLUTIONS 1. Let A be the vertex with affix z1. There are two possibilities and 2π either in clockwise can be obtained by rotating z1 through n or in anti-clockwise direction. 12 π  n

z2  zi e

and 6

{ z2  z1 } O

6

 3  i   i 3      (iω)6  (iω2 )6 Hence,   2   2       i 6 (ω6  ω12 ) 1(1  1) 2. 4.

Let z 5 x 1 iy. Then, ( x  iy )  i  ( x  iy )  1  1 or x 2  ( y  1)2  ( x  1)  y 2  1 ∴ x2 1 y2 2 2y 1 1 5 x2 1 y2 1 2x 1 1

C(z2)

A(z1)

B(z2)

2. We have, (1 1 ω ) (1 1 ω 2 ) (1 1 ω 4 ) (1 1 ω 8 ) ... to 2n factors 5 (1 1 ω ) (1 1 ω 2 ) (1 1 ω 3 ⋅ ω ) (1 1 ω 6 ⋅ ω 2 ) ... to 2n factors 5 (1 1 ω) (1 1 ω2) (1 1 ω) (1 1 ω2) ... to 2n factors [ ω3 5 ω6 5 ... 5 1] 5 [(1 1 ω) (1 1 ω) ... to n factors] [(1 1 ω2) (1 1 ω2) ... to n factors] 5 (1 1 ω)n (1 1 ω2)n 5 [(1 1 ω) (1 1 ω2)]n 5 (1 1 ω 1 ω2 1 ω3)n 5 (0 1 1)n 5 1 [ 1 1 ω 1 ω2 5 0, ω3 5 1]. 3. We have,

 1  3i  3  i i 3  i2   iω  i    2 2i 2  

i.e., x 5 2 y

...(1)

and x 1 y 22y 1 1 5 1

...(2)

2

2

From (1) and (2), x2 1 x2 1 2x 5 0; or x (x 1 1) 5 0 ∴ x 5 0,21; ∴ y 5 0, 1 ∴ z 5 x 1 iy 5 0,21 1 i. 5. We have, | z21 |2 1 | z 1 1 |2 5 4

...(1)

⇒ (x21) 1 y 1 (x 1 1) 1 y 5 4 [Putting z 5 x 1 iy] 2

2

2

2

⇒ 2 (x2 1 y2 1 1) 5 4, ∴ x2 1 y2 5 1 or | z |2 5 1 ⇒ |z|51

(since | z | cannot be2ve)

Thus, the equation (1) represents all points z on the circle with centre origin and radius unity.

▌ Mathematics

3.36

6. We have,

n

1 1   cos θ  i sin θ . z cosθ  i sin θ zn

∴

 (cos θ  i sin θ ) n  cos nθ  i sin nθ ,

1  (cosθ  i sin θ ) n  cos nθ  i sin nθ zn 1 1 Hence z n  n  2 cos nθ and z n  n  2i sin nθ. z z

and

7. We have, arg 

z 1 π  z 1 3

x  iy  1 π  [ Putting z  x  iy ] x  iy  1 3 y y π  tan1  ⇒ tan1 x 1 x 1 3 ⇒ arg 

  z ⇒  Arg 1  Argz1  Argz2  z2   ⇒ tan1

y y  x 1 x 1 π  3 y2 1 2 x 1

2y π ⇒ 2  tan  3 3 x  y2 1 ⇒ x2  y2 

8. Let

2 3

11.

 z 1   z 1  We have, ω       1  iz   i( z  i )  n

1     (i ) n i ∴ | ω | 5 | (2i)n | 5 | 2i |n 5 1 for all n. ∴ ω lies on unit circle for all n. 2 2 2 2 12. We have, |z|  z    z   . z |z| z z ⇒ |z|  2 

⇒  3  | z | 1  3 ⇒ 1  3  | z |  1  3 That is, the maximum value of |z| is 1 1 3. 13. We have, i  3  and i  3 

⇒ z

1  3i 2 2ω .  i i 2

1  3i 2 2ω 2 .  2 i i

∴ ( i  3 )100  ( i  3 )100  2100 100

 2ω     i 

100

 2ω2      i 

 2100

2100 100 (ω  ω200 )  2100 i100  21000 (ω  ω2 )  2100 

z 1  iy , where y is real z 1

z 1 1  z  1 iy 2 z 1  iy ⇒   2 1  iy

2 ⇒ | z |2  2 z  2 z

⇒ |z|2  2 |z| 1  1  2 ⇒ (|z|  1)2  3

y  1  0, which is a circle.

2100  2100  0. 14. We have,

⇒

n

p q r   1 i a b c 2

[by componendo and dividendo]

1 y2 1  iy ⇒ | z |  1. 1  iy 1 y2

9. Given, i z3 1 z22z 1 i 5 0 ⇒ i z2 (z2i)2(z2i) 5 0 1 ⇒ ( z  i ) ( i z 2  1)  0 ⇒ z  i or z 2  i. i Now z 5 i ⇒ | z | 5 | i | 5 1 and z2 5 2i ⇒ | z2 | 5 | 2i | ⇒ | z |2 5 1 ⇒ | z | 5 1 Thus, in both cases | z | 5 1. 10. Since b > 0, bi represents a point on the positive side of the π imaginary axis on which the argument of every point is . 2

⇒ ⇒ ⇒ ⇒

p q r      1  1  2i  2i  a b c  qr rp pq  p2 q2 r 2   2      2i a 2 b2 c 2  bc ca ab  p 2 q 2 r 2 2 pqr  a b c          2i abc  p q r  a 2 b2 c 2 p2 q2 r 2    2i a 2 b2 c 2

15. The equation | z2(3 1 0i) | 1 | z 2 (23 1 0i) | < 6 represents the interior of ellipse with foci at (3, 0) and (23, 0). So, major axis is along real axis. 16. We have, |z  z |  |z  z | 8 ⇒ 2 | x | 1 2 | y | 5 8 or | x | 1 | y | 5 4

Algebra ▌ 3.37 17. Clearly, cube roots of unity 1, ω, ω 2 satisfy | z | 5 1.

 a  ib  ∴ tan i log   tan 2θ a  ib  

2

3  3  3 Also, |1  ω|2       2   2  2

b 2 2 tanθ a   1  tan 2 θ b2 1 2 a

|1  ω|  3

⇒

|ω  ω2 |  3 i|  3 and

 1 i 3  |1  ω2 |  1      2 2   

3 i 3   3 2 2

Therefore, 1, ω, ω2 form an equilateral triangle.



21. We have, 1  i sin α (1  i sin α )(1  2i sin α )  1  2i sin α (1  2i sin α )(1  2i sin α )

18. As2θ 5 arg (z) < 0, we take z 5 r [cos (2θ) 1 i sin (2θ)] 5 r (cos θ2i sinθ)



1  3i sin α  2 sin 2 α 1  4i 2 sin 2 α



(1  2 sin 2 α )  3i sin α , 1  42 sin α

which is purely real if sin α 5 0 i.e., if α 5 nπ, where n is an integer.

(Z)

r

22.

πθ O

2 z1 is purely an imaginary number, 3z2 ∴ let 5 ib or ib

θ r

∴ let (Z)

⇒2z 5 r (2cos θ 1 i sinθ) 5 r [cos (π2θ) 1 i sin (π2θ)] ∴ arg (2z) 5 π2θ Thus, arg (2z)2arg (z) 5 π2θ 1 (θ) 5 π. 19. Let z 5 x 1 iy, then z2 1 | z |2 5 0 ⇒ (x 1 iy)2 1 | x 1 iy |2 5 0 ⇒ x22y2 1 2ixy 1 x2 1 y2 5 0 ⇒ 2x2 1 2ixy 5 0 ⇒ 2x2 5 0 and 2xy 5 0 ⇒ x 5 0 and xy 5 0 Clearly, y can be any real number. Hence, we will get infinitely many solutions. 20. Let a 5 r cos θ and b 5 r sin θ b ∴ tan θ 5 . a a  ib r (cos θ  i sin θ ) Now   (cos θ  i sin θ )2 a  ib r (cos θ  i sin θ )  cos 2θ  i sin 2θ  e2 i θ ∴i log

2ab . a 2  b2

a  ib  i log e2 i θ  i(2 iθ )  2θ a  ib

...(1)

2 z1 z 3  ib or 1  ib....(1) 3z2 z2 2 4

4 z1 3 1 4 ib 1  z  z2 z  2 ∴ 1  2  z1 3 z1  z2 ib  1 1 2 z2

3 ib  1 2 3 ib  1 2

4

4

4

 9 b2   1   4   1  4 2   9b  1     4 23. x6 5 (423i)5 x6

 4 3i   56     55 (cos θ  i sin θ )5 5 5 

 3 where θ  tan1    55 (cos 5θ  i sin θ )  4 5/6 x  5 (cos 5θ  i sin 5θ )1/ 6   2 kπ  5θ   2 kπ  5θ   55/6 cos    i sin   6 6      x1x2 ... x6 5 55 [cos (5π 1 5θ) 1 i sin (5π 1 5θ)] 5 55 [2cos 5θ2i sin 5θ] 5255 [cos 5θ 1 i sin 5θ]

3.38

▌ Mathematics

24. We have, | 1 1 i |2 ⋅ | 1 1 2i |2 ⋅ | 1 1 3i |2 ... | 1 1 ni |2 5 | α 1 iβ |2 ⇒ (1 1 1) ⋅ (1 1 4) ⋅ (1 1 9) ... (1 1 n2) 5 α2 1 β2 ⇒ 2 ⋅ 5 ⋅ 10 ... (1 1 n2) 5 α2 1 β2 25. We have, 3 a  ib  x  iy a 1 ib 5 (x 1 iy)3 5 (x3 23xy2) 1 i (3x2y2y3) ∴ a 5 x323xy2 and b 5 3x2y2y3. a b ∴  x 2  3y 2 and  3x 2  y 2 x y a b   x 2  3y 2  3x 2  y 2  4( x 2  y 2 ). x y

∴

26. Putting x 5 1, ω, ω2 respectively, 3n 5 a0 1 a1 1 a2 1 ... 1 a2n (1 1 ω 1 ω2)n 5 a0 1 a1 ω 1 a2 ω2 1 ... 1 a2n ω2n (1 1 ω2 1 ω4)n 5 a0 1 a1 ω2 1 a2 ω4 1 ... 1 a2n ω4n Adding these, 3n 1 (1 1 ω 1 ω2)n 1 (1 1 ω2 1 ω4)n 5 3a0 1 a1 (1 1 ω 1 ω2) 1 a2 (1 1 ω2 1 ω4) 1 a3 (1 1 ω3 1 ω6) 1 ... ∴ 3n 1 0n 1 0n 5 3a0 1 3a3 1 3a6 1 ... ∴ 3n21 5 a0 1 a3 1 a6 1 .... 27. We have, z4 1 1 5 0 ⇒ z4 521 ⇒ z 5 (cos π 1 i sin π)1/4 1 1 ⇒ z  cos ( 2 kπ  π )  i sin ( 2 kπ + π ), k  0, 1, 2, 3. 4 4 3π 3π π π  i sin , cos  i sin , 4 4 4 4 5π 5π π 7π 7π cos  i sin , cos  i sin 4 4 4 4

⇒ z  cos



1 2

(1  i),

1 2

, (1  i)

1 2

, ( 1  i )

Hence, the four roots of z4 1 1 5 0 are 28. We have, Arg ( z  i)  Arg (z  i ) 

1 2 1 2

, (1  i ). (± 1 ± i ).

π 2

 z i   z i      z  i   z 1   0 ⇒  z  i    z  i   0     2  z 1   z  i 

⇒ (z 1 i) ( 1 i) 1 (z2i) (2i) 5 0 ⇒ z 1 i (z 1)21 1 z z2i (z 1 )21 5 0 ⇒ 2 (z ) 5 2 ⇒ z 5 1 or | z |2 5 1 ⇒ | z | 5 1. The equation represents a circle centred at origin and radius 1 unit

z1  2 z2 1 2  z1 z2

 z  2 z2   z1  2 z2  ⇒  1     1  2  z1 z2   2  z1 z2  ⇒ z1z1  2 z1z2  2 z1z2  4 z2 z2  4  2 z1z2  2 z1z2  z1z2 z1z2 ⇒ ⇒ ⇒ ⇒

| z 1 | 2 1 4 | z 2 | 2 5 4 1 | z 1| 2 | z 2 | 2 | z1 |2 [12| z2 |2] 5 4 [12| z2 |2] | z1 |2 5 4 [ | z2 | ≠ 1] | z1 | 5 2

30. Let α, β, γ be the roots ∴α 1 β 1 γ 52a ∴ | a| 5 | α 1 β 1 γ | ≤ | α | 1 | β | 1 | γ | 5 1 1 1 1 1 ∴|a|≤3 31. We have, z1  z2 

z  z2 1 1   1 z1z2 z1 z2

 1  ⇒ z1  z2   1  0 z1 z2   ⇒ | z1 z2 | 5 1.  ( z1 ≠2z2) z1  z3 z2  z4   2 2 i.e., the diagonals bisect each other. ∴ It is a parallelogram.

32. We have, z1  z4  z2  z3 or

Also, amp

 z i π     ∴ Re  z  i   0 ⇒Arg      z  i 2 z  i    

⇒

29. Clearly 

z4  z1 π  z2  z1 2

⇒ angle at z1 is a right angle. ∴ It is a rectangle and hence a cyclic quadrilateral. 33. Given : |z| 5 2 and arg (z ) 5 ∴

2π . 3

If z  r (cos θ  i sin θ ), then r  2 and θ 

 1  2π  3 2π  ∴ z  2  cos  i sin   2    i 2  3 3    2 1  i ).  ( 34. Let z1 5 x1 1 iy1 ∴ z1  x1  iy1 Now z2  z1 ⇒ z1  z2  z1  z1  x1  iy1  x1  iy1  2x1 , which is real. Hence, result holds if z2 5 z1 .

2π 3

Algebra ▌ 3.39 35. If a complex number z is rotated through an angle 180º, then it’s new position is 2z. So, 2 24i is 2 1 4i and stretching it 5/2 times means modulus 5/2 times of previous complex number 5 i.e., (22 1 4i) 5 25 1 10i 2 36. We have p q r (1  i )2      q b c ⇒ 1  1  2i 

2

 qr rp pq  p2 q2 r 2    2    a 2 b2 c 2  bc ca ab 

p 2 q 2 r 2 2 abc  a b c  ⇒ 2i  2  2  2      pqr  p q r  a b c  ⇒ 

p 2 q 2 r 2 2 abc    (0) pqr a 2 b2 c 2

p2 q2 r 2    2i a 2 b2 c 2

37. Interpretating according to Coni’s Theorem. Let the angle between the lines joining z1, z2 and z1,2z2 be α ∴

z1  z2  cos α  t sin α z1  z2

Using Componendo and Dividendo, we have 2 z1 1  cos α  i sin α  2 z2 cos α  1  i sin α α  α α 2 cos 2    i 2 sin cos z1 2 2 2  ⇒  z2 α α 2α  2 sin    i sin cos 2 2 2 z α α ⇒ 1  i cot ⇒ iz1  cot z2 2 2 z2

  α  where, k  cot ( say )  2   α Now k cot 2 2k ⇒ cot k ⇒tan α  2 k 1  2k 2 k  1 ⇒ tan α  ⇒α  tan1   2 2   2 tan k 1 k  1 k  iz1  kz2

But

Now,

z1  z2  cos α  i sin α z1  z2

where α is the angle between (z12z2) and (z1 1 z2) ⇒α 5 22tan21k is the angle between (z12z2) and (z1 1 z2). 38. Let z1, z2, z3 be affixes of points A, B, C respectively. Since, z1, z2, z3 are in A.P., therefore 2z2 5 z1 1 z3 z1  z3 ⇒ B is the mid point of the line AC 2 ⇒ A, B, C are collinear ∴ z1, z2, z3 lie on a line ⇒ z2 

39. z3 1 2z2 1 2z 1 1 5 0 ⇒ (z3 1 1) 1 2z (z 1 1) 5 0 ⇒ (z 1 1) (z22z 1 1 1 2z) 5 0 ⇒ (z 1 1) (z2 1 z 1 1) 5 0 ⇒ z 521, ω, ω2 Now equation z1985 1 z100 1 1 5 0 gets satisfied by ω and ω2 but not by 21. So, common roots are ω and ω2.

(

)

40. z z  2i  zz  2iz  2( 2  i )gives x2 1 y222y 5 4 and 2x 5 2, on equating the real and imaginary parts. ∴ y222y23 5 0 giving y 5 3,21 The solutions are 1 1 3i and 1 2i.

NDA BOx 1. The function f(x) = ex, x ∈R is (a) onto but not one – one (b) one – one onto (c) one – one but not onto (d) neither one-one nor onto

3. If f (x )  kx 3  9 x 2  9 x  3 is monotonically increasing in every interval, then which one of the following is correct? (a) k > 3 (b) k ≤ 3 (c) k > 3 (d) k ≥ 3

2. if f : R → R, g : R → R and g (x )  x  3 ( fog ) (x )  ( x  3)2 , then what is the value of f ( 3)? (a) −9 (c) 9

(b) 0 (d) 3

4. The function f (x) 5 x cosec x is (a) continuous for all values of x (b) discontinuous everywhere (c) continuous for all x except at x = nx, where n is an integer (d) continuous for all x except at x = nx/2, where n is an integer

3.40

▌ Mathematics

5. Which one of the following function f : R → R is injective? (a) f ( x )  x for all x ∈ R (b) f ( x )  x 2 for all x ∈ R (c) f ( x )  11 for all x ∈ R (d) f ( x )  x for all x ∈ R 6. What is the maximum value of the function log x 2 x? (a) 21 (b) 0

13. Let A {1, 2, 5, 8}, B {0, 1, 3, 6, 7} and R be the relation ‘is one less than’ from A to B, then how many elements will R contain? (a) 2 (b) 3 (c) 5 (d) 9 14. If n(A) = 115, n(B) = 326, n(A – B) = 47, then what is n ( A ∪ B ) equal to? (a) 373 (b) 165 (c) 370 (d) 394 15. If P(A) denotes the power set of A and A is the void set, then what is number of elements in P{ P{ P{ P( A)}}}?

(c) 1 (d) ∞ 7. Let g: R→R be a function such that, g(x) = 2x + 5. Then what is g21 ( x ) equal to? x 5 2 5 (c) x  2

(b) 2 x  5

(a)

x 5  2 2

(d)

8. What is the minimum value of 2x  3x  5 2

(a) 0 (c) 31/4

(b) 3/4 (d) 31/8

1 9. A function f is defined by f(x)  x  . x Consider the following:

( ) 2. ( f (x) )  f ( x )  3 f(x)

1. ( f (x) )  f x  2 2

2

3

3

Which of the above is/are correct? (a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2 10. For a set A, consider the following statements 1. A ∪ P( A)  P( A) 2. { A} ∩ P( A)  A 3. P( A) { A}  P( A) Where P denotes power get. Which of the statement given above is/are correct? (a) 1 only (b) 2 only (c) 3 only (d) 1, 2 and 3 11. If A = {a, b, c, d}, then what is number of proper subsets of A? (a) 16 (b) 15 (c) 14 (d) 12 12. Out of 32 person, 30 invest in National Savings Certificates and 17 invest in shares. What is the number of person who invest in both? (a) 13 (b) 15 (c) 17 (d) 19

(a) 0 (c) 4

(b) 1 (d) 16

16. The probabilities of two events A and B are given as P(A) 50.8 and P(B) 5 0.7. What is the minimum value of P(A ∩ B)? (a) 0 (b) 0.1 (c) 0.5 (d) 1 17. Given that P ( A) 5 1 / 3, P( B ) 5 3 / 4 and P( A ∪ B ) 5 11 / 12, then what is P( B /A)? (a) (b) (c) (d)

1/6 4/9 1/2 1/3

18. If N a 5{ax | x ∈ N }, then what is N12 ∩ N 8 equal to? (a) N12 (b) N 20 (c) N 24 (d) N 48 n 19. If X {( 4  3n  1) | n ∈ N } and Y {( n  1) | n ∈ N }, then what is X ∪ Y equal to?

(a) X (c) N

(b) Y (d) A null set

20. Sets A and B have n elements in common. How many elements will (A × B) and (B × A) have in common? (a) 0 (b) 1 (c) n (d) n2 21. If f ( x ) 5 e x and g (x ) 5 log x , then what is the value of ( gof )`( x )? (a) 0 (c) e

(b) 1 (d) None of these

22. Consider the equation (x – p) (x – 6) + 1 = 0 having integral coefficients. If the equation has integral roots, then what values can p have? (a) 4 or 8 (b) 5 or 10 (c) 6 or 12 (d) 3 or 6

Algebra ▌ 3.41

23. If

1

is one of the roots of ax2 + bx + c = 0 where,

2 2 22

a, b, c are real then what are the values of a, b, c respectively? (a) 6, −4, 1 (b) 4, 6, −1 (c) 3, −2, 1 (d) 6, 4, 1 24. If α, β are the roots of the quadratic equation x2 – x + 1 = 0, then which one of the following is correct?

(

)

(a) α4  β 4 is real

(

)

(b) 2 α  β 5  (αβ )5 5

( ) (d) ( α  β )  (αβ ) (c) α6  β 6  0 8

8

8

25. If p and q are positive integers, then which one of the following equation has p 2 q as one of its roots?

1/ 3 2/3 31. If x  2  2 x  2 , then what is the value of x 3  6x1  6x ?

(a) 1 (c) 3

32. The roots of the equation (x – p) (x – q) = r2, where p, q, r are real are (a) always complex (b) always real (c) always purely imaginary (d) None of these 33. The equation x  2 ( x  1)1  1  2 ( x  1)1 has (a) no roots (b) one root (c) two equal roots (d) infinite roots 34. If a, b, and c are real numbers then the roots of the equation ( x  a ) ( x  b)  ( x  b) ( x  c )  ( x  c ) (x  a )

(a) x 2  2 px  (p 2  q )  0

 0 are always

(b) x 2  2 px  ( p 2 − q )  0

(a) Real (c) Positive

(c) x  2 px  ( p − q )  0 2

2

(d) x 2  2 px  ( p 2  q )  0

(b) 2 (d) − 2

(b) Imaginary (d) Negative

26. If the product of the roots of the equation x2 – 5x + k = 15 is − 3, then what is the value of k? (a) 12 (b) 15 (c) 16 (d) 18

35. For the two equations x 2  mx  1  0 and x 2  x  m  0, what is/are the value/values of m for which these equations have at least one common root? (a) −2 only (b) 1 only (c) −2 and 1 (d) −2 and −1

27. If the equation x2 – bx + 1 = 0 does not passes real roots, then which one of the following is correct?

36. If sin α and cos α are the roots of the equation px 2  qx  r  0, then which one of the following is correct?

(a)  3  b  3

(a) p2  q 2  2pr  0

(b)  2  b  2

(b) p2  q 2  2pr  0

(c) b  2

(c) ( p  r)2  2(q 2  r 2 )

(d) b  2

(d) ( p  r)2  q 2  r 2

28. If p and q are the roots of the equation x  px  q  0, then what are the value of p and q respectively? (a) 1, 0 (b) 0, 1 (c) −2, 0 (d) −2, 1 2

29. If α , β are the roots of ax 2  bx  b  0, then what is α β



(a) 0 (c) 2

β α



b a

37. If the roots of the equation x 2  bx  c  0 are two consecutive integers, then what is the value of b2 2 4c? (a) 1 (b) 2 (c) 22 (d) 3 38. If a  x  x 2  1 then what is x equal to? (a) (1 / 2) (aa + a 1 )

equal to?

(b) (1 / 2) (a  a 1 ) (b) 1 (d) 3

30. If the roots of ax 2  bx  c  0 are sin α and cos α fro some α, then which one of the following is correct? (a) a2 1 b2 5 2ac (b) b2 2 c2 5 2ab 2 2 (c) b 2 a 5 2ac (b) b2 2 c2 5 2ab

(c) a  11 (d) a  a 1 39. If sum of the roots of 3x 2  (3p  1) x  (p  5)  0 is equal to their product, then what is the value of p? (a) 2 (b) 3 (c) 4 (d) 9

▌ Mathematics

3.42

40. If α and β are the roots of x 2  4 x  6  0, then what is the value of α  β ? (a)  2 / 3 3

3

(b) 2 / 3

47. The geometric mean of three number was computed as 6. It was subsequently found that, in this computation, a number 8 was wrongly read as 12. What is the correct geometric mean?

(c) 4 (d) 8 41. If r and s are roots of x 2  px  q  0, then what is the value of (1/r 2 )  (1/ s 2 )? (a) p2  4q

(a) 4 (c) 2

(b) 3

18

3

5

(d) None of these

48. If (1  3  5  ...  p )  (1  3  5  ...  q )

p2  4q (b) 2 p2  4q (c) q4 (d)

46. If p times the pth term of an AP is q times qth term, then what is the (p + q)th term equal to? (a) p + q (b) pq (c) 1 (d) 0

 (1  3  5  ...  r ) where each set of parentheses where each set of Parentheses consecutive odd integers as shown, what is the smallest possible value of (p + q + r) where p > 6? (a) 12 (b) 21 (c) 45 (d) 54

p2  2q q2

42.The 59th term of an AP in 449 and the 449 term is 59. Which term is equal to 0 (zero)? (a) 501st term (b) 502nd term (c) 508th term (d) 509th term 43. If the AM and HM of two number are 27 and 12 respectively, then what is their GM equal to? (a) 12 (b) 18 (c) 24 (d) 27 44. If angles A, B, C are in AP, then what is sin A + 2 sin B + sin C equal to?  A2C  (a) 4 sin B cos 2   2   A2C  (b) 4 sin B cos   4  2

 A2C  (c) 4 sin ( 2 B ) cos 2   2   A2C  (d) 4 sin ( 2 B ) cos 2   4  45. Let a, b, c be in AP. Consider the following statements. 1 1 1 are in AP. , , ab ca bc 1 1 1 2. are in AP. , , b1 c c1 a a1 b

1.

Which of the statement given above is/are correct? (a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2

49. If x 2 , y 2 , z 2 are in AP, then y + z, z + x, x + y are in (a) AP (c) GP

(b) HP (d) None of the above

50. A class consists of 3 sections A, B and C with 35, 35 and 30 students respectively. The arithmetic means of the marks secured y students of sections A and B, who appeared for a test of 100 marks are 74 and 70 respectively. The arithmetic mean of the marks secured by student of section C, who appeared for a test in the same subject which carried 75 marks is 51. What is the average percentage of marks secured by all the 100 student of the three sections? (a) 70.0 (b) 70.8 (c) 65.0 (d) 67.5 51. If x, 2x + 2, 3x + 3 are the first three of a GP, then what is its fourth term? (a) −27/2 (b) 27/2 (c) −33/2 (d) 33/2 1 1 3 52. Which term of the sequence 20, 19 , 18 , 17 , ..... is the first 4 2 4 negative term? (a) 27th (b) 28th (c) 29th (d) No such term exists 53. In an AP, the mth term 1/n and nth term is 1/m. What is its (mn) th term? (a) 1/(mn) (b) m/n (c) n/m (d) 1 54. If the sum of ‘n’ terms of an arithmetic progression is n 2 2 2n, then what is the nth term? (a) 3n  n 2 (b) 2n  3 (c) 2n  3

(d) 2n  5

Algebra ▌ 3.43 55. If a, 2a + 2, 3a + 3 are in GP, then what is the fourth term of the GP? (a) 213.5 (b) 13.5 (c) 2 27 (d) 27 56. If the nth term of an arithmetic progress on is 2n – 1, then what is the sum upto n terms? (a) n 2

(a) 131 (c) 256

(c) n 2 1 1 1 n(n 1 1) 2

57. What is the 15th term of the series 3, 7, 13, 21, 43, ……..? (a) 205 (b) 225 (c) 238 (d) 241 58. If the AM and GM of two numbers are 5 and 4 respectively, then what is the HM of those numbers? 5 4 9 (c) 2 (a)

(b)

16 5

(d) 9

59. The arithmetic mean of 4 numbers is 15. The arithmetic mean of another 6 numbers is 12. What is the arithmetic mean of the combined 10 numbers? (a) 12.2 (b) 12.8 (c) 13.2 (d) 13.8 60. What is the sum to the 100 terms of the series 9 + 99 + 999 +….? 10 100 (a) (10 2 1) 2 100 9 10 99 (b) (10 2 1) 2 100 9 (c) 100 (10010 2 1) (d)

9 (10100 2 1) 100

61. If the number of terms of An A.P. is (2n+1), then what is the ratio of the sum of the odd terms to the sum of the even terms? n (a) n 11 n2 n 11 n 11 (c) n n 11 (d) 2n (b)

63. A team of 8 players is to be chosen from a group of 12 players. Out of the eight players one is to be elected as captain and another an vice-captain. In how many ways can this is done? (a) 27720 (b) 13860 (c) 6930 (d) 495 64. If C ( n, 12) 5 C ( n, 8), then what is the value of C(22, n)?

(b) n 2 − 1

(d)

62. What is the number of ways of arranging the latter of the word ‘BANANA’ to that no two N’s appear together? (a) 40 (b) 60 (c) 80 (d) 100

(b) 231 (d) 292

65. In a football championship 153 matches were played. Every team played one match with each other team. How many team participated in the championship? (a) 21 (b) 18 (c) 17 (d) 15 66. Two numbers are successively drawn from the set {1, 2, 3, 4, 5, 6, 7} without replacement and the outcomes recorded in that order. What is the number of elementary events in the random experiment? (a) 49 (b) 42 (c) 21 (d) 14 67. How many times does the digit 3 appear while writing the integers from 1 to 1000? (a) 269 (b) 271 (c) 300 (d) None of these 68. From 7 men and 4 woman a committee of 6 is to be formed such that the committee contains at least two women. What is the number of ways to do this? (a) 210 (b) 371 (c) 462 (d) 5544 69. What is the value of 8 C0  8 C1  8 C2  8 C3  8 C 4  8 C5  8 C 6  8 C 7  8 C8 ? (a) 0

(b) 1

(c) 2

(d) 28

70. What is the equivalent binary number of the decimal number 13.625? (a) 1101.111 (c) 1101.101

(b) 1111.101 (d) 1111.111

71. If ω is a complex cube root of unity and x = ω2 − ω − 2, then what is the value of x2 + 4x + 7? (a) − 2 (b) − 1 (c) 0 (d) 1

3.44

▌ Mathematics

72. If x 2  y 2  1, then what is (a) x − iy (c) 2x

(

1  x  iy equal to? 1  x  iy (b) x + iy (d) −2iy n

 1 i  73. What is the least positive integer n for which   1 ?  1 i  (a) 16 (b) 12 (c) 8 (d) 4 74. What is the modulus of (a) 5 (c) 3

)

4n + 3

(

75. What is the value of  1  i 41  i257 n ∈ N? (a) 0 (b) 1 (c) i (d) − i

1  2i equal to? 1  (1  i )2

)

9

where

76. If ω is the cube root of unity, then what is the conjugate of 2ω2 1 3i? (a) 2 ω  3i

(b) 3 ω  2i

(c) 2 ω  3i

(d) 3 ω  2i

77. If z is a complex number such that z + z−1 = 1, then what is the 99 99 value of z  z ? (a) 1 (b) −1 (c) 2 (d) −2

(b) 4 (d) 1

ANSwER KEy 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

(c) (c) (c) (b) (d) (a) (a) (d) (c) (a)

21. 22. 23. 24. 25. 26. 27. 28. 29. 30.

(b) (b) (b) (a) (d) (c) (c) (c) (b) (d)

31. 32. 33. 34. 35. 36. 37. 38. 39. 40.

(b) (a) (a) (c) (b) (a) (b) (a) (a) (c)

(b) (b) (a) (a) (c) (b) (a) (b) (a) (d)

41. 42. 43. 44. 45. 46. 47. 48. 49. 50.

(d) (c) (b) (b) (c) (d) (c) (b) (b) (b)

51. 52. 53. 54. 55. 56. 57. 58. 59. 60.

(a) (b) (d) (b) (a) (a) (d) (b) (c) (a)

61. 62. 63. 64. 65. 66. 67. 68. 69. 70.

(c) (a) (a) (b) (b) (b) (b) (b) (a) (c)

SOLUTIONS 1. It is clear from the graph that f (x 0) 5 e x , ∀x ∈ R is one-one but not onto.

2 2.  fog (x )  ( x  3) and g ( x )  x  3

⇒ f ( x)  x2

y

y e

x

⇒ f (3)  (3)2  9 3.  f (x )  kx 3  9 x 2  9 x  3 On differentiating w.r.t. x , we get

x

x′

∴ f `(x )  3kx 2  18 x  9 For a function to be monotonically increasing. ∆  b2  4 ac  0 ⇒ 36  12 k  0

y′

⇒ k 3

71. 72. 73. 74. 75. 76. 77.

(c) (b) (d) (d) (c) (a) (d)

Algebra ▌ 3.45 4. The function f(x) = x cosec x is discontinuous every. Where. 5. A injective function means one-one. In option (d), f(x) = − x For every values of x, we get a different values of f. Hence, it is injective. 6. Let y = log x – x For maximum value of y. dy 0 dx ⇒

dy 1  1 0 dx x

⇒ x 1 Also 1 d2y  2 dx 2 x (vc, at x  1) So, y is maximum when x  1 ∴ ymax  log 1  1  0  1 1 8. The function is y  2x 2  3x  5 For minimum value of y dy d2y  0 and 2 is  vc dx dx ⇒

dy  4x  3  0 dx

⇒ x

3 4

d2y 40 dx Hence minimum value at 3 x  is 4  3 2  3 y  2    3    5  4  4 9 9  2   5 16 4 9  18  40 8 31  8 

9. (1) (f(x))2 2  1   x   x  1  x2  2  2 x  f(x)2  2

(2) {f (x)}3a 3

 1  x   x   1 1 1 3  x  3  3x   x   x x x  f(x)3  3( f (x)) 10. Statement (1) is correct. 11.  n ( A) 5 4 ∴ Number of subsets = 2n = 24 Hence, number of proper subsets = 24 – 1 = 16 – 1 = 15 12. n ( N ∪ S )  32, n( N )  30, n( S )  17  We know that, n ( N ∪ S )  n( N )  n( S ) n( N ∩ S ) ⇒ 32  30  17  n ( N ∩ S ) ⇒ n ( N ∩ S )  47  32  15 13. Given, A {1, 2, 5, 8} and B {0, 1, 3, 6, 7} ∴ R {(1, 0), ( 2. 3) (5, 6)} Hence, option (b) is correct. 14. Now, n( A  B )  n( A)  n ( A ∩ B ) ⇒ 47  115  n ( A ∩ B ) ⇒ m ( A ∩ B )  68 ∴ n ( A ∪ B )  n( A)  n( B)  n ( A ∩ B ) = 155 + 326 – 68 = 373 15. The number of elements in power set of A is 1. ∴ P{ P( A)} 5 21 5 2 ⇒ P{ P{( A)}} 5 22 5 4 ⇒ P{ P{ P{ P( A)}}} 5 24 516 16. Since, P ( A ∪ B )  1 ⇒ P( A)  P( B )  P( A ∩ B )  1 ⇒ 0.8  0.7  P ( A ∩ B ) 1 ⇒ P ( A ∩ B )  1.5  1 ⇒ P( A ∩ B )  0.5

▌ Mathematics

3.46

17.  P ( A ∪ B )  P( A)  P( B )  P ( A ∩ B ) 1 3 11 ⇒ P ( A ∩ B)    3 4 12 1  6 P ( A ∩ B) ∴ P( B /A)  P( A) 

42 1  36 6

∴ Required equation is x 2  (sum of roots) x + ( product of roots)  0 1 4 x2  x   0 6 6 ⇒ 6x 2  4x 1 0

1/ 6 1  1/ 3 2

Thus, the value of a, b, c are 6, −4, 1 respectively. 2 24.  α and β be the roots of the equation x  x  1  0

18. Given, N 5{ax | x ∈ N |} a

∴ α  β  1 and αβ  1

∴ N12 5{12, 24, 36, 48....}

19.



And N 8 ∩ N12 5{ 24, 48, .....} 5 N 24

Now, α  β  (α  β) 2  4αβ  3i

X {( 4 n  3n  1) | n ∈ N |}

⇒ α

And Y {9( n  1) | n ∈ N |}

1 i 3 1 i 3 and β  2 2

Now, α  cos

⇒ X {0, 9, 54, .....} And Y {0, 9, 18, 27, 36, 54, .....}

π π π π  i sin and β  cos  i siin 3 3 3 3

(a) α4  β 4  cos

∴ X ∪ Y {0, 9, 18, 27, 36, 54, .....}  Y

4π 3

20. The total number of elements common in (A × B) and (B × A) is n2.

 2i sin

21. ( gof ) x 5 g ( f ( x )) 5 g ( e x ) 5 log e x 5 x

Now, (αβ )5  1

(

)

⇒ 2 α5  β5 ≠ ( αβ )5

On differentiatiing w.r.t. x, we get (gof )'(x ) 5 1

6π 6π  i sin 3 3 6π 6π  cos  i sin  2i sin 2 π  0 3 3 Hence, optioon (c) is correct. (c) α6  β 6  cos

22. The given equation can be rewritten as x 2  ( p  6) x  (6q  1)  0 Now, b2  4 ac  0

25. If any equation has p 2 q as a root, then another root will

⇒ ( p  6)2  4 (6 p  1) = 0

be p 1 q .

⇒ p 2  12 p  32  0 ⇒ ( p  4) ( p  8)  0

Sum of roots 5 p  q  p  q  2p

⇒ p  4, 8 23. ∴Given, root 

4π 4π 4π 4π  i sin  cos  i sin 3 3 3 3

(

and product of roots  p  q 1 2  2



1 2  2i



2  2i 2  2i

2  2i 2  2i   42 6 ∴Another root will be

2

Now, required equation is x2 − (sum of roots) x + (product of roots) = 0

(

)

⇒ x 2  2 px + p 2  q  0 2  2i 6

2  2i 2  2 2i 4   6 6 6     2  2i   2  2i  and product of roots      6  6     Thus, sum of roots 

)( p  q )  p  q

26. Let α and β be the roots of the equation x 2  5x  k  15  0 ∴ αβ  k  15 But αβ  3 ⇒  3  k  15 ⇒ k  15  3  12

Algebra ▌ 3.47 27.  Equation x 2  bx  1  0 has no real roots ie, it has imaginary roots which is possible only, if b2  4  0 ( B 2  4 AC  0)

34. Given, ( x  a ) ( x  b)  ( x  b) ( x  c )  (x  c ) ( x  a )  0 ⇒ 3x 2  2( b  a  c) x  ab  bc  ca  0

⇒ b2  4 ⇒2 b 2

Now, D  4( a  b  c ) 2  12 ( ab  bc  ca )

28.  p and q are the roots of x 2  px  q  0 ∴ p  q  p, pq  q ⇒ q( p  1)  0

 2 a 2  b2  c 2  ab  bc  ca 1 {( a  b)2  ( b  c )2  ( c  a)2 } 2

2

⇒ q  0, p  1 2 29. Since α and β one the roots of ax  bx  b  0 b b ∴ α  β  and αβ  a a

α

∴ 

β



b / a b/ a

β α 



b a



α β αβ



b a

b b b   a a a

0 30. Since, sin α and cos α are the roots of ax 2  bx  c  0 b c and sin α cos α  ∴ sin α  cos α  a a b2 2 2 ⇒ sin α  cos α  2 sin α cos α  2 a 2 c b2 ⇒ 1  2 a a ⇒ b2  a 2  2 ac 31. Given, x  2  21/ 3 (1  21/3 ) On cuddling both sides we get x 3  8  6 x 2  12 x  2 (1  2  3.21/ 3  3.22/ 3 ) ⇒ x 3  6 x 2  6 x  14  6.21/ 3  6.22/ 3  6 x  14  6.21/ 3  6.22/ 3  6 ( 2  21/ 3  22/ 3 )  2 32. Given, (x  p ) ( x  q )  r 3 ⇒ x  ( p  q ) x  pq  r 0 2

2

Now, D  ( p  q )2  4 ( pq  r 2 )  ( p  q ) 2  4r 2  0 Hence, roots arre always real. 33. Given, x  2 ( x  1)1  1  2 ( x  1)1 2 2 1 x 1 x 1 ⇒ x 1 ⇒ x

But x = 1 is not satisfied the given equation. Hence, no roots exist.

0 35. Let given equations have common root α. Then, α2  mα  1  0 and α2  α  m  0 ⇒

1 α2 α   m 1 1 m 1 m

⇒

1 α  ⇒ α 1 1 m 1 m

2

Also,

α2 1  m 1 1 m 2

⇒ 1  m  m2  1

( α  1)

⇒ m  m20 2

⇒ ( m  2) ( m  1)  0 ⇒ m  1 and  2 36. Comparing the given equation with standard quadratic equation ax 2  bx  c  0 b Sum of the roots  , a Product of roots 

c a

q Sin α  cos α  .......................(1) p r and sin α cos α  ......................(2) p Squaring both sides of equationn (1) we get Sin 2 α  Cos 2 α  2 Sin α Cos α  ⇒ 1 2 

r q2  2 q p

⇒ p 2  2 pr  q 2 ⇒ p 2  q 2  2 pr  0

q2 p2

3.48

▌ Mathematics

37. Let the roots be α, α  1 where α is integer. Sum of the roots α  α 1  (b)  b

and a  448d  59

on solving Eqs. (i) and (ii), we get

Products of thhe roots  α  ( α  1)  c

38.

...(ii)

b2  4c(2α  1)2  4 ( α  1)α

a  507 and d 1

 4α2  1  4α  4α 2  4α

Now, assume that nth term will be zero.

1

∴0  507  (n  1) ( 1) ⇒ 507  n  1

x 2  1  (a  x)

⇒ n  508

Squaring both sides we get x 2  1 = a 2 + x 2  2ax ⇒ a 2ax  1  0 2

⇒ 2ax  a 2  1 a 2 1 2a 1 1 1   a    [a  a 1 ] 2 a 2 ⇒ x

(3p  1) 3 (p  5) Product of the rootts  3 (3p  1) (p  5) ⇒  3 3 ⇒ 3p  1  p  5 ⇒ 2p  4 ⇒ p2

39. Sum of the roots 

40. α and β are the roots ⇒ α + β  4 and αβ  6 ( α  β )3  α3  β 3  3αβ ( α  β )

43.  AM = 27 and HM = 12 And we know that (GM)2 = (AM) (HM) = 27 × 12 ⇒ GM = 3 × 3 × 2 = 18 44.

1 1 1 , , are in AP. ab ca bc 1 1 2. , , b1 c c1 a

1.

sin A  2 sin B  sin C AC AC  2 sin  2 sin B cos 2 2   AC  2sin B cos  1 2     AC    2sin B 2 cos 2    4    AC   4 sin B cos 2    4  45. Let

⇒ α3 + β 3  ( α  β )3  3αβ(α  β )

1 1 1 , , are in AP. ab ca bc

⇒ α3 + β 3  (4)3  3  6(  4)

⇒

 64  72 8

1 1 1 1    ca ab bc ca

⇒

bc a b  abc abc

41. x 2  px  q  0 Sum of the roots r  s  p Product of the roots rs  q Now 1 1  r 2 S2 S2  r 2 (s  r )2  2sr p2  2q  2 2   Sr q2 S2 r 2 42. Let a and d be the first term and common difference of an AP. ∴ a + 58 d = 449 …(i)

1

⇒ b  c  a  b ⇒ 2b  a  c ⇒ a, b, c are in AP. w, Now ∴

b c 2

c a

⇒ 2

(

1

(



b c

1

,

c a 1 b c

)(

a 2 b  c

,



1 a b 1 a b

) (

a b 

)

are in AP.

c a

)

Algebra ▌ 3.49

)

⇒2

(



ab  2 bc  c  a  2 ab  ac

ab  b ac  bc

2 2 2 49. Since, 2y  x  z (a) Let 2(z  x )  ( y  z )  ( x  y )

 2 ac  2 bc  ab  c  a

⇒ 2z  2 x  2 y  z  x ⇒z  x2y ⇒ x , y and z in AP. (b) Let y  z , z  x , x  y in HP.

⇒ a, b, c are in AP.

∴ zx =

⇒ 2 ab  2b  2 ac  2 bc

Hence, both the statements are correct. 46. Let a and d be the first term and common difference of an AP. According to question. p  a  ( p  1) d   q  a  ( q  1) d 

(

)

(

)

⇒ pa  p  p d  qa  q  q d 2

2

(

)

⇒ ( p  q)a  q2  p2  p  q d ⇒ ( p  q ) a  ( p  q ) ( p  q  1) d ⇒ a  ( p  q  1) d Now, Tp q  a + ( p  q  1) d ( p  q  1) d  ( p  q  1) d 0

2( y  z ) ( x  y ) y  z  x y 2( y  z ) ( x  y ) ⇒zx = 2y z  x ⇒ 2 yz  z 2  zx  2 xy  xz  x 2  2 yx  2 y 2  2 zx  2 yz ⇒ z2  x2  2 y2 ⇒ x 2 , y 2 and z 2 are in AP. 50. Since, section C carried 51 average marks of 75. 51 ∴C carried   100  68 average marks out of 100. 75 ∴ Average percentage marks 35  74  35  70  300  68  100 2590  2450  2040   70.80 100 51. Since, (2x  2)2  x (3x  3) ⇒ x 2  5x  4  0

47. Required correct geometric mean 5 2 3 18

⇒ x 1,  4 Now,, first term a  x

48. Since, Tn  a  ( n  1) d

Second term, ar  2(x  1) ⇒ r 

∴ p  1  ( n  1) 2 ⇒ n 

p 1 2

p 1   p 1    1 2 ∴ 2 2  1   2   1    q 1    2   q 1     1 2 2  1   2 2      r 1   r 1    1 2 2  1   4   2   p 1 q 1 [2  ( p  1)]  [2  ( q  1)] ⇒ 4 4 r 1 [2  r  1]  4 ⇒ ( p  1)2  ( q  1)2  ( r  1)2 Total the possible onlyy when p  7, q  5, r  9 ∴ p  q  r 7 59  21

2( x  1) x

 x ( 2 x  1)  ∴ Fourth term ar 3  x   x   Put x  4, we get 3

3

 2(4  1)   3 T4  4    4     4   2 27  2

3

52. Given series can be rewritten as 20, 77 , 37 , 71 ........ 4 2 4 This is an AP series 3 Here, a  20, d  4 ∴Tn  a  ( n  1) d  3  20  ( n  1)     4 83 3   n 4 4

3.50

▌ Mathematics

For first negative term Tn  0 83 3  n  0 ⇒83  3n 4 4 83 ⇒n  3 So, n should be 28. ⇒

Hence, 28th term is first negative term. 53. Given, Tm = a  (m  1) d 1 ⇒  a  (m  1) d ......( i) n and Tn  a (n  1) d 1  a  (n  1) d m On solving Eqs. (i) and (ii), we get ⇒

1 ad  mn 1 1 1 ∴Tmn   ( mn  1) mn mn 54. It is given that sum of term Sn  n 2  2 n n ( n  2) Hence sum of (n  1) terms Sn 1  (n  1) (n  3)

56. nth term  2 n  1 1st term  2  1  1 2nd term  4  1  3 3rd term  6  1  5 This series has First term  a  1 Common difference  2 Sum up to nth term n Sn  {22  1  ( n  1)  2} 2 n n  {2  2 n  2}   2 n  n 2 2 2 57. [This series] is 3, 7, 13, 21,31, 43, 57, 73, 91, 111, 133, 157, 183, 211, 241 58. G 2  A  H ⇒ 16  5  H or H 

59. Sum of four numbers  4  15  60 Sum of fix numbers  6  12  72 A.M. of ten numbers

nth term = Sn  Sn 1  n 2  2 n {(n  1) (n  3)}  n 2  2 n {n 2  4 n  3}  n2  2n  n2  4n  3  2n  3

16 5



60  72 10

132  13.2 10

60. S100  9  99  999  ....................  (10  1)  (100  1)  (1000  1)

55. Since the three terms are in GP, (2a  2)2  a(3a  3) ⇒ 4(a 2  1  2a)  3a 2  3a ⇒ 4a 2  8a  4  3a 2  3a  0 ⇒ a 2  5a  4  0 5  25  16 5  3 ⇒a =   4, 1 2 2 ⇒ a  4 Common ratio 8  2 3  2 4 4th term  ar 3 r

  4     13.5

3  27   4  2 8

 ..............100 term  10 +1000  1000 + ........100 terms  100 This is a G.P. with a = 10 and r =10. Hence S100 

10(10100  1) 100 10  1

62. Number of ways that can be formed by using the words 6! 5 60 ‘BANANA’ 3!2! 5! Number of ways in which two N comes together = 5 20 3! ∴ Required number of ways = 60 – 20 = 40 63. Number of ways to choose 8 players from 12 players 5 12C8 5

12! 5 495 8!4!

And number of ways to choose captain and a vice –captain

3

 8C8  7C1 = 8 × 7 = 56 Hence, required number of ways = 495 × 56 = 27720

Algebra ▌ 3.51 64. Given C ( n, 12)  C ( n, 8) n

0.250 × 2 = 0.5 0 0.5 × 2 = 11 ∴ 13.625 = 1101.101

n

⇒ C12  C8 ⇒

n! n!  ( n  12)!12! ( n  8)!8!

⇒ ( n  8) ( n  9) ( n  10) ( n  11)

Now

 12  11  10  9 ⇒ n  20

(i  3) ( 3  i )( 3  i)

i  3  2 3i 3 1 1  3  2 3i 1  3i  ω2  4 2 

∴C ( 22, n)  C20 22! 22  21   231 2!20! 2

(i  3)  3 i  3 ( )

2

65. Let total team participated in a championship is n.

and

n( n  1) ∴ n C2  153 ⇒  153 2



⇒ n( n  1)  306 ⇒ n  18

i 3 i

2

2

i 2  3  2i 3 2  2i 3  4 4

i 2  3  2i 3 2  2i 3  4 4 1  i 3  ω 2 200 200  i 3   i 3   1   ∴  i 3   i  3       (  ω)200  ω200 1  ω31331  ω3662  1  0 

66. Total number of elementary events  7C1  6C1  42 67. No. of time 3 occurs = (when 3 occurs exactly at one place) + (when 3 occurs exactly at two places) + (when 3 occurs exactly at three places)  3C1  9  9 3 C2  9 3 C3  243  27  1  271

71.  x  ω2  ω  2

68. The committee can consists of 2W + 4M, 3W + 3M, 4W + 2M

⇒ x  2  ω2  ω On squaring both sides, we get

Total number of ways

( x  2)  (ω  ω) 2

⇒ P ( A ∩ B )  1.5  1

3

2

⇒ x 2  4 x  4  ω4  ω2  2ω3

4 3 7 6 5 7 6 5 7 6   4 1 3 2 3 2 2 2  210  140  21  371 69.

1  3



2

22



1 3

⇒ x 2  4 x  4  3  ω ω2  2  3 ⇒ x 2  4 x  7  1  ω ω2

(

⇒ x 2  4 x  7  0 1  ω ω2  0 8

8

8

8

8

8

8

8

8

C0  C1  C2  C3  C4  C5  C6  C7  C8

1 8 

87 87 6 87 6 5 8 7 6    2 3 2 4  3 2 3 2

87  8  1 11  8  28  56  70  56  28  8  1 2  128  128  0 

70. 13.625 2

13

2

6

1

2

3

0

1

1

0.625 × 2 = 1.2501

)

72. 1  x  iy (1  x  iy ) (1  x  iy )  1  x  iy (1  x  iy )(1  x  iy ) 

(1  x )

2

 iy (1  x )  iy (1  x )  y 2



1 x2  2x  y2 1  x  2 x  y 2  2iy (1  x )



1  y 2  2 x  x 2  2iy (1  x )



2 x 2  2 x  2iy (1  x )

2

2 (1  x )

 x  iy

2 (1  x )

2 (1  x )

(∴ x

2

)

 y2 1

3.52

▌ Mathematics

1  i (1  i ) 1  i 2  2i   i 73. 2 1 i 1 1 2

n

 1 i  n ∴   i 1  1 i  Which is possible for n  4. 74. Let z 

1  2i 1  (1  i )2

75.

1  (1  i )2

( 1)

(

(1 3)  3i 2

( )  z  1  (3  3 ) i

1  3  3 i

 2ω  3i

1  2i

4n + 3

2

1  3i  3i

1  2i z 1  (1  i )2 ⇒ z

76. Let z  2 ω2  3i



 i i

1  2i 1 1  2i

257

41

9

)

9

77. Given, z  z1  1 ⇒ z 2  z 1 0 ⇒ z  ω,  ω2 When, z ω ∴ z 99  z99  ( ω)99 1  1  2

 1  (i )4 n3   i   ( i 4  1) i  4 n 3 4n  ( i ) ( i )3  ( i  i )9  (1)

∴ z 99  z99  (ω2 ) 99  (ω2 )99

(1) (i )  0  i

1  1  2

When, z ω2

C

H

2

A

P

T

E

R

Matrices and determinants Matrix addition and Multiplication by nuMber A matrix is a rectangular array of real numbers enclosed within brackets. Each number in the array is called an element of the matrix. A m ´ n matrix (read “m by n matrix”) is a matrix with m rows and n columns. If a matrix has the same number of rows and columns, is called a square matrix. A matrix with only one column is called a column matrix, and a matrix with only one row is called a row matrix. Two matrices are equal if they have the same dimension and their corresponding elements are equal. The sum of two matrices of the same dimension is a matrix with elements that are the sums of the corresponding elements of the two given matrices. Matrix addition is commutative and associative. A matrix with all zero elements is called the zero matrix. The negative of a matrix M, denoted by –M, is a matrix with elements that are the negatives of the elements in M. If A and B are matrices of the same dimension, then we define subtraction as follows: A – B = A + (–B). The product of a number K and a matrix M, denoted by kM, is a matrix formed by multiplying each element of M by k.

The dot between the two matrices is important. If the dot is omitted, the multiplication is called a matrix product. The product of two matrices A and B is defined only on the assumption that the number of columns in A is equal to the number of rows in M. If A is an m ´ p matrix and B is a p ´ n matrix, then the matrix product of A and B, denoted by AB, is an m ´ n matrix whose element in the ith row and jth column is the dot product of the ith row matrix of A and the jth column matrix of B. Matrix multiplication is not commutative, and the zero property does not hold for matrix multiplication. That is, for matrices, A and B, the matrix product AB can be zero without either A or B being the zero matrix. Matrix multiplication does have other general properties, some of which are similar to the properties of real numbers. Assuming all products and sums are defined for the indicated matrices A, B and C, then for k a real number: Associative Property

1. A(BC) = (AB)C 2. A(B + C) = AB + AC

Left Distributive Property

3. (B + C)A = BA + CA

Right Distributive Property

4. If A = B, then CA = CB

Left Multiplication Property

Matrix Multiplication

5. If A = B, then AC = BC

Right Multiplication Property

The dot product of a 1 ´ n row matrix and an n ´ 1 column matrix is a real number given by

6. k(AB) = (kA)B = A(kB).

n 1

 b1    b  a1 a2  an  .  2   a1b1  a2 b2  ...  an bn     bn  1 n

A real number.

inverse of a Square Matrix The identity matrix for multiplication for the set of all square matrices of order n is the square matrix of order n, denoted by I, with 1’s along the principal diagonal (from upper left corner to lower right corner) and 0’s elsewhere. If M is a square matrix of order n and I is the identity matrix of order n, then IM = MI = M

3.54

▌ Mathematics

If M is a square matrix of order n and if there exists a matrix M–1 (read “M inverse”) such that M–1M = MM–1 = I then M–1 is called the multiplicative inverse of M or, more simply, the inverse of M. If the augmented matrix [M|I] is transformed by row operations into [I|B], then the resulting matrix B is M–1. If, however, we obtain all 0’s in one or more rows to the left of the vertical line, then M–1 does not exist.

Matrix equations and System of linear equations A system of linear equations with the same number of variables as equations such as a21 x1  a22 x2  a23 x3  k2

A

X

B

a13   x1   k1      a23   x2    k2  a33   x3   k3 

a22 a32

If the inverse of A exists, then the matrix equation has a unique solution given by After multiplying B by A–1 from the left, it is easy to read the solution to the original system of equations.

determinants Associated with each square matrix A is a real number called the determinant of the matrix. The determinant of A is denoted by det A, or simply by writing the array of elements in A using vertical lines in place of square brackets. For example, a det  11  a21

a12   a11  a22   a21

a12   a22 

A determinant of order n is a determinant with n rows and n columns. The value of a second-order determinant is the real number given by  a11   a21

1. If each element of any row (or column) of a determinant is multiplied by a constant k, the new determinant is k times the original. 2 a 2b a b 2 c d c d 3

a b 3a b  c d 3c d

a b 0 0 0

a31 x1  a32 x2  a33 x3  k3 a12

The use of the following five determinant properties can greatly reduce the effort in evaluating determinants of order 3 or greater:

2. If every element in a row (or column) is 0, the value of the determinant is 0.

a11 x1  a12 x2  a13 x3  k1

 a11   a21 a  31

properties of determinants

a12    a11a22  a21a12 . a22 

The value of a third-order determinant is the sum of three products obtained by multiplying each element of any one row (or each element of any one column) by its cofactor. The cofactor of an element aij (from the ith row and jth column) is the product of the minor of aij and (– 1)i + j. The minor of an element aij is the determinant remaining after deleting the ith row and jth column. A similar process can be used to evaluate determinants of order higher than 3.

0 b 0 0 d 3. If two rows (or two columns) of a determinant are interchanged, the new determinant is the negative of the original. a b c d  c d a b a b b  c d d

a c

4. If the corresponding elements are equal in two rows (or columns), the value of the determinant is 0. a b 0 a b a a 0 c c 5. If a multiple of any row (or column) of a determinant is added to any other row (or column), the value of the determinant is not changed. a b a b  c d c  ka d  kb a b a  kb b  c d c  kd d

cramer’s rule Systems of equations having the same number of variables as equations can also be solved using determinants and Cramer’s rule. Cramer’s rule for three equations and three variables is as follows:

Matrices and Determinants ▌ 3.55 Given the system, a11x + a12 y + a13 z = k1 a21x + a22 y + a23 z = k2 with

a11 D  a21 a31

a12 a22 a32

Cramer’s rule can be generalized completely for any size linear system that the same number of variables as equations. The formulas are easily remembered if you

a13 a23 ≠ 0 a33

a31x  a32 y  a33 z  k3 then, k1 a12 a13 a11 k2 a22 a23 a21 k3 a32 a33 a31 x y D a11 a12 k1 a21 a211 k2 a31 a32 k3 z D

k1 k2 k3

a13 a23 a33

D

Observe the following: 1. Determinant D is formed from the coefficients of x, y, and z keeping the same relative position in the determinant as found in system of equations. 2. Determinant D appears in the denominators for x, y, and z. 3. The numerator for x can be obtained from D by replacing the coefficients of x(a11, a21, a31) with the constants k1, k2 and k3, respectively. Similar statements can be made for the numerators for y and z. Cramer’s rule is rarely used to solve systems of order higher than 3 by hand, since more efficient methods are available. Cramer’s rule, however, is a valuable tool in more advanced theoretical and applied mathematics.

practice exerciSe MatriceS 1. Which of the following is correct? (a) skew symmetric matrix of even order is always singular (b) skew symmetric matrix of odd order is non-singular (c) skew symmetric matrix of odd order is singular (d) none of the above 2. Let A be an invertible matrix, which of the following is not true? (a) (A′ )21 = (A21)′ (b) A21 = | A |21 (c) (A2)21 = (A21)2 (d) None of these 3. If A is a singular matrix, then adj A is (a) non-singular (b) singular (c) symmetric (d) not defined 4. If B is a non-singular matrix and A is a square matrix, then det (B 21AB) is equal to (a) det (A21) (b) det (B21) (c) det (A) (d) det (B) 5. If A is a 3 × 3 skew-symmetric matrix, then trace of A is (a) | A | (b) 1 (c) 21 (d) none of these 6. If A is an orthogonal matrix, then | A | is (a) 1 (b) 21 (c) 0 (d) none of these

 2 2 4 4  is 7. The matrix A   1 3  1 2 3 (a) Nilpotent (b) Idempotent (c) Orthogonal (d) Involutary  i i   1 1 8 8. If A    and B    the A equals i i  1 1  (a) 64 B (b) 264 B (c) 126 B (d) 128 B 9. If A and B are two matrices such that A + B and AB are both defined, then (a) A and B are two matrices not necessarily of same order (b) A and B are square matrices of same order. (c) number of columns of A = number of rows of B (d) none of the above 10. The value of a for which the system of equations ax + y + z = 0, x + ay + z = 0, x + y + z = 0, possess non-zero solutions are given by (a) 1, 2 (b) 1, 2 1 (c) 1 (d) none of these  0 2b c  11. Let A   a b c  be an orthogonal matrix then the    a b c    values of a, b, c are 1 1 1 1 , c  ( b) a  , c  (a ) b  3 2 6 6 1 1 (c ) a  , b  (d ) all off these 2 6

▌ Mathematics

3.56

 31    1 1 T T 2005 12. If P   2 2  , A    and Q  PAP , then P (Q ) 1 3  0 1    2 2  P is equal to

1 2005 (a )   1  0  1 2005 (c )   1   3 / 2

 3 / 2 2005 ( b)   0   1 1 3 / 2 (d )   0 2005 

13 If A  then A64 is  1 32  (a )   32 1  1 32  (c )   0 1 

 1 0 ( b)   32 1 (d ) none of these

1 0 0  1 0 0      A  0 1 1  , I  0 1 0  0 0 1  0 2 4      1 A1  [ A2  cA  dl ] 6 where c, d ∈ R, the pair of values (c, d) are (a) (6, 11) (b) (6, 2 11) (c) (26, 11) (d) (2 6, 2 11)

14. If

 1 0 100 15. If A    , then A is equal to 1 / 2 1  1 0  1 0 ( b)  (a )    110 1 50 1  1 0 (c )  (d ) noone of these  25 1 16. If A and B are symmetric matrices and AB = BA, then A21B is a (a) symmetric matrix (b) skew-symmetric matrix (c) identity matrix (d) none of these    2 6 4    17. If the product of the matrix B   1 0 1     1 1 1    with a matrix    −1 0 1    A has inverse C   1 1 3  , then A1 equals    2 0 2   

   23 25 5    (a)  0 9 14    2 2 6      23 25 25    (c)  0 9 2    2 14 6   

  23 5  (b)  0 0   2 14    23 23  (d)  0 9   2 14 

5 9 16

      

25 2 6

      

α 2  3 18. If A   and A = 125 then the value of α is 2 α   (a) ± 1 (b) ± 2 (c) ± 3 (d) ± 5 19. If A is an involutory matrix and I is unit matrix of the same order then, (I 2 A)(I + A) = (a) 0 (b) A (c) I (d) 2A 20. Matrix A is such that A2 = 2A 2 I, where I is unit matrix then for n ≥ 2, An = (a) nA 2 (n 2 1)I (b) nA 2 I (c) 2n 2 1 A 2 (n 2 1)I (d) 2n 2 1A 2 I 21. If AB = A and BA = B, where A and B are square matrices, then (a) B2 = B and A2 = A (b) B2 = A and A2 = B (c) AB = BA (d) none of these 22. If A is a square matrix, B is a singular matrix of same order, then for a positive integer n, (A21BA)n equals (a) A2n BnAn (b) AnBn A2n 21 n (c) A B A (d) n (A21 B A) 23. If A is a matrix of order 3 × 4, then each row of A has (a) 3 elements (b) 4 elements (c) 12 elements (d) 7 elements 24. If A = [aij] is a scalar matrix of order n × n such that aij = k for all i, then trace of A is equal to (a) kn (c) nk

n k (d) none of these (b)

 1/ 2 1/ 2   is 25. The matrix A   1 / 2 1 / 2  (a) unitary (b) orthogonal (c) nilpotent (d) involuntary 26. The value of θ in [0, 2 π] such that the matrix  2 sin θ  1 sin θ cosθ     sin(θ  π ) 2 cosθ  3 tan θ  is skew-symmetric, is cos(θ  π ) tan(π  θ ) 0   

Matrices and Determinants ▌ 3.57 (a) π/2 (c) π/4

(b) π/3 (d) π/6

 cos 2 θ cosθ sin θ  27. If E(θ)    and θ and φ differ by an cos θ sin θ sin 2 θ   π odd multiple of , then E(θ) E(φ) is a 2 (a) null matrix (b) unit matrix (c) diagonal matrix (d) none of these 28. If A and B are two square matrices such that B = 2 A21BA, then (A + B)2 = (a) 0 (b) A2 + B2 (c) A2 + 2AB + B2 (d) A + B  1 tan θ  29. If A    and AB  I , then B  tan θ / 2 1    θ (a ) cos 2 . A 2 2θ ( b) cos . AT 2 2θ (c ) cos . I 2 (d ) none of these 30. For each real number x such that 21 < x < 1, let A(x) be the −x x y 1  1 . Then matrix (1  x )   and z  1  xy 1  x   (a) A(z) = A(x) + A(y) (c) A(z) = A(x) A(y)

(b) A(z) = A(x)[A(y)]21 (d) A(z) = A(x) A(y)

31. The inverse of a skew symmetric matrix of odd order is (a) a symmetric matrix (b) a skew symmetric matrix (c) diagonal matrix (d) does not exist 32. The number of solutions of equations x2 2 x3 = 1, 2 x1 + 2x3 = 2, x1 2 2x2 = 3 is (a) zero (b) one (c) two (d) infinite α β  33. If   is to be the square root of two-rowed unit matrix,  γ 2α  then α, β and γ should satisfy the relation (a) 1 + α2 + βγ = 0 (b) 1 2 α2 2 βγ = 0 (c) 1 2 α2 + βγ = 0 (d) α2 + βγ 2 1 = 0 34. If A is a matrix of order 2 3 3 and B is of order of 3 3 2 then AB is of order (a) 2 3 3 (b) 2 3 2 (c) 3 3 3 (d) 3 3 2

35. If in be the identity matrix of order n, then (a ) does not exist ( b)  I n (d )  n I n

(c )  0 36. If A is symmetric, then A is (a) symmetric (c) scalar 3

(b) skew-symmetric (d) None of these

37. If A and B are Hermitian matrices of the same order, then (AB 2 BA) is: (a) a null matrix (b) a Hermitian matrix (c) a skew-Hermitian matrix (d) None of these α β  38. If A    the adj A is equal to; γ δ   δ γ   δ β  (a )  ( b)     β α    γ α  δ β  δ β  (c )  (d )    α  γ  γ α   3 3 4    39. If A  2 3 4  , then A-1 equal to:  0 1 1    (a) A (b) A2 3 (c) A (d) A4 1 40. If the rank of the matrix, A  1 of λ is; (a) 1 (b) (c) 1 (d)

λ 1

1  is 1, then the value λ

21 None of these

41. For what value of λ, the system of equation x 2 2i 2 2y + z = 0, 2x 2 y + 3z = 0, λx + y 2 z = 0 has the trivial solution as the only solution 4 24 (a) λ  (b) λ ≠ 5 5 (d) None of these (c) λ ≠ 2 cos α sin nα  42. If A α    is a non-singular matrix and the α  sin cos nα   eigen values of A are 1, 2, 3 then the eigen values of A21 are; (a) 1, 2, 3 (b) 1, 1 2 , 1 2 (c)

A , 2 A ,3 A

(d) None of these  cos β sin β   cos α sin α  43. If Aα    then  , Aβ   sin β cos β  sin α cos α  onsider the following statement

3.58

▌ Mathematics (a ) Aα . Aβ  Aαβ ( b) Aα . Aβ  A( α β )  cos n α sin n α  (c) ( Aα ) n    n n sin α cos α   cos nα sin n α  (d ) ( Aα ) n    sin nα cos n α 

 (1 / 2 (1 / 2 )   is; 44. The matrix, A   (i / 2 (1 / 2 ) (a) orthogonal (b) unitary (c) involuntary (d) nilpotent 1 2  45. A    is the zero of the polynomial? 3 4  (a ) f ( x )  x 2  3x  10

( b) f ( x )  x 2  3x  10

(c ) f ( x )  x 2  3x 10

(d ) None of these

2x 0   1 0 1 46. If A    and A    then the value of x is; x x   1 2  (a) 1 (b) 2 (c) 1

2

(d) None of these

2 5 7    47. Let A   0 1 3  , then A21 is equal to; 0 0 0    1 / 2 25 / 2 4    (a )  0 1 23  0 0 0    1 25 8  1  ( b) 0 2 23 2 0 0 22    (c )

 1 25 8  1  0 22 6  2 0 0 2   

(d ) Noone of these 48. The system of equations x + 2y + 3z = 1, 2x + y + 3z = 2, 4x + 5y + 9z = 4 has (a) no solution (b) a unique solution (c) an infinite number of solutions (d) more than two solution

49. If A is a 2 lowed square matrix and | A | = 3, the A (adj A ) is equal to 3 0 2 0  (a )  ( b ) 3.    0 3   0 2  1 2 0  1 1 0  (d )  (c ) .    3 0 2  3 0 1   1 1  a 1  50. If A   , B   and then  2 1  b 1 (a ) a  2, b  2 ( b) a  3, b  2 (c ) a  1, b  4 (d ) a  4, b  7  2 1 3    51. If the rank of matrix A   4 7 λ  is 2. Then, the value 1 4 5  of λ is   35 (a) (b) 213 4 (c) 13 (d) None of these 52. A matrix A [ aij ] is said to be square matrix if (a) m = n (c) m ≥ n

(b) m ≤ n (d) m < n

53. A matrix A  [ aij ]m n is said to be rectangular if (a) m = n (c) m = p

(b) m ≠ n (d) m < r

54. A column matrix has only (a) One row and one column (b) One row with one or more columns (c) One column with one or more rows (d) One element 55. A row matrix has only (a) One element (b) One row with one or more columns (c) One column with one or more rows (d) One row and one column 56. A matrix is a (a) A collection of real numbers (b) A collection of real or complex numbers (c) An array of real numbers (d) An array of real or complex numbers 57. Which of the following matrices is not invertibles? 1 (a )  0 2 (c )  4

1  1 3  6

21 21 ( b)   21 2   2 22  (d )   1 1 

Matrices and Determinants ▌ 3.59

anSwer Key 1. (c)

9. (b)

17. (c)

25. (c)

33. (b)

41. (b)

49. (a)

2. (b)

10. (c)

18. (c)

26. (d)

34. (b)

42. (b)

50. (c)

3. (b)

11. (d)

19. (a)

27. (a)

35. (b)

43. (d)

51. (c)

4. (c)

12. (a)

20. (a)

28. (b)

36. (a)

44. (b)

52. (a)

5. (a)

13. (c)

21. (a)

29. (b)

37. (c)

45. (c)

53. (b)

6. (a, b)

14. (c)

22. (c)

30. (c)

38. (b)

46. (c)

54. (c)

7. (b)

15. (b)

23. (b)

31. (d)

39. (c)

47. (d)

55. (b)

8. (d)

16. (a)

24. (c)

32. (a)

40. (b)

48. (b)

56. (d)

57. (c)

SolutionS 1.

Since the determinant of a skew symmetric matrix of odd order is zero, ∴ the matrix is singular.

2.

A–1 = | A |–1 is not true, as L.H.S. is a matrix and R.H.S. is a number.

3.

Let A be of order n × n.

(iii) If A2 = I then A is involutary matrix.

Since A ⋅ adj A = | A | I

(iv) If A2 = A then A is said to be idempotent and by checking we get A2 = A.

7. In order to get the result, we go to the definition of these matrices and check

∴ | A ⋅ adj A | = | A |n ⇒ | A | | adj A | = | A |n Since A is singular, ∴ | A | = 0.

4.

8.

det (B21AB) = det (B21) det A det B = det (B21) ⋅ det B ⋅ det A = det (B21B) ⋅ det A = det (I) ⋅ det A = 1 ⋅ det A = det A.

5.

AA′ = 1 for othogonal matrix.

(ii)

If A2, A3 ... or Ak is zero i.e. A2 = 0 or A3 = 0 or Ak = 0 then A is said to be nilpotent.

∴ A is idempotent matrix.

∴ | adj A | = | A |n 2 1 ∴ | adj A | = 0. Hence, adj A is singular.

(i)

 i  i  1  1 We have, A   i    iB  i i   1 1   1  1  1  1 ⇒ A2  (iB )2  i 2 B 2  B 2      1 1   1 1   2 2     2 B.  2  2

As A is a skew symmetric matrix

⇒ A4 = (2B)2 = 4B2 = 4(2B) = 8B

⇒ a ii  0 ∀ i ⇒ trace (A)  0

⇒ A8 = (A4)2 = (8B)2 = 64B2 = 128B.

Also, | A | = | A′| = |2A| = (21)3|A| ⇒ 2| A | = 0 ⇒ | A | = 0 ∴ trace(A) = | A |. 6. Since the matrix A is orthogonal A′A = AA′ = In ⇒ | A′A |

= | AA′| = | In |

⇒ | A′| | A | = 1 ⇒|A||A| ∴ | A | = ±1.

= 1 or | A |2 = 1

9. Both matrices are of same order for A + B to be defined. Now let both are of order m × n. But AB is defined only when number of columns in A is same as number of rows in B. So both are square matrices of same order. 10.

The set of homogeneous equations will have a non-zero solution if ∆ = 0 a 1 1 i.e., 1 a 1  0 ⇒ a  1 1 1 1

3.60

11.

▌ Mathematics

As A is orthogonal ∴ AA′ = I  0 2b c   0 a a  1 0 0        a b c   2b b b  0 1 0   a b c   c c c  0 0 1      2 2 2 2   4 b2  c 2 2b  c 2b  c   ⇒  2 b2  c 2 a 2  b2  c 2 a 2  b2  c 2  −2b2  c 2 a 2  b2  c 2 a 2  b2  c 2    1 0 0    0 1 0  0 0 1   ∴ Using definition of equality of two matrices 4 b2 + c2 = 1, 2 b2 2 c2 = 0, a2 + b2 + c2 = 1. On solving them, 1 1 1 a  , b  , c  2 6 3 12.

1 0 0   0 1 5   0 10 14   d 0 0 0 0  c    , dI   0 c c   0 d 0   0 2c 4c   0 0 d 1 ∴ A− 1  [ A2  CA  dI ] 6 ⇒ 6 = 1 + c + d, (By equality of matrices) ∴ (26, 11) satisfy the relation         cA    

 0 2b c  0 a a    A   a b c and A'   2b b b  a b c   c c c 

We have, Q = P A PT ⇒PT Q = APT (Q as PTP = I = PPT) ∴ PT Q2005 P = A PT Q2004 P = A2 PT Q2003 P = A3 PT Q2002 P = A2004 PT (QP) = A2004 PT (PA)

15.

 1 0  1 0 A100     100(1 / 2) 1  50 1  16. We have, AB = BA = B′A′ = (AB)′ ⇒ AB is symmetric. Also, ABA21 = BAA21 = B [Q AB = BA] ⇒ A21ABA21 = A21B ⇒ BA21 = A21B Therefore, (A21B)′ = (BA21) = (A21)′B′ = A′B [Q A21 and B are symmetric] Thus, the matrix A21B is symmetric.

1 2005   1  0 13.

 1 A2    0

1 1 2   1  0  1 1  1 A4    0 1 0 1 2  1 A8    0 1  0

1  1 1 2     0 1 1   1  1 2     1 0 1 

18.

2  1 4     1  0 1   1 32  Similarly, A64    0 1    14. Given A       A2    

 1 0 0    , A1  1  0 1 1  6   0 2 4   1 0 0   1 0  0 1 1  0 1  0 2 4   0 2

6 0 0   0 4 1   0 2 4  0   1   4 

 1 0   1 0  1 0   1 0 A2       / ) 2 1 2 1 ( 1 2 1 1 2 1 1 1 / /        1 0 0  1 0  1 A3  A2 ⋅ A      3 ( 1 / 2 ) 1 ( / ) 2 1 2 1 / 1 2 1      Continuing in this way, we get

17.

(Q = P A PT ⇒ Q P = P A) = A2005

     

We have (BA)21 = C ⇒ A21B21 = C ⇒ A21 = CB 1 0 1  2 6 4  ∴ A1   1 1 3  1 0 1      2 0 2 1 1 1 3 5 5  0 9 2    2 14 6  α 2  Given: A    and | A3 | = | A |3 = 125. 2 α Now A = α2 2 4 ⇒ (α2 2 4)3 = 125 = 53 ⇒ α2 2 4 = 5 ⇒ α=±3

19.

Since A is an involutory matrix ∴ A2 = I ⇒ I 2 A2 = 0 ∴ (I 2 A) (I + A) = 0

20.

A2 = 2A 2 1 ... (1) Multiplying by A, we have A3 = 2A2 2 A = 2(2A 2 I) 2 A {using (1)} ⇒ A3 = 3A 2 2I Again, multiplying by A, we get A4 = 3A2 2 2AI ⇒ A4 = 3(2A 2 I) 2 2A {using (1)}

Matrices and Determinants ▌ 3.61 ⇒ A4 = 6A 2 3I 2 2A ⇒ A4 = 4A 2 3I Hence, by induction, we have An = nA 2 (n 2 1)I 21.

22.

We have AB = A ⇒A(BA) = A [Q BA = B] ⇒(AB) A = A ⇒AA = A [∴ AB = A] ⇒A2 = A Again BA = B ⇒B (AB) = B [Q AB = A] ⇒(BA) B = B ⇒BB = B ⇒ B2 = B Consider n=2 ∴ (A21 BA)2 = (A21BA)(A21BA) = (A21B)(A21)(BA) = A21B2A Again for n = 3, we have (A21BA)3 = (A21B2A)(A21BA) = A21B3A ∴ generalizing the case (A21BA)n = A21BnA

 π π  cos θ sin φ( 2 n 1) cos θ cos φ cos( 2 n 1) 2 2  ⇒ S  π π   cos φ sin θ cos( 2 n 1) 2 sin θ sin φ cos( 2 n 1) 2  0 0 π ∴ S  {Qθ  φ  (2 n  1) ( given)} 2 0 0   28.

Thus, (A + B)2 = A2 + B2. 29.

23. Q these are 4 columns ∴ each row has 4 elements. 24.

Trace of A = a11 + a22 + ... ann = k + k + ... + k = nk.

25.

 1  A2   2  1  2 

1  1  2  2 1  1    2 2

1   2  1    2

Given, B = 2 A21BA ∴ AB = 2 AA21BA = 2 IBA = 2 BA ∴ AB = 2 BA Now (A + B)2 = (A + B)(A + B) = A2 + AB + BA + B2 = A2 + B2 [∴ BA = 2AB]

30.

θ θ  sec 2 2 2 21 AB = I ⇒ B = IA  θ  tan  1 0   1 2    0 1   tan θ 1   2   cos 2 θ . AT  θ 2 sec 2 2 A  1  tan 2

 x y    1  xy A( z )  A     1  xy   (1  x ) (1  y ) 

  x y  1     1  xy        x  y  1    1  xy 

0 0  A2   O 0 0  ∴ A is nilpotent 26.

The matrix can be written as  2 sin θ 21 sin θ cosθ    2 cosθ 2 3 tan θ  2sin θ 2cosθ 2 tan θ 0    The above matrix is skew symmetric if 2 sin θ 2 1 = 0 and 2 cos θ 2 3  0

π The simultaneous equations hold in [0, 2π ] if θ  6 27.

Let  E (θ ) E (φ )  ⇒ S   cos 2 θ cos θ sin θ    sin 2 θ  cos θ sin θ    cos 2 φ cos φ sin φ    sin 2 φ  cos φ sin φ

 cos θ cos φ cos(θ  φ cos θ sin φ cos(θ  φ)  ⇒ S   cos φ sin θ cos(θ  φ) sin θ sin φ cos( θ  φ) 

∴ A(x). A(y) = A(z). 31.

Let A be a skew symmetric matrix of order n. By definition A′ = 2A ⇒| A′ | = |2 A | ⇒ | A | = (2 1)n | A | ⇒| A | = 2 | A | [Q n is odd] ⇒2| A | = 0 ⇒ | A | = 0. ∴ A1 does not exist.

32.

The system is 0x1 + x2 2 x3 = 1 2x1 + 0x2 + 2x3 = 2 x1 2 2x2 + 0x3 = 3 0 1  1  x1   1       2   x2    2  or AX  B ⇒  1 0       1 2 0   x3   3  Clearly, A  0 4 2 2   Now Adj A   2 1 1    2 1 1 ∴ (Adj A) B ≠ 0 ⇒ system is inconsistent

3.62

▌ Mathematics

α β  33. Since   is a square root of I2 i.e., two rowed unit  γ 2α  matrix.

43.

 cos(α  β ) sin(α  β )     Aα β sin(α  β ) cos (α  β ) 

2

α β  1 0  ∴    γ  α   0 1   1 0  α β  α β  ⇒     0 1   γ  α   γ  α   1 0   α 2  βγ α β α β  ⇒  2   0 1  γ α  γ α γ β  α   1 0  α 2  βγ 0  ⇒   2 0 1  0 α βγ     ∴ α 2  βγ  1 ⇒ 1  α 2  βγ  0 34. Number of columns of A = 3 Number of rows of B = 3 Since both are equal ∴ AB is of order 2 × 2. 35. ∴ I n  I n  I n ∴ ( I n )1  I n [ AB  I , B  A1 ] 36. Every positive integral power of a symmetric matrix is always symmetric. So, A3 is symmetric.

Also, it is easy to prove by inducting that  cos nα sin nα  ( Aα ) n    sin nα cos nα 

44.

 BA  AB  ( AB  BA) 38. Minors of the elements of | A | are M11 = S, M12 = r, M21 = β, M22 =α ∴ Co-factors of the elements of | A | are C11 = S, C12 = 2r, C21 = 2 β and M22 = α t

 δ  γ  δ β ∴ Adj. A      β α   γ a  39. By direct multiplication, we find that A4 = I ∴ A, A3  I , Hence. A1  A3 40. Since P(a) = 1, every 2-rowed minor is zero. 1 λ ∴ ⇒ 1  λ  0 ⇒ λ 1 1 1  1 2 1    41. Let A   2 1 3  for zero solution as the only solution  λ 1 1   we must have P(a) = 3 So, So | A | ≠ 0. 4 ∴ 2  2(2  3λ )  1.( 2  λ ) ≠ 0 or λ ≠ 5 42. Since the eigen values of A21 are the reciprocals of the eigen 1 1 values of A, so the required values are 1, and . 2 3

 1    2  AA°    i    2 

 i    1       2    2     1   i    2    2 

 i    2   1 0     1   0 1    2  

 I 2 Hence, A is un nitary. 45. The characteristic equation of A is A  λ I 2  0 1  λ 2  ∴ 0   03 4  λ  or, (1 · I) (2 4 · I) · 6 = 0 or I2 + 3I 2 10 = 0 By Cayley Hamilton theorem, A satisfies x2 + 3x 2 10 =0 Hence, A is the zero of polynomial, f(x) = x2 + 3x 2 10 = 0.

37. Let A and B be Hermitian matrix. Then, Aθ = A and B = B ∴( AB  BA)θ  ( AB )θ  ( BA)θ  Bθ Aθ  Aθ Bθ

 cosα sinα   cosβ sinβ  Aα , Aβ     sinα cos α  sinβ cos β 

2 x 0   1 0  1 0  46.      x x  1 2  0 1  2 x 0  1 0  ⇒    0 2 x  0 1  1 So, 2x  1 ⇒ x  2 47. Since one of the diagonal elements of the upper triangular matrix is O, So | A | = 0 and therefore A-1 does not exist. 48. ( A : B )      1 2 1 : 3   1 2 1 : 3       2 2 3 : 2  0 6 5 : 4   3 1 2 : 1  0 7 5 : 8     1 2 1 : 3    0 6 5 : 4  5 10 0 0 : 3   6 ∴ P(A : B) = P(a) = 3 numbers of unknowns. Hence, the given system has a unique solution. 1 0   3 0  49. A(adj A)  A  A . I 2  3    0 1  0 3 50.

(A  B )2  A2  B 2 ⇒ AB  BA  0   ⇒  1 1   2 1 

 a 1   a 1   0 0             b 1   b 1   0 0 

Matrices and Determinants ▌ 3.63

  ⇒   a  b 2   a 2 a  1  2 a  b 3   b  2 b  1    

  0 0  0 0   

  ⇒   2a  b  2 a  1 0 0   2a  2 b  4  0 0   ⇒  a  1  0,  b  4 ⇒ a  1, b  4 51. Since P(a) every 3-rowed miner must vanish,

2 1 3   ∴ A  0 ⇒ 4 7 λ   0 ⇒ 1 4 5   

0 9 7    0 9 λ  20 1 4 5  

∴ 9 λ  20  63  0 λ  13  2 3 57. Since    12  12  0 4 6  2 3 ∴  is not invertible. 4 6

deterMinantS a1 1. If D  a2 a3

b1 b2 b3

c1 c2 and c3

a1  pb1 , b1  qc1 D'  a2  pb2 b2  qc2 a3  pb3 b3  qc3 (a) (b) (c) (d)

c1  ra1 c2  ra2 , then c3  ra3

D’ = D (1 + pqr) D’ = D D’ = D (1 – pqr) D’ = D (1 + p + q + r)

r 1 2. If ∆ r  ( r 1)2 ( r 1)3

n 2 n2 3n3

(a) 1 (c) 3

6 4 n  2 , then is equal to 3n2  3n (b) 2 (d) 0

3. If A1B1C1, A2B2C2 and A3B3C3 are three three-digit numbers, each of which is divisible by k, then

5. Let λ and α be real. The set of all values of λ for which the system of linear equations λx + (sin α) y + (cos α) z = 0 x + (cos α) y + (sin α) z = 0 –x + (sin α) y – (cos α) z = 0 has a non-trivial solution, is (a ) [0, 2 ]

( b) [2 2 , 0]

(c ) [2 2 , 2 ]

(d ) none of these

6. a, b, c are in G.P. with common ratio r1 and α, β, γ are in G.P. with common ratio r2. If the equations ax + αy + z = 0, bx + βy + z = 0, cx + γy + z = 0 have only trivial solution, then (a) a, α = 0 (b) r1, r2 = 1 (c) r1, r2 ≠ 1 (d) r1 = r2 1 1 1 , and are respectively the pth, qth and rth terms of a b c

7. If

an A.P., then the value of the determinant 8

C3

9

C5

10

C7

9

C6

10

C8 is

A1

B1

C1

8

∆  A2

B2

C2 is

C4

9

A3

B3

C3

Cn

(a) (b) (c) (d)

divisible by k divisible by k2 divisible by 2k none of these

4. If the three digit numbers A28, 3B9 and 62C, where A, B and C are integers between 0 and 9, are divisible by a fixed integer A 3 6 k, then the determinant 8 9 C is 2 B 2 (a) divisible by k (b) divisible by k2 (c) divisible by 2k (d) none of these

(a) (b) (c) (d)

10

Cn 2

11

Cn  4

abc pqr 0 none of these

8. If [x] denotes the greatest integer less than or equal to x, then the value of the determinant [e]

[π ]

[π 2 2 6]

[π ]

[π 2 2 6]

[e]

[π 2 6]

[e]

[π ]

2

(a) (b) (c) (d)

–8 8 0 none of these

, then

3.64

▌ Mathematics

1 n n n 2 2 9. If Dk  2 k n  n 2 n  n and ∑ Dk  48, k 1 2k  1 n2 n2  n  2 then n equals (a) 4 (c) 8

(b) 6 (d) none of these

10. If α, β, γ are different from and are the roots of ax3 + bx2 + cx + 25 d = 0 and (β 2 γ) (γ 2 α) (α 2 β) = , then the determinant 2 α β γ 1 α 1 β 1 γ ∆ α β γ equals

α2

(a )

β2

γ2

25d 2a

( b)

25d (c ) 2( a  b  c  d )

(c )

1

1

(d ) none of these

1 0

(d )

0 1

1 2005 0 1

2005

2005

1

(b) pq + qb + rc (d) none of these

13. If A + B + C = π, eiθ = cos θ + i sin θ and

(a) (b) (c) (d)

1 a bc

x3 sin x then sin 2 x

∆ (x) is divisible by x ∆ (x) = 0 ∆′ (x) = 0 none of these 1 bc a

1

1

1

16. If ∆1  a

b

c , ∆ 2  1 ca

a

b

c

2

2

b then

1 ab c

2

( b) ∆1  2 ∆ 2  0 (d ) ∆1  2 ∆ 2 17. If A  B  C  π, sin(A  B  C ) sin B cos( A  B )

(a) 0 (c) 2 sin B tan A cos C

sin B

sin C

0 tan A is equal to  tan A 0

(b) 1 (d) None of these

18. If 1  I , and ω is a non-real cube root of unity then the 1 ω2 1 i  ω 2 1 1  i  ω equal to value of i 1 i ω 2 1 1

(a) 0 (c) 1 eiC e2 iB eiA

(a) (b) (c) (d)

1

qb rc ra pb is pc qa

e2 iA z  eiC eiB

(d )

x 1+ x2 2 ex 15. If ∆ ( x )  log(1 + x ) cos x tan x

then the value of

12. If p + q + r = 0 = a + b + c, then the value of the determinant pa qc rb

ab a bc

(c ) ∆1  ∆ 2

( b)

2005 1

( b)

( a ) ∆1  ∆ 2  0

1   2  , A   1 1 and Q  PAP' , then p ' Q 2005 P is   3 0 1  2 

 3  11. If P   2  1   2 (a )

25d a

cb a bc b a (c ) a bc (a )

e− iB eiA then e2 iC

Re (z) = 4 Ιm (z) = 0 Re (z) = 2 4 Im (z) = 2 1

14. (b + c) (y + z) 2 ax = b 2 c, (c + a) (z + x) 2 by = c 2 a, (a + b) (x + y) 2 cz = a 2 b, where a + b + c ≠ 0, then x =

(b) i (d) o

(a) 1 (c) ω 1

2iπ / 3

19. The value of determinant e e2iπ / 4

e iπ / 3 1 e22 iπ / 3

e iπ / 4 e2 iπ / 3 is 1

(a ) 2  2

( b) [2  2 )

(c )  2  3]

(d ) 2  3

0 cos x sin x 20. If sin2x  1, the sin x 0 cos x equal cos x sin x 0 (a) 3 (c) 1

(b) 2 (d) None of these

Matrices and Determinants ▌ 3.65

cos x sin x cos x 21. Let f (x )  cos 2 x sin 2 x 2 cos 2 x Then, f `(π /2)  cos 3x sin 3x 3cos 3x (a) 8 (c) 4

22. The system of equation 3x  y  z , 5x  2 y  3z  2, 15x  6 y  9 z  5 has (b) Two distinct solutions (d) Infinitely many solutions

1 23. The determinant ∆  m C1 m

C2

(a) m(m + 1) (c) 0 1 24. The determinant a a2 (a) a + b (c) b c

1 C1

1 C1 is equal to

m 1

m 2

m 1

m 2

C2

C2

(b) 1 (d) None of these 1 b b2

1 c is divisible by c2 (b) b 2 c a (d) c

a 2  x ab ac 25. The determinant ∆  ab is divisible by b2  x bc 2 ac bc c x (a) x (c) x3

(b) x2 (d) None of these

0 1 1 26. Let d  3 0 2  6 then the value of is independent 3 4 0 of

m

∑∆ r 1

r

(a) x (c) z

(b) y (d) All of these

27. The system of equation ax + by = P, cx + dy = q, ex + fy = r will be consistent if a b p a b p ( b) c

d

q 0

r

e

f

r

b

p

a

b

p

(c ) c

f

r 1

(d ) c

d

q 1

e

d

q

e

f

r

(a ) c

d

q 1

e

f

a

2 cos x 1 0 π 1 2 cos x 1 then f `  equal to  3 0 1 2 cos x

(a) – 5 (c) 2 3

(b) 6 (d) 2

(a) A unique solutions (c) No solutions

29. If f ( x ) 

28. The system of equations x + 4y 2 2z = 3, 3x + y + 5z = 7, 2x + 3y + z = 5 has (a) a unique solution (b) no solution (c) a trivial solution (d) None of these

(b) – 4 (d) – 2

30. If x is a positive integer,  x! ( x  1)! ( x  2)!   then ∆   ( x  1)! ( x  2)! ( x  3)! is equal to ( x  2)! ( x  3)! ( x  4)!   (a ) 2 x !( x  1)!

( b) 2 x !( x  1)! ( x  2)!

(c ) 2 x !( x  3)!

(d ) 2 x !( x  1)! ( x  2)!

31. The system of equation x  y  z  1, 3x  6 y  z  8,∞x  2 y  3z  1 has a unique solutions for (a) a ≠ 0 (b) all real ∝ (c) all integral ∝ (d) all rational ∝ 32. The system of equation x  2 y  11,  2 x  4 y  22 has (a) only one solution (b) finitely many solutions (c) no solutions (d) infinitely many solutions 33. The system of equation x  y  2 and 2 x  2 y  3 has (a) no solution (b) a unique solution (c) finitely many solution (d) infinitely many solutions 34. The equation x + 2y + 2z = 1 and 2x + 4y + 4x = 9 have (a) only one solution (b) a unique solution (c) only two solutions (d) all rational µ 35. If the three linear equations x  4 ay  az  0, x  3by  bz  0 & x  2cy  cz  0 have a non-trivial solution, then a, b, c are in (a) A. P (b) G. P (c) H. P (d) None of these 3µ µ 1 µ 1 36. Let aµ 3  bµ 2  cµ  d  µ  3 2µ µ  2 µ  3 µ  4 5µ be an identity in µ, where in a, b, c, d are non-constants, then the value of d is (a) 5 (b) 2 6 (c) 9 (d) 0 37. If the equations ax + by + cz = 0, bx + cy + az = 0 and cz + ay + bz = 0 and have a non-zero solution, then which one of the following is true? (a) a + b + c ≠ 0 (b) a = b = c or (a 2 b)2 + (b 2 c) + (c 2 a)2 = 0 (c) ( a  b)2  ( b  c )2  ( c  a )2 ≠ 0 (d) None of these

▌ Mathematics

3.66

anSwer Key 1. (a)

6. (c)

11. (b)

16. (a)

21. (c)

26. (d)

31. (c)

36. (b)

2. (d)

7. (c)

12. (a)

17. (a)

22. (c)

27. (b)

32. (c)

37. (b)

3. (a)

8. (a)

13. (b, c)

18. (d)

23. (b)

28. (b)

33. (a)

4. (a)

9. (a)

14. (a)

19. (b)

24. (b)

29. (c)

34. ( )

5. (c)

10. (d)

15. (a)

20. (d)

25. (a, b)

30. (b)

35. (c)

SolutionS 1.

D'

a1  a2 a3

b1 b2 b3

c1 b1 c2  pqr b2 c3 b3

c1 c2 c3

a1 a2 a3

Now

A1 ∆  A2 A3

[All other determinants will vanish]

A1  A2 A3

= (1 + pqr) D. n

∑ ( r 1)

n

6

n

n

r 1

r 1

(

)

2 n2

4n  2

3n3

3n2  3n

A1  A2 A3

2

3

n

∑ ( r 1)

)

1 ( n 1 n 2 

n

1 ( n 1) n( 2 n 1) 2 n2 6 2 1 ( n 1 n2 3n3 4

)



6 4n  2 3n2  3n

6 1 1 2 n ( n 1) 2( 2 n 1) 2 n 12 3n ( n 1 3n2

)

6 2( 2 n 1 3n( n 1)

[Taking n (n 2 1) common from C1 and n common from C2] = 0. [ C1 and C3 are identical]. 3. Since A 1B 1C 1 , A 2B 2C 2 and A 3B 3C 3 are divisible by k, therefore 100A1 + 10B1 + C1

= n1k

100A2 + 10B2 + C2

= n2k

100A3 + 10B3 + C3

= n3k

where n1, n2, n3 are integers.

B1 100 A1  10 B1  C1 B2 100 A2  10 B2  C2 B3 100 A3  10 B3  C3

B1 B2 B3

n1k A1 n2 k  k A2 n3k A3

B1 B2 B3

n1 n2  k ∆1 n3

⇒ ∆ is divisible by k [since elements of ∆1 are integers, ∴ ∆1 is an integer].

r 1

[Q the terms in C1 are dependent on r whereas the terms in C2 and C3 are constant]

C1 C2 C3

[Applying C3 →C3  10C2  100C1 ]

r 1

2. We have, ∑ ∆ r  ∑ r 1

B1 B2 B3

4.

Since A28, 3B9 and 62C are divisible by k ∴ A28 = n1k = 100A + 20 + 8 ...(1) 3B9 = n2k = 300 + 10B + 9 ...(2) 62C = n3k = 600 + 20 + C ...(3) where n1, n2 and n3 are integers. A 3 6 Now,  8 9 C 2 B 2 3 6 A  100 A  20  8 300 10 B  9 600  20  C 2 2 B = [Applying R2 → R2 + 100R1 + 10R3] A  n1k 2

3 n2 k B

A 3  k n1 n2 2 B

6 n3k [Using (1), (2) and (3) 2 6 n3 , which is divisible by k . 2

Matrices and Determinants ▌ 3.67 5. Since the system has a non-trivial solution,

A  ( p  1) D A  ( q  1) D A  ( r  1) D

λ sin α cos α therefore 1 cos α sin α  0 1 sin α cos α ⇒ λ (2cos2 α 2 sin2 α) 2 (2 sin α cos α 2 sin α cos α) 2 (sin2 α 2 cos2α) = 0 ⇒ 2 λ + sin 2α + cos 2α = 0 ⇒ λ = sin 2α + cos 2α  π ⇒ λ  2 cos  2α   4   π Since 1  cos  2α    1 ∀ α ∈ R 4  ∴  2  λ  2 i.e. λ ∈  2 , 2 .   6. Since a, b, c are in G. P. with common ratio r1 and α, β, γ are in G. P. with common ratio r2, therefore a ≠ 0, α ≠ 0, b = ar1, c = ar12 β = αr2, γ = αρ22 Also, the system of equations have only trivial solution, so a α b β   c γ  a  ⇒  ar1  ar 2  1

α ar2

0 0   1 1 ≠0 r2  r1 1  r1 

⇒ r1 ≠ r2 , r1 ≠ 1, r2 ≠ 1.

⇒

1 1 1 , , are pth, qth and rth terms of an A.P. a b c

0 0 0  abc p q r 1 1 1 [Applying R1 → R1 2 (A 2 D)R3 2 ∆R2] = 0 8.

Since 2 < e < 3, 3 < π < 4 and 3 < π2 2 6 < 4, the given determinant reduces to 2 3 3 3 3 2  8. 3 2 3 n

∑1

n

n

∑ 2k

n2  n  2

n2  n

∑ 2k 1

n2

n2  n  2

k 1

n

n

9.  ∑ Dk 

k 1

n n n ⇒ 48  n2  n n2  n  2 n2  n n2 n2 n2  n  2 0 n n 2  n n 2 n  n C2 → C2 → C1 0 n2  n  2 n2 2

= 2 (n3 + n2 + 2n 2 n3) = 2 (n2 + 2n) ⇒ 24 = n2 + 2n ⇒ 25 = (n + 1)2 ⇒ n + 1 = 5 (Q n ∈N) ∴ n = 4. 10. Taking α , β , γ common from c1 , c2 , c3 respectively, we get 1 1 α ∆  αβγ 1 α

1  A  ( p  1) D a

1 1 α  αβγ 1 α

1  A  ( q  1) D b 1  A  ( r  1) D c where A is the first term and D is the common difference. 1 bc ca ab a ∴ p q r  abc p 1 1 1 1

r 1

k 1

⇒ aα ( r2  r1 )(1  r1 ) (1  r2 ) ≠ 0

Since

q 1

n

 1 1 1 1    1 ≠ 0 ⇒ aα  r1 r2 1 ≠ 0  r 2 r 2 1 ar22 1 1  2 1 0 0    ⇒ aα  r1 r2 2 r1 1 2 r1  ≠ 0 r 2 r 2 2 r 2 12 r 2  1 2 1 1 

7.

p 1

k 1

1 1 ≠ 0 1

1  ⇒ aα ( r2  r1 )(1  r1 )  r1  r12

 abc

1 b q 1

1 c r 1

1 α β 1 β

1 1 γ 1 γ

1 1  1 β 1 α 0 β α

1 1  1 γ 1 α 0 γ α

[using C2 → C2  C1 and C3 → C3  C1 ] 

αβγ (  1)( β  α )(γ  α ) 1  γ (1  α )(1  β )(1  γ ) 1



αβγ (α  β )( β  γ )(γ  α ) (1  α )(1  β )(1  γ )

1 β 1

3.68

▌ Mathematics

As α , β , γ are the roots of ax 3  bx 3  cx  d  0,

)

ax 3  bx 2  cx 2  d  a ( x  α ) ( x  β ( x  γ

14.

Adding all three equations, we get (a + b + c) (x + y + z) = 0 ⇒ x + y + z = 0 since , a + b + c ≠ 0. From the first equation (b + c) (2 x) 2 ax = bc cb ∴ x a bc

15.

Let ∆ (x) = A + Bx + Cx2 + Dx3 +...

)

and αβγ  dla

( d / a )( 25 / 2)  25d (a  b  c  d ) / a 2(a  b  c  d )

Thus, ∆  11.

We have,  3  P 'P   2  1   2

1  3 1    2  2 2 0 0   3 1 3 0 1   2  2 2 

0 1 0 ∆(0)  0 1 0  0 ⇒ A  0 1 0 0 1 0 0

⇒ P′P = I or P′ = P1 As, Q = PAP′ ∴ P′Q2005P = P′[(PAP′) (PAP′) ... 2005 times]P

⇒ ∆ ( x )  x  Cx 2  Dx 3  ... ⇒ ∆ ( x ) is divisible by x

2005Times

 1 1 2  1 1  1 1  1 2  Now, A   , A      0 1 0 1 0 1 0 1   1 2   1 1  1 3 2005  1 2005 A3       ...A   1  0 1  0 1 0 1 0  1 2005 ∴ P ' Q 2005 P   . 1  0 12.

1 16. Q ∆1  a a2

13. z  e . e . e iB

qb rc ra pb pc qa

eiA iC

ei ( C  A)

i ( C  B )

e

ei ( B C )

e

iB

ei ( AC )

a 1 a2 1 a a2  b 1 b2 1 b b2 c 1 c2 1 c c2 1  a a2

ei ( B  A) ei ( A B )

eiB − eic

 eiA  eiB

eiC

Since e i (A  B  C )  eiπ  cos π  i sin π ) 1 2eiB

0 ⇒ z  2e

iA

eiA

0

0

0

eiB

eiC

(Using R1 → R1 + R3, R2 →R2 + R3) ⇒ z = 2 2eiB {2 2 ei (A + C)} ∴ z = 4 ei(A + B + C) = 4 eiπ = 2 4

1 b b2

1 c c2

 ∆1

eiC

∴ ∆1  ∆ 2  0

eiA  eiB  eic ⇒ z  1  eiA

1 c  ( a  b) ( b  c ) ( c  a ) c2

a abc a 2 1  b abc b2 abc c abc c 2

⇒ ∆ = pqr (a3 + b3 + c3) 2 abc (p3 + q3 + r3) ∴ ∆ = pqr (3 abc) 2 abc(3pqr) = 0

iA

1 b b2

1 bc a & ∆ 2  1 ca b 1 ab c

Since p + q + r = 0 = a + b + c (given) ⇒ p3 + q3 + r3 = 3pqr or a3 + b3 + c3 = 3 abc pa Let ∆  qc rb

0 1 0

∆ '(0)  0 1 0  0 1 1  0 1 0  1 1 0 0 1 0 0 0 1 0

 ( P ' P ) A( P ' P ) A( P ' P )...( P ' P ) A( P ' P )     IA2005  A2005

0 1 0

17. The given determinant is skew-symmetric of odd order [∴ cos (A + B) 2 cos(π 2 C) = cos C and sin (A + B + C) sin π = 0] ∴ if D is the value of the determinant, then ∆  (1)3 ∆ ⇒ ∆  ∆ ⇒ 2 ∆  0 ⇒ ∆  0 Let ∆  18.

1 i  ω 2 1 1  i  ω 1 i ω 2 1 1 1 i

ω2

Applying R3 → R3  (R1  R2 )

0

Matrices and Determinants ▌ 3.69

1 ω 2 1 i  ω 2 Then, ∆  i 1 1  i  ω  0 0 0 0 eiπ / 3 1 e2 iπ / 3

1 19. Let ∆  eiπ / 3 e iπ / 4

21.

cos x cos x cos x  cos 2 x 2 cos 2 x 2 cos 2 x cos 3x 3cos 3x 3cos 3x

eiπ / 4 ei 2 π / 3 ei 2 π / 3

cos x cos x cos x  cos 2 x sin 2 x  4 sin 2 x

Taking eiπ / 3common from R2 iπ / 3

∴∆ e

eiπ / 3 eiπ / 4 iπ / 3 e eiπ e2 iπ / 3 1

1 1 eiπ / 4

cos 3x

1 1 e iπ / 4 ∆ 1 1 ei / π eiπ / 4 e2 iπ / 2 1

1

2 1

1

1

1 11

∴ 3(5x + 2y – 3z) = 6, i.e. also, 15x + 6y – 9z = 5 and 6 ≠ 5; ∴ the system has no solution. 23. Put m = 1 and solve the determinant.

 1 i  1 i  1 i   1 (1  1) 11   1    2  2  2  1 i 1 i 2    0 1 2 2 2

24. ∆ = (a 2 b)(b 2 c)(c 2 a) (by factor theorem), therefore, the given determinant is divisible by (b 2 c). 25. Put a  1, b  1, c  1 we get D  x 2 ( x  3) which is divisible by x and x2

 2  2 26.

 (2  2 ) 20. Qsin 2 x  1 ∴xπ / 4



0

2 1

1

2

2

r 1

m



1.∑ 2r r 1 m

(3m  1) 2 y 2.∑ 4r

2m  1 x 2m  1 3m  1 2 y 3m  1 4 m  1 3z 4 m  1

0

r 1

m

( 4  1)

3z

m

3.∑ 4r r 1

which is independent of x , y , z.

2 1

x

2

1

2 1

m

A, B, C ; ∑ ∆ r  ( 2 m  1)

0 cos x sin x then sin x 0 cos x cos x sin x 0 1

1 0 9

3 1 1 3 1 1 22. ∆  5 2 3  3 5 2 3  0 15 6 9 3 2 3

2

0

1 0 1

 2 (1  3)  0 1 (9  1) 1  8  4

1 i

1

sin 3x  9 sin 3x

1 1 0 0 f ′( π / 2)  0 0  2  0  1 3 1 1 0

Multiplyiing eiπ/3 in C2 then

1

 sin x sin x cos x f ′( x )  2 sin 2 x sin 2 x 2 cos 2 x  3sin 3x sin 3x 3cos 3x

2

27. The given equations can be written as ax  by  p  0

2 0

1 1 2 0  1 1 1      1 0 1  2 2 2 1 1 0 1  {0  1(0  1)  1(1)}2 2 0

cx  dy  q  0 2

ex  fy  r  0 Which are three equation in two variables and consistent either.

a c e

b p a d  q  0 or c f r e

b d f

p q 0 r

3.70

▌ Mathematics

1 4 2 28. ∆  3 1 5  0 and any one of ∆1 , ∆ 2 , ∆ 3 ,  0 2 3 1 the system of equation have no solutions. 29. f `(x )∆ 

2 sin x 0 0 1 2 cos x 1  0 1 2 cos x

 4  4  0 ⇒ ( 2 x  4 y  22) ⇒ the given equation have no solutions. 33. Let, A 

1 2  ( 2  2)  0; and 2 x  2 y  4 1 2

⇒ 2x + y + 3 the given equation has no solution. 34. Now, x + 2y + 2Z = 1 ⇒ 2(x + 2y + 2z) = 2 i.e., 2x + 4y + 4z + 1 also, 2x + 4y + 4z + 9 (given), ∴the equation has no solutions.

2 cos x 0 0 2 sin x 1 0  0 1 2 cos x 2 cos x 1 0 1 2 cos x 1 2 sin x 0 0

35. The system has a non-trival solution if

 2sin x ( 4 cos 2 x  1)  2 cos x ( 4 sin x cos x )  2 sin x ( 4 cos 2 x  1)  8sin x cos 2 x  2 sin x  8 sin x cos 2 x  8 sin x cos 2 x  8 sin x cos 2 x  2 sin x  24 sin x cos 2 x  4 sin x f '( π / 3)  24 

 1  1 2  2  32. Let A   A   2 4  2  4     

3 1 3  4 2 4 2

1 4a a 1 3b b  0 1 2c c Operate R 1 → R 2 , and R 2 → R 2  R 3 1 4 a  3b a  b 2 ac 1 3b  2c b  c  0 ⇒ b  ⇒ a, b ac 1 2c c C are in H.

 3 3  2 3  3 30. Put x = 1 in the determinant and from the given option we will see that the value of the determinant and the option (b) coincide.

0 1 1 36. Put µ = 0 on the sides, we get, d   3 0 2  6 3 4 0 37. The system has a non-zero solution if a b c b c a 0 c a b

1 1 1 31. For unique solution 3 6 1 ≠ 0 α 2 3 ⇒ (18  2)  (9  ∞)  1(6  6 a ) ≠ 0 ⇒ 16  9  α  6  6α ≠ 0 13  5α  13 ≠ 0 ⇒ 5α ≠ 13 ⇒ ≠ 5

1  [( a  b  c ) ( a  b)2  ( b  c )2  ( c  a )2 ] 0 2 (Simplify it) ⇒ a + b + c = 0 or (a – b)2 + (b – c)2 = 0 ⇒ a = b =c.

nda box 5 6 1 1. Let A    . Let there exist a matrix B such that 2 1 5  35 49  AB    . what is B equal to? 29 13  5 (a )  2 5  (c )  1 4 

1 4  6 3 2  6 3 

2 ( b)  5 2  (d )  6 3 

6 3  1 4 5  1 4 

2. Consider the following statements 1. If A' = A; then A is a singular matrix, what A' is the transpose of A. 2. If A is a square matrix such that A3 = I then A is nonsingular. Which of the statements given above is/are correct? (a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2 3. What is the value of cos 15 cos 15 cos 45 cos 15  ? cos 45 cos 45 sin 45 sin 15

Matrices and Determinants ▌ 3.71

(a )

1 4

(c ) 2

3 2 3 (d ) 2 4 ( b)

1 4

4. Let A be an n × n matrix. If det (λA) = λs det (A), what is the value of s? (a) 0 (b) 1 (c) −1 (d) n x2 y2   1 and C be the circle 9 4 2 2 x  y  9. Let P = (1, 2) and Q = (2, 1). Which one of the

5. Let E be the ellipse

following is correct? (a) Q lies inside C but outside E (b) Q lies outside both C and E (c) P lies inside both C and E (d) P lies inside C but outside E. 6. If A be a real skew-symmetric matrix of order n such that A2  I  0 , I being the identity matrix of the same order as that of A, then what is the order of A? (a) 3 (b) Odd (c) Prime number (d) Even 1 2  7. Let A      aij  , where i, j = 1, 2. If its inverse matrix 3 4 

1 2 3   A  10. If 2 1 2  , then which one of the following is 3 2 1   correct? (a) A is symmetric matrix (b) A is anti-symmetric matrix (c) A is singular matrix (d) A is non-singular matrix

ω 0  100 11. If A    where ω is cube root of unity, then what is A 0 ω   equal to? (a) A (b) 2 A (c) Null matrix (d) Identity matrix 12. A matrix X has (a + b) rows and (a + 2) columns; and a matrix Y has (b + 1) rows and (a + 3) columns. If both XY and YX exist, then what are the values of a, b respectively? (a) 3, 2 (b) 2, 3 (c) 2, 4 (d) 4, 3 2 4 0    13. If 0 5 16   20, then what is the value of p? 0 0 1  p    (a) (b) (c) (d)

is bij  what is b22 ?   (a ) 2 2 3 (c ) 2

( b) 1 (d ) 2

1 2

8. If X and Y are the matrices of order 2 × 2 each and 2X – 3Y 7 9 13 0  =   and 3 X  2Y    , then what is 7  13   4 13 Y equal to ?  1 3 (a )   2 2 1  1 3 ( b)    2 1  3 2 (c )   2 1 5  3 2  (d )   1 2 5  9. If a matrix A is symmetric as well as anti-symmetric, then which one of the following is correct? (a) A is a diagonal matrix (b) A is a null matrix (c) A is a unit matrix (d) A is a triangular matrix

14.

0 1 2 5

a b c l m n  2, then what is the value of the determinant p q r 6 a 3b 15c 2l m 5n ? 2 p q 5r (a) (b) (c) (d)

10 20 40 60

15. If a, b, c are non-zero real numbers and 1 a 1 1 1 ab 1  0 then what is the value of 1 1 1 c 1 1 1   ? a b c (a) 2 (c) − 1

(b) 1 (d) 0

3.72

▌ Mathematics

2a 3r x a r 16. A  4b 6s 2 y  λ b s 2c 3t z c t

x y , then what is the z

value of λ? (a) 12 (b) − 12 (c) 7 (d) − 7

1 i 17. What is the value of

ω2

ω

ω

i

ω i 2

1  2i  ω

2

, where

ω ω i ω 2

ω is the cube root of unity? (a) − 1 (b) 1 (c) 2 (d) 0

anSwer Key 1. 2. 3. 4.

(c) (b) (c) (d)

5. 6. 7. 8.

(d) (a) (d) (c)

9. 10. 11. 12.

(b) (d) (a) (b)

13. (b) 14. (d) 15. (c)

16. (b) 17. (d)

SolutionS  5 6 1 1. A     2 1 5 5 2   And let B   1 6  4 3   5 2  5 6 1   ∴ AB    1 6   2 1 5  4 3   

det (λA)  λ n det ( A) But det (λA)  λ s det A ⇒ sn 5. For a point P(1, 2) 4 (1)2  9 ( 2)2  36  40  36  0 and 12  22  9  5  9  0 ∴ Point P lies outside of E and inside of C.

 25  6  4 10  36  3   10  1  20 4  6  15 

6. If A be a real skew-symmetric matrix of order n such that A2 + I = 0, then order of A is 3.

 39 49     29 13 

1 2  7. Q A   3 4 

Hence, opption (c) is correct. 2. Only statement (2) is correct. 3.

4. If A is an n × n matrix, then

cos 15 cos 15 cos 45 cos 15  cos 45 cos 45 sin 45 sin 15 (sin 45 cos 15  cos 45 sin 15 )  ( cos 45 sin 15  sin 45 cos 15 )  sin( 45  15 )  ( 45  15 ) = - sin 30  sin 30 1 1 1 =    2 2 4

1  4 2  ∴ A−1    2 3 1   4 2  ⇒  bij     3 1  ⇒ b22 

1 2

7 0  8. Given, 2 X  3Y =    7 13 9 13 3 X  2Y    4 13

...(i) ...(ii)

on multiplying Eq, (i) by 3 and Eq. (ii) by 2 and subtracting Eqs. (i) from (ii) we get

Matrices and Determinants ▌ 3.73

9 13 7 0  13Y  2    3  4 13  7 13  39 26  ⇒ 13Y    13 65  3 2 ⇒Y   1 5  9. Since, A′  A and A′  A ⇒ A  A ⇒ A  0

a b c 14. Q l m n  2 p q r 6 a 3b 15c a b c ∴ 2l m 5n  30 l m n  30  2  60 2 p q 5r p q r 1 a

1

1

1 b

1

1

1

1 c

15. Given

10. Here we see that its diagonal elements are not zero, so it is not ant symmetric matrix. Now, | A |  1(1  4)  2( 2  6)  3 ( 4  3)  5  16  3  18 ≠ 0 Hence, it is non-singular matrix. ω 0  11. Given, A     0 ω

0  ω2 

ω 2 A3   0 ω 2  0

0  ω 0    ω2   0 ω 

0  ω2 

Applying C2 → C2  C1 and C3 → C3  C1 1  a a a →

1

b

0

1

0

c

0

1( ab)  c ( b  ab  a )  0 ⇒ ab  bc  ca  abc  0 ⇒

1 1 1   1 a b c

2a 3r x a r 16. Given, 4b 6s 2 y  λ b s 2c 3t z c t

x y z

Taking 2 common from C1 and 3 from C2 from LHS a ∴2  3 2b

Similarly, ω100 A100    0

0

⇒ Expanding along R3 .

ω 0  ω 0  Now , A2      0 ω  0 ω ω 2  0

1

0   ω100 

ω 0     0 ω

x 2y

a r  λ b s

c t z

c

t

x y z

Taking 2 common from R2 and 1 from R3 from LHS

(Q ω 3  1)

A 12.  X ( ab )  ( a2 ) and Y 

r 2s

a r ∴ 12 b s c

( b  1)  ( a  3)

⇒ a  2, b  3

a r  λ b s

t z

1 i

⇒ 2 [5 (1  p)  0]  0  0  20 ⇒ 10 (1  p)  20 ⇒ p  1

t

17. Let ∆

ω2

ω

ω

i

ω i 2

1  2i  ω

2

2 4 0  13.  0 5 16   20   0 0 1 p   

c

⇒ λ 12

As  XY  and YX  exist ∴ a  2  b  1 and a  3  a  b

x y

ω ω i ω 2

Applying R3 → R1  R2  R3 1 i  ω2  i 0

ω2 ω ω 0

i  0 0

x y z

This page is intentionally left blank.

C

H

3

A

P

T

E

R

Trigonometry TrigonomeTry There are six trigonometric functions called trigonometric ratios or circular functions. Any trigonometric function can be expressed algebraically in terms of any specified trigonometric function of the same angle. The sign of the trigonometric ratio of any angle depends on the quadrant in which the angle lies. A system of rectangular coordinates axes divides a plane into four quadrants. An angle lies in one and only one of these quadrants. We can see it an shown in Fig. 3.1 II Quadrant

I Quadrant

Only sin θ All tratios and cosec θ ve are positive

Only tan θ Only cos θ ve ve and cot θ and sec θ

(ii) If n is even, the result contains the same trigonometric function as the given function, but if n is odd, the result contains the corresponding co-function, i.e., sine becomes cosine, tangent be comes, cotangent, secant becomes cosecant and vice versa. For instance, to determine sin (540 – q), we note that 540° – q = 6 ´ 90° – q, a second quadrant angle if 0 < q < 90°; in this quadrant sine is +ve. Since, the angle contains an even multiple of 90°, the sine is retained. So, sin (540° – q) = Sin q and to determine cos (630° – q), +ve note (630 – q) = 7 ´ 90° – q, a third quadrant angle if 0 < q < 90°, in this quadrant cosine is negative. Since the angle contains an old multiple of 90°, cosine is replaced by sine. So, cos (630° – q) = – sin q. Table 3.1 Sine, Cosine and Tangent of Some Angles Leess Than 90° sine θ cos θ tan θ 0 0 1 0 15

Fig. 3.1 1  The trigonometric functions of the angle  nπ 2 θ  and 2  1   nπ  θ  , where n is zero or an integer, can be expressed as 2  trigonometric function of q, with either the plus or the minus sign. Such angles are known as ‘allied angles’ or ‘related angles’. The following useful working rule can be verified to determine the trigonometric functions of these angles. (i) Assuming that o < q < 90°, note the quadrant in which the given angle lies. The result has the plus or the minus sign according as the given function is positive or negative in that quadrant.

18 36

6 2 4

6 2 4

5 1 4 1 2

10  2 5 4 3 2

2

36 45 60 90

10  5 4 1

5 1 4 1

2

2

3 2 1

1 2 0

2 3 25  10 5 5 1 3 5 2 5 1 3 not defined

3.76

▌ Mathematics

• An angle made up by the algebraic sum of two or more angles is called a compound angle. • Some formulas and results regarding compound angles are: 2 sin A cos B  sin ( A  B )  sin ( A  B ) 2 cos A cos B  cos ( A  B )  cos ( A  B ) 2 cos A sin B  sin ( A  B )  sin ( A  B ) 2 sin A sin B  cos ( A  B )  cos ( A  B ) 1 1 sin C  sin D  2 sin (C  D ) cos (C  D ) 2 2 1 1 sin C  sin D  2 cos (C  D ) cos (C  D ) 2 2 1 1 cos C  cos D  2 cos (C  D ) cos (C  D ) 2 2 1 1 cos C  cos D  2 sin (C  D ) cos ( D  C ) 2 2 sin 2 A  2 sin A cos A cos 2 A  cos 2 A  sin 2 A  2 cos 2 A 1  1 2 sin 2 A 1 cos 2 A  2 sin 2 A;1  cos 2 A  2 cos 2 A 2 tan A sin 2 A  ; 1  tan 2 A 1 tan 2 A cos 2 A  ; 1  tan 2 A 2 tan A tan 2 A  ; 1  tan 2 A sin 3 A  3sin A  4 sin 3 A; cos 3 A  4 cos 3 A  3cos A; 3 tan A  tan 3 A tan 3 A  ; 1  3 tan 2 A tan( A  B  C )  tan A  tan B  tan C − tan A tan B tan C ; (1  tan B tan C  tan C tan A  tan A  tan A tan B )

identities A trigonometric equation is an identity if it is true for all values of the angle or angles involved. A given identity may be established by reducing either side to the other, or by reducing each side to the same expression, or by any convenient modification of these methods. When the angles A, B, C satisfy some given relation, many identical relations hold between their t-ratios. Some important identities: if A + B + C = p, then tan A + tan B + tan C = tan A tan B tan C cot B cot C + cot C cot A + cot A cot B = 1 B C  C   A  A B tan   tan    tan   tan    tan   tan    1 2 2 2 2 2 2  A B C   A  B C  cot    cot    cot    cot   cot   cot   2 2 2 2 2 2 sin 2A + sin 2B + sin 2C = 4 sin A sin B sin C cos 2A + cos 2B + cos 2C = – 1 – 4 cos A cos B cos C

cos2 A + cos2 B + cos2 C = 1 – 2 cos A cos B cos C  A  B  C sin A  sin B  sin C  4 cos   cos   cos    2  2  2  A  B  C  1  4 sin   sin   sin   .  2  2  2

relation Between the sides and the angles of a Triangle For a triangle ABC, the magnitude of the angles BAC, CBA, ACB will be denoted by A, B, C, and the lengths of the opposite sides BC, CA, AB by a, b, c, respectively. The lengths of the radius of the circumcircle, the inscribed side and the escribed circle opposite to A, B, C will be denoted respectively by R, r, r1, r2, r3. A b

c B

a

C

The area of the triangle will be represented by D, and the semi-perimeter by s, so that 2s = a + b + c sin A sin B sin C 1 5 5 5 (Sine Rule) a b c 2R  b2  c 2  a 2  a 2  b2  c 2  2bc cos A or cos A   2bc  2 2 2  c a b  2 2 2 b  c  a  2ca cos B or cos B   2ca 2 2 2  a b c  c 2  a 2  b2  2 ab cos C or cos C   2 ab (Cosine Rule) a  b cos C  c cos B   b  c cos A  a cos C  ( Projection Rule ) c  a cos B  b cos A  Trigonometric functions of the half-angles of the triangle  A  A ( s  b) ( s  c ) s (s  a) sin    , cos    bc bc 2 2 B B (s  c) (s  c) s ( s  b) sin    , cos    ca ca 2 2 C    s (s  c) ( s  a ) ( s  b) C sin    , cos    ab ab 2 2  Napier's Analogy  B c  bc A  tan  cot   2  2 b c     C  A ca B  tan  cot  2  2  ca   A B  a b C tan   cot  2   2  ab

Trigonometry

Now, ∆  s ( s  a ) ( s  b) ( s  c ) abc 1 1 1  bc, sin A  ca sin B  ab sin C  2 2 2 4R ∆ ∆ ∆ ∆ r  , r1  ,r  ,r  s ( s  a ) 2 ( s  b) 3 ( s  c )  A  B  C r  4 R sin   sin   sin    2  2  2  A  B  C  A r1  4 R sin   cos   cos    s tan    2  2  2  2  A  B  C  B r2  4 R cos   sin   cos    s tan    2  2  2  2  A  B  C  C r3  4 R cos   cos   sin    s tan    2  2  2  2 a cos A  b cos B  c cos C  4 R sin A sin B sin C a cot A  b cot B  c cot C  2( R  r ) r1  r2  r3  4 R  r ; and r1r2  r2 r3  r3 r1  S2

solution of Triangles Three sides a, b, c and the three angles A, B, C are called the elements of the triangle ABC. When any three of these six elements (except the three angles) of a triangle are given, the triangle is known completely and the other three elements can be expressed in terms of the given elements with the help of the relations given above and hence can be calculated.

Period and graph of Trigonometric Functions A function f(x) is said to be periodic, if for all values of x, f (x + a) = f(x), where a is a constant known as a period of the function. Since, Sin (q ± 2np) = Sin q (N Î Z, the set of all integers) cos (q ± 2np) = cos q tan (q ± np) = tan q all trigonometric functions are periodic. The smallest the period of Sine, Cosine, Second and Cosecant is 2p and that of tangent and cotangent is p. Again, the period of cos nq and sin nq is (2p/n) and that of tan n q is (p/n).

general values and solution of Trigonometric equations if sin q = sin q, then a = np + (–1)na if cos q = cos a, then q = 2np ± a if tan q = tan a, then q = np + a (n Î z) if sin q = sin a and cos q = cos a, then q = 2np + a, (n Î z) These formulas help in solving a trigonometric equation, for 1 example, if sin θ 5 , then 2 π sin θ  sin   therefore, θ = nπ + (1) n π / 6,( n ∈ z )  6

▌ 3.77

In the solution of trigonometric equations, the general values of the angle should be given unless the solution is required in any specified interval or range.

The inverse Trigonometric Functions if sin q = x then q = sin–1 x or arc sin x if cos q = x then q = cos–1 x or arc cos x if tan q = x then q = tan–1 x or arc tan x. The value which is numerically lead is called the principal value of the function. If x is +ve, the principal value of sin–1 x, π cos–1 x and tan–1 x lies between 0 and . If x is –ve, the principal 2 π –1 –1 value of sin x and tan x lies between 2 and 0 and that of cos–1 2 π x lies between and π . 2 In general, the principal value should be taken for every inverse trigonometric function unless it is indicated or implied that the general value is required. Thus,

π π   sin1 x  2 2 0  cos1 x  π π π   tan1 x  2 2 0  cot1 x  π π π 0  sec1 x  or  sec1 x  π 2 2 π π   cosec1 x  0 orr 0  cosec1 x  2 2 Note: cos–1 x and sin–1 x are defined for real values of x : – 1 £ x £ 1. cot–1 x and tan–1 x are defined for all real values of x. sec–1 x and cosec–1 x are defined for real value of x : x £ – 1 or x ³ 1.

some relation/results on inverse Trigonometric Functions π π sin1x  cos1 x  ; tan1 x  cot1 x  ; 2 2 π 1 1 sec x  cossec x  2  x y  1 1 tan x  tan y  tan1    1  xy  (xx  0, y  0, xy  1)  x y  1 1 tan x  tan y  π  tan1    1  xy  ( x  0, y  0, xy  1) 1 1 sin x  sin y  sin1  x 1  y 2  y 1  x 2    ( x  0, y  0, x  y  1) sin1 x  sin1 y  π  sin1

3.78

▌ Mathematics

2x 1 x 2  cos1 2 1 x 1 x 2 2x  tan1 (| x |  1) 1 x 2

 x 1  y 2  y 1  x 2  ( x  0, y  0, x 2  y 2  1)   2 tan1 x  sin1  tan1

π  2 tan1 x  sin1

2x 1 x2  cos1 2 1 x 1 x2

2x (| x |  1) 1 x2

PracTice eXercise TrigonomeTric raTios and idenTiTies 1. If x = a cos q + b sin q and y = ¥ sin q – b cos q then x2 + y2 is equal to (a) a2 – b2 (b) b2 – a2 (c) a2 + b2 (d) a2 + 2ab

10.

(a )

(c )

(c ) 2 2 1 3 4 (d ) 3

1

7 , then θ can lie in 6 first or fourth quadrant. second or fourth quadrant. third and fourth quadrant. first and second quadrant.

4. (1 – sin2 q) tan2 q is equal to (a) cos2 q (c) tan q

(a ) 1 (c )

(d ) 2

sec 2 A  cosec 2 A is equal to (a) sec A + cosec A (c) sec A + tan A

(b) tan A + cot A (d) cot A + cosec A

7. If 4 sin x – 1 = 0 and 0 < x < 2p, then the positive values of x are (a) 30°, 120°, 210°, 300° (b) 30°, 150°, 210°, 330° (c) 30°, 120°, 150°, 210° (d) 30°, 160°, 210°, 320°

9. The value of (a ) zero

tan 30  tan 60 is 1  tan 30 tan 60 1 ( b) 2 (d ) infinity

3

88 160 92 (c ) 160 (a )

2

(d ) 2

91 160 93 (d ) 160 ( b)

13. If sin (30° + q) = cos(60° – q f), their (a) q + f = 0 (b) q – f = 0 (c) q + f = 90° (d) q – f = 90° 14. The value of tan 40° tan 50° tan 60° is (a ) 0 ( b) 1 (c )

2

1 1 , then the angles A and B, 8. If sin (A  B )  and sin A  2 2 respectively are: (a) 15°, 45° (b) 45°, 15° (c) 30°, 60° (d) 60°, 30°

( b)

3 sin θ tan θ  1 12. If cos θ  , then the value of is 5 2 tan 2 θ

(b) sin2 q (d) none of the above

5. sec6 q – tan6 q – 3 sec2 q tan2 q is equal to (a) 1 (b) 0 (c) 2 (d) 6

(c ) 1

1

11. If sec θ  tan θ  1  2 , then sec θ is equal to

3. If sec θ 

6.

( b)

( b)

3

(a) (b) (c) (d)

2

2

2. The value of (1 + tan2 30°) is (a ) 3

cos x  sin x cos x  sin x is  2, then one of the values of cos x sin x

3

(d )

1 3

15. cos2q + 2cosq is always (a) Greater then –3/2 (b) less then or equal to 3/2 (c) Greater than or equal to –3/2 and less than or equal to 3 (d) None of these 16.

cot A  cosec A  1 is equal to cot A  cosec A  1 1  cos A sin A 1  cos A (c ) sin A (a )

1  sin A sin A 1  sin A (d ) sin A ( b)

Trigonometry

2 2 4 17. If sin θ  sin θ  1, then cos θ  cos θ is equal to (a) 4 (b) 3 (c) 2 (d) 1

(1  sin x ) (1  sinn x ) is (1  cos x ) (1  cos x )

4 18. If tan x  , then the value of 3 9 16 4 (c ) 3

3 4 16 (d ) 9

(a )

( b)

19. The value of (a ) (c )

(d ) ∞

20. If cos q = 1 – sin q, then sin q is (a ) 1  cos θ ( b)

sin θ ( 2  sin θ )

(c )

cos θ ( 2  cos θ )

21. The value of x satisfying the equation sin x 

22.

(b) 45° (d) 90°

1 7  is sin x 2 3

sin θ cos θ  is equal to 1  cot θ 1  tan θ (a) sin q + cos q (b) sin q – cos q (c) tan q + cot q (d) tan q – cot q

23. If cosec q – cot q = 2, then cosec q + cot q is equal to 1 (a ) 2 ( b) 2 3 (c ) 1 (d ) 2 24. If x = a sec q + b tanq y = b sec q + a tan q then x2 – y2 is equal to (a) 4ab sec q tan q (b) b2 – a2 (c) a2 – b2 (d) a2 + b2 25. If cos x 5 12 and x is an acute angle, 13 then

115 12

(c )

119 12

(d )

1 2

27. The minimum value of sin x 2 cos x is (a ) 2 2

( b)

(c ) 2 2 2

(d ) 0

2

(a ) 4

( b) 2  3

(c ) 2  3

(d ) 2 3

31. If A  130 and x  sin A  cos A, then (a) x > 0 (b) x < 0 (c) x = 0 (d) x ≤ 0 32. The value of tan 15  cot 15 is (a )

( b)

116 12

(d )

117 12

3

( b) 2 3 (d ) 2 4

(c ) 4

2 12 33. It sin x  sin x  1, then the value of cos x  3

cos10 x  3 cos8 x cos6 x  1 is equal to (a) 2 (c) 0

(b) 1 (d) 21

34. If sinθ  cosθ  a then the value of |sinθ  cosθ| is (a )

2  a2

( b)

(c )

a2  2

(d ) None of these

2  a2

35. The value of sin 2 5  sin 2 10  sin 2 15  ..... sinn 2 90is 1 2 1 (c ) 1 2 (a ) 7

 sin x  1  (1  tan x ) is cosx  

(a )

2

30. Tan 75º – Cot 75º is equal to

(d ) 1  cos θ

(a) 60° (c) 30°

(c )

29. If tan θ  4 , then sin θ is 3 4 4 4 4 (a ) 2 but not ( b) 2 or 5 5 5 5 4 4 (c ) but not 2 (d ) None of these 5 5

( b) 1 3

26. The maximum value of sin x  cos x is (a ) 1 ( b) 2

28. The value of tan 9  tan 27  tan 63  tan 81 is (a) 4 (b) 3 (c) 2 (d) 1

sin 30  cos 60  tan 45 is cos 0  tan 45  sin 90

1 2

▌ 3.79

1 2 1 (d ) 9 2 ( b) 8

36. cos 52  cos 68  cos172  (a) 0 (b) 1 (c) 2 (d) None of these

3.80

▌ Mathematics

37. If tan 2 θ  2 tan 2 φ + 1, then cos 2θ  sin 2 φ  (a) 2 (b) 1 (c) 0 (d) None of these

θ 1 t 2 46. If tan  t , then is equal to 2 1 t 2 (a) cosq (c) sec q

38. sin 47   sin61  sin 11  sin 25 is equal to (a ) sin 36 º

( b) cos 36 º

(c ) sin 7 º

(d ) cos 7º

39. f sin 2θ  sin2φ  1 , cos 2θ  cos2φ  3 , then cos 2 ( θ  φ)  2 2 3 5 (a) ( b) 8 4 3 5 (c ) (d ) 4 8

47. If x  sec φ  tan φ and y  cosec φ  cot φ, then

(c) tan A tan B tan C

(d) 1.

41. If cos (A  B )  3 and tan A tan B  2, then 5 1 1 (a) cos A cos B  (b) cos A cos B  5 5 1 1 (c) sin A sin B  (d) sin A sin B  5 5 42. For A  133, 2 cos A is equal to 2 (a )

1  sin A  1  sin A

( b)

1  sin A  1  sin A

(a ) xy  1  x  y

( b) xy  1  x  2 y

(c ) xy  1  x  y

(d ) xy  1  y  x

48. If tanθ  (a )

1 5

and θ lies in the Ist quadrant, then cosθ is

1

( b) 2

6 (c )

40. If A  B  C  180, then the value of (cot B  cot C ) (cot C  cot A) (cot A  cot B ) will be (a) sec A sec B sec C (b) cosec A cosec B cosecc

(b) sinq (d) tanq

1 6

5

(d ) 2

6

5 6

49. If cos 20 = k and cos x = 2 k 2  1, then the possible values of x between 0 and 360 are (a ) 1408 ( b) 408 and 1408 (c ) 508 and 1308

(d ) 408 and 3208

50. The value of expression 3 cos θ  4 sin θ lie between (a) 27 and 7 (b) 225 and 25 (c) 21 and 1 (d) 25 and 5 51. The value of cos 15  sin 15 is (a )

1

( b) 2

2 (c ) 2

1 2

1

(d ) 0.

2 2

(c )  1  sin A  1  sin A

52.

(d )  1  sin A  1  sin A 43. The value of cot A  tan (180  A)  tan (90  A) 

2  2  2 cos 4θ is equal to (a ) cos θ

( b) cos 2θ

(c ) 2 cos θ

(d ) 2 cos 2θ

tan (360  A) is (a) 0 3 (c) 2

(b) 1 (d) 2

2 3 6 4 2 44. sin x  sin x  sin x  1, then cos x  4 cos x  8 cos x  (a) 3 (b) 4 (c) 2 (d) 1

45.

sec 8 A  1  sec 4 A  1 tan 2 A (a ) tan 8 A cot 8 A (c ) cot 2 A

π  π  53. The value of tan   θ   tan   θ  is 4 2     (a ) tan 2θ ( b) 2 tan θ (c ) 2 tan 2θ

54. If A  sin 20 θ  cos 48 θ, then for all values of θ:: (a ) A  1 ( b) 0  A  1 (c ) 1  A  3

( b)

tan 8 A tan 2 A

(d ) None of these

(d ) None of these

55.

(d ) None of these

cos 21  sin21 is equal to cos 21  sin21 (a ) tan 218

( b) tan 668

(c ) tan 248

(d ) tan 698

Trigonometry ▌ 3.81 56. If tan x  2b / ( a  c )( a ≠ c),

58. If y 

Y  a cos 2 x  2b sin x cos x  c sin 2 x and

, then one of the values of y is 1  sin 4 A  1 (a ) tan A ( b) cot A

Z  a sin 2 x  2b sin x cos x  c cos 2 x , then (a ) y  z ( b) y  z  a  c (c ) y  z  a  c

π  (c ) tan   A  4  

( d ) y  z  ( a  c ) 2  4 b2

57. The values of q lying between c and p/2 and satisfying the equation 1 sin 2 θ cos 2 θ 4 sin 4θ 2 2 sin θ 1 cos θ 4 sin 4θ  θ are sin 2 θ cos 2 θ 1 4 sin 4θ (a ) 7 π / 24

( b) 5π / 24

(c ) 10 π / 24

(d ) π / 24

1  sin 4 A  1

π  (d )  cot   A  4  

59. The value of the expression is sin 98cos98 sin 488sin128 (a) 1 (c) 2

(b) 0 (d) 3

60. 2 sin x cos 2 x  sin x , if (a ) x  nx  π / 6 ( b) x  nx  2 π / 6 ( c ) x  nπ

(d ) x  nπ  π / 2( n ∈ I ).

answer Key 1. (c)

9. (d)

17. (d)

25. (c)

33. (c)

41. (a)

49. (d)

57. (a)

2. (d)

10. (a)

18. (b)

26. (c)

34. (a)

42. (b)

50. (d)

58. (c)

3. (b)

11. (b)

19. (b)

27. (a)

35. (d)

43. (a)

51. (a)

59. (a)

4. (b)

12. (d)

20. (b)

28. (a)

36. (a)

44. (b)

52. (c)

60. (c)

5. (a)

13. (b)

21. (a)

29. (b)

37. ()

45. (b)

53. (c)

6. (b)

14. (c)

22. (a)

30. (d)

38. (d)

46. (a)

54. (b)

7. (b)

15. (c)

23. (b)

31. (a)

39. (d)

47. (d)

55. (c)

8. (b)

16. (a)

24. (c)

32. (c)

40. (b)

48. (c)

56. (c)

soluTions 1.

x  a cos θ  b sin θ y  a sin θ  b cos θ Squaring and adding both x 2  y2  (a cos θ  b sin θ) 2  (a sin θ  b cos θ) 2  ( a 2 cos 2 θ  b2 sin 2 θ  2 ab cos θ sin θ)  ( a 2 sin 2 θ  b2 cos 2 θ  2 ab sin θ cos θ)  a 2 (cos 2 θ  sin 2 θ)  b2 (sin 2 θ  coos 2 θ)  a 2  b2 2

 2  4 2. 1  tan 2 30  sec 2 30     3  3 3. sec q is negative in 2nd and 4th quadrants. sin 2 θ 4. (1  sin θ ) ⋅ tan θ  cos θ ⋅  sin 2 θ . cos 2 θ 2

2

2

5. sec6 q – tan6 q – 3 sec2 q tan2 q = (sec2 q)3 – (tan2 q)3 – 3 sec2 q tan2 q

(sec2 q – tan2 q)

(\ sec2 q – tan2 q = 1)

= (sec2 q – tan2 q)3 = 13 =1. 6.

sec 2 A  cose c 2 A  (1  tan 2 A)  (1  cot 2 A) ⋅ (tan A  cot A)2  tan A  cot A.

7. 4 sin2 x – 1 = 0 1 4 since  x  2π or, sin 2 x 

1 2 ∴ x  30, 150, 210, 330

∴ sin x 

1 1 8. sin( A  B )  and sin A  2 2 \ A – B = 30°, A = 45° \ B = A – 30° = 45° – 30° = 15°.

3.82

9.

▌ Mathematics 15. We have,

tan 30  tan 60  tan(30  60 ) 1  tan 30 tan 60  tan 90  ∞.

10. Given that

cos 2θ  2 cos θ  2 cos 2 θ 1 2 cos θ 2

 1  3 3 ∴ for all 2 θ  cos θ     for all 2 2 2  3 ⇒ cos 2θ  2 cos θ  for all θ 2

cos x  sin x  2 cos x

or, cos 2 x  sinx  2 cos x squaring,

Also, maximum value of this expression is 3.

(cos x  sin x )2  2 cos 2 x

2

 1 3  2  cos θ    2 2 

or, cos x  sin x  2 sin x cos x  2 cos x 2

2

2

or, cos 2 x  sin 2 x  2 sin x cos x

2

(cos x  sin x ) × (cos x  sin x ) 2 cos x sin x cos x  sin x 2 or, 2⋅ sin x

 1 Now 2  cos θ    0 for alll θ 2 

or,

or,

cos x  sin x 2 2 2 2 2      2. sin x 2 2 2 2

11. sec θ  tan θ  1  2 or,

sec θ  1  2  tan θ

squaring, (sec θ  1)2  ( 2  tan θ )2 or ,

sec 2 θ  1  2 sec θ  2  tan 2 θ  2 2 tan θ

or,

sec θ + 1  2sec θ  2  (sec θ  1)  2 2 tan θ

or,

sec θ = 2 tan θ

2

2

squaring, sec 2 θ  2 tan 2 θ  2(sec 2 θ  1)  2 sec 2 θ  2 or,

sec 2 θ  2 ∴ sec θ  2 .

3 12. Here, cos θ  : sin θ  1  cos 2 θ 5 9 16 4   25 25 5 45 4  tan θ   35 3  1

4 4  1 sin θ tan θ  1 5 3  16 2 tan 2 θ 2 9 16  15 9 93  ×  15 32 160 13. sin(30° + q) = cos(60° – f) = sin[90° – (60° – f)] = sin (30° + f) = 30° + q = 30° + f or q – f = 0. 14. = tan 40° tan 50° tan 60° = tan 40° tan (90° – 40°) tan 60° = tan 40° cot 40° tan 60° 1 3  3

16.

cot A  cosec A  1 cot A  cosec A  1 (cot A  cosec A)  (cosec 2 A  cot 2 A) cot A  cosec A  1 (cot A  cosec A) (1  cosec A  cot A)  cot A  cosec A  1 1 cos A cos A  1  cos A  cosec A    . sin A sin A sin A 

17. sin q + sin2 q = 1 or, sin q = 1 – sin2 q = cos2 q squaring, sin2 q = cos4 q or, 1 – cos2 q = cos4 q or, cos2 q + cos4 q = 1. 18. we have, (1  sin x ) (1  sin x ) 1  sin 2 x  (1  cos x ) (1  cos x ) 1  cos 2 x  

cos 2 x  cot 2 x  cot x sin 2 x 1 1 3   . tan x 4 / 3 4

1 1 1 sin 30  cos 60  tan 45 2  2  19. 111 cos 0  tan 45  sin 90 1   1. 1 20. By hypothesis, cos q = 1 – sin q Now, sin2 q = 1 + cos2 q = 1 – (1 – sin q)2 = 1 – (1 + sin2 q – 2 sin q) = ( – 1 – sin2 q + 2 sin q = sin q (2 – sin q) or, sinθ  sin θ ( 2  sin θ ).

Trigonometry

21. sin x 

 1  tan 2 x  1  (sec 2 x  1)

1 7  sin x 2 3

or, 2 3 sin 2 x  2 3  7 sin x or 2 3 sin 2 x  7 sin x  7 sin x  2 3  0 ∴ sin x 

7  49  4  2 3  2 3



4 3 7  49  48 4 3



7 1 4 3

2

3 or ∴ sin x  2 3 2 But sin x  is not possible, for sin x 1 3 sinx 

3  x  60. 2

cosec θ  cot θ  2 1 cosec θ  cot θ  . 2

13 12

   1  sin x  (1  tan x )   cos x  

 (1  tan x ) (1  tan x )

288  169 144

119 119 .  144 12

 π 26. sin x  cos x  2 sin  x    2 4  ∴Max, value 5 2 .  3π  27. Since sin x  cos x  2 sin  x   4  

sin 90 sin 90  cos 9 cos 81 cos 27 cos 63 1 1 2 2     cos 9 sin9 cos 27 cos 27 sin 18 sin 54  2.

sin 54  sin 18 2 cos 36 sin 18  2. 4 sin 18 . cos 36 sin 18 . cos 36

29. Since tan θ 

4 4 4 ∴sin θ  or 3 5 5

5

cosec 2 θ  cot 2 θ  1

24. x2 – y2 = (a sec q + b tan q)2 – (b sec q + a tan q)2 = (a2 sec2 q + b2 tan2 q + 2ab tan q sec q) – (b2 sec2 q – a2 tan2 q + 2ab tan q sec q) = (a2 – b2) sec2 q – (a2 – b2) tan2 q = (a2 – b2) (sec2 q – tan2 q) = (a2 – b2) 1 = (a2 – b2).

Now,



169  144

∴ Min, value  2

sin 2 θ cos 2 θ  sin θ  cos θ cos θ  sin θ sin 2 θ cos 2 θ   cos θ  sin θ cos θ  sin θ cos 2 θ  sin 2 θ  cos θ  sin θ (cos θ  sin θ ) (cos θ  sin θ )  cos θ  sin θ  (cos θ  sin θ )

or,

 2

28. We have, tan 9 tan 27 tan 63 tan 81  tan 9 tan 81(tan 27 tan 61 )



23. We know that, Given, cosec 2 θ  cot 2 θ 1  ⇒ cosec θ  cot θ 2

 13  2  2  sec 2 x  2     12 

Which lies between  2 and 2

sin θ cos θ 22.  1  cot θ 1  tan θ sin θ cos θ   sin θ cos θ 1 1 cos θ sin θ

25. Here, sec x 5

▌ 3.83

5

4

θ

4

θ 3

3

[Positive if q lies in IInd quadrant and negative if q lies in IVth quadrant]. 30. tan 75  tan (45  30 )  1

tan 45  tan 30 1  tan 45 tan 30

3 1

1

3 ⇒ 3 ⇒ ⇒ 1 3 1 11 × 3 3 cot 75 

3 1 3 1

3 1 3 1

tan 75  cot 75 

3 1 3 1



3 1 3 1

3.84 ▌ Mathematics

( ⇒

) ( 3 1) ( 3 1)( 3 1) 2

3 1 

2



1 1  8  1 9 2 2

  3 1 2 3   3 1 2 3   ⇒ 3 1 ⇒

36. cos52  cos 68  cos172  cos 52  cos 68  cos (180  8 )

42 342 3 4 3 ⇒ ⇒2 3 2 2

31. x  sin 130  cos130  cos40  cos130 [ sin 130  sin(90  40 )  cos 40]  2cos 85 cos 45  0 [ cos 85  0, cos 45  0] 1 tan 15    1 2 2    4.   2 tan15 sin 30 1  2 1  tan 2 15 

32. tan 15  cot 15  tan 15    1  tan 2 15     tan 15   

2 33. Since, sin x  sin x  1 (Given )

∴sin x 5 cos 2 x

( i)

Now, the given expression

(

(1)  (1)  (1)  ...  (1) 1  1 8times 2

)

 cos 52  cos 68  cos 8 68  52 52  68  cos 8  2 cos cos 2 2  2 cos 60 cos 8  cos 8  cos 8  coos 8  0. 38. Given value  ( sin 47  sin 61 )  ( sin 11  sin 25 )  2 sin 54 cos 7  2 sin 18 cos 7  2 cos 7 (sin 54  sin 188 )  2 cos 7 2 cos 36 sin 18 2 sin 18 cos 18  2 cos 7 cos 18 sin 36 cos 36  cos 7 cos 18 sin 72  cos 7 [sin 72  cos 18].  cos 7 cos 18

 cos6 x  cos6 x  3 cos 4 x  3cos 2 x  1  1

(

)

3

 cos x cos x  1  1 6

2

 sin 3 x ( sin x  1)  1[using (i)]

(

)

39. sin 2θ  sin 2φ 

⇒ 2 sin( θ  φ) cos ( θ  φ) 

3

 sin 2 x  sin x  1  (1)3  1  1  1  0 34. sin θ  cos θ  ( sinθ  cos θ ) 1  sin2θ 2

2

And, sinθ  cos θ  a  1  sin 2θ  a 2 ⇒ sin 2θ  a 2  1  1  sin 2θ  2  a 2  sin θ  cos θ  2  a 2 . 35. sin 2 5  sin 2 10  sin 2 15  ...  sin 2 85  sin 2 90  (sin 2 5  sin 2 85 )  (sin 2 10  sin 2 80)  (sin 2 15  sin 2 75 )  ...  (sin 2 40  sin 2 50 )  sin 2 45  sin 2 90 = [sin 2 5  sin 2 (90º  5º )]  [sin 2 10  sin 2  (90  10)] [sin 2 15º  sin 2 (90  15)]  ...  (1)2  (sin 2  cos 2 5)  (sin 2 10  cos 2 10)  (sin 2 15  cos 2 15) 1  ... (sin 2 40  cos 2 40)   1 2

1 2

cos 2 θ  cos 2φ 

1 2

3 2

⇒ 2cos( θ  φ) cos ( θ  φ) 

3 2

Squaring and adding, wee get 4 sin 2 ( θ  φ)  cos 2 ( θ  φ)  cos 2 ( θ  φ) 1 9 10 5 ( θ  φ) 5     ⇒ cos 2  . 2 4 4 4 2 8 40. (cot B  cot C ) (cot C  cot A) (cot A  cot B )  cos B cos C   cos C cos A   cos A cos B    ,            sin A   sin A sin B   sin B sin C   sin C sin ( B  C ) sin (C  A) sin ( A  B )  . . sin B sin C sin C sin A sin A sin B sin (180  A) sin (180  B ) sin (180  C ) .  . sin A sin B sin C sin A sin B sin C sin A sin B sin C  sin B sin C ⋅ sin C sin A ⋅ sin A sin B

Trigonometry

1 sin A sin B sin C 5 cos ec A cos ec B cos ec C .

45.

5

41. cos ( A  B) 

sec 8 A  1 1  cos 8 A cos 4 A  . sec 4 A  1 cos8 A 1  cos 4 A 

3 5

2sin 2 A.cos 4 A cos8 A.2 sin 2 A ( 2. sin 4 A cos 4 A)( 2 sin 2 A cos 2 A)

cos 8 A ⋅ 2 sin 2 2 A sin 8 A cos 2 A tan 8 A . .   cos8 A sin 2 A tan 2 A

3 ⇒ cos A cos B  sin A sin B  (1) 5 tan A tan B  2 sin A sin B  2 (2) ⇒ cos A cos B By (1) and (2), 3 3 cos A cos B  5 1 ⇒ cos A cos B  . 5

2 θ 1 t 2 1 t 2   cos θ. 46. θ 1 t 2 1 t 2 2

2  A A 42. 1  sin A   cos  sin  2 2  2  A A 1  sin A   cos  sin  2 2  A A ∴cos  sin  1  sin A 2 2  A     66 1 ∴cos A , sin A are positive.   2 2 2  2  A A ∴cos  sin  1  sin A 2 2     cos A  sin A  0 as sin 66 1  cos66 1    2  2 2 2 

A ∴2 cos  1  sin A  1  sin A. 2 43. cot A  tan(180  A)  tan(90  A)  tan(360  A)  cot A  tan A  cot A tan A  0.

47. We have, xy  sec φ cosec φ  sec φ cotφ  tan φ cosec φ  1  xy  1 = sec φ cosec φ  cosec φ  sec φ 1  cosec φ  sec φ  sin φ cos φ  cosec φ  sec φ 

sin 2 φ  cos 2 φ sin φ cos φ

 cosec φ  sec φ  tan φ  cot φ  (cosec φ  cot φ)  (sec φ  tan φ)  y  x. 1 6 2 2 48. sec θ  1  tan θ  1   5 5 5 5 ∴cos 2 θ  ⇒cos θ  6 6 [because θ lies in the Ist quadrant]. 2 2 49. cos x  2 k  1  2 cos 20  1  cos 40 [becau se cos 2θ  2cos 2 θ  1]  cos(360  40 )  cos 320 ∴x  40 and 320

50. put 3  r cos α 2  ⇒ r  9  16  25 4  r sin α  ⇒ r 5

44. By the given equation,

∴3cos θ  4 sin θ  r ( cos θ cos α  sin θ sin α)

sin x  sin 3 x  1  sin 2 x  cos 2 x 2

 r cos( θ  α)  5cos (θ  α)

⇒ sin 2 x  sin 6 x  2 sin 4 x  cos 2 x

Its maximum value = 5(1) = 5 And minimum value = 25.

(

)

⇒ sin x  sin 3 x  cos 4 x

(

)

3

(

)

2

⇒1  cos 2 x  1  cos 2 x  2 1  cos 2 x  cos 4 x

(

⇒1  cos 2 x  1  cos6 x  3 cos 2 x 1  cos 2 x

(

)

 2 1  cos x  2 cos x  cos 4

2

4

⇒ cos6 x  8cos 2 x  4 cos 4 x 4 ⇒ cos 6 x  4 cos 4 x  8 cos 2 x  4.

▌ 3.85

)

51. cos15  cos(45  30 )  cos 45cos30  sin 45sin30  

3 1 1 .  2 2 2 2

1

.

3 2 2



1 2 2

⇒

3 1 2 2

▌ Mathematics

3.86

sin15  sin (45  30 ) = sin 45 cos 30  cos 45 sin 30  

57. Applying R1 → R1 2 R3 and R2 → R2 2 R3 on the LHS, the given equation can be written as

3 1

1 0 1 0 1 1 0 2 2 sin θ cos θ 1  4 sin 4θ

2 2

Expanding the LHS along R1.

3 1 1   2 2 2 2

1

.

3

1



2 2



2 2

cos15  sin15 

3 1



we get 1 4 sin 4θ  cos 2 θ  sin 2 θ  0 ∴ 4 sin 4θ  02 ⇒ sin 4θ 1/ 2 ∴ 4 θ  7 π / 6 or 11 π / 6 [because 0  θ  ⇒ 0  4 θ  2 π] ∴ θ  7 π / 24 or 11 π / 24.

3 1

2 2 2 2 2 1 ∴cos15  sin15   2 2 2 52.

2  2  2cos 4 θ  2  2 (1  cos 4 θ)

(cos 2 A  sin 2 A)  1 y (cos 2 A  sin 2 A)  1  ( cos 2 A  sin 2 A)  1 ⇒ Yy  Which gives us fourr  ( cos 2 A  sin 2 A)  1 2

 2  2.2 cos 2 2 θ  2  2 cos 2θ  2(1  cos 2θ)  2.2 cos 2 θ  2cos θ 53. The given value 

1  tan θ 1  tan θ  1  tan θ 1  tan θ

(1  tan θ)2  (1  tan θ)2 1  tan 2 θ 4 tan θ   2 tan 2θ. 1  tan θ



54. Because 0  sin 20 θ  sin 2 θ and 0  cos 48 θ  cos 2 θ ∴0  sin 20 θ  cos 48 θ  1 [because cos θ , sin θ cannot be simultaneously zero] ∴0  A  1. 55. The given value 

1  tan 21  tan( 45  21 )  tan 24 1  tan 21

56. Adding the expression for y and z, we get

( ) cos x  c ( cos x  sin x )

y  z  a cos 2 x  sin 2 x  4b sin x  2

2

 a cos 2 x  2b sin 2 x  c cos 2 x  ( a  c ) cos 2 x  2b sin 2 x Now, cos2 x  

1  tan 2 x ( a  c )2  4b2 and sin 2x  1  tan 2 x ( a  c )2  4b2

2 tan x 4b( a  c )  1  tan 2 x ( a  c )2  4b2

yz

( a  c ) ( a  c )2  4b2   8b2 ( a  c ) ( a  c )  4b 2

( a  c ) ( a  c )  4b2 

2

2



( a  c ) 2  4 b2

As a ≠ c, we get y ≠ z.

ac

58.

2

values of y , say y1 , y2 , y3 and y4 . We have, coos 2 A  sin 2 A  1 (1  cos 2 A)  sin 2 A  y1  cos 2 A  sin 2 A  1 ( cos 2 A  1)  sin 2 A 2 cos 2 A  2 sin A cos A 2 sin 2 A  2 sin A cos A cos A (cos A  sin A)  sin a (cos A  sin A)  cot A  (cos 2 A  sin 2 A)  1 y2   (cos 2 A  sin 2 A)  1 (cos 2 A  sin 2 A)  1   (1  cos 2 A)  sin 2 A 2 sin 2 A  2 sin A cos A  2 cos 2 A  2 sin A cos A  tan A (cos 2 A  sin 2 A) 1 y3  (cos 2 A  sin 2 A) 1 (1  cos 2 A)  sin 2 A  (1  cos 2 A)  sin 2 A 

2 cos 2 A  2 sin A cos A cos A  sin A  2 sin 2 A  2 sin A cos A cos A  sin A π  1  tan A   tan   A  1  tan A 4 



π   cot   A 4  (cos 2 A  sin 2 A) 1 y4  (cos 2 A  sin 2 A) 1 (1 cos 2 A)  sin 2 A  (1  cos 2 A)  sin 2 A

Trigonometry

2 sin 2 A  2 sin A cos A 2 sin 2 A  2 sin A cos A π  cos A  sin A 1  tan A   tan   A   cos A  sin A 1  tan A) 4  

59.

2 sin 9 cos9 sin 18  2 sin 48 sin12 cos 60  cos 36

▌ 3.87

5 1 5 1 5 1 4    1 1 5  1 2  5  1 5 1   2 4

(

)

60. The given equation can be written as sin x ( 2 cos 2 x  1)  0. That is, either be written as cos 2 x 5 1 / 2 5 cos( π / 3). Hence, x 5 nπ.

TrigonomeTry equaTions 1. The equation cos 2x  a sin x  2 a  7 possesses a soluution, if (a ) a  2 ( b)  2  a  6 (c ) a  6

(d ) a is any integer

where 0 < α, β < π /2, then 2β is equal to π α 4

( b)

3π α 4

(c )

π α/ 2 8

(d )

3π α/ 2 8

4  3 ( m2  1)2 4

(a ) for all real values fo m

(c ) for no real value of m (d ) for m2  2 7. If Sin α  Cos α  m , then sin 9θ sinθ sin 3θ  then, And k2   cos3θ cos99θ cos 27 θ (a ) k1  k2 ( b) k1  2 k2 (c ) k1  k2  2 (d ) k2  2 k1 8. If a 2  2a cos x  1  674 and tan ( x / 2)  7, then the

3. If sin A, cos A and tan A are in geometric progression, then cot 6 A 2 cot 2 A is equal to (a) 21 (b) 0 (c) 1 (d) None of these 4. If the value of sin

sin 6 α  cos 6 α 

( b) for m2  2

2. If tan α  1/7 and β  1 / 10

(a )

6. If sin α  cos α  m then

integral value of a is (a) 25 (c) 67

(b) 49 (d) 74

9. If 0 ≤ x ≤ 2p and | cos x | ≤ sin x, then

π 3π 5π 7π 9π 11π sin sin sin sin sin 14 14 14 14 14 14

13π is equal to k 2 , then k equal to 14 21 1 (a ) ( b) 8 8 1 (c ) (d ) 1 64

sin

 π (a) x ∈  0,   4  π π (b) x ∈  ,   4 2   π π   π 3π  (c) x ∈  ∪  ∪ ,   4 2   2 4  (d) none of these 10. If

sin 3 θ  cos3 θ cos θ   tan θ cot θ   1 sin θ  cos θ 1  cot 2 θ

0 ∈ [0, 2π], then 5. If k 5 sin π /18sin 5π /18 sin 7π /18, then the numerical value of k is equal to (a ) 1/2 (c ) 1/8

( b) 1 / 4 (d ) 1 / 18

 π  π (a ) θ ∈  0,      2  4  3π   5π  (c ) θ ∈  π,      2  4

 π   3π  ( b) θ ∈  , π     2   4  π π (d ) θ ∈ ( 0, π)   ,  4 2

3.88

▌ Mathematics

11. The equation (cos p 2 1) x2 + (cos p) x + sin p = 0, in the variable x has a real root. Then p can take any value in the interval  2π π  (b)  ,   2 2 (d) (0, 2p)

(a) (0, p) (c) (2 p, 0)

 π 3π  (a )  2 ,   10 10 

 π  ( b)  2 , π   10 

(c ) (2 π, π)

 3π  (d )  2 π,  10  

5 is given

15. The least positive non-integral solution of the equation sin p (x2 + x) = sin px2 is (a ) rational ( b) irrational of the form

p

(c ) irrational of the form

p 1 , where p is an odd integer 4

(d ) irrational of the form

p 1 , wheere p is an even integer 4

16. The number of solutions of the equation sin x = cos 3x in [0, p] is (a) 1 (b) 2 (c) 3 (d) 4 17. If cos x

3 1 sin 2 x  sin x  2 2

 1, the possible values of x are:

(a ) n π or n π  (1) n ( b) n π or 2 n π  (c) n π  (1) n

π , n∈ I 6

π π or n π  (1) n , n ∈ I 2 6

π , n∈ I 6

(d ) n π , n ∈ I 18. The number of solutions of x 1 π 2cos 2   sin 2 x  x 2  2 ,0  x  2 2 x  

(b) 1 (d) none of these

19. The number of solutions of the equation

12. The number of all possible triplets (a1, a2, a3) such that a1 + a2 cos 2x + a3 sin2x = 0 for all x is (a) 0 (b) 1 (c) 3 (d) infinite 13. Solution of the equation 4cot 2q = cot2q 2 tan2q is π π (a ) θ  nπ  ( b) θ  nπ  2 3 π (c ) θ  nπ  (d ) none of these 4 14. The set of all x in (2 p, p) satisfying | 4sin x 2 1 | < by

(a) 0 (c) infinite

 π sin 5x cos 3x = sin 9x cos 7x lying in the interval 0,  is  4 (a) 5 (b) 4 (c) 3 (d) 2

20. The number of solutions of cosq = | 1 + sin q |, 0 ≤ q ≤ 3p, is (a) 2 (b) 3 (c) 4 (d) none of these 21. The number of real solutions of sinex. cosex = 2x–2 + 2–x–2 is (a) 0 (b) 1 (c) 2 (d) none of these. 22. The general solution of the equation 1

2sin θ  2cosθ  2 π (a) n π  4 π (c) 2 n π  4

1 2

is (b) n π 

π 4

(d) none of these.

23. The general solution of the equation 2 Cosθ  1  3 .2Sin θ 2

π ( b) n π  , 2 nπ, n ∈ Z 2

π (a ) 2 n π  , nπ, n ∈ Z 2 π (c ) n π  , nπ, n ∈ Z 2

(d ) none of these.

24. The general solution of a cos x + b sin x = c, where a, b, c are constants, is   (a) x  nπ  cos1   

 (b) x  2 nπ  tan1  

   2 2  a  b  b   a  c

   b  c 1     cos  a  a 2  b2          b c  (d) x  2 nπ  tan1    cos1   a 2  b2   a    (c) x  2 nπ  tan1  

25. The equation 3sin 2 x2 cos

2

x31sin 2 x2 sin

satisfied for the values of x given by (a) cos x = 0 (b) tan x = 21 (c) tan x = 1 (d) none of these

2x

 31sin 2 x2 sin x  28 is 2

Trigonometry

▌ 3.89

answer Key 1. 2. 3. 4.

5. 6. 7. 8.

(b) (a) (c) (a)

9. 10. 11. 12.

(c) (b) (b) (a)

13. 14. 15. 16.

(c) (d) (a) (d)

(c) (a) (c) (c)

17. 18. 19. 20.

(c, d) (a) (a) (b)

21. 22. 23. 24.

25. (a, b)

(a) (a) (c) (d)

soluTions 1. The given equation can be written as 1  2 sin 2 x  a sin x  2 a  7 ⇒ 2 sin 2 x  a sin x  2 a  8  0 a  a 2  8( 2 a  8) 4 a  a 2  16 a  64) a  ( a  8)   4 4 Hence sin x = (a 2 4)/2 (the other value is not passible as |sin x |  1). This value is possible only when a4 1   1 or  2  a  4  2 ⇒  2  a  6. 2 sin x =

⇒

1 3  ⇒ tan β  1 / 3 10 10 10 2 tan β 2.1 / 3 3  tan 2β  ∴ = 1  tan 2 β 1  1 / 9 4 tanα  taan2β And, tan(α  2β )  1  tan α tan 2β

2. sinβ 

1

⇒ cos β 1 

1 / 7 + 3 / 4 25 = =1 1 − 1 / 7.3 / 4 25 Since 0  β  π /2 and tan 2 β = 3 / 4  0, we get 0  2β  π/2. Also, 0  a  π / 2. Hencee, 0  α  2β  π tan ( α  2 β )  1, So thatα  2β  π / 4 ⇒ 2β  π  4  α. 

3. Since sin A, cos A and tan A are in G.P., we have cos A  sin A tan A ⇒ cos A  sin A ⇒ cot A  sec A 2

3

2

2

⇒ cot 4 A  1  tan 2 A ⇒ cot 6 A  cot 2 A  1. 4. The given expression can be written as sin

π 3π 5π π  5π  sin sin sin sin  π 2  14 14 14 2  14 

 3π   π sin  π   sin  π    k 2 14   14   π 3π 5π Where k  sin sin sin 14 14 14  π 3π   π 5π  π π   cos    cos    cos    1 4 2 14 2    2 14   

π π x 2 sin cos π 7 7 2 sin 7 2π 4π 1 4π 4π  cos cos  sin cos 7 7 π 7 7 4 sin 7 1 8π 1  x sin  . π 7 8 8 sin 7  cos

π 3π 2π cos cos  7 7 7

1

5. k  sin π / 18 sin5π / 18 sin7 π / 18  sin10 sin50 sin70 1  [cos 40  cos 60] sin 70 2 1 1  cos 40 sin 70  sin 70 4 2 1 1  [sin 110  sin 30]  sin 70 4 4 1 1 1 1 1  sin (180  70 )    sin 70  . 4 4 2 4 8 6. sin α  cosα  m ⇒ 1 + 2 sin α cos α = m2 ⇒ sin2α  m2  1 which is possible if  1  m2  1  1 or m2  2. Again, sin 6 α  cos 2 α  (sin 2 α  cos 2 α)3  3 sin 2 α cos 2 α(sin 2 α  cos 2 α) 3  1  3sin 2 α cos 2 α  1  (sin 2α)2 4 3 2 4  3( m2  1)2 2  1  ( m  1)  4 4 But this is valid only when m2  2. 7. K1  tan 27 θ  tan9θ  tan 9θ  tan 3θ  tan 3θ  tan θ But tan 3 θ  tan θ 

sin θ cos θ  cos 3θ sin θ cos 3θ cos θ

sin 2θ 2 sin θ  cos 3θ cos θ cos 3θ  sin 9θ sin θ  sin 3θ  ∴ k1 = 2     2k2 .  cos 27 θ cos 9θ cos 3θ 

3.90

▌ Mathematics 1  tan 2 ( x / 2) 1 1  tan 2 ( x / 2) 1  49 1  a 2  2a  1  49

8. 674  a 2  2 a

⇒ ⇒ ⇒



a3  a2 2 k k or a1  , a2  , a3  k 2 2 for any k ∈ R. Hence, the required number of triplets is infinite.

25a 2  48a  673  25  0 ( a  25) ( 25a  673)  0 a  25 (taking the integral value of a).

if

9. We have, | cos x | ≤ sin x ⇒ sin x ≥ 0 ( cos x  0) ⇒x ∉ (π , 2π )

cos x  sin x ⇒  cos x  sin x ⇒ tan x 1



⇒ (1 2 tan2q) [2tan q 2 (1 + tan2q)] = 0



⇒ (1 2 tan2q) (tan2q 2 2tan q + 1) = 0



⇒ (1 2 tan2q) (tan q 2 1)2 = 0





11. Discriminant = cos2p 2 4 (cos p 2 1) sin p For equation to have real roots, cos2p 2 4 (cos p 2 1) sin p ≥ 0 ⇒ cos2p ≥ 4 (cos p 2 1) sin p ⇒ cos 2 p  8 sin 2

For p 

p sin p 2

3π π or , R.H.S.  1 but L.H.S.  1, 2 2

∴ the choices (b), (c) and (d) are ruled out. The correct alternative is (a)

12. We have, a1 + a2 cos 2x + a3 sin2 x = 0

⇒

π tan θ  1,  1. ∴θ  nπ  . 4

14. We have, | 4sin x 2 1 |
2

2 θ 25. If 5cos 2θ  2cos  1  0,  π  θ  π, then θ is equal to 2

(a )

π 3

 3 (c ) cos1   5

( b)

π 1  3  cos   3 5

(d )

 3 π , π  cos1   3 5

Trigonometry

▌ 3.99

answer Key 1. (a, b, c, d)

2. (d) 3. (d)

4. (d)

8. (c)

12. (c)

16. (c)

20. (c)

24. (d)

5. (a,d)

9. (b)

13. (c)

17. (b)

21. (b)

25. (d)

6. (c)

10. (c)

14. (a)

18. (b)

22. (b)

7. (d)

11. (b)

15. (b)

19. (c)

23. (d)

soluTions Also, we, know

1 2 1 2 2 1. c = l  a  2  l  cos θ 4 2 1 1 b2  l 2  a 2  2  l  a cos (1180 º − θ) 4 2 1 So that b2  c 2  2l 2  a 2 2 or 4l 2 = 2b2  2c 2  a 2 But b2  c 2  a 2  2bc cos A Therefore,4l 2  b2  c 2  2bc cos A  a 2  4bc cos A 1 Also, ( 2s  a )2  4bc sin2 ( )  (bb  c )2  2bc (1  cosA) 2  b2  c 2  2bc cos A  45º

1 3 then sin A + sin B + sin C    1. 2 2

b

c

2. ∆ 



θ N

C

D

If the triangle is right-angled and isosceles then one of the angles is 90° and the remaining two are 45° each, so that sin A + sin B + sin C = 1 + 2. and, cos A + cos B + cos C =

1 1 ah  a ⋅b sin c 2 2

2.

c 2  a 2  b2 2ca 1 c 2  a 2  b2  2 2ca

4. cos B 

a sin B 1  ⋅a ⋅ ⋅ sin C sin A 2

or,

sinn B sin C 1  a2 . sin ( B  C ) 2

or, c2 + a2 = b2 + ca or (c – a)2 = b2 – ac 2 2 2 or, a + b + c = 2b2 + ca. 5. If the angles A, B, C of the triangle are in A.P. then A + B + C = 180° and 2B = A + C Þ B = 60°. Since the greater angle of a triangle has greater side opposite to it. \ b = 9, c = 10 and let a = x. Then, by cosine formula

A c

B

3 . 2 If the triangle is isosceles, let A =30°, B = 30°, C = 120°,

3. If the triangle is equilateral sin A + sin B + sin C = 3

 3 then sin A + sin B + sin C = 1 +  .  2    If the triangle is right-angled, let A = 30°, B = 60°, C = 90°,

A

B

∆  s ( s  a ) ( s  b) ( s  c ) 1  ( a  b  c )( b  c  a )( c  a  b)( a  b  c ) 4 1  2 b2 c 2  2 c 2 a 2  2 a 2 b2  a 4  b4  c 4 4 1  4 ( b2 c 2  c 2 a 2  a 2 b2 )  ( a 2  b2  c 2 ) 2 4 1 Again, we know, ∆  rs  ( a  b  c )r . 2

h a

b

C

cos B 

c 2  a 2  b2 2ca

3.100

▌ Mathematics

1 100  x 2  81 ⇒  2 2 ⋅ 10 ⋅ x or, x 2  10 x  19

 0 ⇒ x  5  6.

6. Let ‘l ’ be the length of the arc of circle of redius ‘r ’: ∴l  r  r  π r ∴l  r (π  2) But θ 

1 l

∴θ  π  2.

2π 7. Angle moved in 20 minutes  120  3 l Here, r  10 cm. Use θ  r  2 π  20 π ⇒ l  r θ  10    . 3  3  8. C 60 9. obtuse-angled Now, 2 cos 2 A  3cos 2 B



⇒

2(1  sin 2 A)  3(1  sin 2 B )

⇒

2(1  sin 2 A)  3(1  sin A) [because sin 2B  sin A ]

⇒

2(1  sin A) (1  sin A)  3 (1  sin A)  0

⇒

(1  sin A) ( 2 sin A  1)  0

⇒

sin A  1 or sin A 

1 2

π ∴ 3θ  nπ   θ 2 nπ π π ⇒ θ  2 4 8 π  ( 2 n  1) . 8

⇒ 4θ  nπ 

2 13. sin θ  3 cos θ  3 1  cos 2 θ  3cos θ  3 ⇒ ⇒ cos 2 θ  3cos θ  2  0 (cos θ  1)(cos θ  2)  0 ⇒ ⇒ cos θ  1  0 [∴cos θ  2 ≠ 0] ⇒ cos θ  1 ⇒ θ  0 in [ π, π]

1 5

∴ sin 2 x  cos 2 x  2 sin x cos x 

Which is not possible for any ∆ABC.

∴ A  30 or 150, B  45 or 135 But A ≠ 150 [because in that case B  C  30] ∴either A  30, B  135, C  15 Hence, in both the cases, ∆A ABC is obtuse-angled. 10. The given equation can be written as

(

⇒ 1  sin x = 2 cos 2 x  2 1  sin 2 x

1  sin x  2 cos x cos x

)

2 sin x  sin x  1  0 2

⇒

(1 sin x ) ( 2 sin x  1)  0

⇒

sin x  1 or 1 / 2

Now, sin x 1 ⇒ x  π / 6 or 5 p / 6. ∴ The required number of solutions is 2.

1 25

1 1 24 or sin 2 x  1  25 25 25 2 tan x 24 ⇒ 25 tan 12 12 tan 2 x x     25 1  tan 2 x ⇒ 12 tan 2 x  25 tan x  12 = 0 ⇒ 12 tan 2 x  16 tan x  9 tan x + 12  0 ⇒ 4 tan x (3tan x  4) + 3 (3 tan x  4)  0 ⇒ (4 tan x  3) (3 tan x  4)  0 3 4 or ∴tan x  4 3 Or 1  sin 2 x 

1 1 ∴ sin A  and ∴sin B  2 2



12. tan 3θ tan θ  1 ⇒ tan 3 θ  cot θ π   tan   θ  2  

14. Since sin x  cos x 

But sin A  1 ⇒ sin B  1

⇒

11. The given equation is equivalent to 2 sin2 ((( π / 2) cos 2 x )  2 sin 2 (( π / 2) sin 2 x ) ⇒ cos 2 x  sin 2 x ⇒ cos x (cos x  2 sin x ) = 0 ⇒ 1  2 tan x = 0 as cos x ≠ 0, x ≠ 0, x ≠ ( 2 n  1) π / 2 ⇒ tan x  1 / 2 1  tan 2 x 3  . ⇒ cos 2 x  1  tan 2 x 5

1 2 15. sin θ  2 cos θ   0 4 .4 sin 2 θ  8 cos θ  1  0 ⇒ 4 1  cos 2 θ  8 cos θ  1  0

(

)

⇒ 4 cos 2 θ  8 cos θ  5  0 ⇒ ( 2 cos θ  1) ( 2 cos θ  5)  0 1 ⇒ cos θ  [∴ 2 cos θ  5 ≠ 0] 2 π ⇒ θ . 3

Trigonometry ▌ 3.101

16. For the expression to be valid x ( x  1)  0, x 2  x  1  0

20.

2 sec (2 α)  tan β  cot β

and x  x  1  1 2

 1 3 Now, x 2  x  1   x     0 for all real x. 2 4  Again, x 2  x  1  1 ⇒ x  0 or  1 Thus, x ( x  1)  0 ⇒ x  0 or  1

⇒

Hence, number of real solution  2. π 17. sin θ  cos θ  0 ⇒ tan θ  1 ⇒ θ  . 4 18. cos x  cos y 

3 2

x y x y 3  cos 2 2 2 2π  π x y 3 ⇒ 2 cos cos  ∴ x  y   3 2 3  3 ⇒ 2 cos

x y 3  ⇒ cos 3 2  3  Which is not possible ∴ 1  2   Hence the System of equations has no solutions. 19. Let For ∴

∴

1 0  x  , tan x  x 2  π f ( x )  0 in  0,   2  π f ( x )  0 has no root in  0,   2

π  x  π, tan x is negative, ∴ f ( x )  0 2 π  ∴ f ( x )  0 has no root inn  , π  2  3π For  x  2 x , tan x is negative, 2 ∴ f ( x)  0  3π  ∴ f ( x )  0 has no root in  , 2 π   2  Again, f (x )  tan x  x ⇒ f ( x )  tan π  π  0  π  0  3π  3π 3π f    tan  0 And, 2 2 2   ∴ f ( x)  0 3π Has at least one root between π and . 2 For



f ( x )  tan x  x

1 1  tan 2 β  tan β tan β 1  tan 2 β  cosec (2β ) sec (2α)  2 tan β π  cos 2α  sin 2β  cos   2β  2   tanβ 

∴ ⇒

2α 

π π  2β ⇒2 ( α  β )  2 2

π α β  . 4

21. ( 2 cos x  1) ( 3  2 cos x )  0 ⇒

2 cos x  1  0 [ 2 cos x  3  0 iss not possible]

⇒

cos x 

1 2

 cos

 π π or cos  2 π   3 3 

 cos

5π π or cos 3 3

π 5π ⇒ x  , ∈ (0.2 π). 3 3 1 22. cos θ   cos120 or cos 240 [0  θ  360] 2 ∴ θ  120, 240. 23. Put 3  r cos α 4  r sin α ⇒ 9  16  r 2 ⇒ 25  r 2 4 3 Which is not possible for any values of x [  cos (x  α)  1]

∴r  5, tan α 

Hence, given equation has no solution.

(

)

24. Clearly, x  4  x  4 is real so that

x is also real.

( ) cos ( π x  4 )  1 and if coos ( π x )  1, then cos ( π x  4 )  1 [ their product  1] Again, if cos π x  1, then

3.102

▌ Mathematics

But both of these are not possible (  cos θ cannot be greater than 1)

25. The given equation can be written as 5[2 cos2 θ  1]  (1  cos θ)  1  0

∴

cos π x  4  1 cos π x  1

⇒ ⇒

∴

x  4  0 and x  0

⇒

(

)

( )

But x  0 is not possible ∴

x  4 is the only solution.

⇒

10 cos 2 θ  cos θ  3  0 (5 cos θ  3) (3 cos θ  1)  0 1 3 cos θ  , cos θ  2 5 π 3 θ  or θ  π  cos1 in  π  θ  π. 3 5

HeigHT and disTance 1. The angle of elevation of the top of a tower from a point 20 metres away from its base is 45º The height of the tower is

5. The angle of elevation of the sun, if the length of the shadow of a tower is 3 times the height of the tower, is

(a) 10 m

(a) 308

(b) 20 m

(b) 608

(c) 40 m

(c) 458

(d) 20 3 m

(d) 1508

2. The angle of elevation of the top of a hill from each of the vertices A, B, C of a horizontal triangle is α . The height of the hill is (a ) b tan α coses B 1 a tan α cosec A 2 1 (c ) a tan α cosec C 2 (d ) None of these ( b)

3. A vertical lamp post 6m height stands at a distance of 2m from a wall, 4m high. A 1.5 m tall man starts to walk away from the wall on the other side of the wall in line with the lamp post. The maximum distance to which the man can walk, remains in the shadow is 5 (a) m 2 3 (b) m 2 (c) 4 m (d) None of these 4. If a flag staff of 6 metres high placed on the top of a tower throws a shadow of 2 3 metres along the ground, then the angle (in degress) that the sun makes with the ground is (a) 608 (b) 308 (c) 458 (d) None of these

6. Two posts are 25 metres and 15 metres high and the line joining their tops makes an angle of 45º with the horizontal. The distance between these posts is (a ) 5 m

( b)

10

m

2 (c ) 10 m

(d ) 10 2 m

7. The angle of elevation of the top of a tower at a distance of 500 metres from its foot is 308. The height of the tower is (a )

500 3 m 3 500

(c )

(

500 ( b)

)

(

)m

3 1 3

3 1 3

(d ) 500 m

8. From the top of a house 32 metres high, the angle of elevation of the top of a tower is 45º and the angle of depression of the foot of the tower is 30º The distance of the tower from the house is (a) 30 3 m (b) 32 3 m (c) 35 3 m (d) None of these 9. If two towers of heights h1 and h2 subtend angles 608 and 308, respectively at the midpoint of the line joining their feet, then h1 : h2 = (a) 1 : 2 (c) 2 : 1

(b) 1 : 3 (d) 3 : 1

Trigonometry

▌ 3.103

answer Key 1. (b) 2. (b)

3. (a) 4. (a)

5. (a) 6. (c)

7. (a) 8. (b)

9. (d)

soluTions 1. Clearly,

4. Let OP be the tower of height h m and PQ be the flag staff of height 6m. Q

h  tan 45  1 20 ∴ h  20 m

6m P h h

x

O

AB 5 2 3 be the shadows of the tower and the flag staff,

M

20m

2. The distance of the foot from each vertex = h cot α ∴ the foot is at the circumcentre of the triangle ∴

R 5 h cot α

∴

h 5 R tan α 5 5

θ θ 2 3m A

Let the sum make an angle q with the ground. Let OA 5 x and

45º \ O

B

respectively. From right-angled ∠ d ∆ OAD, tan θ 5 Also, from right-angled ∆ OBQ,

a tan α 2 sin A

a tan α.cosec α. 2

3. BD is the required distance. P

⇒ tan θ

∴

h6 h  ⇒ hx  2 3h  hx  6 x x x2 3

⇒

2 3h  6 x ⇒

⇒

tan θ  3 ∴ θ  60.

h 6   3 x 2 3

5. Let q be the angle of elevation

A

∴

C

6m

h6 x2 3

h 3h

5 tan θ

4m 3 m 2 man Q 2m B

D

From similar triangles 6 QR 3 2  BR  i.e.  4 BR 2 BR ⇒ BR  4 4 3 BR 4 ∴ DR    2 3 / 2 DR DR 3 5 ∴ BD  BR  DR  4   . 2 2

R

h

θ 3h

h x

3.104

∴

▌ Mathematics

tan θ 

1 3

8. Here, x is required.

∴ θ  30.

6. 10  tan 45  1 ⇒ x  10 x

Clearly,

x  cot 30  3 32

∴

x  32 3m. Q

D 10 45º

B

x

R

32m

15m A

30º

O

C

x

Thus, BE = 10m. 7.

45º 30º

A

E 25m

x

P

9. h1 5 x tan 60 º 5 3x

h 1  tan 30  500 3

h2  x tan 30 

x 3

h1 : h2  3x :

∴

x

 3:

1

3

 3:1.

3

h

h1

θ

30º

500 h5

∴

500 3

5

h2 60º

500 3 m. 3

A

x

30º B

x

nda BoX 1 1 1. If tan A  and B  , then what is the value of ( A  B) ? 2 3 π (a) 0 (b) 4 π (c) (d) π 2 2. What is the value of (a ) 1

1  tan 15 ? 1  tan 15 ( b)

1 2

(cc )

1 3

(d )

3

3. What is the value of 3 cos ec 20  sec 20 ? (a ) 1 / 4

( b) 4

(c ) 2

(d ) 1 

 1 4. What is tan  7  equal to ?  2 (a )

6  3 − 2 2

( b)

6  3 2 2

(c )

6  3 2 2

(d )

6  3 2 2

C

Trigonometry

5. What is the value of

cos15  cos 45 ? cos3 15  cos3 45

1 4 1 (c ) 3 (a )

( b)

List I

(d ) None of these

7. What is the value of tan 15º + cot 15º? (a )

3

( b) 2 3 (d ) 2

(c ) 4

8. If p = sin (989º) cos (991º) then which one of the following is correct? (a) p is finite and positive (b) p is finite and negative (c) p = 0 (d) p is undefined 41π 1  3 tan 2 A , then what is the value of ? 9. If A  12 3 tan A  tan 3 A (a) 2 1 (c) 1/3

(b) 1 (d) 3

(c )

(

2 3

)

2 3

( b) (d )

1  2

 2 3  

2 3

12. What is the value of cos(π / 9)  cos(π / 3)  cos(5π / 9)  cos(7π / 9)? (a) (b) (c) (d)

1 –1 – 1/2 1/2

13. What is the value of (a) 4 (b) 3 (c) 2 (d) 1

tan 15

1.

2  3

B.

tan 75

2.

2 3

C.

tan 105

3.

2  3

4.

2 3

Code: A (a) 4 (b) 4 (c) 3 (d) 2

B 1 2 2 1

C 2 1 1 4

15. If sec A+ tan A = p, than what is the value of sin A? (a )

p2  1 p2  1

(c ) 1

( b)

p2  1 p2  1

(d ) None of these

16. What is the value of sin (1110)º? (a ) 1 (c )

( b) 1 / 2

1

(d )

3 2

17. What is the maximum value of sinq cos q?

11. What is the value of cos 15º? 1 2

List II

A.

2

10. Consider the following statements I. If q = 1200º, then (sec q + tanq)-1 is positive. II. If q = 1200º, then (coses q 2 cot q) is negative. (a) I only (b) II only (c) Both I and II (d) Neither I nor II

(a )

14. Match List I with List II and select the correct answer using the code given below the lists

1 2

6. The angle A lies in the third quadrant and it satisfies the equation 4 (sin2 x + cos x) = 1. What is the measure of the angle A? (a) 225º (b) 240º (c) 210º (d) None of these

▌ 3.105

3 cosec 20º - sec20º?

(a ) 1 ( b) 1 / 2 (c ) 1 / 2 (d )

3/2

18. If sin x + cosec x = 2, then what is the value of sin 4x + cosec4 x? (a) 2 (b) 4 (c) 8 (d) 16 19. If A + B + C = p/2, then what is the value of tan A tan B + tan B tan C + tan C tan A? (a) 0 (b) 1 (c) 2 1 (d) tan A tan B tan C 20. If angles of a triangle are in the ratio 1 : 2 : 3, then what is the ratio of its corresponding sides? (a) 3: 2 :1

(b) 1: 2 : 3

(c) 1: 3 : 2

(d) 2 : 3 : 4

21. If (sin x  cos ec x )2  (cos x  sec x )2 = k + tan2x + cot2 x, then what is the value of k? (a) 8 (b) 7 (c) 4 (d) 3

3.106

▌ Mathematics

22. If cot q = 2 cos q, where (p/2) < q < p, then what is the value of q? (a) 5p/6 (b) 2p/3 (c) 3p/4 (d) 11p/12

( )

27. What is the principle value of cosec21 2 2 ? π 4 π ( b) 2 (a )

23. If in a DABC, cos B = (sin A)/(2 sin C), then the triangle is (a) Isosceles triangle (b) Equilateral triangle (c) Right angled triangle (d) Scalene triangle 24. If sin x + sin y = a, cos x +cos y = b, then what is the value of cos(x –y)?

(c ) 2 (d ) 0

1 28. If sin

(a ) a 2  1 29.

1 2 (a  b 2  2) 2 1 (d ) (a 2  b 2 ) 2 (c )

(a ) Cos A ( b) Cos(2A)

2 cos A

 4x  dy If y = sin1  then what is equal to? 2  dx x 1  4  

(c )

(a ) 0

( b) 1 / 5

(c ) 2 / 5

(d )

3/2

(a ) 0

( b) 1 / 5

(c ) 2 / 5

(d )

3/2

32. Looking from the top of a 20 m high building the angle of elevation of the top of a tower is 60º and the angle of depression of its bottom is 30º. What is the height of the tower? (a) 50 m (b) 60 m (c) 70 m (d) 80 m

1 1 + 4x2

( b) 

If sin1 x  cot1 (1 / 2 )  π / 2, then what is the value of x?

31. The angle of elevation of the lop of a flag post from a point 5 m away from its base is 75º. What is the approximate height of the flag post? (a) 15 m (b) 17 m (c) 19 m (d) 21 m

(c ) 2 Cos ( A / 2)

(a )

(b) 7 (d) 17

30. If sin1 x  cot1 (1 / 2)  π / 2, then what is the value of x ?

25. What is 2  2  2  2 cos( 4A) equal to ?

26.

5 12 π  sin1  , then what is the value of x? x x 2

(a) 1 (c) 13

( b) b 2  1

(d )

π 4

1 1 4x 2

(

)

(

If angle A is 30º, then what is the angle B? (a) 120º (b) 90º (c) 75º (d) 60º

4x (d ) 1 4x 2

answer Key 1. 2. 3. 4. 5.

(b) (d) (b) (c) (d)

)

33. If a D ABC, a + b = 3 1  3 cm and a  b  3 1  3 cm.

4 1 4x 2

6. 7. 8. 9. 10.

(c) (c) (b) (b) (d)

11. 12. 13. 14. 15.

(b) (d) (a) (b) (a)

16. 17. 18. 19. 20.

(b) (b) (a) (b) (c)

21. 22. 23. 24. 25.

(b) (a) (a) (c) (c)

29. 30. 26. 27. 28.

(b) (b) (c) (c) (c)

31. (c) 32. (d) 33. (d)

Trigonometry

▌ 3.107

soluTions 1 1 1.  tan A  and tan B  2 3 tan A  tan B ∴ tan( A  B )  1  tan A tan B 1 1 5  = 2 3  6 1 1 1 5 1 . 2 3 6  tan  π / 4

=

cos15  cos 45 (cos15  cos 45 )(cos 2 45 + cos 2 15 − cos 45 cos15 )

=

1 (cos 2 45  cos 2 15  cos 45 cos15 )

=

1 1 cos15  (cos45  cos 30  sin 45 sin 30 )2  2 2



⇒ A B  π / 4 2. 1  tan 15 = tan 45  tan 15 1  tan 15 1  tan 45 tan 15  tan( 45  15 )  tan 60  3 3.



1  2  1  3 1  1  3  1       2  2 2 2 2 2  2 2    1

1 3 1  2 3 3 1   2 8 4 8 4 1    4 4 2 3  2 3 2 6 3 8

3 cos ec 20  sec 20 3 1 3 cos ec 20  sec 20   sin 20 cos 20 sin 20 cos 20  3  4 1   cos 20  sin 20   2 sin 20 cos 20  2 2  4 (sin 60 cos 20  cos 60 sin 20 )  sin 40 

4 4 sin(60  20 )   Sin 40  4 sin 40 Sin 40

   3 1 1    2 2     2( 3  1)  (2  3) 4.  3 1 2 2

6.  4 sin 2 x  4 cos x  1  0 ⇒ 4  4 cos 2 x  4 cos x  1  0 ⇒  4 cos 2 x  4 cos x  1  0 ⇒ 4 cos 2 x  4 cos x  3  0 ⇒ 4 cos 2 x  6 cos x  2 cos x  3  0 ⇒ (2 cos x  3)(2 cos x  1)  0 ⇒ cos x 

3 2

Which is not possible and cosx 

1 2

1 ⇒ cos A   cos 210 2 ⇒ A  210

 6 2 4 3  6  3 2 2  1 tan  7   2



5.

 1  2 sin 2  7    2   1   1  2 sin  7  cos  7   2  2

1 cos15 1  cos(45  30 )  sin15 sin(45  30 )

cos15  cos 45 cos3 15  cos3 45

7. tan15  cot15  

sin15 cos15  cos15 sin15

sin 2 15  cos 2 15 1 4  1 cos15 sin15 sin 30 2

8. Given, p = sin (989º) cos (991º) = sin (1080º 2 91º) cos (1080º 2 89º) = -sin91º cos89º = -cos1º cos89º As cos 1º and cos 89º are positive Hence, p is the finite and negative.

3.108

▌ Mathematics

1  3tan 2 A 1 9. 3tan A  tan 3 A  tan 3 A

  3 1 cos 20  sin 20   2 2  13. 3 cos ec 20  sec 20  2    sin 20 cos 20     2[sin(60  20 )]  1 sin 40 2 4 sin 40 4  sin 40

1 41π tan 4 1 1     π π tan  10π   tan 4 4  1 

14. Atan 15  tan( 45  30 )

10. I. (sec θ + tan θ )1  (sec 1200  tan 1200 )1

1



[sec(6π  120 )  tan(6π  120 )]1 (sec 120  tan 120 )1  (2  3 )1 

1 2 3

3 1 2 3 2 3 2 B.tan 75  tan( 45  30 )

 cos ec 120  cot 120

1



 sec 30  tan 30 2

 3, positive

Hence, both staatements are incorrect.

=

11. We have Cos2A  2 cos 2 A  1

cos15 

1 2

3 2 4 3 2

π  π   5π   7π  12. cos    cos    cos    cos   9 3 9        9   cos( 20 ) cos(60 )  cos(100 )  cos(140 )

3 1 1 3



1 3 1 3

42 3  2  3 2 Hence, option (b) is correct.

2 cos 2 15  cos330  1

Cos 2 15 

3  3 1  3 1  2  3 1 3 1 3 1 1 3



Cos30  2 cos 2 15  1

3 2 3 + 1 2 2

1

C .tan(105 )  tan(60  45 )

3



3  3 1  3 1 1 3 1 3 1 1 3



negative

II. cosecθ  cot θ  cos ec(6π  120 )  cot(6π  120 )



1

15.

1 sin A  p cos A cosA 1 sinA p cos A 1 sinA  pcosA 1  sin 2 A  2sinA  p2 cos 2 A ⇒ 1  sin 2 A  2 sin A  p2 (1  sin 2 A ) ⇒ sin 2 A(1  p2 )  2 sin A  1  p2  0 ⇒ sin A

1  cos 20   2 cos120 cos 20 2 1  cos 20   2 sin 30 cos 20 2



2  4  (1  p4 ) 2(1  p2 )



2  2 1  1  p4 2(1  p2 )

1 1  cos 20   cos 20  2 2



2  2 p2 p2  1  2(1  p2 ) 1  p2

Trigonometry

16. Sin 1110  sin(3  360  30 )  sin 30 )  sin 30 

1 2

17. Let P = sin q cos q 1 5 sin 2θ 2 The maximum value of sin 2q is 1 ∴P 5

1 2

18. Given sin x  cosec x  2 ∴ sin 4 x  cos ec 4 x  (sin 2 x  cos ec 2 x )2  2  (sin xc  cos ecx)2  2)   2 2

 ( 4  2) 2  2  2 19. Given A  B  C 

π 2

π  ⇒ tan ( A  B  C )  tan   2 ⇒

22. Given, cot θ = 2cos θ ⇒ cot θ(1  2 sin θ)  0 π cot θ ≠ 0 For  θ  π, 2 ∴ 1 − 2 sin θ  0 1 ⇒ sin θ  2 5π ⇒ θ 6 23. Hence, it is an isosceles tringle siin A sin B sin C   k Using sine rule, a b c sin A Given, cos B  2 sin C a 2  c 2  b2 a  ⇒ 2 ac 2c ⇒ a 2  c 2  b2  a 2 ⇒ b2  c 2 ⇒ bc 24. a  sin x  sin y a 2  sin 2 x  sin 2 y  2 cos x sin y b2 cos 2 x  cos 2 y  2 cos x cos y a 2  b2  sin 2 x  sin 2 y  2 sin x sin y  cos 2 x  cos 2 y  2 cos x cos y

1 tan A  taan B  tan C  tan A tan B tan C  1  tan A tan B  tan B tan C  tan C tan A 0

 2 + 2 cos(x  y) ⇒ 2 cos(x  y) = a 2  b2  2

⇒ tan A tan B  tan B tan C  tan C tan A  1

1 cos(x  y)  (a 2  b2  2) 2

20. Let the angles of a triangle be x, 2x and 3x. ∴ ⇒

x  2 x  3x  180 x  30

By using sine rule, a b c   sin 30 sin 60 sin 90

25.

2  2  2 cos(4A) 2  2  2  2 (2 cos 2 A  1) 2  2  2 cos 2A a

a b c ⇒   1 1 3

2  2  2  ( 2 cos 2 A  1)

a b c ⇒   1 2 3

2  2(cos 2 A/2  1)

21. Given, (sin x  cosec x )2  (cos x  sec x )2  k  tan 2 x + cot 2 x ⇒ sin 2 x  cosec 2 x  2 + cos 2 x + sec 2 x  2  k  tan 2 x  cot 2 x ⇒ 1  cosec 2 x  cot 2 x  sec 2 x  tan 2 x  4  k ⇒ 1 1 1  4  k ⇒ k  7

2  2 cos A 4 cos 2 A / 2  2 cos A/2  4x  26. ∴ y  sin1  2   1 4x  Put 2x  tan θ  2 tan θ  ∴ y  sin1   2  1  tan θ   sin1 (sin 2θ)  2θ  2 tan1 2 x

▌ 3.109

▌ Mathematics

3.110

AB h  BC 5 tan 45  tan 30 h ⇒  1  tan 45 tan 30 5

On differentiating w.r.t. x, we get. 4 dy 2  .2  dx 1  ( 2 x )2 1 4x 2

)

(

1 27. Principle value of cosec  2 

tan 75 

π 4

2

2

2

5



5 18.660 5 19 m(approx) 32. In ∆ABD, tan 30 

AD AB

⇒ AB = 20 3m.

1 π 29. sin x  cot    2 2 1 π 1 ⇒ sin x  cos1  5 2 π ∴ sin1 θ  cos1 θ  2 1 ∴ x 5 1

E

60º

D

30º A

1

B

In ∆DCE, tan 60 

(

h DC

⇒ h  3  20 3 ∴ AB  DC  20 3 ⇒ h  60 m. ∴Height of tower, BE  60  20  80m

31. Let h be the height of the flag post.

(

33. Given, a  b  3 1  3

In D ABC,

(

And a  b  3 1  3 A

On solving sine rule, a b  sin A sin B h

75º C

C

20

1 π 30. sin x  cot    2 2 1 π ⇒ sin1 x  cos1  5 2 π 1 1 ∴ sin θ  cos θ  2 1 ∴ x 5 1

)

3 1

( 3)  (1)  3  = 5(2  3) 

  3 1 2 ⇒ h 5   3 1   5  3.732

5 12 π  sin1  x x 2 x 2  144 π 1 5 1 ⇒ sin  cos  x x 2 π 1 1 But sin y  cos y  2 x 2  144 5 ∴  x x ⇒ 25  x 2  144 ⇒ x 2  169 ⇒ x  13

28. sin1

1

(

h  ⇒ h ⇒ 3 1 5 1 3

5

B

3 3 3  sin 30 sin B 1 ⇒ sin B  3  2 ⇒ B  60 ⇒

)

)

)

C

H

4

A

P

T

E

R

Analytical Geometry in 2D and 3D Point The smallest geometrical figure in a plane is a point.

Some Results Regarding Points in a Plane 1. Distance Formula The distance between two points P(x1, y1) and Q(x2, y2) is given by PQ  ( x1  x2 )  ( y1  y2 ) . 2

2

2. Section Formula If R(x, y) divides the joint of P(x1, y1) and Q(x2, y2) in the ratio m : n (m, n > 0) then mx2  nx1 my  ny1 , y 2 mn mn positive sign is taken for internal division, and negative sign is taken for external division. x

3. Centroid of a triangle If G(x, y) is the centroid of the triangle with vertices A(x1, y1), B(x2, y2), C(x3, y3), then x1  x2  x3 y  y2  y3 , y 1 3 3 a, b, c being the lengths of the sides BC, CA and AB, respectively, of the triangle ABC. x

4. Incentre of a triangle If I(x, y) is the incentre of the triangle with vertices A(x1, y1), B(x2, y2), C(x3, y3), then ax1  bx2  cx2 , a bc ay  by2  cy3 y 1 a bc

x

a, b, c being the lengths of the sides BC, CA and AB, respectively, of the triangle ABC. 5. Area of the triangle ABC with vertices A(x1, y1), B(x2, y2), C(x3, y3) is

x1 1 x 2 2 x3

y1 1 y2 1 5 y3 1

1 [x1 (y2 2 y3) 1 x2 (y2 2 y1) 1 x3(y1 2 y2)] 2

6. the Condition of Collinearity of three points A(x1, y1), B(x2, y2), C(x3, y3) is x1

y1 1

x2

y2 1 5 0.

x3

y3 1

Definition (slope) If (x1, y2) and (x2, y2) (x1 ¹ x2) are any two points on a line then the slope of the line joining these two points is defined as, y  y2 m 1  tan θ x1  x2 where, 0 is the angle which this line makes with the positive direction of x-axis, 0° £ q £ 180°, except when 0 5 90°. If q 5 90°, m does not exist and the line is parallel to y-axis. The slope between any two points of a line is called the slope of the line.

Some Standard Forms of the Equation of a Line 1. Line parallel to x-axis : y 5 k, x-axis : y 5 0 Line parallel to y-axis : x 5 h, y-axis : x 5 0. 2. Line passing through the origin and (a) making an angle q with the positive direction of x-axis: y 5 x tan q. (b) having slope m : y 5 mx. (c) though the point (x1, y1) : x1 y 5 y1 x.

3.112

Mathematics

3. Slope Intercept Form Line with slope m and making an intercept c on y-axis: y 5 mx 1 c 4. Point Slope Form Line with slope m and passing through (x1, y1): y 2 y1 5 mx 1 c. 5. Two-point Form (x1, y1), (x2, y2) :

Line passing through the points

y  y1 x  x1  . y1  y2 x1  x2 6. Intercept Form Line making intercepts a and b on x-axis and y-axis, respectively: x y  1 a b 7. Normal Form Line with length of the perpendicular form the origin on it as p and the angle which this perpendicular makes with the positive direction of x-axis as a : x cos a 1 y sin a 5 p. 8. Parametric Form Line passing through a fixed point A(x1, y1) and making angle q with the positive direction of x-axis: x  x1 y  y1  r cos θ sin θ where, r is the distance of any point P(x, y) on the line from the point A, and x 5 x1 1 r cos q, y 5 y1 1 r sin 0. In general, we observe that the equation of a straight line is of the form ax 1 by 1 c 5 0, where a, b, c are real numbers and a and b cannot be zero simultaneously. This is known as the General Form of the equation of a straight line. From this general form of the equation of the line ax 1 by 1 c 5 0, we can calculate the following constants: a (i) slope   (a ≠ 0) b (ii) Intercept on x -axis  c  (a ≠ 0) b c Intercept on y -axis   (b ≠ 0) b a | c| , cosα  , (iii) p  2 2 a 2  b2 a b sin α 

b a 2  b2

,

positive sign is taken if c is negative and vice-versa. (iv) If p1 denotes the length of the perpendicular from (x1, y1) on the line, then | ax  by1  c | p1  1 a 2  b2

(v) The points (x1, y1) and (x2, y2) lie on the same side of the line if the expressions ax1 1 by1 1 c and ax2 1 by2 1 c have the same sign and on the opposite sides if these expressions have opposite sign.

Some Results Regarding two or More Lines 1. Two line given by the equations ax 1 by 1 c 5 0 and a¢x 1 b¢y 1 c ¢ 5 0 are: a b (i) parallel if 5 , (slopes are equal) a′ b′ (ii) at right angles if aa¢ 1 bb¢ 5 0 (product of their slopes is 2 1) a b c 5 5 (iii) identical if a′ b′ c′ (compare with conditions in (i)) (iv) not parallel then, (a) the angle q, between them at their point of intersection is given by tan θ 

m  m′ a′b  ab′  1  mm′ aa′  bb′

(m, m ¢ being the slopes of the two lines) (b) the coordinates of their point of intersection are  hc′ 2 b′ c ca′ 2 c′a   ab′ 2 a′b , ab′ 2 a′b  (c) equation of any line through their point of intersection is (ax 1 by 1 c) 1 l (a¢x 1 b¢y 1 c¢) 5 0, where l is a real number. 2. Equation of a line parallel to the line given by ax 1 by 1 c 1 0 is ax 1 by 1 c ¢ 5 0 at a distance | c  c′ | d . a 2  b2 3. Three lines a1x 1 b1y 1 c1 5 0, a2x 1 b2y 1 c4 5 0, a3x 1 b3y 1 c3 5 0 are concurrent (intersect at a point) if a1

b1

c1

a2

b2

c2 5 0.

a3

b3

c3

4. The equations of the bisectors of the angles between two intersecting lines ax 1 by 1 c 5 0, a ¢x 1 b¢y 1 c¢ 5 0 are ax  by  c a 2  b2



a′x  b′ y  c′ a 2  b′ 2

If f is the angle between one of the bisectors and the line ax 1 by 1 c 5 0 such that |tan f| < 1 or 2 p/4 < f < p/4 then that bisector is the acute angle bisector of the two lines. The other equation then represents the obtuse angle bisector. 5. The equations of the lines passing through (x1, y1) and making an angle f with the line ax 1 by 1 c 5 0 are

Analytical Geometry in 2D and 3D ▌ 3.113 m  tan φ 1  m tan φ m  tan φ and, y  y1  m2 (x  x1 ) where m2  1  m tan φ y  y1  m1 (x  x1 ) where m1 

7. Some particular forms of the equations of a circle: (i) (x 2 r)2 1 (y 2 r)2 5 r2 is the equation of a circle with centre (r, r), radius r and it touches the coordinates axes. (ii) (x 2 x1)2 1 (y 2 r1)2 5 r2 is the equation of a circle with centre (x1, r), radius r and it touches the axis of x. (iii) (x 2 r)2 1 (y 2 y1)2 5 r2 is the equation of a circle with centre (r, y1), radius r and it touches the axis of y.

(m 5 tan q 5 2 a/b being the slope of the given line). Note that m1 5 tan(q 2 f), m2 5 tan (q 1 f). Circle is the locus of a point which moves in a plane so that its distance from a fixed point in the plane is always constant. The fixed point is called the centre of the circle and the constant distance is called the radius of the circle.

(iv) x3 1 y2 2 ax 2 by 5 0 is the equation of a circle  a b a 2  b2 with centre  ,  , radius  which 4  2 2 passes through the origin (0, 0) and has intercepts a and b on the axes of x and y, respectively.

Results and Some Equations Regarding Circles 1. The equation of a circle with centre (h, k) and radius r is (x 2 h)2 1 (y 2 k)2 5 r2. In particular, if the centre is at the origin, the equation of the circle is x2 1 y2 5 r2.

8. Let S º x2 1 y2 1 2gx 1 2fy 1 c,

2. The equation of the circle on the line segment joining (x1, y1) and (x2, y2) as diameter is (x 2 x1) (x 2 x2) 1 (y 2 y1) (y 2 y2) 5 0.

S¢ º x2 1 y2 1 2g¢x 1 2f¢y 5 c¢, and, L º ax 1 by 1 k, then (i) If S 5 0 and S¢ 5 0 intersect in real and distinct points, S 1 lS¢ 5 0 (l 5 21) represents a family of circles passing through these points. S 2 S¢ 5 0 (for l 5 2 1) represents common chord of the circles S 5 0 and S¢ 5 0.

3. The general equation of a circle is x2 1 y2 1 2gx 1 2fy 1 c 5 0 where, g, f, c are constants. The centre is (2 g, 2 f) and the radius is

(ii) If S 5 0 and S¢ 5 0 touch each other, S 2 S¢ 5 0 is the equation of the common tangent to the two circles at the point of contact.

g 2  f 2  c. (g 2  f 2  c ) Note. A general equation of second degree ax2 1 2hxy 1 by2 1 2gx 1 2fy 1 c 5 0 in x, y represents a circle if (i) coefficient of x2 5 coefficient of y2, i.e., a 5 b ¹ 0 (ii) coefficient of xy is zero, i.e., h 5 0.

(iii) If S 5 0 and S¢ 5 0 intersect each other orthogonally, (the tangents at a point of intersection to two circles include a right angle) then 2 gg¢ 1 2ff ¢ 5 c 1 c¢. (iv) If S 5 0 and L 5 0 intersect in two real distinct points, S 1 lJ 5 0 represents a family of circles passing through these points.

4. The point (x1, y1) lies outside, on or inside the circle S º x2 1 y2 1 2gx 1 2fy 1 c 5 0, according as S1 º x1 2 1 y12 1 2gx1 1 2fy1 1 c > 1 5 or < 0. 5. The parametric coordinates of any point on the circle (x 2 h)2 1 (y 2 k)2 5 r2 are given by (h 1 r cos q, k 1 r sin q), (0 £ q £ 2 p). In particular parametric coordinates of any point on the circle x2 1 y2 5 r2 are (r cos q, r sin q), being a parameter, 0 £ q £ 2p. 6. The equation of the tangent to the circle x2 1 y2 1 2gx 1 2fy 1 c 5 0 at the point (x1, y1) on the circle is xx1 1 yy1 1 g(x 1 x1) 1 f(y 1 y1) 1 c 5 0 and that of the normal is y f y  y1  1 (x  x1 ). x1  g In particular, the equation of the tangent to the circle x2 1 y2 5 r2 at the point (x1, y1) is xx1 1 yy1 5 r2 and that of the normal is x y 5 . x1 y1

(v) If L 5 0 is a tangent to the circle S 5 0 at P, S 1 lL 5 0 represents a family of circles touching S 5 0 at P having L 5 0 as the common tangent at P. 9.

(i)

y 5 mx 1 c is a tangent to the circle x2 1 y2 5 r2 if c2 5 r2(1 1 m2).

(ii) y 5 mx ± r 1 2 m2 is a tangent to the circle x2 1 y2 5 r2, m being the slope of the tangent. If m is infinite, the tangents are x ± r 5 0. 10. The equation of the chord of the circle S º x2 1 y2 1 2gx 1 2fy 1 c 5 0 in terms of its middle point (x1, y1) is T 5 S1 where T º xx1 1 yy1 1 g(x 1 x1) 1 f (y 1 y1) 1 c S1 º x12 1 y12 1 2gx1 1 2fy1 1 c. In particular, the equation of the chord of the circle x2 1 y2 5 r2 in terms of its middle point (x1, y1) is xx1 1 yy1 5 x12 1 y12.

3.114

Mathematics

11. The equation of the chord of contact of the tangents drawn from a point (x1, y1) outside the circle S 5 0 is T 5 0 (S and T defined above). 12. Length of the tangent drawn from a point (x1, y1) outside the circle S 5 0 to the circle is

S1 (S and S1 defined in 10).

13. Two circles with centres C1(x1, y1), C2(x2, y2) and radii r1, r2, respectively, touch each other. (i) Externally, if |C11 C2| 5 r1 1 r2, and the point of contact is  r x r x r y r y   1 2 2 1 2 1 , 1 2 .  r1  r2   r1  r2

(ii) Internally, if |C12C2| 5 |r1 2 r2| and the point of contact is

 r x r x r y r y   1 2 2 1 2 1 , 1 2  .  r  r r  r 1 2 1 2   14. (x 2 x1) (x 2 x2) 1 (y 2 y1) (y 2 y2) 1 l [(y 2 y1) (x2 2 x1) 2 (x 2 x1) (y2 2 y1)] 5 0 is the family of circles passing through the points (x1, y1) and (x2, y2). 15. (x 2 x1)2 1 (y 2 y1)2 1 l[(y 2 y1) 2 m(x 2 x1)] 5 0 is the family of circles which touch y 2 y1 5 m(x 2 x1) at (x1, y1) for any fixed m. If m is infinite, the family is (x 2 x1)2 1 (y 2 y1)2 1 l(x 2 x1) 5 0.

PARAboLA The plane curves obtained by intersecting a right circular cone with a plane are called conic sections. If the plane cuts clear through one nappe, then the intersection curve is called a circle if the plane is perpendicular to the axis and an ellipse if the plane is not perpendicular to the axis. If a plane cuts only one nappe, but does not cut clear through, then the intersection curve is called a parabola. If a plane cuts through both nappes, but not through the vertex, the resulting intersection curve is called a hyperbola. A plane passing through the vertex of the cone produces a degenerate conic—a point, a line, or a pair of lines. The figure illustrates the four nondegenerate conics.

Directix: x 5 2a Symmetric w.r.t. x-axis Axis the x-axis y

y

F

F

x

0

0

x

a  0 (opens left) a  0 (opens right) Fig. 4.2 2. x 5 4ay Vertex: (0, 0) Focus: (0, a) Directix: x 5 2 a Symmetric w.r.t. y-axis Axis the y-axis 2

Circle

Ellipse

y

y

F Parabola

Hyperbola Fig. 4.1

The graph of Ax2 1 Bxy 1 Cy2 1 Dx 1 Ey 1 F 5 0 where A, B, and C are not all 0, is a conic. The following is a coordinate-free definition of a parabola: A parabola is the set of all points in a plane equidistant from a fixed point F and a fixed line L in the plane. The fixed point F is called the focus, and fixed line L is called the directrix. A line through the focus perpendicular to the directrix is called the axis, and the point on the axis halfway between the directrix and focus is called the vertex. Standard equation of a Parabola with a vertex at (0, 0) 1. y2 5 4ax Vertex: (0, 0) Focus: (a, 0)

0

x

0

x

F a  0 (opens down) a  0 (opens up) Fig. 4.3 The following are some standard results for the parabola y2 5 4ax: 1. The parametric equations of the parabola or the coordinates of any point on it are x 5 at2, y 5 2at. 2. The tangent to the parabola at (x¢, y¢) is yy¢ 5 2a(x 1 x¢) and that at (at2, 2at) is ty 5 x 1 at2. 3. The condition that the line y 5 mx 1 c is a tangent to the parabola is c 5 a/m and the equation of any tangent to it (not parabola to the y-axis) is therefore y 5 mx 1 (a/m). 4. The chord of contact (defined as in circles) of (x¢, y¢) w.r.t. the parabola is yy¢ 5 2a(x 1 x¢).

Analytical Geometry in 2D and 3D ▌ 3.115 5. The polar (defined as incircles) of (x¢, y¢) w.r.t. the parabola is yy¢ 5 2a(x 1 x¢).

2.

6. The chord with mid-point (x¢, y¢) of the parabola is T 5 S¢, where T ≡ yy¢ 2 2a(x 1 x¢) and S¢≡ y¢2 2 ax¢. 7. The equation of the pair of tangents from (x¢, y¢,) to the parabola is T2 5 SS¢. 8. The normal at (at2, 2at) to the parabola is y 5 2 tx 1 2at 1 at3. If m is the slope of this normal, then its equation is y 5 mx 2 2am 2 am3, which is the normal to the parabola at (am2, 22am).

x2 y2   1, a  b  0 b2 a 2 x : intercepts : b y intercepts : a (vertices) Foci: F¢(0, 2c), F(0, c) c2 5 a2 2 b2 major axis length 5 2a minor axis length 5 2b y a c F

9. A diameter of the parabola is the locus of the middle points of a system of parallel chords of the parabola and the equation of a diameter is y 5 2a/m where m is the slope of the parallel chord which are bisected by it.

b

c

Ellipse An ellipse is the set of all the points P in a plane such that the sum of the distances of P from two fixed points in the plane is constant. Each of the fixed points, F¢ and F, is called a focus, and together they are called foci. Referring to the figure, the line segment V¢V through the foci is the major axis. The perpendicular bisector B¢B of the major axis is the minor axis. Each end of the major axis, V¢ and V, is called a vertex. The mid-point of the line segment F¢F is called the centre of the ellipse. d1  d2  Constant B V d1 P F d2 B Fig. 4.4

F

V

b c0

0 F′

x

a Fig. 4.6 Note: (Both graphs are symmetric w.r.t. the x-axis, y-axis, and origin. Also, the major axis is always longer than the minor axis).

Hyperbola A hyperbola is the set of all points F in a plane such that the absolute value of the difference of the distances of P of two fixed points in the plane is a positive constant. Each of the fixed points, F¢ and F, is called a focus. This intercepts points V ¢ and V of the line through the foci and the two branches of the hyperbola are called vertices, and each is called a vertex. The line segment V¢V is called the transverse axis. The mid-point of the transverse axis is the center of the hyperbola. |d1  d2|  Constant

Standard equations of an ellipse with centre at (0, 0) 1.

P d 2

x2 y2   1, a  b  0 a 2 b2 x : intercepts:  a ( vertices)

d1 A

y interceps:  b; Foci : F ′ (c, 0), F ( c, 0)

F V V′

F′

c 2  a 2  b2 major axis length  2a minor axis length  2b

Fig. 4.7 y

Standard equations of a hyperbola with centre at (0, 0)

b 1.

d F′ a c 0 b Fig. 4.5

F x ca

x2 y2  1 a 2 b2 x intercepts ± a (vertices) y intercepts: none Foci: F¢(2c, 0), F(c, 0) c2 5 a2 1 b2 Transverse axis length 5 2a Conjugate axis length 5 2b

3.116

Mathematics

y

F′ c

xx ′ yy ′  2 1 a2 b

b c a b

b a

F c

x

y  mx  a 2 m2  b2 6. An equation of the chord of contact of (x¢, y¢) w.r.t. the ellipse is xx ′ yy ′  2 1 a2 b

Fig. 4.8 2.

y2 x2  1 a 2 b2 x intercepts: none y intercepts:  a (vertices) Foci: F¢(0, 2c), F(0, c) c2 5 a2 1 b2 Transverse axis length 5 2a Conjugate axis length 5 2b

7. An equation of the polar of (x¢, y¢) w.r.t. the ellipse is xx ′ yy ′  2 1 a2 b 8. An equation of the chord of the ellipse whose mid-point is (x¢, y¢) is T 5 S¢, where xx′ yy′ x ′2 y ′2 T ≡ 2  2  1 and S ′ ≡ 2  2  1 a b a b 9. An equation of the pair of tangent from a point (x¢, y¢) to the ellipse is T2 5 SS¢, where T and S¢ are as above, and x2 y2 S ≡ 2  2 1 a b

y F c a c a b

x

b

a F′ c Fig. 4.9 Note:

5. The condition that the line y 5 mx 1 c touches the ellipse is c2 5 a2m2 1 b2, so that the equation of any tangent to the ellipse (not parallel to the y-axis) can be written as

Both graphs are symmetric with respect to the x axis, y axis, and origin.

The following are some standard results for an ellipse and a hyperbola: 1. The parametric equations of an ellipse (hyperbola) or the coordinates of any point of the ellipse (hyperbola) are x 5 a cos q, y 5 b sin q (x 5 a sec q, y 5 tan q). The point is denoted “q”. 2. An equation of the tangent at the above point “q” is x  x y y cos θ  sin θ  1  sec θ  tan θ  1  a b a b   3. An equation of the normal at the same point “q’’ is ax by   a 2  b2 cos θ sin θ  ax  by   a 2  b2   sec θ tan θ   4. An equation of the tangent at the point P(x¢ y¢) on the ellipse is

10. Director circle of an ellipse in the locus of the point of intersection of tangents to the ellipse which intersect at right angles and its equation is x2 1 y2 5 a2 1 b2. 11. Auxiliary circle of an ellipse is the circle on major axis of the ellipse as diameter and its equation is x2 1 y2 5 a2. If P is a point on the ellipse and Q is a point on the auxiliary circle such that Q lies on the ordinate produced of the point P, then ÐACQ (where CA is the semi-major axis of the ellipse) is called the Eccentric angle of the point P on the ellipse and the coordinates of P are (a cos f, b sin f) where f 5 ÐACQ. 12. A diameter of an ellipse is the locus of the middle points of a system of parallel chords of the ellipse and its equation is b2 y  2 x , am where m is the slope of the parallel chords of the ellipse which are bisected by it. This is a line through the centre of the ellipse. Two diameters of an ellipse are said to be conjugate when each bisects the chord parallel to the others. Thus, two diameters y 5 mx and y 5 m¢x of the b2 ellipse are conjugate if mm¢  2 . a 13. An asymptote to a curve is a line which touches the curve at infinity. Equation of the asymptotes of the hyperbola x2 y2   1 is a 2 b2 x2 y2 x y   0 or   0. a b a 2 b2

Analytical Geometry in 2D and 3D ▌ 3.117 14. A hyperbola whose asymptotes are perpendicular is called rectangular hyperbola and its equation is x2 2 y2 5 a2. By taking the asymptotes of the rectangular hyperbola as the coordinate axis, its equation can be written as xy 5 c2 (where c2 5 a2/2) and the parametric equation of this rectangular hyperbola is x 5 ct, y 5 c/t, t being the parameter.

(x  h)2  4a(y  k) Vertex (h, k) y Focus (h, k  a) a  0 opens up a  0 opens down F

translation of Axes

a

In the last three sections, we found standard equations for parabolas, ellipses, and hyperbolas located with their axes on the coordinate axes and centred relative to the origin. We now move the conics away from the origin while keeping their axes parallel to the coordinate axes. In this process, we obtain new standard equations that are special cases of the equation Ax2 1 Cy2 1 Dx 1 Ey 1 F 5 0, where A and C are not both zero. The basic mathematical tool used is translation of axes. A translation of coordinate axes occurs when the new coordinate axes have the same direction as and are parallel to the original coordinate axes. Translation formulas are as follows:

V(h, k) x (y  k)2  4a(x  h) y Vertex (h, k) Focus (h  a, k) a  0 opens left a  0 opens right

a

(h, k) V

F

1. x 5 x¢ 1 h y 5 y¢ 1 k

x

2. x¢ 5 x 2 h Fig. 4.12

y¢ 5 y 2 k where, (h, k) are the coordinates of the origin O¢ relative to the original system. y

y

Ellipses

(x  h)2 (y  k)2 1  a2 b2 y

y

Centre (h, k) Major axis 2a Minor axis 2b

P(x, y) P(x′, y′)

y′

b (h, k) (0, 0) 0

x′

0 (0, 0)

x Fig. 4.10

Standard Equations for translated Conics Circles (x 2 h)2 1 (y 2 k)2 5 r2 y

Centre (h, k) Radius r r

x′

a

(h, k)

x x ab0 (x  h)2 (y  k)2 1  b2 a2 Centre (h, k) Major axis 2a Minor axis 2b

y

a (h, k)

b

c(h, k) x Fig. 4.11

Fig. 4.13

3.118

Mathematics

ab0 (y  k)2 (x  h)2 1  b2 b2 y

(x  h)2 (y  k)2 1  a2 b2 Centre (h, k) Transverse axis 2a Conjugate axis 2b

y

a

Centre (h, k) Transverse axis 2a Conjugate axis 2b

b a

b

(h, k)

(h, k)

x

x Fig. 4.14

PRACtiCE EXERCiSE Point on A PLAnE 1. Two vertices of a triangle are (6,4) and (2,6). If its centroid is (4, 6) the coordinates of the third vertex is one of the following: (a) (4,8) (b) (3,6) (c) (2,8) (d) (4,4) 2. Find the distance between the point A(22, 1,23) and B(4, 3, 26). (a) 8 (b) 7 (c) 9 (d) None of these 3. If A and B are two points whose position vectors are →

(

)

→

(

)

r1  3iˆ  ˆj  2 kˆ and r2  iˆ  3 ˆj  kˆ respectively, find the position vector of the point dividing A internally in the ratio 2:3. 11 ˆ 3 ˆ 8 ˆ i  j k 5 5 5 11 ˆ 3 ˆ 8 ˆ (c) i  j k 5 5 5

11 ˆ 3 ˆ 8 ˆ i  j k 5 5 5 11 ˆ 3 ˆ 8 ˆ (d) i  j k 5 5 5

(a) −

(b)

4. If A and B are two points whose position vectors are →

(

)

→

(

r1  3iˆ  ˆj  2 kˆ and r2  iˆ  3 ˆj  kˆ

)

respectively, find

the position vector of the point dividing AB externally in the ratio 3:1. ? 1 ?   ? 1 ? (a)   5 j  k  (b)  5 j  k  2  2    ? 1 ?  (c)  5 j  k  2  

? 1 ?  (d)  5 j  k  2  

5. Find the cordinates of the point which divides the joint of the points P(5, 4, 2) and Q (21, 22, 4) in the ratio 1:3.

 13 8 14  (a)  , ,   5 5 5

 13 8 214  (b)  , ,  5 5 5 

 7 5 5 (c)  , ,   2 2 2

 27 5 25  , , (d)   2 2 2 

6. Find the ratio in which the point of A(2,1,5) and B(3,4,3) is divided by plane 2x 1 2y 2 2z 5 1. Also, find the cordinates of the point of division.  29 9 25   29 29 25  (a)  , ,  (b)  , ,  12 4 6   12 4 6   29 9 225  (c)  , ,  12 4 6 

(d) None of these

(

)

7. Find the angles at which the vector 4iˆ 1 8 ˆj 1 kˆ is inclined to the coordinate axes.  4  8  1 (a) cos21   , cos21   , cos21    9  9  9  4  9 (b) cos21   , cos21   , cos21 (9)  8  8  8  1  1 (c) cos21   , cos21   , cos21    4  8  9 (d) None of these →

8. A vector r is inclined at equal acute angles to x-axis, y-axis →  and z-axis. if r 5 6 units, find r . ? ? ? (a)  2 3 ( i  j  k ? ? ? (c) 2 3 ( i  j  k

) )

? (b) 2 3 ( i  ? (d) 2 3 ( i 

? ? jk ? ? jk

) )

Analytical Geometry in 2D and 3D ▌ 3.119 9. Find the angles of triangle ABC whose vertices are A(21,3,2), B(2, 3, 5) and C (3, 5, 22).  2  1  (a) ∠A  90, ∠B  cos 1   , ∠C  cos 1    3  3  2  1 (b) ∠A  90, ∠B  cos1   , ∠C  cos1    3  3

 2  1 (c) ∠A  90, ∠B  cos1   , ∠C  cos1    3  3  (d ) None of these 10. Find the angle between the vectors with direction ratios 3,4,5 and 4,23,5. (a) 60º (b) 120º (c) 45º (d) 0º

AnSwER KEy 1. (a)

3. (d)

5. (c)

7. (a)

9. (a)

2. (b)

4. (a)

6. (a)

8. (d)

10. (a)

SoLutionS 1. Let (x, y) be the coordinates of a third vertex. Then, we have 62 x 4 ; x4 3 46 y 6 ; y  8. 3 2.

→

8. Suppose, that r make an angle a with each one of the x-axis, y-axis and z-axis. l2 5 cos a, m2 5 cos a, n2 5 cos a l2 5 m2 1 n2 35 cos2 a

( 4  2)2  (3  1)2  (66  3)2 2

2

 6  2  (  3)

∴

→

→

→

(

1 3

)

r  r liˆ  mjˆ  nkˆ .

 49  7.  3. Let C be internally in the ratio position vector r dividing AB internally in the ratio 2:3    2 ⋅ r2  3 ⋅ r1 r 23 11 3 8  iˆ  ˆj  kˆ. 5 5 5

3

l mn

2

 36  4  9

1

cos α 

( x2  x1)2  ( y2  y1)2  ( z2  z1)2 .

9. Direction ratios of AB and AC, i.e., (3, 0, 3) and (4, 2, 24), respectively. A(1, 3, 2)

→

1   3 r  1 r1     5 ˆj  kˆ  . 4. By section formula r  2 3 1 2  

B (2, 3, 5)

C (3, 5, 2)

5. Let P(x, y, z) be the required point, then apply section formula. 6. Suppose, the given plane intersects AB at a point C and let the required sation by l:1 coordinates of C  3λ 1 2 4λ 1 1 3λ 1 5   λ 1 1 , λ 1 1 , λ 1 1  , since C lies on the place. 7. Find the direction ratios, b find the direction cosines of   4 8 1 r 5 , , Let α , β , γ , be the angle made by r with x-axis, 9 9 9 y-axis and z-axis, respectively. 4 8 1 ∴ cos α 5 ; cos β 5 ; cos γ 5 . 9 9 9

∴ cos A 

(

(3  4)  (0  2)  (3  (4)) 32  02  32

)(

42  22  (4)2

)

so, Ð A 5 90°. Now, again direction ration of BA and BC are (23, 0, 23) and (1, 2, 27) respectively and get Ð B 5  2  1  1 . cos1   , similarly, ∠C  cos    3  3 10. cos θ 

a1a2  b1b2  c1c2 a  b12  c12 a22  b22  c22 2 1

.

3.120

Mathematics

EquAtion oF A LinE oR MoRE tHAn two StRAiGHt LinES 1. Find the foot of the perpendicular drawn from the point A (1,0,3) to the join of the point B(4,7,1) and C(3,5,3).  25 7 217  (a)  , ,  3 3 3 

 5 27 17  (b)  , ,  3 3 3 

 5 7 17  (c)  , ,  3 3 3 

 25 27 217  (d)  , , 3   3 3

2. Find the equation of the line passing through the point (1,2,3) x 6 y 2 z 17   and parallel to the line . 12 4 5 x 1 y  2 z 1   4 12 5 x 1 y  2 z  3 (c)   4 5 122

(a)

3. Find the point where the line place 2x 1 4y 2 z 5 3. (a). (3, 21, 21) (c). (3, 1, 1)

(b)

x 1 y  2 z  3   12 4 5

(d) None of these x 1 y  2 z  5   meets the 12 3 4 (b) (3, 21, 1) (d). (3, 1, 21)

4. Find the equations of the line passing through the point (21, 3, 22) and perpendicular to the lines. x y z x  2 y 1 z 1   and   . 1 2 3 3 2 5 x 1  2 x 1  (c) 2

(a)

y 3 z 2  7 4 y 3 z 2  7 4

(b)



x 1 y  3 z  2   2 7 4

x  4 y 1  . 3 5

z 3 x 1 y  4 z  5   and 4 1 1 2

 8 3 (a) cos21    15 

 28 3  (b) cos21    15 

 8 3  (c) cos21    3(15) 

(d) None of these

6. Find the angle between the lines, one of which has direction ratio 2, 2, 1 while the other one is obtained by joining the points (3, 1, 4) and (7, 2, 12). (b) cos21(22/3) (a) cos21(2/3) 21 (c) cos (3/2) (d) cos21(23/2) 7. Find the angle between the lines

( ) ( λ ( 3 ˆj 1 2 ˆj 1 6 kˆ ) →

)

 219  (b) cos21   21 

 21 (c) cos21    19 

 221 (d) cos21   19 

8. Find the equation of a plane passing through points P(2, 2, 21), Q(3, 4, 2) and R (7, 0, 6). (a) 5x 2 2y 2 3y 5 17 (b) 5x 1 2y 1 3z 5 17 (c) 5x 1 2y 2 3z 5 17 (d) None of these 9. Reduce the equation of the plane 2x 2 4y 1 3z 5 12 to intercept from and find its intercepts on the coordinate axes. (a) (6, 23, 4) (b) (6, 23, 24) (c) (26, 3, 4) (d) (26, 23, 4) 10. Find the equation of the plane passing through the point (2,3,4) and parallel to the plane, 5x 2 6y 1 7z 5 3. (a) 5x 1 6y 2 7z 5 20 (b) 5x 2 6y 2 7z 5 20 (c) 25x 1 6y 1 7z 5 20 (d) 5x 2 6y 1 7z 5 20 11. Find the angle between the line

(d) None of these

5. Find the angle between the lines

 19  (a) cos21    21

→

(

)

r 3iˆ  2 ˆj  4 kˆ  λ iˆ  2 ˆj  6 kˆ and r  5 ˆj  2 kˆ 

x  2 y 1 z  3   and 3 1 2

the plane 3x 1 4y 1 z 1 5 5 0.  27  (a) sin21   2 91 

 7  (b) sin21   2 91 

 14  (c) sin21   91 

 214  (d) sin21   91 

12. Find the equation of the sphere with centre (3,6,24) and touching the plane. 2x 2 2y 2 z 2 10 5 0 (a) x2 1 y2 1 z2 1 6x 1 12y 2 8z 1 45 5 0 (b) x2 1 y2 1 z2 2 12x 1 6y 2 8z 1 45 5 0 (c) x2 1 y2 1 z2 2 6x 2 12y 1 8z 1 45 5 0 (d) None of these π 13. Find the slope of a line with inclination 30º or . 6 21 1 (a) (b) 3 3 (c)

3

(d) 2 3

14. Transform x2 1 3y2 1 4x 1 18y 1 30 5 0 when the axes are translated to (22,23). (a) x2 2 3y2 5 1 (b) x2 1 3y2 5 21 (c) 2 x2 1 3y2 5 21 (d) x2 1 3y2 5 1

(1)

Analytical Geometry in 2D and 3D ▌ 3.121 15. The slope of the line joining A (23, 5) and B (4, 2) is 3 23 (a) (b) 7 7 7 27 (c) (d) 3 3 16. The ratio in which the line segment joining the points P(24, 2) and Q(8, 3) divided by y-axis (a) 3:1 (b) 1:3 (c) 2:1 (d) 1:2 17. A(23,b) and B(1,b 1 4) are two points and the co-ordinates of middle point of AB are (21,1). The value of B is (a) 1 (b) 21 (c) 2 (d) 0 18. In order that the graph of y 5 2x2 2 4x 1 3m crosses the x-axis in two distinct points, the value of m must be such that 4 2 (a) m  (b) m  3 3 2 3 (c) m  (d) m  3 2 19. The line 3x 2 7y 5 10 meets x - axis at the point  10  (a)  , 0  3 

 210  (b)  0, 7  

 210  (c)  0, 3  

(d) None of these

20. The coordinates of centroid of the triangle whose sides are 12x2 2 20xy 1 7y2 5 0 and 2x 2 3y 1 4 5 0 are  8 28  (a)  ,  3 3 

 8 8 (b)  2 ,   3 3

 8 8 (c)  ,   3 3

 8 8 (d)  2 ,2   3 3

21. If the pairs of straight lines x2 2 2pxy 2 y2 5 0 and x2 2 2qxy 2 y2 5 0 be such that each pair bisects the angle between the other pair, then (a) pq 5 1 (b) pq 5 2 1 (c) pq 5 2 (d) None of these 22. The angle between the lines joining the origin to the points of intersection of the line x 3  y  2 and the curve x2 1 y2 5 4 is π π (a) (b) 6 4 π π (c) (d) 3 2 23. If the ratio of gradients of the lines, represented by ax2 1 2hxy 1 by2 5 0 is 1:3, then the value of the ratio h2:a b is 1 3 (a) (b) 3 4 4 (c) (d) 1 3

24. Two pairs of straight lines have the equations y2 1 xy 2 12x2 5 0 and ax2 1 2hxy 1 by2 5 0. One line will be common among them if (a) a 1 8h 216h 5 0 (b) a 2 8h 1 16h 5 0 (c) a 2 6h 1 9b 5 0 (d) a 1 6h 1 9b 5 0 25. If the angle between the two lines represented by 2x2 2 5xy 1 3y2 1 6x 1 7y 1 4 5 0 is tan21m, then m 5 1 5 7 (c) 5

(a)

(b) 1 (d) 7

26. Given the system of straight line a(2x 1 y23) 1 b(3x 1 2y25) 5 0, the line of the system farthest from the point (4, 3) has the equation (a) 3x24y 1 1 5 0 (b) 4x 1 3y25 5 0 (c) 7x2y 1 4 5 0 (d) None of these 27. A rectangle has two opposite vertices at the points (1, 2) and (5, 5). If the other vertices lie on the line x 5 3, then the coordinates of the other vertices are (a) (3,21), (3,26) (b) (3, 1), (3, 5) (c) (3, 2), (3, 6) (d) (3, 1), (3, 6) 28. A line joining two points A (2, 0) and B (3, 1) is rotated about A in anti-clockwise direction through an angle 15º. If B goes to C in the new position, then the coordinates of C are  3 (a)  2,  2  

 3 (b)  2   2  

 3 1 (c)  2  ,  2 2  

(d) none of these

29. Without changing the direction of coordinates axes, origin is transferred to (α, β ) so that the linear terms in the equation x2 1 y2 1 2x 1 4y 1 6 5 0 are eliminated. The point (α, β ) is (a) (21, 2) (b) (1,22) (c) (1, 2) (d) (21,22) 30. P is a point on either of the two lines y 2 3 x 5 2 at a distance of 5 units from their point of intersection. The coordinates of the foot of the perpendicular from P on the bisector of the angle between them are  1   1  (a) 0, ( 4  5 3  or 0, ( 4  5 3  2 2     depending on which line the point P is taken  1  (b) 0, ( 4  5 3   2   1  (c) 0, ( 4  5 3   2  5 5 3 (d)  ,   2 2 

3.122

Mathematics

AnSwER KEy 1. 2. 3. 4.

(c) (b) (a) (a)

5. 6. 7. 8.

9. 10. 11. 12.

(a) (a) (a) (c)

(a) (d) (b) (c)

13. 14. 15. 16.

17. 18. 19. 20.

(a) (d) (b) (d)

21. 22. 23. 24.

(a) (c) (b) (c)

25. 26. 27. 28.

(b) (c) (c) (d)

(a) (a)

29. (a) 30. (b)

(d) (c)

SoLutionS 1. Let D be the foot of the perpendicutar from A on B; suppose that D divides the joint of B (4,7,1) and C (3,5,3) in 10th ratio l:1 Then coordinates of D  3λ 1 4 5λ 1 7 3λ 1 1 (1)  λ 1 1 , λ 1 1 , λ 1 1  Now, the direction ratios of BC are (3 2 4),(5 2 7), (3 2 1), i.e., 21, 22, 2. A(1, 0, 3)

B (4, 7, 1)

D

C (3, 5, 3)

 2λ  3 5λ  7 2  Similarly, direction ratio of AD are  , ,   λ 1 λ 1 λ 1  since AD⊥BC we have,   2λ  3   5λ  7   2   7 2   2 . 1   0 or λ     4  λ 1   λ  1    λ 1  Putting λ 

7 , we get the coordinates of the required foot 4

 5 7 17  of the ┴ as  , ,  . 3 3 3  2. Direction ratios of equation ⇒12, 4, 5 are passing through the point (1,2,3). x 1 y  2 z  5    λ . Find values of x, y, z in terms 2 3 4 of λ. Thus (2l 1 1, 2 3l 1 2, 4l 1 5) is a general point on the given line. If this point lies on the plane (2l 1 1, 2 3l 1 2, 4l 1 5) we have 2(2l 1 1) 1 4(23l 1 2) 2 (4l 2 5) 5 3, ⇒ l51 The direction ratio of the required line be a, b, c. Then, a 1 2b 1 3c 5 0 (I) 23a 1 2b 1 5c 5 0 (II) Solving (I) and (II) by class multiplication, we get: a b c a b c   and   k 4 14 8 2 7 4 ∴  a 5 2k, b 5 27k, and c 5 4k

3. Let

4.

 

the required equations of the line are: x 1 y  3 z  2 x 1 y  3 z  2   or   . 2k 7 k 4k 2 7 4 5. The direction ratios of the given lines are 3,5,4 and 1,1,2. Thus, if θ is the angle between the lines, then cos θ 

(3  1  5  1  4  2)

θ  cos1

8 3 15

. 32  52  42 12  12  22 16 (3  5  8) ⇒ cosθ  cosθ  9  25  16 1  1  4 50 6 16  500 16 3 16 3 8 3     30 15 100 5 3

6. Direction ratio of one line are (2,2,1). Direction ratio of another line are (7,23), (2,21), (12,24) i.e., (4,1,8). Now, apply formula for angle between two lines  θ   

 . a22  b22  c22 

a1a2  b1b2  c1c2 a12  b12  c12 ⋅

  m ⋅m 7. Using cos θ  1 2  | m1 | | m2 | ˆ iˆ  2 ˆj  2 k , 3iˆ  2 ˆj  6 kˆ

(

)(

iˆ  2 ˆj  2 kˆ 3iˆ  2 ˆj  6 kˆ

)  19

21

 19  θ  cos1   .  21  8. Any plane through P (2,2, 2 1) is a(x 2 2) 1 b(y 2 2) 1 c(z 1 1) 5 0 (I) If it passes through Q (3, 4, 2), we have a(3 2 2) 1 b(4 2 2) 1 c(2 1 1) 5 0 a 1 2b 1 3c 5 0 (II) Solving 5a 2 2b 1 7c5 0 (III) Solving II and III we get, a,b,c a 5 5k, b 5 2k, c 5 3k, put in (I) we get 5x 1 2y 2 3z 5 17.

Analytical Geometry in 2D and 3D ▌ 3.123 9. The given equation of the plane is 2x 2 4y 1 3z 5 12 (I) x y z  1 On dividing (1) throughout by 12, we get  6 3 4 which is the equation in requisite form clearly, intercepts of the plane with x- axis y-axis and z-axis are respectively 6, 23 and 4. 10. Clearly, the direction ratios of the normals to the parallel planes are proportional. So, the equation of a plane parallel to the given plane is of the form, 5x 2 6y 1 7z 5 k. Since, it passes through (2,3,4), we have 10 2 18 1 28 5 k or k 5 20. Hence, the equation of the required plane is 5x 2 6y 1 7z 5 20. 11. sin θ 

(3  3  1  4  2  1) 2

3  (1)2  22 32  42  12

1

12. Radius of the sphere length of perpendicular from (3, 6,24) to the plane. 2x 2 2y 2 z 2 10 5 0 Þ 4 The required equation of the sphere is ( x  3)2  ( y  6)2  ( z  4)2  42. x 2  9  6 x  y 2  36  12 y  z 2  16  8 z  16 x 2  y 2  z 2  6 x  12 y  8 z  45 = 0 13. m 5 tan 30° 5

1

.

3 14. The tranformation for shifting origin to (22, 23) are x 5 X 1 (22) y 5 Y 1 (23). Put in (1) and get x2 1 3y2 5 1. 15. Slope  

y2  y1 x2  x1

2 5 3  43 7

16. Let the ratio K:1 The abscissa of point lying on y-axis is 8 k  4 1 1 ∴  0 ⇔ 8k  4  0 ⇔ k  k 1 2 17.

bb4 1 2

18. Equation of x-axis is y 5 0. Solving, y 5 2x2 4x 1 3m 5 0 and y 5 0, Simultaneously, we get 2x2 2 4x 1 3m 5 0. 2 For distinct roots, we have, m  . 3 19. The equation of the y-axis is x 5 0. solving 3x 2 7y 5 10 and  210   x 5 0, we get the point  0,   7    210  ∴ The required point is  0, .   7   20. We have, 12x2 2 20xy 1 7y2 5 0 Þ (2x 2 y)(6x 2 7y) 5 0

so the equations of three sides of the triangles are 2x 2 y 5 0, 6x 2 7y 5 0, 2x 2 3y 1 4 5 0. The coordinates of vertices of the triangle are (0,0), (7,6) and (1,2). Centroid of the triangle is  0 1 7 11 0 1 6 1 2  8 8 ,   i.e,  ,  . 3 3    3 3 21. The equation of the bisectors of the angles between the lines x2 2 2pxy 2 y2 5 0 is x2  y2 xp  1  (1)  p

or

x2  y2 xp  2 p

2 (1) xy  y 2  0 p (2) also, x2 2 2pxy 2 y2 5 0 is the equation of the bisectors of the angles between the same lines (given). From (1) and (2), by comparing coefficients, we get 1 2 / p 1 1   i.e.1  or pq 1. 1 2 p 1 pq i.e., x 2 

22. Making the equation of the curve x2 1 y2 5 4 homogeneous, using the equation of the line x 3  y  2, we get x2  y2  4 ⇒

(

)

2 1 x 3  y  0 or x 2  3 xy  0 4

∴ a  1, b  0, h  3 / 2. so, the angle between the two lines is 2 h2  ab π  tan1 3  . ab 3

θ  tan1

23. Let m, 3m be the gradients of the lines represented by ax2 1 2hxy 1 by2 5 0. 2h ∴ m  3m  4m  (1) b a and, m  3m  3m 2  b Substituting the value of m from (a) and (2), we get 2

 h a h2 4 3  ⇒   . b ab 3  2 b 24. The equation y2 1 xy 2 12x2 5 0 can be rewritten as ( y 1 4x)(y 2 3x) 5 0 y ⇒  4, 3. x The two pairs will have a line common if 24 or 3 is a root of  b 

2

 y  2h   x 

y  a 0 x 

\ 9b 1 6h 1 a 5 0 or 16b 2 8h 1 a 5 0. 25. Comparing the given equation with ax2 1 2hxy 1 by2 1 2gx 1 2fy 1 c 5 0, we get a 5 2,

3.124

Mathematics

h 5 5/2 and b 5 3. Let the angle between the lines be q, then

AB 

2

tan θ  

2 h2  ab 2 ab 2 5

5   23 2 23

a∴ ∠BAX  45 Given ∠CAB  15 ∴ ∠CAX  60.

1 2 1 1    . 4 5 2 5

∴ Slope of the AC  tan 60  3.

26. The given system of lines pass through (1, 1) So, the required line passes through (1, 1) and perpendicular to the join of (1, 1) and (4, 2 3).    

∴ The equation of line is

y 1 3  , i.e., 3x  4 y  1  0 x 1 4

27. Let A ≡ (1, 2) and C ≡ (5, 5). Since the vertices B and D lie on the line x 5 3, therefore, let B ≡ (3, y1) and D ≡ (3, y2). Since AC and BD bisect each other, so they have same middle point D(3,y2)

(1,2)A x3 C(5,5)

Now, line AC makes an angle of 60º with positive direction of x-axis and AC  AB  (3  2)2  (1  0)2  2. ∴ i.e.,

Coordinates of C are (2  2 cos 60, 0  2 sin 60 )   3 1  ,  2 .  2 2  

29. The given equation is x2 1 y2 1 2x 1 4y 1 6 5 0 ...(1) Putting x 5 x′ 1 α, y 5 y′ 1 β in (1), we get x′ 2 1 y′ 2 1 x′ (2α 1 2) 1 y′ (2β 2 4) 1 (α2 1 β2 1 2α 2 4β 1 6) 5 0 To eliminate linear terms, we should have 2α 1 2 5 0 and 2β 2 4 5 0   ⇒ α 5 2 1 and β 5 2   ∴ (α, β) ≡ (21, 2). 30. Equation, of two lines are

B(3,y1)

 

0 1 1 23  tan 45

y1  y2 2  5 i.e.,  2 2 or y1 1 y2 5 7 ...(1) Also, BD2 5 AC2 ⇒ (y12 y2)2 5 (125)2 1 (225)2 5 25 or y1 2 y2 5 ± 5 ...(2) Solving (1) and (2), we get y1 5 6, y2 5 1 or y1 5 1, y2 5 6. Thus, the other vertices of the rectangle are (3, 1) and (3, 6).

28. Slope of line

y  3x  2, and

if x  0

y  3x  2, if x  0.

Clearly y  2. Y Q

P2

30º

5

P1 5

A(0,2) 2 O

Y C

Also, y-axis is the bisector of the angle between the two lines. P1, P2 are two points on these lines, at a distance 5 units from A. Q is the foot of the ⊥ from P1 and P2 on the bisector ( y-axis).

B(3,1)

Then, the coordinates of Q are (0, 2 1 5 cos 30º) 15º 45º O

X

X

   5 3   1    0, 2     0, (4  5 3  .   2 2    

Analytical Geometry in 2D and 3D ▌ 3.125

CiRCLE 1. If the lines 12x 1 5y 1 16 5 0 and 12x 1 5y 2 10 5 0 are tangents to the same circle, then the radius of this circle is (a) 1 (b) 2 (c) 4 (d) None of these 2. A circle C and the circle x2 1 y2 5 1 are orthogonal and have radical axis parallel to y-axis, then C can be (a) x2 1 y2 1 1 5 x (b) x2 1 y2 1 1 5 y (c) x2 1 y2 1 1 5 2x (d) x2 1 y2 2 1 5 2x 3. A point moves such that the sum of the squares of its distances from the sides of a square of side unity is equal to 9. The locus of the point is a circle such that (a) centre of the circle coincides with that of square. 1 1 (b) center of the circle is  ,  2 2 (c) radius of the circle is 2 (d) all the above are true 4. If the two circles x2 1 y2 2 8x 1 2y 1 8 5 0 intersect in two distinct points, then (a) 2 < r < 8 (b) r < 2 (c) r 5 2 (d) r > 2 5. If (α, β) is a point on the chord PQ of the circle x2 1 y2 5 19, where the coordinates of P and Q are (3,2 4) and (4, 3) respectively, then (a) α ∈ [3, 2 4], β ∈ [2 4, 3] (b) α ∈ [2 4, 3], β ∈ [3, 4] (c) α ∈ [3, 3], β ∈ [2 4, 4] (d) None of these 6. If the point (k 1 1, k) lies inside the region bounded by the curve and y-axis, then k belongs to the interval x  25  y 2 . (a) (b) (c) (d)

(21, 3) (24, 3) (2∞,24) ∪ (3, ∞) None of these

7. If the equations of four circles are (x ± 4)2 1 ( y ± 4)2 5 42, then the radius of the smallest circle touching all the four circles is (a) 4( 2 1) (b) 4( 2 1) (c) 2( 2 1) (d) none of these

8. The intercept on the line y 5 x by the circle x2 1 y222x 5 0 is AB. Equation of the circle with AB as a diameter is (a) x2 1 y2 1 x 1 y 5 0 (b) x2 1 y22x2y 5 0 (c) x2 1 y2 1 x2y 5 0 (d) None of these 9. The equation of the circle whose centre is (3,21) and which cuts off an intercept of length 6 from the line 2x25y 1 18 5 0, is (a) x2 1 y226x 1 2y 1 28 5 0 (b) x2 1 y2 1 6x 1 2y228 5 0 (c) x2 1 y226x22y 1 28 5 0 (d) x2 1 y226x 1 2y228 5 0. 10. The line y 5 mx 1 c intersects the circle x2 1 y2 5 r2 at the two real distinct points if (a)  r 1  m2  c  r 1  m2 (b)  c 1  m2  r  c 1  m2 (c)  r  c  r 1  m2 (d) none of these 11. If a circle passes through the points of intersection of the coordinate axes with the lines λx2y 1 1 5 0 and x22y 1 3 5 0, then the value of λ is (a) 2 (b) 1 (c) 21 (d) 22 12. The locus of the mid point of the chord of the circle x2 1 y222x22y22 5 0 which makes an angle of 120º at the centre is: (a) x2 1 y222x22y 1 1 5 0 (b) x2 1 y2 1 x 1 y21 5 0 (c) x2 1 y222x22y21 5 0 (d) None of these 13. The area of an equilateral triangle inscribed in the circle x2 1 y2 1 2gx 1 2f y 1 c 5 0 is (a)

3 3 2 (g  F 2 C) 2

(b) 

3 3 2 (g  F 2 C) 4

3 3 2 (g  F 2 C) 4 (d) None of these (c)

3.126

Mathematics 18. The sum of the square of the length of the chords intercepted by the line x 1 y 5 n, n ∈ N on the circle x2 1 y2 5 4 is (a) 11 (b) 22 (c) 33 (d) None of these

14. The tangent to the circle x2 1 y2 5 5 at the point (1,22) also touches the circle x2 1 y228x 1 6y 1 20 5 0. Then its point of contact is (a) (3,21) (b) (23, 0) (c) (21,21) (d) 22, 1) 15. 3x 1 y 5 0 is a tangent to a circle with centre (2,21). Then the other tangent to the circle from the origin is (a) x 1 3y 5 0 (b) 3x2y 5 0 (c) x23y 5 0 (d) x 1 y 5 0

19. The equation (x 1 y2 6) (xy23x2y 1 3) 5 0 represents the sides of a triangle then the equation of the circumcircle of the triangle is (a) x2 1 y225x29y 1 20 5 0

16. If a circle passes through the points (1, 0), (5, 0) and touches the y-axis at (0, h), then h 5 (a)

5

(c) 3 5

(b) x2 1 y224x28y 1 18 5 0

(b) 2 5

(c) x2 1 y223x25y 1 8 5 0

(d) 4 5

(d) x2 1 y2 1 2x23y21 5 0 20. ABCD is a square, the length of whose side is a. Taking AB and AD as the coordinate axes, the equation of the circle passing through the vertices of the square, is (a) x2 1 y2 1 ax 1 ay 5 0

17. If the circle x 2 1 y 2 1 2gx 1 2fy 1 c 5 0 bisects the circumference of the circles x2 1 y2 1 2g′x 1 2f′y 1 c′ 5 0, then (a) 2g′(g2g′) 1 2f′(f2f′) 5 c2c′ (b) g′(g2g′) 1 f′(f2f′) 1 c2c′ 5 0 (c) 2g(g2g′) 1 2f (f2f′) 5 c2c′ (d) None of these

(b) x2 1 y22ax2ay 5 0 (c) x2 1 y2 1 2ax 1 2ay 5 0 (d) x2 1 y222ax22ay 5 0

AnSwER KEy 1. (a) 2. (a, c) 3. (b)

4. (b) 5. (a) 6. (a)

7. (b) 8. (b) 9. (d)

10. (a) 11. (a) 12. (a)

13. (b) 14. (a) 15. (c)

16. (a) 17. (a) 18. (b)

19. (b) 20. (b)

SoLutionS 1. Since the given two lines are parallel and distance between two parallel tangents is equal to the diameter of the circle

4. Centre of the circle (x 2 1)2 1 ( y 2 3)2 5 r2 is (1, 3) and radius is r. Centre of the circle x2 1 y2 2 8x 1 2y 1 8 5 0 is (4, 21)

1 16  (10) ∴ Radius  ⋅ 2 122  52

and its radius  16  1  8  3. \ Distance between their centres is

13   1. 13

(1  4)2  (3  1)2  5 and sum of the radii  r  3.

2. The radical axes of the circle x2 1 y2 5 1 and the circles given in (a) and (c) are x 5 2 and x 5 22, respectively, which are parallel to y-axis. Also, the given circle intersects the circles given in (a) and (c) orthogonally. 3. Let the sides of the square be y 5 0, y 5 1, x 5 0 and x 5 1. Let the moving point be (x, y). Then, y2 1 ( y 2 1)2 1 x2 1 (x 2 1)2 5 9 is the equation of the locus. Þ 2x2 1 2y2 2 2x 2 2y 2 7 5 0  1 1 which represents a circle having centre  ,  (which is also  2 2 the centre of the square) and radius.

Since the two circles intersect, \ 50

Analytical Geometry in 2D and 3D ▌ 3.127

Y

x 25y2

(3,1) C

(k1,k)

B

X

O

L

         

A 2x5y180

⇒ 2k2 1 2k224 < 0 and k >21 ⇒ k2 1 k212 < 0 and k >21 ⇒ (k 1 4) (k23) < 0 and k >21 ⇒ 24 < k < 3 and k >21 ⇒  21 < k < 3.

CL  length of the ⊥ from C on AB 

7. Clearly, from the figure, the radius of the smallest circle touching the given circles is

(2)2  (5)2

 29

∴ radius of the circle AC  AL L2  CL2  32  19  38 .

Y

Thus, equation of the required circle is (x23)2 1 ( y 1 1)2 5 38 or

4 4 O

2  3  5 (1)  18

X

10. Given line is y 5 mx 1 c ...(1) and the given circle is x2 1 y2 5 r2 ...(2) Solving (1) and (2), we get (1 1 m2) x2 1 2mcx 1 c22r2 5 0 ...(3)

4

 42  42  4  i.e., 4 2  4. 8. Equation of any circle passing through the point of intersection of x2 1 y222x 5 0 and y 5 x is x2 1 y222x 1 λ (y2x) 5 0 or x2 1 y22(2 1 λ) x 1 λy 5 0.  2  λ λ  Its centre is  , . 2   2 For AB to be the diameter of the required circle, the centre must lie on AB, i.e. 2  λ λ ⇒ λ 1.  2 2 Thus, equation of required circle is x2 1 y22x2y 5 0. 9. Let C be the centre of the circle, then C ≡ (3,21). Equation of line AB is 2x25y 1 18 5 0 and AB 5 6.   ∴   AL 5 3.

x2 1 y226x 1 2y228 5 0.

For two real distinct points of intersection, both the roots of (3) must be real and distinct.  

∴  4m2c224 (1 1 m2) (c22r2) > 0

 

⇒ c2 < r2 (1 1 m2)

 

⇒  r 1  m2  c  r 1  m2

11. Let the lines cuts the x-axis at A and B, then OA 5 2 1 and OB 523. λ Also, if the lines cut the y-axis at C and D, then 3 . 2 Now if the circle passes through A, B, C and D then

OC 5 1 and OB 5

 1 3 OA × OB 5 OC × OD ⇒    (3)  1  2  λ

⇒ λ 5 2.

12. Given equation of circle is x2 1 y222x22y22 5 0. Let mid point of chord AB be (h, k) Its centre is (1, 1) and radius 5

1  1  2  2.  OB.

3.128

Mathematics

15.

5 10



2m 1 m2  1

y0 3x

O A    

B

P

ymx

(0,0)

In ∆OPB, ∠OBP 5 30º. ∴ sin 30º 5 OP/2 or OP 5 1. Since, OP 5 1 ⇒ (h21)2 1 (k21)2 5 1 or h2 1 k222h22k 1 1 5 0 ∴ Locus of mid point of chord is x2 1 y222x22y 1 1 5 0.

C(2,1)

13. Given Circle is ...(1) x2 1 y2 1 2gx 1 2fy 1 c 5 0 Let C be its centre and PQR be an equilateral triangle inscribed in the circle, then C ≡ (2g,2f ) and radius of the circle

1 3

∴

m  3,

But

m ≠ 3 x y  or x  3y  0 3

∴

16. Let the equation of the given circle be x2 1 y2 1 2gx 1 2fy 1 c 5 0 Y

CQ  g 2  f 2  c . P

(0,h)

60º C 60º Q

R

L

From ∆ QLC , QL  CQ sin 60 

3 2

Now, area of ∆PQR  3 3 4

g 2  f 2  c.

3 3 ⋅ 3 g2  f 2  c .QR 2  4 4

∴  

4 f 2  20  0 ⇒ f  5

g2  f 2  c

14. Equation of tangent to the circle x2 1 y2 5 5 at (1,22) is x22y25 5 0 ...(1) Let this line touches the circle x2 1 y228x 1 6y 1 20 5 0 at (x1, y1)   ∴ Equation of tangent at (x1, y1) is xx1 1 yy124 (x 1 x1) 1 3 ( y 1 y1) 1 20 5 0 or x (x124) 1 y ( y1 1 3)24x1 1 3y1 120 5 0 Now (1) and (2) represent the same line

X (5,0)

Since it passes through (1, 0) and (5, 0), therefore 1 1 2g 1 c 5 0 and 25 1 10g 1 c 5 0. On solving the two equations, we get g 5 23 and c 5 5. So, the circle is x2 1 y226x 1 2fy 1 5 5 0 Since it touches y-axis at (0, h), therefore h2 1 2fh 1 5 5 0. Discriminant of this quadratic must be zero, i.e.,

QR  2QL  3. g 2  f 2  c .

∴



O (1,0)

∴ h2  2 5h  5  0  

...(2)

x1  4 y1  3  4 x1  3 y1  20 =  1 2 5

⇒22x1 1 8 5 y1 1 3 or 2x1 1 y12 5 5 0. Only the point (3, 2 1) satisfies it. Hence the point of contact is (3,21).

⇒ h5 5

17. The given circles are S1 : x2 1 y2 1 2gx 1 2fy 1 c 5 0 ...(1) and S2 : x2 1 y2 1 2g′x 1 2f′y 1 c′ 5 0 ...(2) The equation of common chord of (1) and (2) is S12S2 5 c i.e., 2(g2g′)x 1 2(f2f′)y 1 (c2c′) 5 0 ...(3) Since (1) bisects the circumference of (2), therefore, common chord will be the diameter of circle (2)   ∴ Centre (2g′,2f′) of circle (2) lies on (3).   ∴ 2(g2g′)g′22(f2f′)f′ 1 c2c′ 5 c or 2g′(g2g′) 1 2f′(f2f′) 5 c2c′.

Analytical Geometry in 2D and 3D ▌ 3.129

18.

 n2  AB 2  4 AM 2  4  4    2(8  n2 ) 2 

(0,6) (1,5) (3,3)

B xyn

(y3)

M O (0,0)

A 2

O

(6,0)

x1

Shaded triangle is right angled at (1, 3). ∴ the circumcircle is the circle on (3, 3) and (1, 5) as ends of a diameter and its equation is (x23)(x21) 1 (y23)(y25) 5 0 i.e., x2 1 y224x28y 1 18 5 0. Hence, required sum 5 2(8212 1 8222) 5 2 × 11 5 22 19. We have, x 1 y 5 6 and xy2y23x 1 3 5 0  

⇒ y(x21)23(x21) 5 0

 

⇒ (x21)(y23) 5 0

20.

Equations of the sides of the triangle are x1y56

...(1)

y53

...(2)

x51

...(3)

C

D

a a A

(1,5)

B

According to the figure  a a A(0, 0), B(a, 0) and D(0, a) and centre is  ,  . . Therefore,  2 2 the equation of circle is 5 2

(1,3)

(3,3)

 

2

 a  a a2  x    y    2 2  2  2 2 ⇒ x 1 y 2ax2ay 5 0

3-D GEoMEtRy 1. Chord AB is a diameter of the sphere r  2iˆ  ˆj  6 kˆ  18 . If the coordinates of A are (3, 2, 22), then the coordinates of B are (a) (1, 0, 10) (b) (1, 0, 210) (c) (21, 0, 10) (d) None of these. 2. The angle between the lines x 5 1, y 5 2 and y 5 21, z 5 0 is (a) 30° (b) 60° (c) 90° (d) 0° 3. If the axes are rectangular and P is the point (2, 3, 21) the equation of the plane through P at right angles to OP is (a) 2x 1 3y 1 z 5 14 (b) 2x 1 3y 2 z 5 14 (c) 2x 2 3y 1 z 5 14 (d) None of these.

4. A sphere of constant radius k passes through origin and meets axes in A, B, C. The centroid of the triangle ABC lies on the sphere. (a) 9(x2 1 y2 1 z2) 5 4k2 (b) 3(x2 1 y2 1 z2) 5 4k2 (c) x2 1 y2 1 z2 5 4k2 (d) None of these 5. The co-ordinates of the point of intersection of the line x 1 y  3 z  2   with the plane 3x 1 4y 1 5z 5 25 are 1 3 2 (a) (5, 15, 10) (c) (5, 215, 10)

(b) (5, 15, 210) (d) None of these

3.130

Mathematics

6. The perpendicular distance of P (1, 2, 3) from the line x 6 y 7 z 7   is 3 2 2 (a) 7 (c) 0

(b) 5 (d) None of these.

7. Distance between two parallel planes 2x 1 y 1 2z 5 8 and 4x 1 2y 1 4z 1 5 5 0 is 7 2 3 (c) 2

(a)

5 2 9 (d) 2 (b)

8. The position vector of the centre of the circle r  5, r ⋅ ( iˆ  ˆj  kˆ )  3 3 is 3( iˆ 1 ˆj 1 kˆ ) (c) 3( iˆ 1 ˆj 1 kˆ )

(a)

(b) iˆ 1 ˆj 1 kˆ (d) None of above.

9. The direction cosines of a line equally inclined to the axes are 1 1 1 , , 3 3 3 1 2 1 , , (c) 3 3 3

(a)

1 1 1 , , 3 3 3 1 1 1 (d)  ,  ,  3 3 3 (b)

10. If A (21, 4, 3) is one end of diameter AB of the sphere x2 1 y2 1 z2 2 3x 2 2y 1 2z 2 15 5 0, the coordinates of the other end B are (a) (4, 2, 1) (b) (4, 22, 21) (c) (4, 22, 1) (d) None of these. 11. The equation of the sphere whose centre is the point (2, 3, 1) and which touches the plane x 1 y 1 z 5 0 (a) x2 1 y2 1 z2 2 4x 2 6y 2 2z 1 2 5 0 (b) x2 1 y2 1 z2 1 6y 1 2z 2 2 5 0 (c) x2 1 y2 1 z2 2 4x 2 6y 2 2z 2 2 5 0 (d) None of these 12. The equation of a sphere having centre at (0, 0, 0) and passing 1 1  through  , , 0 is 2 2  (a) x 2 1 y2 1 z 2 5 1 (c) 4x 2 1 4y2 1 4z 2 5 1

(b) 2x 2 1 2y2 1 2z 2 5 1 (d) None of these.

13. The coordinates of the point where the perpendicular from the origin meets the join of (29, 4, 5) and (11, 0, 21) are (a) (1, 2, 2) (b) (2, 1, 2) (c) (2, 2, 1) (d) None of these. 14. The equation of the plane through (2, 3, 4) and parallel to the plane x 1 2y 1 4z 5 5 is (a) x 1 2y 1 4z 5 24 (b) x 1 y 1 2z 5 2 (c) x 1 2y 1 4z 5 3 (d) x 1 2y 1 4z 5 10

15. The position vectors of points A and B are iˆ  ˆj  3kˆ and 3iˆ 1 3 ˆj 1 3kˆ respectively. The equation of a plane is r. (5iˆ  2 ˆj  7 kˆ ) 1 9 5 0. The points A and B (a) (b) (c) (d)

lie on the plane are on the same side of the plane are on the opposite side of a plane None of these.

16. Perpendicular distance of the point (3, 4, 5) from the y-axis, is (a)

34

(c) 4

(b)

41

(d) 5

17. The raito in which the line joining (1, 2, 3) and (23, 4, 25) is divided by xy-plane is (a) 5:3 (b) 3:5 (c) 2:3 (d) None of these 18. The distance of the point (3,4,5) from x-axis is (a) 3 (b) 5 (c)

34

(d)

41

19. The direction cosines of x-axis are (a) (b) (c) (d) 20. The radius of the circular section of the sphere | r| 5 5 by the plane r. (iˆ  ˆj  kˆ )  3 3 is (a) 16 (b) 8 (c) 4 (d) None of these 21. If l,m,n are direction cosines of a line then (b) l2 1 m2 1 n2 5 1 (a) l2 1 m2 1 n2 5 0 (c) l 1 m 1 n 5 1 (d) l 5 m 5 n 5 1 22. The number 3,4,5 can be (a) direction cosines of a line (b) direction numbers or rations of a line in space (c) coordinates of a point on the line y 5 4, z 5 0. (d) coordinates of a point on the plane x 1 y 2 7 5 0. 23. The distance of the plane (x,y,z) from the xy plane is (a) d (b) y (c) z (d) |z | 24. The locus of x2 1 y2 1 z2 5 0 is (a) (0,0,0) (b) a sphere (c) a circle (d) None of these 25. From every point (x,y,z) on the xy plane (a) x 5 0 (b) y 5 0 (c) z 5 0 (d) x 5 0, z 5 0 26. For every point (x,y,z) on x-axis (a) y 5 0, z 5 0 (b) x 5 0, z 5 0 (c) x 5 0, y 5 0 (d) x 5 0. 27. The graph of the equation in three-dimensional space is (a) x-axis (b) y-axis (c) z-axis (d) xy-plane

Analytical Geometry in 2D and 3D ▌ 3.131 28. The angle between the lines x 5 1, y 5 2, y 5 2l and z 5 0 is (a) 90º (b) 30º (c) 60º (d) 0º x y z 29. The plane x 2 2y 1 z 2 6 5 0 and the line 5 5 are 1 2 3 related as (a) (b) (c) (d)

30. The angle between the two planes 3x 2 4y 1 5z 5 0 and 2x 2 y 2 2z 5 5 is

π 3 π (b) 2 π (c) 6 π (d) 4 (a)

parallel to the plane at right angle to the plane lies in the plane meet the plane obliquely

AnSwER KEy 1. 2. 3. 4.

5. 6. 7. 8.

(b) (c) (b) (a)

(b) (a) (a) (a)

9. 10. 11. 12.

(d) (c) (a) (b)

13. 14. 15. 16.

(a) (a) (c) (a)

17. 18. 19. 20.

21. 22. 23. 24.

(b) (d) (a) (c)

(b) (b) (d) (a)

25. 26. 27. 28.

(c) (a)

29. (a) 30. (b)

(c) (a)

SoLutionS 1. The equation of the sphere is r  2iˆ  ˆj  6 kˆ  18 Þ

Its centre is at the point ( 2iˆ 1 ˆj 1 6 kˆ ), i.e., at (2, 1, 26)

co-ordinates of A are (3, 2, 22). Lets the co-ordinates of B be (a, b, g ) 2  γ 3 α 2β  2,  1 and  6 2 2 2 ⇒  = 1,   0,  10 Hence B  (1,0, 10). Then,

2. Lines are

x 1 y  2 z x y 1 z   and   0 0 0 0 1 1

\ cos q 5 0.1 1 0.0 1 1.0 5 0 \ q 5 90° 3. P is (2, 3, 21) and 0 is (0, 0, 0); direction cosines of OP are proportional to [2, 3, 21]. Therefore these will be the coefficients of x, y, z in a plane which is perpendicular to OP. So let the plane be 2x 1 3y 2 z 5 l. The plane also passes through P(2, 3, 21) \ 419115l \ l 5 14. Hence the required plane is 2x 1 3y 2 z 5 14. 4. Let the co-ordinates of the points A, B, C be (a, 0, 0), (0, b, 0), (0, 0, c) respectively. Then equation of the sphere ABC is x2 1 y2 1 z2 2 ax 2 by 2 cz 5 0. Radius of this sphere is equal to k.

\ a2 1 b2 1 c2 5 4k 2 (1) co-ordinates (x, y, z) of the centroid of DABC are given by 1 1 1 x 5 a, y 5 b, z 5 c, so that 3 3 3 a 5 3x, b 5 3y, c 5 3z Eliminating a, b, c from (1) and (2), we have. (3x)2 1 (3y)2 1 (3z)2 5 4k2 9(x2 1 y2 1 z2) 5 4k2 x 1 y  3 z  2    r. 1 3 2 Then any point on the line is (r 2 1, 3r 2 3, 22r 1 2). If this point lies on the plane 3x 1 4y 1 5z 5 25, then 3(r 2 1) 1 4(3r 2 3) 1 5(22r 1 2) 5 25 or 5r 5 30, i.e., r 5 6. Putting this value of r in the co-ordinates of the point, the required point is (5, 15, 210).

5. Let

6. The point A(6, 7, 7) is on the line. Let the perpendicular from P meet the line in L. Then AP 2 5 (6 2 1)2 1 (7 2 2)2 1 (7 2 3)2 5 66 Also AL 5 projection of AP on line   2 3  actual d.c1s , ,  17 17  3  (6  1).  (7  2). 17 2 17

 17.

 −2   17  2  (7  3) 17

3.132

Mathematics

\ ^ distance d of P from the line is given by. d 2 5 AP2 2 AL2 5 66 2 17 5 49, so that d 5 7. P

11. Since the required sphere touches the plane x 1 y 1 z 5 0…(i) its radius 5 length of perpendicular from the centre (2, 3, 1) on (1)

A(6, 7, 7) 7. Given planes are 4x 1 2y 1 4z 2 16 5 0 and 4x 1 2y 1 4z 1 5 5 0 Distance between planes (1) and (2) is 

16  5 16  4  16



(1) (2)

21 7  . 6 2

8. The equation of ON is r 5 l (i 1 j 1 k) (i) Since is passes through the origin and is parallel to the vector (i 1 j 1 k), any pt. on it is l (i 1 j 1 k). If this pt. lies on the plane r. (i 1 j 1 k) 5 3 3 then λ (i  j  k )(i  j  k )  3 3 or λ  (1  1  1)  3. ∴λ  3 Putting this value is (i) we get the position vector N i.e., centre of the circle as

1  x1 3 4  y1  , 1 2 2 2 3  z1 1. and 2 So, x1 5 4, y1 5 22 and z1 5 1. Hence, the coordinates of B are (4, 22, 1). ∴

3 (i 1 j 1 k)



2  3 1 2

2

2



1 1 1

6

2 3

3

\ The required equation of sphere is: (x 2 2)2 1 ( y 2 3)2 1 (z 2 1)2 5 ∴Foci of Þ x2 1 y2 1 z2 2 4x 2 6y 2 2z 1 2 5 0. 12. The centre of the sphere is (0, 0, 0) and it passes through

1 1   2 , 2 , 0 . Its radius 5 distance between (0, 0, 0) and 1 1   , , 0 2 2  2

2

1  1  1    0     0   (0  0) 2  2 2  2  ∴ The required equation of the sphere iss 2

 1  (x  0)  ( y  0)  ( z  0)    2   1 2 2 2 2 2 ⇒ x  y  z  or 2x  2 y  2 z 2  1. 2 2

O S P

N

9. If a line makes angles a, b, g with the axes, we have a5b5g \ cos a 5 cos b 5 cos g 5 l 5 m 5 n, l 2 1 m2 1 n2 5 1 \ l2 1 m2 1 n2 5 1, \ l2 1 l2 1 l2 5 1 Þ 3l2 5 1, 1 1 ∴l 2  or l  3 3  1 1 1  1  ∴ The d.c s of the line are   , ,  3 3  3  10. The given equation of the sphere is x2 1 y2 1 z2 2 3x 2 2y 1 2z 2 15 5 0. This is of the form, x2 1 y2 1 z2 1 2mx 1 2ny 1 2wz 1 d 5 0; where 2m 5 23, 2n 5 22, 2w 5 2 3 and d 5 215, i.e. µ  , 2 n 5 21, w 5 1 and d 5 215. 3  \ centre of the sphere is (2m, n, w), i.e., c  ,1,21 . 2  Now, let the coordinates of B be (x, y, z,). Then, clearly C is the midpoint of AB.

2

2

13. Let D be the foot of the perpendicular from the origin to the join of A (29, 4, 5) and B (11, 0, 21).

0 (0, 0, 0)

k A (9, 4, 5)

1 D

B (11, 0, 1)

Suppose, D divides AB in the ratio k : 1, then coordinates of

 11 k  9 0 k  4  k  5 

, , D are given by   k  1 k  1 k  1   11 k  9 4 5  k   , ,    k k k 1 1 1      So the direction ratios of OD are given by 11 k  9 4 5 k  0,  0, 0 k 1 k 1 k 1

Analytical Geometry in 2D and 3D ▌ 3.133

or

11 k  9 4 5 k , , k 1 k 1 k 1

Also direction ratios of AB are 11  (  9), 0  4,  1  5 or, 20,  4,  6 or 10,  2,  3 Since OD ⊥ AB,  11 k  9        2  4  3  5 − k   0 ∴ 10        k 1   k  1  k  1 ⇒ 110k  90  8  15  3k  0 ⇒ 113k  113 or k  1. Thus, the coordinates of D are  11  9 4 5  1  , ,    (1, 2, 2)  11 11 11  14. Any plane parallel to the plane x 1 2y 1 4z 5 5 is x 1 2y 1 4z 5 k This passes through (2, 3, 4), \ 2 1 6 1 16 5 k

Þ

18. The distance of point (x,y,z) from x-axis y 2  z 2  16  25  41 19. l 5 cos µ 5 cos 0 5 1; m 5 cos b 5 cos 90 5 0 n 5 cos g 5 cos 90 5 0 \  direction cosines are 20. The sphere | r| 5 5 has centre at the origin and radius 5. Distance of the plane r ⋅ (iˆ  ˆj  kˆ )  3 3 from the origin 

Thus, in figure OP 5 5, ON 5 3 \ NP2 5 OP2 2 ON 2 5 (5)2 2 (3)2 5 16, \ NP 5 4 Hence the radius of the circular section 5 NP 5 4.

k 5 24

\ Required plane is x 1 2y 1 4z 5 24.

N

15. The position vectors of two given points are a  iˆ  ˆj  3kˆ and b  3iˆ  3 ˆj  3kˆ and the equation of the given plane is r . (5 iˆ  2 ˆj  7 kˆ )  9  0 or r. n  d = 0 We have, a ⇒ n  d  (iˆ  ˆj  3kˆ ). (5iˆ  2 ˆj  7 kˆ )  9  5  2  21  9  0 and b. n  d  (3iˆ  3ˆj  3kˆ ). (5iˆ  2ˆj  7kˆ )  9  15  6  21  9  0 So, the points a and b are on the opposite sides of the plane. 16. Distance of (a, b, g ) from y-axis is given by d α 2  γ 2 \ Distance (d) of (3, 4, 5) from y-axis is d  32  52  9  25  34 . 17. Let A º ( 1, 2, 3) and B º (23, 4, 25). Let the point P where the line AB meets the xy-plane divides it in the ratio k : 1 \ The co-ordinates of P are  3 k  1 4 k  2 5 k  3  , ,   k 1 k 1   k 1 Since the point P lies upon xy-plane \ z co-ordinate of P must be zero. −5k + 3 3 ∴ = 0 ⇒− 5k  3  0 ⇒ k  k +1 5 Hence required ratio is

3 3 3 3  3 ˆ 2 ˆ ˆ (i  j  k ) 1  12  12

3 : 1 i.e, 3 : 5 internally. 5

P

O

21. l2 1 m2 1 n2 5 1; an important result. 22. 32 1 42 1 52 ≠ 1; \ 3,4,5 are direction numberss or direction ration of a line. 23. equation of x 2 y plane is z 5 0;   \ direction of (x,y,z) from z50; 5

z 1

5z

24.  x 2 1 y 2 1 z2 5 0 is satisfied by x 5 0, y 5 0, z 5 0 only 25. Equation of my plane is z 5 0; every point (x,y,z) on xy plane has z 5 0 26. As every point (x,y,z) on x 2 axis is y 5 0, z 5 0 27. Graph of x2 1 y2 5 0 is y 5 0, x 5 0 which is z 2 axis. 28. Lines are x 1  y  2  z and x  y 1  z ; 0 0 0 0 1 1 ∴cos θ  0.1  0.0  1.0  0 θ  90. 29. Since 1.11 2(22) 1 3(1) 5 0 ; \ line is parallel to the plane. 30. cos θ  ⇒θ 5 π

2

3( 2)  (4)(1)  (5)(2) 9 16  25 4 1 4

0

3.134

Mathematics

ConiC SECtionS 1. The equation of a directrix of the ellipse (x2/16)1 (y2/25) 5 1 is 25 (a) y  (b) x  3 3 25 (c) x  3 (d) y  3 2. The equation of a tangent to the hyperbola 3x2 2 y2 5 3, parallel to the line y 5 2x 1 4 is (a) y 5 2x 1 3 (b) y 5 2x 1 1 (c) y 5 2x 2 1 (d) y 5 2x 1 2 3. Two of the straight lines given by 3x3 1 3x2y 2 3xy2 1 dy3 5 0 are at right angles if 1 1 (a) d  (b) d  3 3 (c) d  3 (d) d  3 4. Equation of the tangent to the parabola y2 5 4x which makes an angle q with its axis is (a) y 5 x tan q 1 cot q (b) y 5 x tan q 1 sec q (c) x 5 y cot q 2 cot2 q (d) x 5 y cot q 1 tan q 5. The focus of parabola y 5 8x and the equation of directrix are respectively (a) (2, 0), x 5 2 (b) (22, 0), x 5 22 (c) (2 ,0), x 5 22 (d) None of these 2

6. The vertex and the focus of the parabola y2 2 4y 2 4x 2 8 5 0 are respectively (a) (3, 22), (2, 22) (b) (23, 2), (2, 22) (c) (23, 2), (22,2) (d) (23,22), (2,2) 7. The vertex and the focus of a parabola are respectively, (0, 4) and (0, 2). The equation of the parabola is, (a) y2 1 8x 2 32 5 0 (b) y2 2 8x 2 32 5 0 2 (c) x 1 8y 2 32 5 0 (d) x2 2 8y 1 32 5 0 8. The latus reactum of the conic 3x2 1 4y2 2 6x 1 8y 2 5 5 0 is (a) 3 (c)

(b)

2

3 2

(d) None of these.

3 9. The angle between the tangent drawn from the origin to the parabola y2 5 4a(x 2 a) is (a) 30º (b) 45º (c) 90º (d) tan21(1/2) 10. The locus of the mid-point of a focal chord of a parabola is (a) a circle (b) a parabola (c) an ellipse (d) a hyperbola

11. The distance between the foci of the ellipse is (a) 1 (b) 2 (c) 2 5

(d) 4

12. The eccentricity of the ellipse 9x2 1 5y 2 2 30y 5 0 is 1 2 (a) (b) 3 3 3 (c) (d) None of these. 4 13. The vertices of an ellipse are (± 5, 0) and its foci are (± 4, 0). The equation of the ellipse is x2 y2  1 16 25 x2 y2  1 (c) 25 9

(a)

(b)

x2 y2  1 25 16

(d) None of these

14. The angle between the pair of tangents drawn to the 3x2 1 y2 5 5 ellipse from the point (1,2) is (a) tan21 21

(c) tan

12 5 6

(b) tan21

12 5

(d) tan21

6 5

5

15. With respect to the ellispe 4x2 1 7y2 5 8, the pole of the straight line 12x 1 7y 1 16 5 0 is given by  3 21  23 21 (a)  ,  (b)  ,  2 2    2 2  23 1  (c)  ,   2 2

 3 1 (d)  ,   2 2

– 16. The eccentricity of the hyperbola is √ 2 and the distance between its foci is 16. The equation of the hyperbola is (a) x 2 2 y2 5 16 (b) x2 2 y2 5 32 (c) x2 2 y2 5 4 2 (d) None of these. 17. The length of the latus rectum of the hyperbola 16x 2 2 3y 2 2 32x 2 12y 2 44 5 0 is (a) (c)

16 3 32

(b) 16 2 (d) None of these

3 18. The line x 2 y 5 1, touches the hyperbola 3x 2 2 4y2 5 12 at the point (a) (24, 1) (b) (2 2 3) (c) (4, 3) (d) (24, 23)

Analytical Geometry in 2D and 3D ▌ 3.135 25. The number of real tangents that can be drawn to the ellipse 3x2 1 5y2 5 32 and 25x2 1 9y2 5 450 passing through (3, 5) is (a) 0 (b) 2 (c) 3 (d) 4

19. The length of the latus rectum of the parabola 25 [(x 22)2 1 ( y 24)2] 5 (4x23y 1 12)2 is 16 5 8 (b) 5 12 (c) 5 (d) none of these (a)

20. If the two parabolas y2 5 4a (x 22) and x2 5 4a ( y23) touch each other, then their point of contact lies on a (a) circle (b) parabola (c) ellipse (d) hyperbola

26. If the extremities of a line segment of length l move in two fixed perpendicular straight lines, then the locus of that point which divides this line segment in the ratio 1 : 2, is (a) a parabola (b) an ellipse (c) a hyperbola (d) None of these 27. An ellipse slides between two lines at right angles to one another. The locus of its centre is (a) x 2  y 2  a 2  b2 (b) x 2  y 2  a 2  b2

21. If the line x21 5 0 is the directrix of the parabola y22kx 1 8 5 0, then one of the values of k is (a)

1 8

(c) x 2  y 2  a 2  b2 (d) x 2  y 2  a 2  b2

(b) 8

(c) 4

(d)

28. If (5, 12) and (24, 7) are the foci of an ellipse passing through the origin, then the eccentricity of the conic is

1 4

22. The locus of a point whose sum of the distances from the origin and the line x 5 2 is 4 units, is (a) y2 5212 (x23) (b) y2 5 12 (x23) (c) x2 5 12 ( y23) (d) x2 5212 ( y23) 23. A circle has its centre at the vertex of the parabola x2 5 4y and the circle cuts the parabola at the ends of its latus rectum. The equation of the circle is (a) x2 1 y2 5 5 (b) x2 1 y2 5 4 (c) x2 1 y2 5 1 (d) none of these x2 y2   1 at the a 2 b2 end of the latus rectum in the first quadrant, is (a) x 1 ey2ae3 5 0 (b) x2ey 1 ae3 5 0 (c) x2ey2ae3 5 0 (d) none of these

24. The equation of the normal to the ellipse

(a)

386 12

(b)

386 13

(c)

386 25

(d)

386 38

x2 y2  1 14 5 intersects it again at the point Q(2θ), then cos θ is equal to:

29. If the normal at the point P(θ) to the ellipse (b) 2 2 / 3 1 (d) 2 3

(a) 2 / 3 1 (c) 3

30. The eccentricity of the hyperbola 9x2216y2 1 72x232y216 5 0 is (a) 5/4 (b) 4/5 (c) 9/16 (d) 16/9

AnSwER KEy 1. 2. 3. 4.

(a, d) (b, c) (c) (c)

5. 6. 7. 8.

(c) (c) (c) (a)

9. 10. 11. 12.

(c) (b) (b) (b)

13. 14. 15. 16.

(c) (a) (c) (b)

17. 18. 19. 20.

(c) (c) (b) (d)

21. 22. 23. 24.

(c) (a) (a) (c)

25. 26. 27. 28.

(c) (b) (c) (d)

29. (b) 30. (a)

3.136

Mathematics

SoLutionS 1. Equation of the ellipse is of the form (y 2/a 2) 1 (x 2/b 2) 5 1, where a2 5 25 and b2 5 16(a > b) and the equation of the directrix is y 5 ± a/e, where e 2 5 (a 2 2 b 2)/a 2 9 5 (25 2 16)/25 5 , thus the required equations are, 25  3 25 y  5 /    . 3  5

3. Let the lines represented by the given equations be y 5 m1 x, y 5 m2x and y 5 m3x. Then, m1m2m3 5 2 3/d. If two of the lines are perpendicular. Let m1m2 5 21 Þ m3 5 3/d. Þ y 5 (3/d ) x satisfies the given equation Þ d (3/d )3 2 3(3/d )2 1 3(3/d ) 1 3 5 0 Þ d 5 23. 4. Equation of the tangent at any point (t2, 2t) on the given parabola is

Since

1 y  x t t

1  tan θ , t  cot θ , the required t

equation is y 5 x tan θ 1 cot θ or, y cot θ 5 x 1 cot2 θ Þ x 5 y cot θ 2 cot2θ. 5. Given equation is y 2 5 4ax, where a 5 2. ∴ Focus of directrix is x 5 2a, i.e. x 5 22. 6. Given equation is: (y 2 2)2 5 4(x 1 3) Þ y2 5 4ax, where Y 5 y 22, X 5 x 1 3 and a 5 1. ∴ Vertex is (X 5 0, Y 5 0 i.e. (x 1 3 5 0, y 2 2, i.e., (23, 2)). Focus is (x 5 a , y 5 0 i.e. (x 1 3 5 1, y 2 2 5 0) i.e., (22, 2) 7. Let A(0, 4) be the vertex and S(0, 2) be the focus of the parabola. Let z (a, b) be the foot of the directix. Then, A is the mid-point of 7S.

α 0 β 2  0 and  4 Þ α 5 0, β 5 6 Thus, the 2 2 coordinates of 7 are (0, 6). ∴ Directrix is y 5 6. Let P (x, y) be any point on the parabola. Then, (x 2 0)2 1 (y 2 2)2 5 (y 2 6)2 Þ x2 1 8y 2 32 5 0.

∴ 

( x  1)2 ( y  1)2  1 4 3 2 x y2 Comparing with 2  2  1, we get a  2, b  3 a b Hence, latus reaction 

2. The equation of the hyperbola is x2 y2 x2 y2   1 or 2  2  1 1 3 a b where a2 5 1 and b2 5 3. And, y 5 mx 1 c is a tangent to the hyperbola if c2 5 a2 m2 2 b2. Here, m 5 2, whence, c2 5 1. m2 2 3 5 1, i.e. c 5 ± 1.

ty  x  t 2 ⇒

8. The given equation can be written in the form

2 b2 2 ( 3 ) 2  3 2 a

9. Any line through origin is y 5 mx, it will cut y2 5 4a (x 2 a) in two coincident points, if the roots of m2x2 2 4ax 1 4a2 5 0 are equal. This happens when 16a2 2 4m2 ´ 4a2 5 0 or m2 5 1, so m 5 1 or 21 Hence, the angle between the tangents is 90°. 10. Let P(x1, y1) be the mid-point of a focal chord of parabola y2 5 4ax. Equation of the chord is yy1 2 2a(x 2 x1) 5 y21 2 4ax1 yy1 2 2ax 5 y21 2 2ax1 Þ Since it passes through the focus (a, 0), we have, 22a2 5 y21 2 2ax1 Þ y21 5 2a(x1 2 a) ∴  locus of P(x1, y1) is y2 5 2a(x2a), which is a parabola. x2 y2   1. 5 4 So, a2 5 5 and b2 5 4, therefore, b2 5 a2(1 2 e2) 1 Þ 4 5 5(1 2 e2) Þ e2 5 5 1 ∴ Distance between the foci 5 2ae 5 2 5   2. 5 12. The given equation of the ellipse can be written in the form

11. Given ellipse is

x 2 ( y  3)2  1 5 9 2 Here a 5 9 and b2 5 5 Eccentricity of the ellipse is given by a 2  b2 9  5 4   9 9 a2 2 e 3 e2  ∴

13. We have a 5 5 and ae 5 4, so, e 5 (4/5)  ∴  b2 5 a2 (1 2 e2) 5 25(1 2 16/25) 5 9, Thus, a2 5 25 and b2 5 9. 14. The tangent is SS, 5 T2 ∴ (3x2 2 2y2 2 5) (3 ´ 12 1 2 ´ 22 2 5) 5 (3x 1 4y 2 5)2 Þ 9x2 2 4y2 2 24xy 1 30x 1 40y 2 55 5 0. 1  12 2  9  4  2 2 ( ) ( )  h  ab  ∴ tan θ  2   2  (9  4) ( a  b) 2 12 12   180  Hence, θ , tan1 . 5 5 5

Analytical Geometry in 2D and 3D ▌ 3.137 15. Let the pole be P(x1,y1), Then, the equation of the Polas is 4xx1 1 7yy1 5 8 Þ 16 2 8xx1 2 14yy1 5 0. Compasing it with 12x 1 7y 1 16 5 0, we get 28x1 5 12 and 3 1 and y1  . 2 14y1 5 7 Þ x1 5 2 2 16. e 5 2 5 2ae 5 16 ; Now, b2 5 a2(e2 2 1) 2

 16  ⇒ b2 5 a 2 5   5 32 2 2  x2 y2 ∴  Required equation is  1 32 32 i.e. x2 2 y2 5 32. 17. Given equation is 16(x 2 1)2 23(y 1 2)2 5 48. ( x  1)2 ( y  2)2   1. 3 16 Here, a2 5 3 and b2 5 16 2b2 32 ∴ latus rectum 5 5 . a 3 ⇒

Distance of the point from the line x22 5 0 is h22. According to the given condition, h2  k 2  h  2  4 ⇒

x2 y2   1. 4 3

Line is y 5 x 2 1. ∴ m 5 1, a2 5 4 and b2 5 3. So, the point of contact is (4, 3). 19. The given equation of the parabola can be written as 2

 4 x  3 y  12  ( x  2)2  ( y  4)2    .  ( 4)2  (3)2  ∴ The coordinates of focus are (2, 4) and the equation of directrix is 4x23y 1 12 5 0. The distance of the focus from the directrix 4( 2)  3(4)  12 2

4  (3)

2

Shifting the origin to (8/k, 0), the parabola becomes Y 2 5 kX where X 5 x28/k and Y 5 y. 2k Directrix of this parabola is X 5 4 8 k or x   . k 4 8 k This will coincide with x 5 1 if   1 k 4 ⇒ 322k2 5 4k  ⇒ k2 1 4k232 5 0 or (k 1 8) (k24) 5 0 ⇒ k 528, 4. Thus, k 5 4.

 ( h  0)2  (k  0)2  h2  k 2 .

  a2m b2 . ,  2 2 2 2 2 2   a m b a m b 



21. The given parabola is  8 y 2  kx  8  k  x   . k 

22. Let the coordinates of the point be (h, k). Distance of the point from origin

18. The point of contact is

Given hyperbola is

∴ point of contact (x1, y1) lies on the curve xy 5 4a2. which is a hyperbola.

8  . 5

8 16 ∴The length of latus rectum 5 2   . 5 5 20. Let P (x1, y1) be the point of contact of the two given parabolas y2 5 4a (x22) ...(1) and x2 5 4a ( y23) ...(2) Equation of tangent at P to (1) is yy1 5 2a (x 1 x1)28a or 2ax2y1 y 1 (2ax128a) 5 0 ...(3) Equation of tangent at P to (2) is xx1 5 2a ( y 1 y1)212a or x1x22ay2(2ay1212a) 5 0 ...(4) Since (3) and (4) represent the same line, 2 a  y1  ∴ ⇒ x1 y1  4 a 2 . x1 2 a  

h2  k 2  6  h.

Squaring both sides, we have, h2 1 k2 5 36 1 h2212h or k2 5 212 (h23). ∴ The path of the point is y2 5212(x23). 23. Coordinates of the vertex of the parabola x2 5 4y are (0, 0) and the ends of latus rectum are (2, 1) and (22, 1). ∴ Centre of the circle is (0, 0) and radius of the circle is ( 2)2  (1)2  5. ∴ Equation of circle is x2 1 y2 5 5.

 b2  24. The end of the latus rectum in the first quadrant is  ae,  . a   b2  Equation of normal at  ae,  is a   a 2 x b2 y  a2 x b2 y  2  a 2  b2    a 2  b2  ae b / a x y  1  1 or

 a 2  b2  a x  ay  a 2 e2  e2   e a2  

or

x  ey  ae 3  0.

25. Since 3 3 32 1 5 3 52232 > 0, the point (3, 5) lies outside the ellipes 3x2 1 5y2 5 32. ,25 3 32 1 9 3 522450 5 0, ∴ the point (3, 5) lies on the ellipse 25x2 1 9y2 5 450. So the required number of tangents is 3.

3.138

Mathematics

26. Let the two fixed ⊥ straight lines be the coordinates axes. Let P(h, k) be the point whose locus is required. Y B

28. Let S(5, 12), S′(24, 7) be the two foci. P(0, 0) is a point on the conic.

θ h

P(h,k) k

O

θ A

SP  25  144  13 S ' P  576  46  625  25

X

SS '  (24  5)2  (7  12)2  192  52  386

Let PA : PB  1 : 2 l 2l Then PA  and PB  . 3 3 l ...(1) k  or 3k  l sin θ 3 2l 3h h  cosθ or  l cosθ ...( 2) 3 2 Squaringg and adding (1) and (2) we get

If the conic is an ellipse, then SP 1 S′P 5 2a and SS′ 5 2ae ∴ e

SS ' 386 386   . SP  S ' P 13  25 38

29. Equation of normal at (a cos 2θ, b sin 2θ) is ax sec θ  by cosec θ  a 2  b2

9l  l 2 or 9 x 2  36 k 2  4l 2 4 ∴ loues of P(h, k) is 9 x 2  36 y 2  4l 2 Which is an ellipse 9k 2 

Since OX and OY are mutually perpendicular tangents to sliding ellipse for all its positions, therefore, O (0, 0) will lie on circle (1) ∴ α2 1 β 2 5 a2 1 b2. Hence, locus of C (α, β) is x2 1 y2 5 a2 1 b2.

2

27. Let the two given lines be taken as the coordinate axes. Let C (α, β ) be the centre of the ellipse in any position. Here the position of centre C changes as the ellipse slides. Let a and b be the semi major and semiminor axes of the ellipse. Equation of the director circle of the ellipse is (x2α)2 1 (y2β)2 5 a2 1 b2 ...(1) Y

⇒

14 x sec θ  5 y cos ec θ  9

It passes through (a cos 2θ , b sin 2θ ) ⇒

14 14cos 2θ .sec θ  5 sin 2θ cos ec θ  9

⇒ 14

sin 2θ cos 2θ 5 9 sin θ cos θ

⇒ 14(2 cos 2 θ  1)  10 cos 2 θ  9(0)θ ⇒ 18 cos 2 θ  9 cos θ  14  0 ⇒ cos θ 

2 3

30. The given hyperbola can be written in the form

C( α, β)

M O

( x  4)2 ( y  1)2   1. 16 9 Here a 2  16 and b2  9.

L

X

∴ e2  1 

b2 9 25  1   ⇒ e  5 / 4. 16 16 a2

nDA boX 1. Under what condition do the planes bx  ay  n, cy  bz  l , az  cx  m intersect in a line? (a) a + b+ c = 0 (b) a = b = c (c) al + bm + cn = 0 (d) l + m + n = 0 2. If the lines 3y  4 x  1, y  x  5 and 5y  bx  3 are concurrent, then what is the value of b? (a) 1 (b) 3 (c) 6 (d) 0

3. If (−5, 4) divides the line segment between the coordinate axes in the ratio 1 : 2, then what is its equation?

(a) 8 x  5 y  20  0

(b) 5x  8 x  7  0

(c) 8 x  5 y  60  0

(d) 5x  8 y  57  0

4. What is the value of n so the angle between the lines having direction ratios (1, 1, 1) and (1, 21, n) is 60º? (a) (c) 3

3

(b)

6

(d) None of these

Analytical Geometry in 2D and 3D ▌ 3.139 5. What is the product of the perpendicular formt he two points

( b  a ,0) to the line ax cos φ + by sin φ = ab? 2

2

(a) a2 (c) ab

(b) b2 (d) a/b

6. The middle point of the segment of the straight line joining the points ( p, q ) and ( q, 2 p ) is ( r / 2, s / 2). What is the length of the segment? 1/ 2  1/ 2    (a)  s 2 + r 2  / 2 (b)  s 2 + r 2  / 4    

(

(

(c) s 2 + r 2

)

)

1/ 2

(

)

3

(c) 2 2

15. Consider the following statements in respect of circles x 2  y 2  2 x  2 y  0 and x 2  y 2  1 1. The radius of the first circle is twice that of the second circle. 2. Both the circles pass through the origin. Which of the statements given above is/are correct? (a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2

(d) s 1 r

7. The direction cosines of a line are proportional to (2, 1, 2) and the line intersects a plane perpendicularly at the point (1, −2, 4). What is the distance of the plane from the point (3. 2. 3)? (a)

14. What are the coordinates of the point equidistant from the four points (0, 0,0), (2,0,0), (0,4, 0,) (0,0,6)? (a) (1, 2, 3) (b) (2, 3, 1) (c) (3, 1, 2) (d) (1, 3, 2)

(b) 2 (d) 4

8. The foot of the perpendicular drawn from the origin to a plane is the point (1, −3, 1). What is the intercept cut on the x-axis by the plane? (a) 1 (b) 3 (c) 11 (d) 11 9. A line makes the same angle α with each of the x and y axes. If the angle θ, which it makes with the z-axis, is such that sin2θ = 2 sin2 α, then what is the value of α? (a) π/4 (b) π/6 (c) π/3 (d) π/2 10. What is the locus of a point which is equidistant from the point (m + n, n – m) and the point (m – n, n + m)? (a) mx = ny (b) nx = − my (c) nx = my (d) mx = − ny 11. What does the equation x 3 y  xy3  xy  0 represent? (a) A pair of straight lines only (b) A pair of straight lines and a circle (c) A rectangular hyperbola only (d) A rectangular hyperbola and a circle 12. The point of intersection of the two lines 2x  3y  4  0 and 4 x  3y  2  0 is at a distance d from origin. What is the value of d? (a)

2

(b)

3

(c)

5

(d)

7

13. What is the locus of a point which is equidistant from the points (a + b, a –b) and (b – a, a + b)? (a) bx  ay  0

(b) bx  a 

(c)  ax  by  0

(d) ax  b

16. The circle x 2  y 2  4 x  4 y  4  0 touches (a) Only the x-axis (c) Both the axes 17.

(b) Only the y-axis (d) Neither of the axes

The equation of the circle which touches the axes at a distance 5 from the origin is y  x 2  2αx  2αy  α2  0. what is the value of α ? (a) 4 (b) 5 (c) 6 (d) 7

18. The planes px + 2y + 2z – 3 = 0 and 2x – y + z + 2 = 0 intersect π at an angle . What is the value of p2? 4 (a) 24 (b) 12 (c) 6 (d) 3 19. What is the equation of the plane through z-axis and parallel x 1 y  2 z  3   ? to the line cos θ sin θ 0 (a) x cot θ + y = 0 (b) x tan θ − y = 0 (c) x + y tan θ = 0 (d) x – y tan θ = 0 20. What is the equation of the sphere which has its centre at (6, −1, 2) and touches the plane 2x – y + 2z – 2 = 0? (a) x 2  y 2  z 2  12 z  2 y  8 z  16  0 (b) x 2  y 2  z 2  12 z  2 y  8 z  16  0 (c) x 2  y 2  z 2  12 x  2 y  8 z  16  0 (d) x 2  y 2  z 2  12 z  2 y  8 z  25  0 21. What are the direction ratios of the line determined by

the planes x – y + 2z = 1 and x + y – z = 3? (a) (2 1, 3, 2) (c) ( 2, 1, 3)

(b) (2 1, 2 3, 2) (d) ( 2, 3, 2)

22. If the radius of the sphere x 2  y2  z 2  6 x  8 y  10 z  λ  0 is unity, what is the value of λ? (a) 49 (c) 2 49

(b) 7 (d) 2 7

3.140

Mathematics

23. If (4, 0) and (−4, 0) are the foci of an ellipse and the semi-minor axis is 3, then the ellipse passes through which one of the following points? (a) (2, 0) (b) (0, 5) (c) (0, 0) (d) (5, 0)

x2 y2 24. A circle is drawn with the two foci of an ellipse 2  2  1 a b at the end of the diameter. What is the equation of the circle? (a) x2 + y2 = a2 + b2 (b) x2 + y2 = a2 2 b2 (c) x2 + y2 = 2(a2 + b2) (d) x2 + y2 = 2(a2 2 b2) 25. What are the equation of the directrices of the ellipse 25x 2  16 y 2  400 ? (a) 3x  25  0 (b) 3y  25  0 (c) x  15  0 (d) y  25  0

x2 y2   1 has the same eccentricity as the 169 25 2 x y2 ellipse 2  2  1. What is the ratio of a to b? a b

26. The ellipse

(a) 5/13 (c) 7/8

(b) 13/5 (d) 8/7

27. Equation of the hyperbole with eccentricity 3/2 and foci at (2,0) is 5x 2  4y2  k 2 . What is the value of K?

4 3  4  (c)   5  3

3 4  3  (d)   5  4

(a)

(b)

28. What is the area of the ellipse 4x 2 + 9y2 = 1? π (a) 6π (b) 36 π π (c) (d) 6 6

AnSwER KEy 1. (c)

5. (a)

9. (a)

13. (c)

17. (b)

21. (a)

25. (b)

2. (c)

6. (c)

10. (c)

14. (a)

18. (a)

22. (a)

26. (b)

3. (c)

7. (b)

11. (d)

15. (d)

19. (b)

23. (d)

27. (c)

4. (b)

8. (d)

12. (c)

16. (c)

20. (c)

24. (b)

28. (c)

SoLutionS 1. The planes bx – ay = n, cy – bz = l and az – cx = m intersect in a line, if al + bm + cn = 0. 2. The equation of given lines are 3y + 4x = 1 …(i) y=x+5 …(ii) and 5y + bx = 3 …(iii) On solving Eqs. (i) and (ii), we get x = −2 and y = 3 If there lines are concurrent, then these values must satisfy the third equation. 15 – 2b = 3 ⇒ 2b 5 12

⇒ b56 3. Let A(a, 0) and B(0, b) are also two points on respective coordinate axes and (−5, 4) divides AB in the ratio 1 : 2. 1 0  2  a ∴ 5 3 15 ⇒ a 2

1 b  2  0 3 ⇒ b  12

and 4 

Hence, equation of line joining   15   2 , 0 and (0, 12) is (y  0) 

12  0  15  . x  15  2 0 2

4 ⇒ y  ( 2 x  15) 2 ⇒ 5 y  (8 x  60) ⇒ 8 x  5 y  60 0 4. ∴ cos 60 

1  1  1  (1)  1  n 12  12  12  12  12  n2

Analytical Geometry in 2D and 3D ▌ 3.141 9.  l 2  m2  n2  1

1 n  2 3 2  n2

⇒

⇒ cos 2 α  cos 2 α  cos 2 θ  1 ...( i) Also, sin 2 θ  2 sin 2 α

⇒ 3( 2  n2 )  4 n2

⇒ 1  cos 2 θ  2(1  cos 2 α)

⇒ n2  6 ⇒ n  6

⇒ cos 2 θ  2 cos 2 α  1 ∴ From Eq. (i)

5. Given, ax cos φ  by sin φ = ab = 0

(

At point  b2  a 2 , 0

d1 

⇒ 4 cos 2 α  2 ⇒ cos 2 α 

a 2 cos 2 φ  b2 sin 2 φ

(

2

2

⇒ cos α 

)

a 2 cos 2 φ  b2 sin 2 φ

(

2

)

∴ [ x1  ( m  n)]  [ y1  ( n  m)]2 2

a 2 cos 2 φ  b2 sin 2 φ

 [ x1  ( m  n)]2  [ y1  ( n  m)]2

a 2 ( − b sin 2 φ  a 2 cos 2 φ) a 2 cos 4 φ  b2 sin 2 φ

⇒ x12  ( m  n)2  2 x1 ( m  n)  y12  ( n  m)2  2 y1 ( n  m) ⇒ 2 x1 ( m  n  m  n)  2 y1

 pq q p  , 6. Midpoint of (p, q ) and (q,  p ) is   2   2  r s which is given  ,  .  2 2 pq r q p s  and  2 2 2 2

Now, length of segment 

1

10. Let the locus of a point be ( x1 , y1

)

[ a 2 b2  a 2 cos 2 φ  a 2 b2 ]

a2

∴

1 2

π 3π ⇒ α , 4 4

a b2  a 2 cos φ  ab

∴ d1d2  

2cos2 α  2 cos 2 α  1  1

a b2  a 2 cos φ  ab

At point  b  a , 0 d2 

)

( p  q)  (q  p) 2

2

 s2  r 2 7. Equation of plane passing through (1, -2, 4) and whose normal (2, 1, 2) is 2(x  1)  1 (y  2)  2 (z  4)  0 ⇒ 2x  y  2 z  8  0 ∴ Required distance 2(3)  1(2)  2(3)  8  4 1 4 6  2 3 8. Equation of plane passing through (1, −3, 1) and whose normal (1, −3, 1) is 1( x  1)  3( y  3)  1( z  1)  0 ⇒ x  3 y  x  11 = 0 z x y  0 ⇒  11 11 / 3 11 The above plane intercept the x -axiss at 11.

( n  m  n  m)  0 ⇒ 4 x1n  y1m  0 ⇒ my1  nx1 Hence, locus of a point is nx = my 11.

x 3 y  xy3  xy  0 xy(x 2  y2  1)  0 ⇒ xy  0 and x 2  y2  1  0 or x 2  y2 1

12. Let the point of intersection be (a, b). Then 2a + 3b + 4 = 0 ………(1) 4a + 3b + 2 = 0 ……….(2) Subtracting (1) from (2) 2a  a  0 ⇒ a 1 4  2 b 2 3 Hence point of intersection is (1,  2) d  (1  0)2  (2  0)2 d  1 4  5 13. The locus will be a line perpendicular to the line joining the two points and passing through its mid-point = (b, a)

Slope of the line joining given points size

3.142

Mathematics

m

a  ba  b b  b  a  a  b a

a Slope of the line perpendicular  b Le the equation be a y x c b

a ⇒ a   bc b ⇒ c 0 a ∴ y x b

⇒  ax  by  0

14. Let the coordinates of the point to be a, b, c a 2  b2  c 2  (a  2)2  b2  c 2 or a 2  b2  c 2  a 2  4  4a  b2  c 2 or 4a  4 a 1 Similarly b  2, c  3 15. The equation of first circle is x 2  y 2  2 x  2 y  0

Radius of this circle  (1)2 + (1)2  2 and equation of second circle is x 2  y 2  1 radius of this circle  1 From above it is clear that the radius of first circle is not twice that of second circle. Also, first circle passes through the origin while second circle does not pass through the origin. Hence, neither 1 nor 2 statement is correct. 16. Given equation is x2  4x  4  y2  4 y  0 ⇒ ( x  2)  ( y  2)  22 2

2

Here, we see that the values of centre and radius are same. Hence, it touches both axes. 17. The radius of the circle is 5 ⇒ 5 α2  α2  α2 ⇒ α2 ⇒ α  5

18. We know that the angle between the planes a1x  b1 y  c1z  d1  0 and a2 x  b2 y  c2 z  d2  0 is given by cos θ 

∴ cos

a1a2  b1b2  c1c2 a  b12  c12 a22  b22  c22 2 1

π p  2  2  (1)  2  1  4 p2  4  4 4 11

⇒

1



2

2p p2  8 6

⇒

1 4 p2  2 2 p 8 6

(

)

⇒ p  24 2

19. Equation of plane through z-axis is ax + by = 0 …(i) This plane is parallel to the line

x 1 y  2 z  3   cos θ sin θ 0 ⇒ a cos θ  b sin θ  0 ⇒ a  b tan θ ∴b tan θx  by  0 ⇒ x tan θ  y  0 Which is required eqquation of plane. 20. Given centre of sphere is (6, 1, 2) 2(6)  1(  1)  2(2)  2 ∴ Radius  4 1 4 15 5 3 ∴ Equation of sphere is 

(x  6)2  ( y  1)2  (z  2)2  52 x 2  36  12 x  y 2  1  2 y  z 2  4  4 z  25 x 2  y 2  z 2  12 x  2 y  4 z  16  0 21. The intersection of given plane is x  y  2 z  1  λ ( x  y  z  3)  0 ⇒ x (1  λ )  y (λ  1)  z ( 2  λ )  3λ  1  0 DR's of normal to the above plane is (1  λ , λ  1, 2  λ ) By taking option (a) 1(1  λ )  3((λ  1)  2( 2  λ )  0 ⇒ 1  λ  3λ  3  4  2λ  0 ⇒ 00 Hence, option (a) is correct. 22. x 2  y2  z 2  6 x  8 y  10 x  λ  0 Comparing it with standard equation of the sphere x 2  y2  z 2  2gx  2 hy  2fz  c  0 We have h  3, h  4, f  5, c  λ Radius g2  h2  f 2  λ ⇒ 1  9  16  25  λ ⇒ 1  50  λ ⇒ λ  49

Analytical Geometry in 2D and 3D ▌ 3.143 23.  Foci of an ellipse are (4, 0) and (−4, 0). ∴ 2ae = 8 ⇒ ae = 4 and b = 3

we know that, e  1 

b2 a2

Also ellipse is

e  1

∴

2

⇒ ⇒

Thus, the equation of an ellipse is x2 y2  1 25 9 which is satisfied by (5, 0). Hence, the ellipse passes through (5, 0). x2 y2   1 are (ae, 0) and ( ae, 0) a 2 b2 ∴ Equation of circle whose end points of diameter, is

24. ∴Foci of an ellipse

(x  ae) (x  ae)  ( y  0) ( y  0)  0 ⇒ x 2  a 2 e2  y 2  0  b2  ⇒ x 2  y 2  a 2  1  2   0  a   2 2 2 2 ⇒ x  y  a b 0 ⇒ x 2  y 2  a 2  b2 25. The equation of ellipse can be rewritten as x2 y2  1 16 25 Here, a = 4 and b = 5 ( b > a ) ∴ Equations of the directories are b y   e

5 1

 16 25

25 3

⇒ 3 y  25  0 x2 y2  1 169 25 25 12 e  1  169 13

26. Given, ellipse is ∴

b2 a2

b2 12  1 2 13 a 2 b 144 25 1  169 169 a2 a 13  b 5

∴

4  9  ⇒    1 2  a a     16 a 2  9 ⇒ 2 a a2 2 ⇒ a  25 ⇒ a  5

x2 y2 + 1 a 2 b2

27. 5x 2  4 y2  k 2 5 2 4 2 ⇒ x  2 y 1 k2 k 3 Eccentricity e  2 Foci  (a, e), 0  (  2, 0) ⇒ ae = 2 3 ⇒ a.  2 2 4 ⇒a 3 Comparing with standard equation k2 5 16 k 2 ⇒  5 9 16 5 ⇒ K2  9 4 ⇒ k 5 3 a2 

28. 4x 2  9y2  1 ⇒

x2



y2

1 2 1 1     2  3 1 1 ⇒ a  , b 2 3 Area of the ellipse 2

1 1 π  πa  b π   2 3 6

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C

H

5

A

P

T

E

R

Differential Calculus FUNCTIONS Let D  φ be any subset of R, (the set of real numbers). A rule which associates with each element of D, a unique element of R is called a function on D, D is called the domain of this function and the set of all associates (or images) of the elements of D is called its range.

Notice: |sec x|  1 and |cosec x|  1 whatever x may be in their domains.

(

)

4. The domain and the range of In x 5 log x are (0, ∝) and (2 ∝, ∝) (i.e. whole of R), respectively.

Example

5. The domain of ex is R and its range R1

The domain of the function. f ( x )  ( 2  x )(x  3)

6. The domain of are sin x (also written as sin21 x); also of are cos x is [21, 1] whereas their ranges, respectively are

is 2  x  3 or [2, 3] as of f(x) to be meaningful, either 2  x  0 and x 2 3 < 0 or 2 2 x  0 and x 2 3  0; the second alternative  1 is obviously absurd. Also its range is 0,  as the equation,  2 1 ( 2  x )(x  3)  k with k  , does not admit of real solutions. 2 The domains and the ranges of some of the elementary functions are given below: 1. The domain of sin x (also of cos x) is R and the range is [1;1], i.e. { x ∈ R : 1  x ∈ 1}.

π : m ∈ I} and its 2 range is R where I stands for the set of all integers. The

2. The domain of tan x is R  {( 2 m 1 1)

domain of cot x is R  {mπ : m ∈ I} and its range is R. 3. The domain of sec x is R  {( 2 m 1 1)

π : m ∈ I} and its 2

range is [(1, ∞) ∪ (– ∞, –1)] do Also, the domain of cosec x is R 2{mπ : m ∈I} and its range is the same as that of sec x.

 π π 2 2 , 2  and [0, π].  

 π π 7. The domain of tan x is R and its range is  2 ,  .  2 2 The domain of are cot x is R and its range is (0, π). 8. The domain of both are sec x and are cosec x is

(

)

R  21,1 and their ranges ranges, respectively are  π π   π   π 0,  ∪  , π and 2 , 0  ∪  0,  . 2 2      2   2 Let f(x) be any function with domain D. It is said to be even (or odd) if f(x) 5 f(x) (respectively, f(2x) 5 2f(x)) for every x in D (to be usually written as ∀x ∈ D. Clearly, sin x is an odd function and cos x in an even function. A function f(x) on D is said to be periodic if there exists a positive number T s.t.f(x 1 T) 5 f(x) ∀ x ∈D. The least such positive number, T, is called the period of this function. For example, the period of the function tan x is π.

3.146

▌ Mathematics

IDEA OF LIMIT Let a function, f(x) be defined in the neighborhood of a point c not necessarily at the point, c. Then, by lim f ( x )  l , we mean the

3. Both lim f(x) and lim f(x) exist (as finite numbers) but x → c2

right hand limit of f(x) as x tends to c, through values of x larger than c; the difference between f(x) and l can be made arbitrarily small. One can define the left hand limit, lim f ( x ) 5l, in the same

The calculation of limits is based on the following theorems: If lim f ( x ) 5 l and lim g ( x ) 5 l ' then x→c

x → c2

way. If lim f(x) 5 lim f(x) 5 lim f(x) then we say that limit of x→ c1

x → c2

are unequal.

x → c

x→ c2

x→c

(a) lim [ f ( x )  g ( x )]  lim f ( x )  lim g ( x )

f(x) as x → c, exits and we write it’s as lim f(x) . The limit of the

x→c

x→ c2

function, f(x) may not exist in any of the following situations:

x→c

x→c

(b) lim [ f ( x ) ⋅ g ( x ) 5 lim f ( x ) ⋅ lim g ( x ) x→c

1. lim f(x) does not exist (i.e. when f(x) becomes infinitely x→ c2

x→c

x→c

f ( x)  f ( x )  lim (c) If l ′ ≠ 0, lim  . 5 x→ c  x→c g( x)  g ( x )  lim x→c

small or large as x→c2). 2. lim f(x) does not exist. x→ c1

SOME IMpOrTANT LIMITS 1. lim x →0

sin x tan x ⇒ lim cos x 5 lim x →0 x →0 x x

(

2. lim 1 + x x →0

)

1 x

4. lim

in (1  x ) x

x →0

x

 1  lim  1    e x →∞  x

(

)

1

m

5. lim 1  x  1  m x →0 x

a x 1  In a x →0 x

6. lim Inx  0( m  0) x →∞ x m

3. lim

CONTINUOUS FUNCTIONS A functions f(x) said to be continuous at x 5 c, if lim f(x) 5 f(c). x→ c

If is not continous at x 5 c, it is said to be discontinuous there at. It can be discontinuous in any of the following situations; 1. lim f ( x ) does not exist. x→c

2. f(x) is not defined at x 5 c.

3. lim f ( x ) exists but is not equal to f(c) x→c

All polynomials, trigonometric, exponential and logarithmic functions are continuous at each point of their respective domains. Also, sum, difference, product and quotient (if lim g(x)  at x→ c x 5 c, is again continuous there at.

DErIVATIVE The derivative (or differential coefficient f`(x) of a function y 5 f(x) at a given point x of its domain is given by dy ∆y f ( x  ∆x )  f ( x )  f ′( x )  lim  lim ∆x → 0 ∆x ∆x → 0 dx ∆x

at x 5 0. Also, a function derivable at a point is always continuous there at but not conversely. We shall denote the second derivative of y 5 f(x) by f’’(x) by f’’(x) or by

d2y d  dy  5 . dx 2 dx  dx 

If this limit does not exist, we say f(x) is not differentiable (or derivable) at the point x; for example, f (x) 2 |x| is not derivable

SOME IMpOrTANT FOrMULAS 1.

d d 1 (are sin x )  (are cos x )  dx dx 1 x2

2.

d d 1 (are tan x )  (are cot x )  dx dx 1 x2

Differential Calculus ▌ 3.147

3.

d d 1 (are sec x )  (are coses x )  dx dx x x2 1

4.

d 2 d ( a ) 5 a x In a : in particular, ( e x ) 5 e x dx dx

5.

1 d ( In x ) 5 dx x

AppLICATIONS OF ThE DErIVATIVE 1. Increasing and Decreasing functions. A continuous function, f(x), with domain [a, b] is said to be nondecreasing (non-increasing) in its domain if x1  x2 ⇒ f ( x1 )  f ( x2 ) ( or x1  x2 ⇒ f ( x1 )  f ( x2 )), ∀x1 , x2 ∈[ a, b] For f(x) to be increasing (decreasing) on [a, b] it is sufficient for it to fulfill the condition: f ′ ( x )  0( or f ′( x )  0) ∀x ∈( a, b) 2. Tangents and Normals. The equation of the tangent at (x0, y0) to the curve, y 5 f(x), is y  y0  m( x  x0 ) dy  Where, m 5  is the slope of the tangent at this dx ( x , y ) 0

0

point. If this tangent is parallel to the y-axis then its equation is, x 5 xn. In case m  0, then the length of the tangent 1 at (x0, y0) to the curve is, y0 1 1 2 Further, if m  0, the m equation of the normal to the curve at the said point, is 1 y  y0  x  x0 . However, if m 5 0, the normal has m

(

)

the equation, x 5 x0. In any case the length of the normal at the point to the curve is, y0 11 m2 . 3. Maxima and Minima A necessary condition for the existence of an extremum (maximum or minimum) point of the function, f(x), is that f`(x) 5 0 at that point or f`(x) does not exist there at. Let x0 be a root of f`(x) 5 0.

Sufficient conditions A. (i) If f`(x) changes sign from positive to negative at x0 i.e. f`(x) > 0 for x > x0 and f`(x) < 0 for x > x0 then the point x0 is a maxima (or maximum point) of the function. (ii) If f´(x) changes sign from negative to positive at x0 then x0 is a minima of the functions. (iii) If f´(x) does not change sign in moving through the point x0 then there is no extre-mum at x0. B. Let f(x) be twice differentiable and x0 be a point obtained from f´(x) 5 0. Then, x0 is a point of maxima (or minima) if f``(x0) < 0 (or f``(x0) > 0).

prACTICE EXErCISE FUNCTIONS, LIMITS, CONTINUITy AND DIFFErENTIAbILITy 1. If 3x 1 2y 5 |y| then y as function of x is (a) defined for all real x. (b) continuous at x 5 0. (c) differentiable for all x. dy (d) such that 3 for x  0. dx   sin[ x ] [ x ] ≠ 0 2. If f ( x ) 5  , where [x] denotes the gratest  [ x0 ] [ x ] 5 0;

 1  1 1 1 4. lim  1 1 1 .... 1  is given by n→∞ 1.3 3.5 5.7 ( 2 n 1 1)( 2 n 1 3)   1 (a) 0 (b) 2 (c) 1 (d ) 2 5. If f ( a )  2, g ( a )  1, f '( a )  1 and g `( a )  2 the value  f ( x ) g ( a)  f ( a) g ( x )  of lim   is x→a xa  

integer ≤ x, then lim f ( x ) is equal to

(a) 5

(a) 2 1 (c) 1

(c) 

x→0

(b) 0 (d) None of these

3. If f ( x )  9  x 2 then lim x→2

(a ) 2 / 5 (c ) 0

f (2 x )  f ( x ) has the value x2 ( b)  2 / 5 (d ) None of these

(b) 1 5

 x2  6. lim   x →∞ x  2   (a) e4 (c ) e2

1 5

(d)  5 x 1

is given by (b) e2 (d) e4

3.148

▌ Mathematics

( 27  2 x )1/ 3  3 7. The value of f(0), so that the function, f ( x )  9  3( 243 5x )1/5 is continuous is given by (a) 2/3 (b) 6 (c) 2 (d) None of these

In(1 ax )  In(1 bx ) , is not defined x at x 5 0. The value which should be assigned to f at x 5 0, so that it is continuous at x 5 0, is (a) a 2 b (b) a 1 b (c) 2 a 1 b (d) 2 a 2 b

8. The function, f ( x ) 

1

 sin x  aα 9. The value of f(α) so that the function, f ( x )   where  sin α  α  mπ (m ∈ I), is continuous is given by

(a) e tan α

(b) ecot α

(c) tan α

(d) cot α

10. If f(x) 5 logx (In x) then f`(e) is (a) e (b) e2 -1 (c) e (d) None of these dy 11. Let x 5 2 In cot t and y 5 tan t 1 cot t. Then, sin 2t 1 1 has dx the value (a) cos 2t (b) 2 cos2 t (c) 2 sin2t (d) None of these 12. If x  of

1 t 2  1 t 2 1 t 2 and y  ; then the value 2 1 t 1 t 2  1 t 2

d2y at t  0, is given by dx 2 1 2 (d) 1

(a) 0

(b)

(c) 1

 π π (a) increase in  ,   4 2  π 5π  (b) decreases in  ,  4 4   π   5π  (c) increase in 0,  ∪  , 2 π 4 4      π π  (d) decreases in  0,  ∪  , 2 π 4 2     18. If y  f ( x )  (a) (b) (c) (d)

x 4 then x 1

x 5 f(y) f(1) 5 2 y is a rational function of x y decreases with x for all x  1.

19. Let f and g be increasing and decreasing function, respectively from [0, ∝) to [0, ∝). Let h(x) 5 f(g(x)). If h (0) 5 0, then h(x) 2 h (1) is (a) always zero (b) always positive (c) always negative (d) strictly increasing 20. The smallest positive root of the equation, tan x 2 x 5 0 lies in  π π  (a)  0,  (b)  , π   2 2   3π   3π  (c)  π,  (d)  , 2 π   2   2  a ∞/ a e  e∞ / a and 2 let p denote the length of the normal to this curve at A. then, a, y0 and p are in (a) A.P (b) G.P (c) H.P (d) None of these

21. Let A(x0, y0) be any point on the curve, y 

13. The derivative of the function, f(x) 5 x|x| is (a) 2x (b) 2 2x (c) 2|x| (d) None of these 14. Let f(x 1 y) 5 f(x)f(y) for all x and y, f(2) 5 3 and f`(0) 5 1. Then f(2) is given by (a) 1 (b) 2 (c) 3 (d) 4 15. Let the equation, a0 x n  a1x n−1  ... an1x  0, have a positive root α. Then the equation, na0xn-11 (n 2 1)a1xn-2 1 …1an-1 5 0, has a positive root β where (a) β < α (b) β 5 α (c) β > α (d) β 5 2α

(

17. Let the function, f(x) 5 sin x 1 cos x be defined in [0, 2π]; they f (x)

)

16. If y 5 2x 1 are cot x 1 In 1 + x 2 2 x , they y (a) increases in [0, ∝) (b) decreases in [0, ∝) (c) neither increases nor decreases in [0, ∝) (d) increases in (2 ∞, ∞).

(

)

22. The equation of the tangents to the curve, (1 1 x2) y 5 1 at the points of its intersection with the curve, (x 1 1) y 5 1 are given by (a) x 1 2y 5 1, y 5 1 (b) x 12y 5 2, x 5 1 (c) x 1 2y 5 2, y 5 1 (d) x 1 2y 5 1, x 5 1 23. If the line, ax 1 by 1 c 5 0 is a normal to the curve, xy 5 1; then (a) a > 0, b > 0 (b) a > 0, b < 0 (c) a < 0, b > 0 (d) a < 0, b < 0 24. The coordinates of the point on the curve, x2y 2 x 1 y 5 0, where a tangent to the curve has the greatest slope are given by (a) (2 3, 1)

(b) (0, 0)

(c) 0, 1

(d) ( 3, 1)

( )

Differential Calculus ▌ 3.149 25. The longest distance of the point (a, 0) from the curve, 2 x 2  y 2  2 x  0 is given by (a)

1 2a  a2

(b)

1 2a  2a2

(c)

1 2a  a2

(d)

1 2a  2a2

 27. Let f x 5 1, if x is rational number be a function from 0, if x is an irrational number

()

R to R where R is the set of real numbers, Find f(1/2). (a) 1 (b) 0 (c) 1 (d) None of these

28. If f ( x )  1 (1 / x ) for all real numbers x except x 5 0 $ g(y) 5 (y2 1 1) for all real numbers y, find f [ig (2 1)]. (b) 2 1

1 (c) 2

(d) 2

1 2

(b) 1 (d) None of these

31. If f(x) 5 (x2 1 1) and g(x) 5 ax, find gf [(x)]. (a) a 2 x

(b) a 2 x  1

(c) a x

(d) None of these

(c) a

x 21

37. Evaluate the limit lim x→0

(a) 0 (c) 2 1 38. Evaluate the limit lim x→0

(a) (b) (c) (d)

0 1 21 None of these

(a) (b) (c) (d)

h→0

(c)

(b) 1 (d) None of these

1 2 cos 2 x . x

0 1 21 None of these

40. Evaluate lim (a)

sin x . x

sin x . x

( x  h)  x h

1 2

(b) 0 1

(d)

2 x

(a) ∞ (c) 1

1 x

x−3 x22 2 4 − x (b) 0 (d) 2 1

42. Evaluate lim tan x loge (sin x). x→π / 2 (a) ∞ (b) 0 (c) 1 (d) 2 1

(b) a x  1 2

(d) None of these

{(

) }

33. Find the domain of the function f ( x )  log 3x 2 x 2 / 2  .   (a) [1, 2] (b) (1, 2) (c) [1, 2) (d) (1, 2] 34. Find the range of the function f(x) 5 x/(1 1 x2). 1 1 (a)   y  2 2 1 (c)   y  0 2

(d) None of these

41. Evaluate lim x →3

32. If f(x) 5 (x2 1 1) and g(x) 5 ax, find g[f(x)]. (a) a 2 x

(c)  1  f ( x )  1

x→∞

3− x  1,. for  1  x  0    30. Given f ( x )   tan ( x / 2), for 0  x  π  . Find f ( 1) .  x / ( x 2  2), for π  x  6   

2 1

(b) 0  f ( x )  1

39. Evaluate lim

29. If f ( x )  1 (1 / x ) for all real numbers x except x 5 0 $g (y) 5 (y2 1 1)for all real number y, find f[g (2 1)]. (a) 1 (b) 21 (c) 0 (d) None of these

(a) 1 (c) 2

(a)  1  f ( x )  0

36. Find the range of the function f ( x )  1 / ( 2  sin 3x ) 1 1 (a)  y  1 (b)  y  1 2 3 (c) 1 / 4  y  1 (d) None of these

26. If y 5 f(x) 5 (x 1 2)/(x 2 1), then x is equal to (a) 1 (b) 0 (c) f(y) (d) None of these

(a) 1

35. Find the range of the function f(x) 5 1/(x2 2 1).

1 (b)  y  1 2 (d) None of these

 sin x  cos x  43. Evaluate the lim  . x→ π/ 4  x  ( π / 4)  (a) 0 (b) 1 (c) 1 44. lim(1 2 x ) tan x →0

(a) 0 2 (c) π

(d)  2 πx . 2 (b) 1 1 (d) π

3.150

▌ Mathematics

45. Find the value of a and b such that x (1 a cos x )  b sin x lim 1 x →0 x3 5 3 (a) a  , b  2 2 3 5 (b) a  , b  2 2 3 5 (c) a  , b  2 2 3 5 (d) a  − , b  2 2

 1 cos10 x , x  0  x2  , then the value of a so that x 0 48. If f ( x )  a,  x  ,x 0  625  x  25 f(x) may be continuous at x 5 0 is (a) 25 (b) 50 (c) 225 (d) None of these 49. Use the formula lim x →0

46. Evaluate lim x log(sin x ) x →0

(a) 1 (c) 2 1

(b) 0 (d) None of these

2 sin x 2 sin 2 x , x ≠ 0 find lim f `( x ) 47. f (x ) is the integral of x →0 x3 (a) 1 (b) 0 (c) 2 1 (d) None of these

a x 1 2 x 1 .  log a to find lim x →0 x (1 x )1/ 2 1

(a) 2 (c) 2 log 8

(b) 2 log 2 (d) None of these

sin 2 x is equal to x (a) 0

(b) 1

50. lim x →0

1 (c) 2

(d) 2

ANSwEr KEy 1. 2. 3. 4. 5. 6. 7.

(a, b) (d) (a) (b) (d) (a) (c)

8. 9. 10. 11. 12. 13. 14.

(d) (b) (c) (b) (b) (c) (c)

15. 16. 17. 18. 19. 20. 21.

(a) (a, d) (b, c) (a, c, d) (a) (c) (b)

22. 23. 24. 25. 26. 27. 28.

(c) (b, c) (b) (d) (c) (a) (c)

29. 30. 31. 32. 33. 34. 35.

(a) (c) (b) (b) (a) (a) (a)

36. 37. 38. 39. 40. 41. 42.

(b) (b) (a) (a) (c) (c) (b)

43. 44. 45. 46. 47. 48. 49.

SOLUTIONS  x , x  0 1. 3x  2 y | y | ⇒ y    3x , x  0 Which is defined for all x. Also, it is continuous everywhere including x 5 0 dy dy for , x  0, 1 and for x  0,  3 dx dx  x 0  at , x  0 f `(0 )  lim  1 x →0 x  0 where, y 5 f(x) and f`(02) 5 2 3 ∴ f is not differentiable at x 5 0 ⇒ Both (c) and (d) do not hold.

2.

lim f ( x )  lim

x → 0

x →0−

sin[ x ] sin( 1)  [ x] 1

As 0  x  1  [x ]  0, therefore lim f ( x )  0

x → 0

⇒ lim f ( x ) does not exist x →0

5  9 x2 f ( x)  f ( x)  lim x→2 x→2 x 2 x 2 2 x 22   lim x→2 5 5  9 x2

3. lim

(d) (c) (a) (b) (a) (b) (b)

50. (d)

Differential Calculus ▌ 3.151

4.

1 sin α    sin x  x α  sin x  sin α  sin x sin α    lim   lim  1    (i) x → α sin α x→α sin α      1 sin x  sin α Where,, t  ⋅ sin α x α  x α   x α  2 cos    sin  1  2   2   cot α  ⋅ sin α x α as x → a

1 1 1   ... 1.3 3.5 ( 2 n 1)( 2 n  3) =

 1 1  1  1  1 1   1       ...   2  3  3 5   2 n 1 2 n  2  

1 1  1  1 → as n → ∞ 2  2 n 3  2  f ( x ) g ( a)  f ( a) g ( x )  5. lim   x→ a xa    ( f ( x )  f ( a )) g ( a )  f ( a )( g ( x )  g ( a ))   lim   x→ a xa    f `( a ) g ( a )  f ( a ) g `( a ) 5  x 2  6. lim   x →∞ x  2  

∴ By (1), f ( α)  ecot α 10.

x 1



x 2     4  4  lim  1   x →∞ x 2   

4 ( x 1) x2

11.  e4

7. For f to be continuous at x 5 0, lim f ( x ) 5 f (0). x →0

1/ 3

 2   1 27 x  1 Now lim f ( x )  lim 1/ 5 x→0 x→0   5   x  3 1 1 243      2 2  x  0( x ) 81  lim 2 x→0  1  5 x  0( x 2 )  3  ⋅  5 243 

In(In x ) In x 1 1 − In( In x ) ⇒ f ( e) 5 . ⇒ f `( x ) 5 e x ( In x )2 f ( x ) 5 log x ( In x ) 5

dx  2 cos ec 2t 2   dt cot t sin t cos t dy cos 2t  sec 2 t  cos ec 2t  2 dt sin t cos t dy ⇒  cot 2t dx ⇒

dy  sin 2t 1  cos 2t 1  2 cos 2 t . dx 1

12.

y

1 t 2 1 t 2



1

where, 0(x2) stands for terms of powers of x greater than or equal to 2.

1 x 1 x

1 t 1 t 2 d2 4 dy 2 ⇒ and 2   2 dx (1  x ) dx (1  x )3 2

At t  0, x  1 ∴

d2y 1  dx 2 t0 2

f ( x)  x | x | 8. For f to be continuous at x  0, lim f ( x )  f (0)) x →0

and, lim f ( x )  lim x →0

x →0

In(1  ax )  In(1  bx ) x

 lim(a ) In(1 ax )

13.  x 2 , x  0  2 x , x  0 for , x  0, f `( x )  2 x  2 | x |

1 ax

for , x  0 f `( x ) 2 x  2 | x |

x →0

for , x  0 f `( x ) 2 x  2 | x |

1

 lim bIn (1 bx ) bx a  b.

x  0 f `( x ) 2 x  2 | x | for all x.

x →0

9. For f to be continuous at x  α , f (α ) should be equal to lim f ( x ) x →α

 sin x  sin α   lim  1 x →α  sin α 

1 xα

14.

f ( x  y )  f ( x ) f ( y ) ⇒ f (0)  f (0). f (0) ⇒ f (0)  0 or f (0)  1 f (0)  0 or f (0)  1 ⇒ f ( x )  0∀x , Which is against f ( 2)  3.

3.152

▌ Mathematics

∴ f (0)  1 Now, f `( 2)  lim

∆x → 0

 lim

∆x → 0

f ( 2  ∆x )  f ( 2) ∆x

f ( 2) f ( ∆x )  f ( 2) f (0) ∆x

 f ( 2) lim

∆x → 0

f ( ∆x )  f (0) ∆x

 f ( 2). f (0)  3 n n1 15. Let f ( x )  a0 x  a1 x  ... an1 x

Then, f(x) 5 0 has two roots, 0 and α. ⇒ f (x) satisfies the conditions of Rolle’s theorem in [0, α] ⇒ There exits β ∈ (0, α) such that f`(β) 5 0 i.e., 0 < β < α and β is a root of n a0 xn 2 1 1 (n 2 1) a1xn 2 2 1…1an 2 1 5 0 16.

  x dy 1 1  2   1 2 2 2 dx 1 x  1  x  x  1  x 2 1 2x 1   2 1 x 1 x2 

x 2  1 x 2 ( 1 x 2 1  0 for all x. 1 x2

⇒ y increases in ( ∞, ∞ ) and in particular in 0, ∞  17.

 π f `( x )  cos x  sin x  2 cos  x   4    π Now , f `( x )  0 ⇔ cos  x    0 4  π π 3π π ⇔ 0  x   or  x   2π 4 2 2 4 π π 5π 7π ⇔   x  or ,x 4 4 4 4 π 5π ⇔ 0  x  or  x  2π 4 4  π   5π  i.e., f ( x ) inncrease in 0,  ∪  , 2 π 4 4      π 5π  in the same way, f ( x ) decreases in  ,  . 4 4 

18.

 x 4  x f ( y )  f    x 1  At x  1, f is not defind dy 5   0 for all x ( x ≠ 1) . dx ( x − 1)2 Clearly, f is a rational function of x.

19. Since g is a decreasing function from [0, ∝) to [0, ∝), we have g ( x )  0, ∀x  0 and g ( x )  0, ∀x  0 ( by hypothesis )

⇒ g ( x )  0, ∀x  0 Now, h( x )  h(1)  f ( g ( x )  f ( g (1))  f (0)  f (0)  0, ∀x  0 i.e., h( x )  h(1) is always zero. 20. Draw the graphs of y 5 x and y 5 tan x in the domain [0, 2π] y

y

0

π 2

π 3π 2

x

2π

x

Note that tan x > x in (0, π/2). As is clear from the graph, their point of intersection lies in the interval (π,3π/2) which corresponds to the smallest positive root of tan x 2 x 5 0. 21. m 

dv 1 x /a x / a  [e 0  e 0 ] dx ( x , y ) 2 0

0

∴ p  the length of the normal at A y  y0 1 m2  y0 0 ⇒ y02 a ⇒ a, y0 p are in G.P.  1 22. Their points of intersection are A (0, 1) and B  1,   2  2 dy x Now, (1 1 x2) y 5 1 ⇒  dx (1 x 2 )2 ⇒ Slopes of the tangents at A and B are 0 and 1 2 respectively. 2 ∴ The equations of the tangents at A and B are 1 1 y 2 1 5 0 (x 2 0) and y   ( x 1) 2 2 i.e., y 5 1 and x 1 2y 5 2. 23. xy  1 ⇒

dy 1  2 dx x

∴ Slope of the normal is x2 Let the line ax 1 by 1 c 5 0 be normal at (x`, y`). a a Then,   x `2  0 ⇒  0 b b ⇒ a  0, b  0 or a  0, b  0

Differential Calculus ▌ 3.153

24. x 2 y  x  y  0 ⇒ y 

x 1 x 2

dy 1 x 2  dx (1  x 2 )2 1 x2 Let f ( x )  (1  x 2 )2 2 x (3 x 2 ) ⇒ f `( x )  (1 x 2 )3

this in equality holds only if 1 ≤ x ≤ 2. Hence, domain is [1, 2] 34. Let y 5 (x/1 1 x2)

⇒

∴ yx 2  x  y  0,

f `( x )  0 ⇒ x  0, 3 At x 5 0, f`(x) changes sign from positive to negative ⇒ f(x) has a maxima at x 5 0. Thus the greatest slope is f(0) 5 1 and it is for the tangent at (0, 0). (At x 5 ± 3, f(x) has minimum points).

∴ S 2  ( x  a )2  y 2  ( x  a )2  2 x  2 x 2 dS 2 ⇒  2( x  a )  2  4 x dx d 2S 2 and,  2  4  2  0 dx 2 2 dS ∴  0 ⇒ x  1  a is a point of maxima dx ∴ The longest distance  S  (1  2 a )2  2(1  a )  2(1  a )2

35. The range of the function f is the set of all real numbers excepts 0 and those lying between 0 and 2 1. 1 2 y 1 or sin 3x  2  sin 3x y 2 y 1 ⇒ 1   1, 1  sin 3x  1 y  y  ( 2 y 1)  y ,  y  0  1 ⇒  Y  1. 3

36. Let y 

sin x  lim x →0 x 3 x  ( x / 3!)  ( x 5 / 5!) − − − − − x 2 x4  x  lim 1  − − − − −   1. x →0 3! 5!  

37. lim x →0

1 cos 2 x  lim x →0 x 2 1 1 ( 2 x ) / 2! ( 2 x )4 / 4!....

38. lim x →0

 1 2a  2a2

⇒ ( y  1) x  y  2 ⇒ x  ( y  2) / ( y  1)  f ( y ).

{

1 1 is a rational number; ∴ f   5 1. 2 2

{

}

39. lim

x →∞

30.

f ( 1)  2.

31.

f [ g ( x )]  a 2 x 1. 2

33. The function is defined for those values of x for which  3x  x 2  3x  x 2 log   0 i.e. 1  2  2  i.e. x 2  3x  2  0 i.e.( x 1)( x  2)  0

sin x sin(1 / h)  lim [ Putting x  1 / h so h → 0 as x → ∞] h→ 0 (1 / h ) x

{ }{

}

x 32. g[ f ( x )]  a 1.

}

x

}

1  lim h sin    lim h lim sin(1 / h) h→ 0 h→ 0 h→ 0 h

2 1 1 f g (1)  f (1) 1  f ( 2)   . 2 2

11  29. g { f (1)}  g    g (0)  0 1  1.  1 

{

  2  lim 2 x  x 3 ...  0. x →0 3  

26. Here y  f ( x )  ( x  2) / ( x 1) ⇒ ( x  1) y  x  2

28.

2

1 1 1 1 4 y 2  0 or y 2  or   y  . 4 2 2

25. Let S be the distance of (a, 0) from any point (x, y) on the curve, y2 5 2x 2 2x2

2

(1 4 y )

2y Since x is real, so we must have,

For the greatest or least slope,

27. Since

x  1

 0  a number lying between 1 and 1  0.

40.

lim h→ 0

 1 h 

{

( x + h)  x  lim x →0 h

}{

}

xh  x    ( x  h)  x 

xh  x

Multiplying numerator and denomiinator by  ( x  h)  x  lim  h→ 0  (x h  x  1 1   x x 2 x

{

}

 1   lim  h→ 0 x  h  x 

{

xh  x

}

▌ Mathematics

3.154

41. ⇒ lim x →3

x 3

5 3 a  and b  2 2

x 2  4  x

( x  3)  x  2  4  x    x →3    ( x 22  ( 4  x ) x 2  4 x    

 lim

( x  3)  ( x  2  ( 4  x    x →3 2( x  3)

⇒ lim  lim x →3

42.

x 2  4 x 2   1. 2 2

lim tan x log ( sin x )  lim

x→ π/ 2

x→ π/ 2

x →0

log(sin x ) cot x

sin x  cos x x→ π / 4 x  ( π / 4)  2 1 / 2 sin x  1 / 2 cos x   lim  x→ π / 4 x  ( π / 4) 

)

{

(

2 sin x  sin 2 x dx , x3 2 sin x  sin 2 x ∴ f `( x )  x3 Using L-Hospital's Rule, we have lim f `( x )  1.

}

48. We have

 

h→ 0

{

h→ 0

625  h  25

 625  h  25    25  25  50, lim h  h→ 0  (625  h  25)    1  cos10 h 2 sin 2 5h  lim h→ 0 h→ 0 h2 h2 2

and f (0)  a

}

sinnce f (x ) is continuous at x  0 ∴ lim f (0  h)  liim f (0  h)  f (0) ⇒ a  50. h→ 0

}

h→ 0

 lim h cot ( π / 2) h h→ 0

h

 sin 5h   2 lim    25  50 h→ 0  5h 

 lim[1 (1 h)]tan ( π / 2)(1 h) , Putting x  1  h

h→ 0

h→ 0

44. lim(1 x ) tan( πx / 2)

{

lim f (0  h)  lim

lim f (0  h)  lim

 2 (1)  2

x→1

∫

x →0

) } 

 sin x  ( π / 4)   lim  2  x→ π / 4 x  ( π / 4)   sin y  2 lim y→0 y

2 x 0   0. sec 2 x 1

47. ∴ f ( x ) 

lim

{(

log sin x  ∞  From  1/ x  ∞

d log sin x  x   0  lim dx  lim    From  x →0 x → 0 tan x d 0   (1 / x ) dx  lim

 0 from  0  cot x lim  lim (cos x sin x )  0 x → π / 2 cos ec 2 x x→ π/ 2 43.

x log(sin x )  lim 46. lim x →0 x →0

 πh / 2 2  cos( πh / 2)   lim  h→ 0 sin( πh / 2) π  2 2  (1).(1).  π π 45. Using the expansions of sin x and cos x, then given limit  lim x→0

(1 a  b)  ( b / 6  a / 2) x 2  ( a / 24  b / 20) x 4  ... x2 For the given limit to be 1, a finite quantity, we must have 1 1 a 2 b 5 0 $ b/6 2 a/2 5 1, Solving these, we get

2 x 1

49. lim

(1 x ) 1

x →0

( 2 1){ x

 lim

} rationalizing the denominator.

(1 x ) 1

(1 x ) 1

x →0

x

 lim x →0

2 1 ,lim  (1 x ) 1  (log 2)2  x x →0 

 2 log 2. 50. We have, lim x →0

 sin 2 x  sin 2 x 5 lim  2  5 1 ⋅ 2 5 2. x → 0 x  2x 

Differential Calculus ▌ 3.155

DIFFErENTATION 1. If µ  sin −1 ( x  y ), x  3t , y  4t 3 then du / dt (b)

(a) 3 3

(c)

1 t 2

2. If x  cos1

1 (t 1) 2

f ( x) 

(d) None of these

1 (a) 8 1 (c) 64

y  sin1

t 1 t

2

cos3 t

π ,y , find dy /dx at t  3. If x = 6 cos 2t cos 2t (b) 

(a) 1 (c)

1 2

1 2

(d) None of these

4. If sin If sin y  x sin ( a  y ), then

dy  dx

(a) sin 2 ( a  y ) sin 2 ( a  y ) sin a sin 2 ( a  y ) (c) sin y (d) None of these (b)

(a) 0

1 , then x 3 y ( dy /dx ) t2 (b) 1

(c) 1

(d) None of these

5. If x 2  y 2  t 1 / t , x 4  y 4  t 2 

x

x

e e is not defined for x 50, the x value that should be assigned to f (0), so that f (x) is continuous at x 50, is (a) 2 (b) 1 (c) 0 (d) None of these

6. The function f ( x )i 

1

1 t 2

the dy / dx is independent of t when dy / dx is equal to 1 (a) 1 (b)  2 1 (c) (d) None of these 2 sin 3 t

8. The value of f (0) so that the function

3

 x 1, x  0  1 7. For the function f ( x )   , x  0. x  f ( x ) and lim f ( x ) are x →0 x →0 4  x 2 , x  0 (a) 0,1

(b) 0, 1

(c) 1,1

(d) None of these

( 256  8 x ) 4  4 16  4(64  3x )

1 3

( x ≠ 0) (b) 

1 8

(d)) None of these

9. Differentiate ( ax 1 b) m is

(

(a) am ax 1 b

)

m−1

(c) 1

(b) 0 (d) None of these

10. Derivative of cos21 (cot x ) is (a) cosec 2 x (c)

12 cot 2 x

cosec 2 x

(b)

12 cot 2 x

(d) None of these

21  12 cox  11. Differentiate tan   is  sin x  (a) 1 (b) 0

(c) 2

1 2

(d)

1 2

 1 cos x  12. Differentiate cos1   is   2   (a) 1 (b) 0 1 1 (c)  (d) 2 2  cos x  sin x  13. Differentiate tan1  is :  cos x  sin x  (a) 1 (c) 1

(b) 1 (d) None of these

dy , when x  a(t  sin t ) and y  a(1 cos t ). dx t t (b)  tan (a) tan 2 2 (d) None of these (c) tan t

14. Find

15. Differentiate e x with respect to x . (a) 2 x

(b) 2e x x

(c) e x

(d) e x x

3.156

▌ Mathematics

16. If y 

sin1 x 1 x

2

(

, then 1  x 2

 xy is equal to ) dy dx

(a) 1

(b) 1

(c) 0

(d) None of these

dy 17. If y  sin x  sin x  sin x  − − −to ∞, then is dx cos x (b) (a) cos x 2 y 1 (c) 2 y 1 (d) None of these 18. If y  cos 2 x 2 , find

dy is dx (b)  2 x sin( 2 x 2 )

(c) 0

(d) None of these

 1 cos x  is 19. Differentiate tan1   1 cos x 

(c)

(d) 

(c) x

2x 12 x

(b)

4

( 12 x ) 4

1 2

(c)

1 1 x

(d) None of these

1 (1 x )2

25. Differentiation of esin x is (a) esin x

(b) esin x cos x

(c) cos x

(d) ecos x sin x 1 a x 2

2

find

(a) x

dy is dx (b) x ( a 2  x 2 )3/ 2

(c) a  x 2

(c)

2

(d) None of these

(log10 )2 (log x )

(b)

2

(log10 x )2 (log10 )

(log x10 )2 (log10 k )2

(d) 2

2

28. If g is the inverse of f and f ' ( x ) 

x 12 x 4

(d) None of these

 2x  dy . , if y  sin1  21. Find 2  dx  1 x  2 (a) 1 x2 (c) 2

(b)

(a) 2

 1 x 2  1 x 2  dy 20. If y  tan1   , is  1 x 2  1 x 2  dx   (a)

1 (1 x )2

27. Differential coefficient of log10 x w.r.t log10 is

(b) 1

1 2

dy dx

(a)

26. If y 

(a)  2 x

(a) 1

24. If x 1 y  y 1 x  0, then

2 (b) 1 x2 (d) None of these

(log x )2 (log10 ) 2

1 , 1 x3

then g' (x ) is equal to 1 1  [ g ( x )]3

(a) 1  [ g ( x )3 ]

(b)

(c) [g ( x )]3

(d) none of these

29. Let y  x 3  8x  7 and x  f (t ). If x  3 at t  0, then

dy  2 and dt

dx at t  0 is given by dt

(a) 1  1 dy 22. If y  cot1   , then is dx  x 1 1 x 1 (c) x

(a)

(b)

(c)

1 1 x 2

(d) None of these

23. If y  cos−1 (4 x 3  3x ) (a)

19 2 2 (c) 19 (d) none of these (b)

3 1 x 3

2

1 x

2

30. Let f be a twice differentiable function such that f '' ( x )  f ( x ) and f ' ( x )  g ( x ). If

dy is dx

h(x ) [ f ( x )]2  [ g ( x )]2 and h (5)  11 then h(10)  (b)

3 1 x

(a) 11 2

(d) None of these

(b) 0 (cc) 1 (d) none of these

Differential Calculus ▌ 3.157

ANSwEr KEy 1. (b)

5. (b)

9. (a)

13. (b)

17. (b)

21. (b)

25. (b)

29. (c)

2. (a)

6. (a)

10. (b)

14. (a)

18. (b)

22. (b)

26. (b)

30. (a)

3. (d)

7. (b)

11. (d)

15. (b)

19. (d)

23. (c)

27. (d)

4. (b)

8. (a)

12. (d)

16. (b)

20. (a)

24. (a)

28. (a)

SOLUTIONS 1. u  sin1 (3t  4t 3 )  sin1 (3sin θ  4 sin 3 θ),

5. We have, t 2 1 / t 2  (t 1 / t )2  2

where t  sin θ  sin sin 3θ  3θ  3sin t . du  3 / (1 t 2 ). dt 1

∴ x 4  y 4  ( x 2  y 2 ) 2  2 or 2 x 2 y 2  2  0

1

∴ x 2 y 2 1, differentiating, 2 xy 2  x 2  2 y

2. Put t 5 tan θ, then x 5 θ 5 y dy ⇒ 5 5 1. dx 3.

dy dy xy 2 or x 3 y x 2 y 2 1 dx dx [ x 2 y 2  1].

∴ x2 y

dx . dt 3sin 2 t cos t cos 2t   

1

2 cos 2t cos 2t

6. (2 sin 2t )sin 3 t

3sin 2 t cos t cos 2t  sin 2t sin 3 t

7.

3

(cos 2t ) 2

3

dy 3cos 2 t sin t cos 2t  sin 2t cos3 t ∴  dt 3sin 2 t cos t cos 2t  sin 2t sin 3 t  dy   dx  tπ / 6 3 1 1 3 3 3 9 9 3. . .  .   4 2 2 2 8 16 16 = 0 3 1 3 1 3 3 1 3 3  3. . .   .  16 4 2 2 2 2 8 16

1

∴

sin y diff. both sides w.r.t. x sin( a  y )

sin( a  y )cos y  sin y cos( a  y ) dy . dx sin 2 ( a  y )

Or, 1 

lim f ( x )  lim f (0  h)  lim h2  0. h→ 0

x → 0+

sin( a  y  y ) dy . sin 2 ( a  y ) dx

dy sin 2 ( a  y )  . sin a dx

h→ 0

h→ 0

x →0

dy 3cos t sin t cos 2t  sin 2t cos t  3 dt (ccos 2t ) 2

4. Here, x 

e x  ex x→ o x e x  e1  lim  2. x→0 1 f (0)  lim

lim− f ( x )  lim f (0  h)  lim( h 1) 1.

Siimilarly, 2

dy  0. dx

h→ 0

8. For f ( x ) to be continuous everywhere we must have f (00)  lim f ( x ) x →0

1

 ( 256  8 x ) 4  4  0  lim form x → 0 16  4 ( 64  3x )1/ 3  0  1 ( 256  8 x )3/ 4 .( 8) 4  lim [using L ′ Hospital′s rule] x →0 4  (64  3x )2/ 3 .(3) 3 1 −3 4 (8) 8 1 4   . 4 −2 64 8  4 .(3) 3 m 9. Let y  ( ax  b)

Put ( ax + b)  t , so that y  t m and t  ( ax  b) dy dt dy  mt m1 and  a so, dt dx dx  dy dt       amt m1  dt dx 

∴

 am( ax  b) m1 .

▌ Mathematics

3.158

10. Let y  cos1 (cot x ). and t  cot x dy dt 1 cos ec 2 x and  2 d x dx 1 t

∴

cos ec 2 x 1 − cot 2 x

[∴ t  cot x ]

16.

d cos ec 2 x Hence, [cos −1 (cot x )]  dx 1 cot 2 x . 11.

d  1  1 cos x   tan   dx   sin x 

17.

d  1  2 sin ( x / 2)   tan   o s( x / 2 ) dx  2 sin( x / 2 )c  

1

(1 x 2 )

1 x 2

y  sin x  sin x  sin x  ...to ∞

dy dy dy cos x  or ( 2 y 1)  cos x dx dx  dx dy cos x ∴  . dx ( 2 y 1)

18.

y  (cos x 2 )2 . Put x 2  t And,

cos x 2  cos t  u,

So that y  u 2 , u  cos t and t  x 2 dy du dt  2u, sin t and  2x du dt dx dy  dy du dt  So,      4ux dx  du dt dx 

∴

sin t 4 x sin t cos t  4 x sin x cos x 2

d  1  1 tan x   tan   dx   1 tan x 

2 x sin( 2 x 2 )

[on diviiding numerator and denominator by cos x ]

14. x  a(t  sin t ) and y  a(1  cos t ) dx dy ∴  a(1  cos t ) and  a sin t dt dt dy  dy dt  ∴   dx  dt dx  a sin t  a(1 cos t ) 2 a sin(t / 2)cos(t / 2   tan t / 2. 2 a cos 2 (t / 2)

y

2 y.

d  1  x   d  x  1 cos  cos       . 2   dx  2  2 dx  

π  d  π  d    tan1 tan   x     x  1. dx  4  dx  4 

2

y 2  sin x  y

d  1  cos x  sin x   tan   dx   cos x  sin x  

x

y  sin x  y Now, squaring both sides

d  1  x  d  x  1  tan  tan      . dx  2  dx  2  2 

d  2 cos 2 ( x / 2)   cos1  dx  2  

13.

1

2

2

 1 cos x  1 12. cos   2  



( 1 x ) y  sin

( 1 x ) dydx  2 12x x 2

⇒

1 x 2

On differentiatin ng both sides of (1) w.r.t. x , we get,

 1 cos x  tan    sin x 



sin1 x

y ⇒

1

⇒

du  du / dx  ex   2e x x   dv  dv / dx   1 x  2 

∴

dy  dy dt  cos ec 2 x    dx  dt dx  1 t 2

So, 

15. Let u  e x and v  x du dv 1 1/ 2 1 Then,  e x and  x  dx dx 2 2 x

19.

∴ [u  cos t ]

2

[∴ t  x 2

1/ 2 d  1  1  cos x    tan    dx   1  cos x    1/ 2 2 cos 2 ( x / 2)  d    tan1 dx  2 sin 2 ( x / 2)   

d  1 tan ( cot x / 2 )  dx    π x  d    tan1  tan    dx    2 2  d  π x  1     . dx  2 2  2 

]

dy  xy  1. dx

Differential Calculus ▌ 3.159 20. Putting x 2  cos 2θ, we get

dy 3  dx 1 x 2 d 3 Hence, cos1 ( 4 x 3  3x )  . dx 1 x 2

∴

 1 cos 2θ  1 cos 2θ  y  tan1    1 cos 2θ  1 cos 2θ   2 cos 2 θ  2 sin 2 θ   tan1   2 2  2 cos θ  2 sin θ   cos θ  sin θ   tan   cos θ  sin θ  1

 1 tan θ   tan1   1 tan θ 

24. x 1 y  y 1 x squaring both sides ⇒ ( x  y )( x  y  xy )  0 ⇒ x  y  xy  0 ⇒ x  y  xy  0 {x  y does not satisfy the giveen equation]

 π   tan1  tan   θ    4 π π 1  θ ⇒  cos −1 x 2 4 4 2 1 ∴ [cos 2θ  x 2  θ  cos −1 x 2 ] 2 dy d  π 1 1 2    cos x  ∴ dx dx  4 2 

∴

25.

21.

x

 2 tan θ  1 y  sin1    sin (sin 2θ) 2  1 θ tan   ⇒ 2θ  2 tan1 x. dy 2  ∴ dx 1 x 2 d  −1  2 x  2  sin  2  dx  x2  1 x 1   

22.

 et cos x  esin x cos x

1 x 4

 2x  y  sin1  2   1 x  Putting x  tan θ , we get

1 y  cot1   Putting x  tan θ, x  1  1 We get , y  cot1    cot ( cot θ)  θ t an θ    tan1 x dy 1 ∴  dx 1 x 2

23. Let y  cos1 ( 4 x 3  3x ).

y  esin x Put sin x  t , so that y  er and t  sin x dy dt ∴  et and  cos x dx dx dy  dy dt  Hence,     dx  dt dx 

1 1  . 2 1 x 4  2x 

x 1 x  (1 x ).1 x ⋅ 1  1 dy .    2 dx x )2 ( 1  ( 1  x )  

⇒ y

26.

y

1

[∴ t  sin x ].

Put ( a 2  x 2 )  t , so that

a x 1 y  t 1/ 2 and t  ( a 2  x 2 ) t 1 dy dt ∴  t 3/ 2 & 2 x , 2 dt dx dy  dy dt     So, dx  dt dx  2

2

 1     t 3/ 2  (2 x )  xt 3/ 2  2   x (aa 2  x 2 )3/ 2 . 27. Let y  log10 x and z  log x10 

1 log10 x

1 dy 1 1  2  2 ∴ y ⇒ z dz z z 1 (log x )2   2 (log10 )2 . ( log x10

)

28. We have, g  inverse of f  f -1

Putting x  θ, we get:

⇒ g ( x )  f −1 ( x ) ⇒ f [ g ( x )]  x.

y  cos1 ( 4 cos3 θ  3cos θ)

Differentiating w.r.t.x , we get

 cos1 (cos 3θ)  3θ  3cos1 x

f 1[ g ( x )] ⋅ g ′ ( x ) 1

3.160

▌ Mathematics

∴ g ′ ( x) 

1  1  [ g ( x )]3 f ′[ g ( x )]

  1 1 , ∴ f '[ g ( x )]   f '( x )  3 3 1  [ g ( x )] 1  x   dy 29. We have, y  x  8 x  7 ⇒  3x 2  8 dx It is given that when t  0, x  3. dy ∴ When t  0   3.32  8  19. dx dy dy / dt Also,  ....(1) dx dx / dt dy dy Since, when t  0,  19 and  2, dx dx dy 2 2 ∴ from (1), 19  ⇒  . dx /dt dt 19 3

2

30. We have, h( x )   f ( x )    g ( x ) 

2

Differentiating both sides w.r.t. x , we get h '( x )  2 f ( x ). f '( x )  2 g ( x ).g '( x )  2 f ( x ) g ( x )  2 g ( x ) g '( x )  f '( x )  g ( x )   2 g ( x )  f ( x )  g '( x ) 

...(1)

But g ( x )  f '( x ) ⇒ g '( x )  f '' ( x )  f ( x ) [∴ f '' ( x )  f ( x )] ∴ From (1), h '( x )  2 g ( x )  f ( x )  f ( x )   0 ⇒ h( x )  constant for all x. Given h(5)  11 , ∴ h(10)  11.

AppLICATIONS OF DErIVATIVES 1. If the normal to the curve y 5 f ( x ) at the point(3,4) makes 3π with the positive x-axis, then f '(3) 5 an angle 4 3 (a ) 21 ( b) 2 4 4 (c ) (d ) 1 3 2. If f (x ) and g (x ) are differntiable functions for 0  x  1 such

6. If f (x ) satisfies the conditions for Rolle’s theorem in [3, 5] then ∫ 5 f(x) dx equals to 3

(a) 2 4 3 (c) 2 1

(b) 2 (d) 0

7. Given that f ( x )  x1/ x , x  0 has maximum value at x  e, then

that f (0)  2, g (0)  0, f (1)  6, g (1)  2 then in

(a) e∂  ∂ e

(b) e∂  ∂ e

the interval (0,1),

(c) e∂  ∂ e

(d) e∂  ∂ e

(a) ( f '( x )  0 for all x (b) f ( x )  2 g '( x ) for at least one x (c) f ( x )  2 g '( x ) for att most one x (d) None of these

3. The equation x log x  3 x has, in the interval (1,3), (a) exactly one root (b) at most one root (c) at least one root (d) no root. 4. The real number x when added to its inverse gives the minimum value of the sum at x equal to (a) 2 (b) 1 (c) 2 1 (d) 2 2

(

)

5. If a  0, the function e ax  eax is a monotonic decreasing

8. If a differentiable function f (x ) has a relative minimum at x  0, then the function y  f y  f (x )  ax  b has a relative minimum at x  0 for (a) all a and all b (b) all b if a  0 (c) all b  0

(d) all a  0

9. The function f ( x )  2 x 3  3x 2 12 x  4 has (a) two maxima (b) two minima (c) one maxima and one minima (d) no maxima and minima 10. If the graphs of a differentiable function y  f (x ) meets the lines y  1 and y  1 then the graph (a) meets the line y = 0 at least one

function for all values of x,where

(b) meets the line y = 0 at least twice

(a) x  0

(b) x  0

(c) x  1

(d) x  1

(c) meets the line y = 0 at least thrice (d) does not meet the line y  0

Differential Calculus ▌ 3.161

11. The angle between the tangents to the curve at the points ( a, 0) and (0, b) is

x2 y2  1 a 2 b2

49 543 1 (c) 52

(a)

π π (b) 4 2 π (c) (d) none of these 3 12. The curves y2 5 2x and 2xy 5 k cut at right angles if (a) k2 5 8 (b) k2 5 4 2 (c) k 5 2 (d) none of these (a)

15. If the curve y 5 x2 1 bx 1 c touches the line y 5 x at the point (1, 1), then the values of x for which the curve has a negative gradient are

(c) x 

(b) x  1 2

1 2

(d) x 

8 89

(b)

(d) none of these 4

14. If the line y 5 2x touches the curve y 5 ax2 1 bx 1 c at the point where x 5 1 and the curve passes through the point (2 1, 0), then the values of a, b and c are 1 1 1 1 (a) a 5 , b 5 1, c 5 (b) a 5 1, b 5 , c 5 2 2 2 2 1 1 (d) none of these (c) a 5 , c 5 , b 5 1 2 2

1 2

n2 , n ∈ N, is n3  200

17. The minimum value of is e( x x x ) (a) e (b) e2 (c) 1 (d) none of these

13. If the line ax 1 by 1 c 5 0 is a normal to the curve xy 5 4, then (a) a > 0, b > 0 (b) a > 0, b < 0 (c) a < 0, b > 0 (d) a < 0, b < 0

(a) x 

16. The largest term in the sequence x n 

1 2

3

2

18. The minimum value of the function f (x) 5 2 | x 2 2 | 1 5 | x 2 3 |, ∀ x ∈ R is (a) 3 (b) 2 (c) 5 (d) 7 19. If f (x) 5 | x | 1 | x 2 1 | 1 | x 2 2 |, then (a) f (x) has minima at x 5 1 (b) f (x) has maxima at x 5 0 (c) f (x) has neither maxima nor minima at x 5 0 (d) f (x) has neither maxima nor minima at x 5 2. 2 20. For all real x, the minimum value of 1 x  x is 1 x  x 2 (a) 0 (b) 1 3 (c) 1 (d) 3

ANSwEr KEy 1. (d)

4. (a)

7. (a)

10. (a)

13. (b, c)

16. (a)

19. (a, c, d)

2. (b)

5. (a)

8. (b)

11. (b)

14. (a)

17. (c)

20. (b)

3. (c)

6. (a)

9. (c)

12. (a)

15. (a)

18. (b)

SOLUTIONS 1. Slope of normal to y  f ( x ) at (3, 4) is

1 . f '(3)

 3π  π π 1 π  tan    tan    cot  1 Thus, f '(3) 4 4 4 2     ⇒ f '(3)  1. 2. Let h( x )  f ( x )  2 g ( x ), x ∈ 0,1 . ⇒ h '( x )  f '( x )  2 g '( x ) Also,

h(0)  f (0)  2 g (0)  2  2.0  2 h(1)  f (1)  2 g (1)  6  2.2  2 ∴

h(0)  h(1).

Since f ( x ) and g ( x ) are differentiate in [0,1], h(x ) is also differentiable in [0,1]. Hence, h(x ) is also continuous in [0, 1]. So, all the conditions of Rolle’s theorem are satisfied Hence, there exists a point c,0  c  1 for which h′( c ) 5 0. ∴ f '( c )  2 g '( c )  0.i.e., f '( c )  2 g '( c ).

3.162

▌ Mathematics

3. Let f ( x )  ( x  3) log x Then, f (1) 2 log 1  0 and f (3)  (3 .3) log 3  0.

10. Since the function y 5 f ( x ) is differentiable and meets the line y 1 and y  1

As, ( x  3) log x  f ( x ) is also continuous and differentiable

∴the function is continuous and hence its graph meets the line y 5 0 at least once.

in [1, 3] Hence by Roll's theorem , there exists a value of x in (1, 3) such that f '( x )  0 ⇒ log x  ( x  3)  0 ⇒ x log x  3  x. 4. We have, f ( x )  x  x −1 ∴ f '( x )  1 0 ⇒ x 1 Now , f '( x )  0 Therefore, f ( x ) will be minimum at x  1 and the minimum value is f (1)  1  11  2. 5. Let f ( x )  e ax  eax f ( x ) is a decreasing function for f '( x )  0 Then, f ( x ) is a decreasing function for x  0. 6. Let f ( x )  ( x  3) ( x  5)  x  8 x 15 2

5

∴

∫ f ( x )dx  ( x

2

 8 x 15)dx

11.

x2 y2  1 a 2 b2 2 x 2 y dy dy b2 x ⇒ 2 2 0 ⇒  2 dx a b dx a y dy  π ∴ tan θ1    ∞ ⇒ θ1  dx ( a ,0 ) 2 We have

And

tan θ2 

Hence, the angle between the twoo tangents is π π θ  θ1 θ2   0  . 2 2 12. Let P(x1,y1)be the point of intersection of the two curves. We have, y2  2x ⇒ 2 y

3

3

 x 3 8x 2    15x  . 3 2  5 125   100  75 [9  36  45]  3 

7.

dy   0 ⇒ θ2  0 dx ( 0 ,b )

 dy  dy 1  2 ⇒ m1     . dx dx y  ( x , y ) 1 1

1

dy and 2 xy  k ⇒ x  y  0 dx y  dy  ⇒ m2     1 dx x  ( x1 , y1 ) 1

 50  50  54 4   18   3 3 3  

 1  y  ∴ m1 m2 1 ⇒      1 ⇒ x1  1  y1   x1 

f ( x )  x1/ x , x  0.since x  e is a point of maxima. ∴ f ( e)  f ( x ) for all x  0 ⇒ f ( e)  f (∂ ) in particular ∴ y has a relative minimum at x  0 If a  0 and for all b.

and hence from y  2 x1, we get y  2

8. Since f ( x ) has a relative minimum at x  0 ∴ f '(0)  0 and f ''(0)  0 Now y  f ( x )  ax  b At ∴ y has a relative minimum at x  0 If a  0 and for all b. 9. Now Since f '' (1) 18  0 ∴ f has maximum value at x 1 and f ''( 2)  18  0 ∴ f has minimum value at x  2 ∴ f has one maximum and one minimum

2

2

1

1

Since ( x1 , y1 ,) also lies on 2 xy  k ∴ k 2  4 x12 y12  4 1 2  8. 13. We have xy  1 ⇒ y  ∴

1 dy  2 . dx x

1 x

∴ The slope of the normal  x 2 . If ax + by + c = 0 is normal to the curve xy = 1 a a then x 2  ∴  0 b b ⇒ a  0, b  0 or a  0, b  0. 14. The given curve is y 5 ax2 1 bx 1 c Since the point (2 1, 0) lie on it ∴a  b  c  0

…(1)

...( 2)

Also, y 5 2x is a tangent to (1) at x 5 1, so that y 5 2. Since the point (1, 2) lies on (1),

Differential Calculus ▌ 3.163

Also

18. We have,

dy    ( 2 ax  b)](1,2 )  2, dx (1,2 )

f ( x)  2 x  2 5 x  3  2(2  x )  5(3 x )  19  7 x , if x  2

∴ 2a  b  2

 5, if x  2  2( x  2)  5(3 x )  11 3x , if 2  x  3

1 1 Solving ( 2),(3) and ( 4) : a  , b  1, c  2 2 15. We have, y 5 x2 1 bx 1 c dy ⇒  2 x  b. dx Since the curve touches the line y 5 x at the point (1,1) ∴ (2x 1 b)(1,1) 5 1 i.e., 2 1 b 5 1 ⇒ b 5 2 1 Also, the curve passes through the point (1, 1) ∴ 1 5 1 1 b 1 c i.e., c 5 2 b 5 1. dy ∴ y  x 2  x 1 ⇒  2x 1 dx dy 1 Now,  0 ⇒ 2 x 1  0 ⇒ x  dx 2. 16. Let f ( n)  xn  ⇒ f '( n)  

n2 ,n∈ N n  200 3

( n2  200) ⋅ 2 n  n2 ⋅ 3n2 ( n3  200)2

n( 400  n3 ) . ( n3  200)2

But f ’(n)  0 for any. n ∈ N . Hence f (n) has no critical point. But the function f (n) is increasing for n < 8 and it starts decreasing for n  8. 49 8 and f (8)  543 89 Clearly, f (7 )  f (8) Here, f (7 ) 

2

19. We have f ( x ) | x |  | x 1 |  | x  2 | 3x  3, x  0  x  3, 0  x  1   x 1,, 1  x  2 3x  3, x  2 3 x0  does not exist x  0 1 0  x 1  ⇒ f '( x )  does not exiist x  1 1 1 x  2  x2 does not exist  3 x2  Clearly, f (x) has minima at x 5 1 and neither maxima nor minima at x 5 0 and x 5 2. 1 x  x 2 1 x  x 2 ⇒ Z  Zx  Zx 2  1 x  x 2

20. Let Z 

⇒ x 2 ( Z 1)  x ( Z 1)  ( Z 1)  0 For x to the reaal, B2  4AC  0

17. e( x  x  x ) is minimum when ( x  x  x ) is minimum Since; x 4  x 3  x 2  x 2 ( x 2  x 1) 2  1  3  x 2  x      0, ∀x ∈ R. 2  4    ∴ The minimum value of x 4  x 3  x 2 is 0 for x  0. 3

Thus, we find that f '(x) has a minimum value 2 at x  3.

⇒ Zx 2  x 2  Zx  x  Z 1  0

49 Hence the largest value is 543 4

 2(3 2)  2, if x  3  2( x  2)  5( x  3)  7 x 19, if x  3

4

(

3

2

)

Hence, the minimum value of e(xx 4  x 3  x 2 ) is e 0  1.

⇒ ( Z 1)2  4( Z 1)( Z 1)  0 ⇒ Z 2  2 Z 1 4 Z 2  8Z  4  0 ⇒  3Z 2 10 Z  3  0 i.e.3Z 2 10 Z  3  0 1 ⇒ ( Z  3)(3Z 1)  0 ⇒  Z  3. 3 1 Therefore, minimum value of Z  3

NDA bOX x 14

 x 16  1. What is the value of lim   ? x →∞  x 11  (a) e (b) e2 (c) e4 (d) e5

2. What is the value of lim x →1

(a) 0 (c) 2 1

( x 2 1)2 ? x 21

(b) 1 (d) The limit does not exist

3.164

▌ Mathematics 3x

 x  3. What is the  lim   equal to? x →∞ 3  x   (a) e (c) e29

2 2 12. If 1  x  1  y  a, then what is

(b) e3 (d) e9

4. Consider the following function f : R → R such that f(x) 5 x if x  0 and f(x) 5 2 x2 if x < 0. Then, which one of the following is correct? (a) f(x) is continuous at every x ∈ R (b) f(x) is continuous at x 5 0 only (c) f(x) is discontinuous at x 5 0 only (d) f(x) is discontinuous at every x ∈ R 5. What is the value of lim x α

(a)

2

(d)

3α x  2 x

?

(3 3) 2

(1 x )(1 y )

(b)

(c)

1 x2 1 y 2

(d) None off these

2

(

2

)(

)(

)

13. If y  1  x1/ 4 1  x1/ 2 1  x1/ 4 , then what is

dy equal to? dx

(a) 0 (c) 0

(c)

(3 3) 1

(b) 2 1 (d) 2 2x

( x  y) (1  log x )

(d)

(log x) (1  log x )2

15. What is the maximum point on the curve x 5 exy? (a) (1, e) (b) (1, e21) (c) (e, 1) (d) (e21, 1)

3 6. What is the least value of f ( x )

x 16. What is the derivative of x a 2  x 2  a 2 sin1   ? a

 2 x 3  3x 2  12x  1 on [  2, 2.5]? (a) 23 (c) 219

1 y 2 1 x2

(a)

14. If x y  e x  y , then dy/dx is equal to which one of the following? y ( x  y) (a) (b) (1  log x ) (1  log x )2

3 1

(b) (c)

α 2x  3x

dy equal to ? dx

(b) 8 (d) 216.5

7. Let f : R → R be defined by f(x) 5 |x| / x, x ≠ 0, f(0) 5 2. What is range of f ? (a) {1, 2}

(b) {1, 21}

(c) {21, 1, 2}

(d) {1}

a2  x2

(b) 2 a 2  x 2

(c)

x 2  a2

(d) 2 x 2  a 2

17. If x 5 t 2 , y 5 t 3 , then what is (a) 1

8. What is the differentiation of logx x with respect to In x? (a) 0 (b) 1 (c) 1/x (d) x 9. The function y 5 tan-1 x - x (a) is always decreasing (b) is always increasing (c) first increases and then decreases (d) first increases and then increases 10. What does the equation x dy 5 y dx represent? (a) A family of circles (b) A family of parabolas (c) A family of hyperbolas (d) A family of straight lines 11. If e y + xy  e, then what is the value of

(a)

d y at x  0 ? dx 2 2

(c)

3 4t

d2 y equal to? dx 2 3 (b) 2t 3 (d) 2

d2y 18. If x 5 log t and yi 5 t2 2 1, then what is at t 5 1 dx 2 equal to? (a) 2 (b) 3 (c) –4 (d) 4 19. What is the derivative of logx 5with respect to log5x? (a) 2 (log5 x )22 (b) (log x )22

(a) e1

(b) e2

(c) 2 (log x 5)22

(c) e

(d) 1

(d) (log x 5)22

Differential Calculus ▌ 3.165

ANSwEr KEy 1. (d) 2. (a) 3. (c)

4. (a) 5. (c) 6. (c)

7. (c) 8. (a)

10. (d)

12. (d)

14. (d)

16. (a)

18. (d)

11. (c)

13. (b)

15. (b)

17. (c)

19. (a)

9. (a)

SOLUTIONS  x 6  1. lim   x →∞  x 1 

x 4

 5   lim  1   x →∞ x  1 

x 1    5  5   lim  1   x →∞ x 1    

5⋅

LHL = RHL = f(x) It is continuous at x = 0. Also, f(x) is continuous in every x R. Hence, f(x) is continuous in every x R

x 4 5 ⋅ ( x 1) 5 x 1

x 4 x 1

5. 6.  f (x )  2x 3  3x 2  12x  1

4 1 x 5 lim x →∞ 1 1 x

⇒ f `( x )  6 x 2  6x  12

e

Put f `(x )  0 for maxima or minima

e

6x 2  6x  12  0

5

2. Let f (x )  ( x  1) x 1

⇒ x2  x  2  0 2

⇒ ( x  2) ( x  1)  0 ⇒ x  2, x 1

  ( x  1) x  1  ( x  1) x  1

Now, f ``(x )  12x  6 ⇒ f `` ( 2)  24  6  18  0

Now, LHL  lim f (1  h)

∴ f ( x ) is minimum at x  2.

 lim[(1  h  1)]

Hence, f (2)  2(2)3  3( 2)2  12  2  1

h→ 0

h→ 0

 lim h  0 and RHL  lim f (1  h) h→ 0

h→ 0

 lim (1  h  1)  lim h  0 h→ 0

h→ 0

 LHL  RHL ( x  1)2 ⇒ lim 0 x →0 x 1 3x

3x

 x   3 3. lim    lim 1  x →∞ 3  x x →∞ x     3  lim  1   x→ x x 

x (9) 3

 16  12  24  1 19 | x |  7. Given, f (x )   x  2,   1, x  0    2, x  0 1, x  0 

∴Range of f ( x ) is { 1, 1, 2}.

⇒

 x, 4. Given, f (x )   2 x

x  0. x0

LHL  lim f ( x )  lim x 2  0 x →0

RHL  lim f ( x )  lim x  0 x →0

and f (0)  0

x 0

8. Let u 5 log x x 5 1 ( log a a 5 1)

 e9

x →0

x≠0

x →0

du 50 dx

and v 5 In x ⇒ ∴

dv 1 5 dx x

du du / dx 5 50 dv dv / dx

9. Given, y = tan−1 x – x On differentiating w.r.t. x, we get

▌ Mathematics

3.166

15. The given equation of curve can be rewritten as y = xe−x On differentiating w.r.t. x, we get dy  xex  ex dx dy On Putting  0 for maxima or minima dx xex  ex  0

1 dy x 2  1 2 dx 1  x 1 x2 dy  0, ∀x ∈ R dx Hence, function is always decreasing. ⇒

10. Given, x d  y dx ⇒

dy dx  y x

⇒ x 1 d2y ex  xex  ex dx 2  d2y  ⇒  2   0  dx  x 1

On integrating both sides, we get

Now,

log y  log x  log c ⇒ y  cx It represents a family of staright lines.

∴ y is maximum at x  1. 12.

1 x 2  1 y 2  a

Hence, maximum point on the curve

(

On differentiating w.r.t. x, we get

x  e x y is 1, e1

1(2 y ) dy 0 (  2x )  2 2 1 x 2 1  y 2 dx 1

⇒

(

(  (1  x

 1  x1/2

)( ) 1 ( x )  )(1 x )

)(

1/2

1/ 4



1/ 2

14. Given Taking log of both sides y log x  x  y ⇒ y (log x  1)  x x ⇒ y 1  log x Differentiating both sides we get,  dy d  x    dx dx  1  log x  (1  log x )

)

dy 1  x. (2 x )  dx 2 a2  x2

2

⇒ y 1 x On differentiating w.r.t x, we get dy 1 dx



x 16. Let y  x a 2  x 2  a 2 sin1   a On differentiating w.r.t. x , wet get

dy x 1 y 2  dx y 1 x 2

13. Given, y  1  x1/ 4 1  x1/ 2 1  x1/ 4

)

 d  dx  x  (1  log x )  dx  dx 

(1  log x))2 1 (1  log x )  1  x. x  (1  log x )2 (1  log x)  1 log x   2 (1  log x ) (1  log x)2

x a x 2

a x 2



2

2

a2  x 2



a2 ⋅ a a x 2

2

⋅

a2

1 ⋅ x2 a 1 2 a

1 a

 a2  x 2

17.  x 5 t 2 , y 5 t 3 dx dy 5 2t , 5 3t 2 dt dt dy dy / dt 3t 2 3 ∴ 5 5 5 t dx dx / dt 2t 2 ⇒

d 2 y 3 dt 3 1 5 ⋅ 5 ⋅ dx 2 2 dx 2 2t 3 5 4t ⇒

18. Given, x  log t and y  t 2 1 ⇒ 2 x  log t 2 ⇒ 2 x  log( y  1) ⇒ e2 x  y  1 On differentiating w.r.t. x , twiccely, we get dy e2x  dx d2y 2x ⇒ 4e  2 dx

Differential Calculus ▌ 3.167

At t 5 1, x 5 0 d2y ∴ 2 5 4e2 ( 0 ) 5 4 dx 19. Let u1  log x and u2  log5 x log e 5 log e x and u2  ⇒ u1  log e x log e 5 On differentiating w.r.t. we get du1 log e 5  dx x (log e x )2

and

du2 1  dx x log e 5

∴

du1 du1 / dx  du2 du1 / dx



log e 5

x (log e x )2

 log e 5     log e x 

2

( log x 5)2 ( log5 x )2

 x log e 5

This page is intentionally left blank.

C

H

6

A

P

T

E

R

Integral Calculus and Differential Equation INTEGRATION A function g(x) is called a primitive or anti-derivative of the function f(x) if f ( x ) if g `( x ) 5 f ( x ). An indefinite integral of f(x) is a collection of all its primitives, i.e.,

c is any arbitrary constant. The process of finding an indefinite integral is called the integration of a function.

∫ f ( x )dx  g ( x )  x where

SOME BASIC INTEGRATION RESULTS 1.

∫ f ( x ) ⋅ f ( x )dx  ∫ f ( x ). f ( x ) dx = f ( x ). ∫ f ( x ) dx − ∫  f ( x ) dx  . f '( x ) dx 1

2

1

2

2

2.

∫

1

2

[ f ( x ) n1 ]  c( n ≠ 1) n +1 1 1 1 dx  arc tan x  c. or  are cot x  c 4. ∫ 2 a a x  a2

3.

5.

∫ [ f `( x )]

n

∫x 

2

f `( x ) dx 

1 1 xa 1  c and ∫ 2 dx  In dx 2a x  a  a2 a  x2

1 ax c In 2a a  x

1 dx dx 6. ∫ 2  +k 2 ax  bx  c a ∫  b b2  4 ac  x  2 a   4 a 2 Now, use the formulas ( 4) or (5).

x ∫ a dx 

8.

∫

9.

∫

10.

∫

11.

∫

a 2  x 2 dx =

12.

∫

a2 1 a 2 + x 2 dx  x a 2  x 2  In x  a 2  x 2  c 2 2

13.

∫

x 2  a 2 dx =

14.

∫ e [ f ( x )  f `( x )]dx  e

1

f `( x ) dx  In f ( x )  c. f ( x)

ax  c; in particular, ∫ e x dx  e x  c. Ina

7.

x x dx  arc sin + c or,  arc cos  c. a a a x 1

2

2

1 a  x2 2

1 x  a2 2

x

(

)

(

)

dx = In x  a 2  x 2  c dx = In x  x 2  a 2  c 1 a2 x are sin  c. x a2  x2 + 2 2 a

1 x 2

( a x  a  In ( x  2 2

2

2

x

f ( x)  c

) x − a ) c 2

2

3.170

▌ Mathematics

INTEGRATION OF TRIGONOMETRIC FUNCTIONS I.

(c) If m 1 n is a negative even integer, then substitute tan x 5 t.

∫ φ(sin x,cos x )dx, where φ is a rational function of sin

x 5 t. 2 However, in some special cases integral can be simplix and cos x can be integrated by substituting, tan

III.

fied as below: (a) φ(sin x ,cos x ) is an odd function w.r.t. sin x, i.e., φ(sin x ,cos x ) φ(sin x ,cos x ) then substitute, cos x 5 t.

Determine the constants A, B, C from the identity a sin x 1 b cos x 1 xc ≡ A ( λ sin x  µ cos x  v )  B( λ cos x  µ sin x )  C Write the given integral into three integrals last of which is to be integrated by the method in I, above.

(b) Similarly, if φ(sin x ,cos x ) φ(sin x cos x ),

IV. tan m x , cos m x , sec m x and cosec m x where m is some positive integer, can be integrated with the help of formulas.

then subsitute, sin x  t . (c) If φ(sin x ,cos x )  φ(sin x ,cos x ) then substitute, tan x 5 t II.

∫ sin

m

a sin x 1 b cos x 1 c

∫ λ sin x 1µ cos x 1 v dx

tan 2 x  sec 2 x 1 and cot 2 x  cosec 2 x  1 or splitting one sec2 x or coses2 x and applying integration by parts.

x ⋅ 2 cos x dx ( m, n ∈ I )

V. If m is any positive integer >1,

(a) If m is an odd integer, substitute, cos x 5 t and if n is odd; substitute, sin x 5 t. (b) If both m and n are positive even integers, use sin 2x 5 2 sin x cos x and cos 2x 5 2 cos2 x 2 1 5 1 2 2 sin2 x and simplify.

π/ 2

then

∫ sin

π/ 2

x dx  ∫ cos m x dx 

m

0

0

( m  1)( m  3)( m  5)... , m( m  2)( m  4)...

π multiply it by   if m is even. 2

DEFINITE INTEGRAL y

b

Let f (x )  0 on [ a, b], then ∫ f ( x ) dx is equal to the area bounded

y  f(x)

a

by the curve, y 5 f(x), then axis of x and the ordinates, x 5 a and x 5 b. Also, 0

b

∫ f ( x )dx   g ( x )  a

b a

 g ( b)  g ( a )

a b Fig. 6.1

x

Where g(x) is an anti-derivative of f(x).

SOME PROPERTIES OF DEFINITE INTEGRALS b

1.

a

2.

3.

b

a

b

∫ f ( x )dx ∫ f ( x )dx b

c

a

a

2a

∫

b

0

∫ f ( x )dx  ∫ f ( x )dx  ∫ f ( x )dx( a  x  b) ∫ f ( x )dx  ∫

f ( x ) dx

0

a

0

0

a

a

0

0

f ( x ) dx  ∫ f ( x ) dx  ∫ f ( 2 a  x ) dx which impllies, 0 if f ( 2 a  x )  f ( x )  f ( x ) dx   a 2 ∫ f ( x ) dx if 2( 2 a  x )  f ( x )  0

c

a

a

∫ 0

a

a

5.

6.

a

b

b

4.

2a

∫ f ( x )dx  ∫ f (t )dt

∫ f ( x )dx  ∫ f ( a − x )dx

7.

t

∞

∫ f ( x )dx 5 lim ∫ f ( x ) dx. a

t →∞

a

Similarly, if lower or both upper and lower limits are arbitrarily large or small.

Integral Calculus and Differential Equation ▌ 3.171

AREAS UNDER CURVES 1. Find the area under the curve f (x)  ( x 2  2)2  2 x between the ordinates x 5 0 to x 5 2. 2. Sketch the region bounded by the curves,

5. The area bounded by the curve, y = x4 – 2x3+x3 + 3. the axis of x and two ordinates corresponding to the points of minimum of this function is 91 30 91 (c) 3

30 91 3 (d) 91

(a)

2

y  5  x and y  x 1 find its area. 3. Find the area bounded by the curve x 2  y2  25, 4 y  4  x 2 , x  0 and above the x-axis. 4. Find the ratio of the areas into which the circle, 2 x 2  y2  64 a 2 , is divided by the curve y y 5 12ax

(b)

6. The area bounded by the curve. y 5 f(x), the x-axis and the ordinates x 5 1 and x 5 b is (b 2 1) cos (3b 1 4). Then, f(x) is n ∈N.

SOLUTIONS b

1. Solution : A  ∫ f (x) dx , here, x ( x 2  2)2  2 x , a

a  0, b  2 b

∫ (x

)

 4 x 2  4  2 x dx

a

2

 x5 4x 3 2x 2     4x   3 2 0 5   25 4  2 3 2 × 22   0 0 0 0    42 3 2   5 35 32   8  4 5 3 96 160 120  60 436 ⇒   29.07. 15 15

(

 x 1, x  1 2. Solution : Now, y  x 11    x 1, x  1 And,

1

1

−1



) 

(1) (2)

y  5 x 2

(

)

1

1 1 4 1  1  5 5 5

1 5   arc siin 1  (5p  2) / 4sq units 2 2 3. Solution : x 2  y2  25 4  x 2 , x 2  4 4y  4  x2   2 2 4  x , x  4 ⇒

(

)

(1)

(

)  4 ( y 1) , ( x  2 or x 2 )

x  4 y 1 .................. 2  x  2 And, x

2

(2) (3)

(3) Y

0

B

A 2

2

1

Y

0

2

)

 2 1 5   arc sin  2 2  5

Now, (3) intersects (1) at A, where x 5 2, now (3) intersects (2) at B, where x 5 21 and (1) and (2) intersects at (1,0).

1

(

2

a

4

2

 ∫ 5  x 2 dx  ∫ x 1 dx  ∫ x 1 dx 1 2  x 5 − x2 5   x2  x  x2    x      x    are sin 2 2 2 1  2  5 1  1   1 5 2 1    arc sin  are sin  2 2 5 5

A  ∫ ( x 2  2)2  2 x  dx 2

The required area

x – – (√2 , √2 ) B

A 1

2 – 2√5

x

Now, (3) and (2) intersect at (2,0) and (2,0) and (22,0) (3) and (1) intersects at (4,3) and (24,3)

▌ Mathematics

3.172

In view of symmetry about x axis and y axis, the required area 2 4 4 4 x2 x2 4    ∫ 25  x 2 dx  ∫ dx  ∫ dx  4 4  0  0 2 4  2     x 25  x  25 are sin x     2 2 5   0    2 4 3 5  1   x  1 x   4 x      4 x   3 0 4  3  4   2  4  4  25 arc sin sq units. 5

Larger area  64πa 2   ∴

(1) ( 2)

1 1 d2y dy  0 ⇒ x 0, ,1at x  , 2  0; 2 2 dx dx

at a point of maximum. At x  0 and 1,

1

0

(

)

)

∫ (x

)

1



8a

8 3 2 .8a  64a 2 sin1 1 / 2 16 3a 2 3 16a 2 . 4 3  3 sq units  3 1 2

d2y  0; point of minimum. dx 2

∴ The required area  ∫ ydx

4a  2 x  4 3a .  x 3/ 2    x 64a 2  x 2  64a 2 sin1  0 3 8a  4 a 

(

)

dy d2y  4 x 3  6 x 2  2 x and 2  12 x 2 12 x  2 dx dx

⇒

Both the curves are symmetrical above x axis and they intersect at A( 4a , 3a ) and B( 4a , 2 4 3a ). The smaller area between the curves is 8a 4a  2  ∫ 12ax .dx  ∫ 64a 2  x 2dx   0  4a



(

Their ratio is 4π  3; 8π  3 .

Now,

y2  12ax

)

16a 2 8π  3 sq units 3

)

5. Solution : y  x 4  2 x 3  x 2  3

4. Solution: x 2  y2  64 a 2

(

(

16a 2 4π  3 3

4

 2 x 3  x 2  3 dx 

0

91 30 b

()

(

1

(

)( ) ( ) Let g( x ) ( x 1) cos ( 3x  4 ) then y f ( x )  g '( x )  cos ( 3x  4 )  3( x 1) sin ( 3x  4 ) b

cos 3b  4  x 1 cos 3x  4  1

b

0

(

)

(

and, A  ∫ ydx  b 1 cos 3b  4

)

1

Hence, (d ) is right answer.

PRACTICE EXERCISE INTEGRATION 1.

∫

1 1 x

2

3.

dx 

(a) sin21 2x (c) sin 21 (sin x) 1 c

(b) sin 21 x (d) None of these

1 dx  2. ∫ 1 x 2 (a) tan 21 x 1 c (c) 2 tan 21 x 1 c

4. (b) tan 21 x (d) None of these

)

6. Solution : The required area  ∫ f x dx  b 1

∫x

1 x2 1

dx 

(a) sec 21 x 1 c

(b) sec 21 x

(c) 2 sec 21 x 1 c

(d) None of these

∫

11sin 2 x is

(a) sin x 1 cos x 1 c (c) cos x 1 c

(b) sin x 1 c (d) None of these

Integral Calculus and Differential Equation ▌ 3.173

5.

14. Find ax 1 bdx ∫

 sin x 

∫  11 sin x  dx (a) x 1 tan x 1 sec x 1 c (c) tan x 1 sec x

(b) x 2 tan x 1 sec x (d) None of these

1 dx 6. ∫ 2 sin x cos 2 x (a) tan x 1 c (c) tan x 2 cot x 1 c 7.

 1 cos 2 x 

∫  1 cos 2 x  dx (a) tan x 1 x 1 c (c) tan x 1 c

8.

9.

sec x ∫ sec x 1 tan x dx (a) tan x 1 c (c) tan x 1 sec x 1 c

(b) tan x 2 x 1 c (d) None of these

(b) tan x 2 sec x 1 c (d) None of these

∫ sin

−1

(cos x ) dx

πx x 2 (a )  c 2 2 2 x (c ) c 2 11. If f `( x )  3x 2  (a ) x 3 

1 x2

(c ) x 3  2 12. If f `( x )  x  x 1 2 x2 1  1 (c ) 2 x 2

(a )

πx ( b) c 2 x2 (d ) 2 2 and f (1)  0, find f ( x ). x3 1 ( b) x 3  2  2 x 1 (d ) 2  2 x

1 1 and f (1)  , find f ( x ). 2 2 x x2 1  ( b) 2 x (d ) None of these

13. Find ∫ ( ax  b) dx , when n ≠ 1. ( ax  b) n+1 c a( n 1)

(c ) ( ax  b) n1

( b)

( ax  b) n1 c a( n 1)

(d ) None of these

2( ax 1 b)3/ 2 1c 3a

(d ) None of these

15. Find ∫ e( 5 x13) dx 1 ( 5 x13) 1c e 5 1 (c ) e( 5 x13) 1 c 5 (a )

16. Find ∫

( b)

∫

1 (5x ) e 1c 5

(d ) None of these

log x dx x

1 (log x ) 1 c 2 1 (c ) log x 2

sin x x

( b)

1 (log x )2 1 c 2

(d ) None of these

dx.

(a ) 2 cos x  c

( b)  2 cos x  c

(c ) 2 sin x cc

(d ) None of these

dx a sin x  b cos x  1 b x  1 log  tan  tan1 ( )    c 2 2 2 2  2 a b  

18. Evaluate ∫ (a )

 1 b x  log  tan( tan1 ( )  )   c 2 2 2   a b  b  1 1 (c ) log  tan( tan1 ( ))   c 2 2 2 2   a b (d ) None of these 1

( b)

2

19. Find ∫

2

3x 2 dx . (1 1 x 6 )

(a) tan 2 1 x 1 c (c) tan 2 1 x 20. Find ∫

( 4 x 1 3) 2 x 2 1 3x 11

(b) tan 2 1 x3 1 c (d) None of these dx

(a ) 2 2 x 2 + 3x + 1 + c

( b) 2 2 x 2 11 1 c

(c ) 2 2 x 2 1 3 1 c

(d ) None of these

21. Find ∫

n

(a )

1 1c 3a

17. Find x2 ( b) 1c 2 (d ) None of these

(c ) x 2 1 c

(c )

( b)

(a )

  1  1 cos 2 x  ∫ tan  1 cos 2 x dx   (a ) x 1 c

10.

(b) 2 cot x 1 c (d) tan x 1 cot x 1 c

(a ) 2( ax 1 b)3/ 2 1 c

(6 x  7 ) dx (3x 2  7 x  5)2 1 c (3x 2  7 x  5)

(a )

1 c (3x 2  7 x )

( b)

(c )

1 c 3x 2  5

(d ) None of these

3.174

▌ Mathematics

∫

22. Find

x 2 tan1 x 3 dx. (1 x 6 )

1 (tan1 x 3 )2  c 6 1 (c ) tan1 x  c 6 (a )

30. Evaluate ∫ 12 x 2 dx ( b)

1 (tan1 x 3 )  c 6

(d ) None of these

tan x 23. Find ∫ dx. (sec x + cos x )

( ) (sin x )  c

( b) − tan1 cos x  c

(c ) tan1

(d )  tan1

24. Find ∫ cot x dx

25.

26.

31.

(a ) log |sin x|1 c

( b) log |cos x|1 c

(c ) log |sin x|

(d ) None of these

(b) 2 x  2 log 1 x  c

(c) 2 log 1 x  c

(d) None of these

∫ sin

2

x dx 2

1 1 x  sin x  c 2 2 1 (c ) x  c 2

∫ sin

1 1 x  sin x  c 2 2 1 (d ) sin x  c 2

( b)

xdx is

(c ) 2 x cos x  c (d ) None of these

(d )

x 1 x 2 c 2

∫ (1 x

dx 2

) 1 x 2

1

32.

Find ∫

tan x dx. sin x cos x

(a ) 2 tan x  c

( b)  2 tan x  c

(c )  2 x  c

(d ) 2 x  c

1 2

dx  a2 )

2

(a )

xa 1 log c xa 2a

( b)

xa 1 log c xa 2

(c )

1 log x  a  c 2

(d )

1 log x  a  c 2a

dx

2

1 1 x 1 x 2 sin x  c 2 2

(c )

∫ (x

∫ (4 1 9 x

(c )

x 2 1 dx x 4 1  1  1 (a ) tan1  c 2  2x 

( b) 2 sin x  c

29.

1 −1 x 1 x 2 sin x  c 2 2

33. Find ∫

(a )  2 x cos x  2 sin x  c

28.

( b)

 1 x 2  c tan1   2 2 x    1 x 2  1 c tan1  ( b)  2 2 x    1 x 2   c (c ) tan1   2x    (d ) None of these

x

(a) 2 x  2 log 1 x  c

1 1 sin x  a 2

(a ) 

dx

∫ 11

(a )

27.

( ) (sin x ) c

(a ) tan1 cos x  c

(a )

tan1 ( x 2 1)  c

(a )

 3x  1 tan   1 c 6  2 

( b)

1 −1  3x  tan   1c 6  2 

(c )

 3x  1 tan   6  2 

(d ) None of these

 x 2 1  tan n1  c  2x  2  

1

(d ) Non ne of these

9 34. Evaluate ∫ x dx.

(a ) 9 x 8 1 c (c )

9x 1c 10

35. Evaluate ∫

)

( b)

x10 1c 10

(d ) None of these 1 dx. x1/ 3

3 2/ 3 x 1c 2 3 (c ) x1/ 3 1 c 2 (a )

( b)

( b) 3x 2/ 3 1 c (d ) None of these

Integral Calculus and Differential Equation ▌ 3.175 x 36. Evaluate ∫ 5 dx

5x ( b) 1c log 5

(a ) 5x 1 c 5x (c ) 1c x

(d ) None of these

(a ) tan x  x  c

( b) 2 tan x  c

(c ) tan x  x  c

(d ) None of these

x dx 2

1 sin x  c 2 1 1 (c ) x  sin x  c 2 2 (a )

(a ) tan1 (cos x )  c ( b) tan1 (sin x )  c

( b)

1 1 x  sin x  c 2 2

(d ) None of these

(d ) None of these 46. Evaluate ∫ sin 3 ( 2 x 1)dx 1 1 (a )  cos( 2 x 1)  cos3 ( 2 x 1)  c 2 6 1 1 ( b)  cos( 2 x 1)  cos 2 ( 2 x 1)  c 2 6 (c )

1 1 cos( 2 x 1)  cos3 ( 2 x 1)  c 2 6

(d )

1 1 cos( 2 x 1)  cos 2 ( 2 x 1) 2 6

39. Evaluate ∫ 1 − sin xdx . 2

(a ) sin x  cos x  c

( b) sin x  c

(c ) cos x  c

(d ) sin x  cos x  c

47. Evaluate cos mn cos nxdx when m ≠ n ∫

 sin x  dx. 40. Evaluate ∫   1 sin x  (a ) tan x  c

( b) tan x  sec x  c

(c ) sec x  c

(d ) x  tan x  sec x  c

41. Evaluate ∫ tan1 ( sec x  tan x ) dx πx 1 2 (a )  x c 4 4 πx 1 (c ) x 2   c 4 4

log x dx. 42. Evaluate ∫ x 1 (a ) log x  c 2 1 (c )  log x  c 2

πx 1 2  x c ( b) 4 4

( b)

1 (log x )2  c 2

( b)

sin( m  n ) x sin( m  n ) x  c 2( m  n ) 2( m  n )

(c )

sin( m  n ) x c 2( m  n )

(d ) None of these

(a )

sin(m  n)x 2(m  n)

( b)

x sin 2 nx  c 2 (4n)

(c )

x sin 2 nx  c 2 4n

(d ) None of these

(d ) None of these 1

3

(a ) tan x  c

( b) tan x  c (d ) tan1 x

(

sin( m  n ) x sin( m − n ) x  c 2( m  n ) 2( m − n )

48. Evaluate ∫ cos mn cos nxdx when m

(c ) tan1 x 3 44. Evaluate ∫

(a )

(d ) None of these

3x 2 43. Evaluate ∫ dx (1 x 6 ) 1

tan x dx (sin x 2 cos x )

(c )  tan1 (cos x )  c

37. Evaluate ∫ tan 2 xdx

38. Evaluate ∫ sin 2

45. Evaluate ∫

1 dx (e x 1)

)

49. Evaluate ∫

sin 2 x. dx (1 cos x )2

(a ) 2 tan( x / 2)  c ( b)  2 tan( x / 2)  x  c

(a ) log | 1 ex | c

( b) log | (1 e x ) | c

(c ) log | (1 ex ) | c

(d ) log | (1 e x ) | c

(c ) 2 tan( x / 2)  x  c (d ) None of thesse

▌ Mathematics

3.176

ANSwER KEy 1. (b)

8. (b)

15. (a)

22. (a)

29. (b)

36. (b)

43. (a)

2. (a)

9. (b)

16. (b)

23. (b)

30. (c)

37. (c)

44. (a)

3. (a)

10. (a)

17. (b)

24. (a)

31. (a)

38. (b)

45. (c)

4. (a)

11. (b)

18. (a)

25. (a)

32. (a)

39. (a)

46. (a)

5. (b)

12. (c)

19. (b)

26. (b)

33. (b)

40. (d)

47. (a)

6. (c)

13. (a)

20. (a)

27. (a)

34. (b)

41. (a)

48. (b)

7. (b)

14. (b)

21. (b)

28. (a)

35. (a)

42. (b)

49. (c)

SOLUTIONS 1. Formula

8.  ∫

2. Formula 3. Formula 4.

∫

1 − sin 2 x  ∫ (cos 2 x  sin 2 x  2 sin x cos)1/ 2 dx

 ∫ sec 2 x dx  ∫ sec tan x dx

 sin x  (cos x )  c  sin x  cos x  c  sin x 

∫  1 + sin x  dx = ∫

(1 sin x ) 1 dx (1 sin x )

 ∫ dx  ∫

1 (1 sin x ) dx  (1 sin x ) (1 sin x )

 ∫ dx  ∫

(1 sin x ) dx cos 2 x

 1 sin x   ∫ dx  ∫    cos 2 x cos 2 x  ⇒ ∫ dx  ∫ sec x dx  ∫ sec x tan x dx 2

 x  tan x  sec x  c 6.

∫ sin

2

 sin 2 x  cos 2 x  1 dx  ∫   dx 2 2 2 x cos x  sin x cos x 

 1 1  ∫  2  dx 2  cos x sin x   ∫ sec 2 x dx  ∫ coses 2 x dx  tan x  cot x  c 7.

2 sin 2 x dx  ∫ tan 2 x dx  ∫ (sec 2 x  1) dx 2 x

∫ 2 cos

 tan x  x  c

(sec 2 x  sec x tan x ) dx (sec 2 x  tan 2 x )

 ∫ (sec 2 x  sec x tan x ) dx

 ∫ (cos x  sin x ) dx  ∫ cos x dx  ∫ sin x dx

5.

sec x (sec x  tan x )  dx sec x  tan x (sec x  tan x )

 tan x  sec x  c 9.

∫ tan

1

 1 cos 2 x    dx  1 cos 2 x 

 2 sin 2 x   ∫ tan1   dx 2  2 cos x   ∫ tan1 (tan x ) dx  ∫ x dx  10.

∫ sin

1

x2 c 2

  π   (cos x ) dx  ∫ sin1 sin   x   dx     2

 ∫ ( π / 2  x ) dx 

π πx x 2 dx  ∫ x dx    c ∫ 2 2 2

  11. f `( x )  3x 2  2 ⇒ f ( x )  3x 2  2 dx ∫  x3 x 3   3∫ x 2 dx  2 ∫ x −3dx x3 x −2 1 2 ⋅  c  ( x 3  2  c) 3 2 x 1 3 Thus, f ( x )  x  2  cf (1)  2  c x ⇒ 2  c  0, c  2 1 ∴ f ( x)  x3  2  2 x 3⋅

Integral Calculus and Differential Equation ▌ 3.177 19. Put x 3  t so that 3x 2 dx  dt

1 x2  1  x2 1 ⇒ f ( x )  ∫  x  2  dx    c 2 x x   1 1 f (1)  1 c   c 2 2 1 x2 1 1 c  f ( x )    2 2 x 2

12. f `( x )  x 

13. Putting ax  b  t , we get a dx  dt or dx  ∴ ∫ ( ax  b) n dx  

1 n 1 t n1 t dt  c ∫ a a n 1

3x 2

∫ (1 + x

1 dt a

1

( 5 x3)

c

1 dx  dt x 1 1 log ∴ ∫ dx  ∫ t dt  t 2  c  (log x )2  c x 2 2

16. Put log x  t so that

1

or

x

∴∫

1 1 t dt  t 2  c 3∫ 6 1  (tan1 x 3 )2  c 6 

∫ e dt  5 e  c  5 e

17. Put x  t so that

23.

dx  2 dt

 θ x  1  ∫ log  tan c r 2     1  b  x  log  tan  tan1     c.   2  a  2  a 2  b2

2

x

dx

1 dt (1  t 2 )

Where, cos x = t and sin xdx =  dt

24.

cos x

1

∫ cot x dx  ∫ sin x dx  ∫ t dt

Where sin x  t and cos xdx  dt  log | t | c

 log | sin x | c ∴ ∫ cot x dx  log | sin x |  c

2 cos x  c dx a sin x  b cos x dx ∫ r cos θ sin x  r sin θ cos x dx 1 1 θ Put abrr sin  ∫  cos ec( x + θ)dx sin θ r sin( x  θ) r ∫

sin x

 tan −1 t  c  tan1 (cos x )  c

dx  2 ∫ sin t dt x  2( cos t )  c  2 cos t  c

18.  ∫

tan x

∫ sec x  cos x dx  ∫ 1 cos  ∫

1 − 1/ 2 x dx  dt 2

sin x



3x 2 dx  dt or (1 x 6 )

1 x 2 tan1 x 3 x2 dx dxx  dt  ∫ 6 3 (1 x 6 ) (1 x )

1 5

t

1 3x 2  7 x  5

22. Put tan1 x 3  t so that

15. Put (5x 1 3) 5 t so that 5 dx 5 dt or dx 5 1/5 dt

1

dt  tan1 t  c  tan1 x 3  c 1 t 3

21. Put (3x 2  7 x  5)  t so that (6 x  7 ) dx  dt . 1 (6 x  7 ) 1 dx  ∫ 2 dt   c ∴∫ t t (3x 2  7 x  5)2 =

1 2 2( ax  b)3/ 2 ax  b dt  ∫ t dt  t 3/ 2  c  c a 3a 3a

t

dx  ∫

 2 2 x 2  3x 1  c

14. Put ( ax  b)  t so that adx  dt

∴ ∫ e( 5 x +3) dx 

)

20. Put ( 2 x 2  3x 1)  t so that ( 4 x  3) dx  dt ( 4 x  3) d ∫ 2 x 2  3x 1 dx  ∫ t  2 t  c

( ax  b) n  1 c a( n  1)

∴∫

6

2 25. Put x  t so that x  t and dx  2tdt . dx 2t 2(1 t )  2 ∴∫ ∫ dt  ∫ dt 1+ t 1 t 1+ x

 2 ∫ dt  2 ∫

dt  2t  2 log 1 t  c 1+ t

 x  2 log(1 x )  c 26.

∫ sin

1



2

1 x x dt 5 ∫ 2 sin 2 dx 2 2 2

1 (1  cos x ) dx 2∫

3.178

▌ Mathematics 31. Put x  1 / t so that dx  1 / t 2 dt

1 1 dx  ∫ cos x dx 2∫ 2 1 1  x  sin x  c 2 2 

27. Put x  t so that

∫ (1  x

1

∫

dx  dt or dx  2tdt

2 x

∴ ∫ sin xdx  2 ∫ t sin t dt  2[t ( cos t )

⇒  2 x cos x  2 sin x  c 1  ( x  a)  ( x  a)   dx ( x  a )( x  a ) 

dx

∫ ( x  a)( x  a)  ∫ 2a  

d dx  1     2a ∫ ( x  a) ∫ ( x  a) 

29.

dx

2

)



2

32.

c

(1 / x 2 ) 1

tan1

2

c

 1 x 2  tan1   c  2 2 x 

1



1 dx  ∫ 9  2  2     x 2   3     1 1 1 x  ⋅ tan c 9 2 2      3  3

1 dx x cos1/ 2 x 1 dx ∫ sin1/ 2 1/ 2 1/ 2 x cos x cos cos1/ 2 1  ∫ 1/ 2 dx tan x cos x dt sec 2  1/ 2 dx  ∫ tan t

∫ sin

1/ 2

Wheere, tan x  t  2 t  c  2 tan x  c

∫ (a

2

 3x  dx 1  tan1   c  x2 ) 6  2 

30. Put x sin θ so that dx  cos θd θ  1 x dx  ∫ 1  sin θ cos θ d θ 2

dt

dt

t 2 1

tan1

2

1 dx 9 ∫ (4 / 9  x 2 )

[using the formula for

1



2

∫ (4  9 x

t

2

1 [log x  a  log x  a  c 2a 1 xa  log c 2a xa dx 1 xa  log c 2 2 a xa x a

2

(t 2 1) t 2  1

1





∴∫

(11 / t )

(1 1 / t 2 )

Put t 2 1  z 2 so that dt  2 zdz z dz dz t  ∫  z ( z 2  2) ∫ z 2  2 1 z  tan1 c 2 2

 2t cos t  2 sin t  c

28.

) 1 x 2 (1 / t 2 )

 ∫

∫ [1(cos t ) dt ] c [integratingg t sin t by parts]

dx 2

33.

(11 / x ) 2

∫

( x 1 / x )

2

1 x 1 x 2  sin1 x  c 2 2

) )

[divviding numerator and denominator by x 2 ]

2

 1  cos 2θ  ∫ cos θd θ  ∫  2 d θ 1 1  ∫ d θ  ∫ cos 2θ d θ 2 2 1 1 sin 2θ  θ c 2 2 2 1 1  θ  sin θ cos θ  c 2 2

( (

1 + 1 / x2  x2 + 1  dx dx   ∫ 4 + 1  x2 + 1 / x2

∫  x

2

dx

2

dt 2 2  t  2      1 1 Where,  x    t and  1 2  dx  dt x   x 

∫

( )

 1 x   t  1 1  x c tan  tan   c    2  2 2 2     1

1

Integral Calculus and Differential Equation ▌ 3.179 43. Hint. put x 3  t so that

  1 a  1 1  x  tan   c 2 2      x 2 1 tan1   c  2x  2

44.

x10

dt 3x 2 dx  ∫  tan1 t  c 1 + x6 1 t 2

∫e

ex 1 1 dx dx  ∫ x dx  x e (1 e ) 1 e x 1

9

1

36.

∫ 5 dx  log

1/ 3

∴∫

3 dx  x 2/ 3  c 2

∫x

5x

x

5

x

Now , put (1 ex )  t So that e − x dx  dt.

∫ x dx  10  c

35.

37.

∴∫

1



34.

3x 2dx  dt

= dg (1  e x )  c

c

45.

tan x

∫

2 x1

2

38.

∫ sin

39.

∫

2

x 1 1 1 dx  ∫ (1 cos x )dx  x  sin x  c 2 2 2 2

46.

47.

∫



1 (1 t 2 ) dt; where cos( 2 x 1)  t 2∫



1 [cos( m  n ) x  cos( m  n ) x ] 2∫

1 1 cos( m  n ) xdx  ∫ cos( m  n ) xdx  2∫ 2

sin( m  n )xx sin( m  n ) x  c 2( m  n ) 2( m  n )

 1 cos( π / 2  x )   ∫ tan1  dx  sin( π / 2  x ) 

 π π 1 π  ∫    dx ⇒ .∫ dx  ∫ xdx 4 2  4 2

2

cos mn cos nexdx

whenm ≠ n

 1 sin x   ∫ tan  dx  cos x 

2 sin ( π / 4  π / 2    ∫ tan   dx  2 sin( π / 4  x / 2)cos( π / 4  x / 2) 

1 dx (1 + t 2 )

3

1

1

dx

1 1  cos( 2 x 1)  cos3 ( 2 x 1)  c. 2 6

1

2

x

∫ sin (2x 1)dx  ∫ (1 cos (2x 1)sin(2x 1) dx 

    1 40.  sin x  dx ⇒ (1 sin x ) − 1 dx   1 ∫  1 sin x  ∫ (1 sin x ) ∫  (1 sin x )  dx  1 sin x  dx  41. ∫ tan (sec x  tan x )dx  ∫ tan   cos x cos x 

2

 tan −1 x  c  tan1 (cos x )  c

1 sin 2 2 xdx  ∫ (cos 2 x  sin 2 x  2 sin x cos x )1/ 2 dx

−1

sin x

∫ (sin x + cos x ) dx  ∫ h cos

∫ tan xdx  ∫ (sec )dx  ∫ sec xdx  ∫ 1dx  tan x  x  c 2

ex 1 1 dx dt  log t  c dx  ∫ 1 e2 ∫ t (e 1) x

cos mn cos nexdx

∫

48.

whenm = n



1  ∫ cos 2 nxdx  2 cos 2 nxdx 2

1 x sin 2 nx (1 cos 2 nx )dx   c 2∫ 2 4n 2



1 π x  x2 c 4 4

log x 1 42. ∫ dx ⇒ Putting log x  t so that dx  dt x x 1 2 1 2 ∫ tdt  2 t  c ⇒ 2 (log x )  c.

49.  ∫

 sin x  sin 2 x dx  ∫  dx  1 cos x  (1 + cos x )2

 2 sin( π / 2)cos( x / 22  ∫  dx 2 cos 2 ( x / 2 )    ∫ tan 2 ( x / 2)dx  ∫ (sec 2 x / 2 1) dx  2 tan( x / 2)  x  c.

▌ Mathematics

3.180

AREA UNDER CURVE 1.

10 π

∫

sin x dx is equal to

π

(a) 18 (c) 40 1 h→ 0 h

2. lim

(b) 20 (d) none of these

x1h

dz

∫

z 1 z 2 11

x

is equal to (b)

(a) 0 (c)

1

1 x 1 x 2 11

(d) none of these

x 11 2

3. Let f (x) be a polynomial of degree 2 satisfying f (0) 5 1, f ' (0) 5 2 2 and f ' (0) 5 6, then

2

∫

f ( x ) dx is equal to

−1

(a) 6 (c) 9

(b) 0 (d) none of these

4. The value of

2

∫ max{(1 x ),(1 x ), 2} dx is

9. If the area bounded by the curve y 5 f (x), x-axis and the ordinates x 5 1 and x 5 b is (b 2 1) sin (3b 1 4), then (a) f (x) 5 cos (3x 1 4) 1 3 (x 2 1) sin (3x 1 4) (b) f (x) 5 sin (3x 1 4) 1 3 (x 2 1) cos (3x 1 4) (c) f (x) 5 sin (3x 1 4) 2 3 (x 2 1) cos (3x 1 4) (d) none of these 10. The area bounded by the circle x2 1 y2 5 8, the parabola x2 5 2y and the line y 5 x in y ≥ 0 is 2 2 (a)  2π (b)  2π 3 3 2 2 (c)  π (d)  π 3 3 11. If u10 5

π /2

∫x

10

sin x dx , then the value of u10 + 90u8 is

0

π  (a) 9   2 π  (c)   2

9

9

π  (b) 10   2

9

8

π  (d) 9   2

2

(a) 8 (c) 9

(b) 2 8 (d) 2 9 a

5. If I 5

∫ (α sin

5

12. The area bounded by the curves x 2  y 2  4; x 2  2 y and x  y is

x 1 β tan 3 x 1 γ cos x ) dx , where α,β,γ are

−a

constants, then the value of I depends on (a) γ, a (b) α, β, γ, a (c) α, β, a (d) none of these 1

6. The value of the integral ∫ e x dx is 2

0

(a) greater than e (c) greater than 1

(b) less than e (d) less than 1

 1 sin  n 1  x 2 7. The value of ∫  dx is x 0 sin   2 π

π 2 (c) π 8. The value of Integral

41π / 4

∫

cos x dx is

0

(a) 20  (c) 19 

1 2 1 2

(b) 20  (d) 19 

1 2 1 2

1 3

(d) None of these

14. If A is the area between the curves y 5 x and x-axis in the  π interval then in the same interval 0,  , area between the  4 curve y 5 cos x and x-axis is

(c) 1 2 A

(d) 2π

(b) π −

13. Area bounded by the curves x 5 1, x 5 3, xy 5 1 and x 2 axis is (a) log 2 (b) log 3 (c) log 4 (d) None of these

(a) A

(b) 0

(a)

1 3 1 (c)  π  3

(a) 2 π 

π 2A 2 (d) None of these

(b)

π 15. Area bounded by the curves y 5 sin4 x, x 5 0, x 5 and 2 y 5 0 is 3 (a) 8 3 (b) 16 3π (c) 16 (d) none of these

Integral Calculus and Differential Equation ▌ 3.181 16. The area of the region bounded by y  | x  1 | and y  1 is (a) 1 (b) 2 (c)

1 2

(d) none of these

17. Area comon to the curve y 5 x 3 and y 5 x is 2 (a) 1 (b) 3 5 (c) (d) None of these 12

19. The area bounded by the curve y  x | x |, x  axis and the ordinates x 5 1, 21 is given by 1 (a) 0 (b) 3 2 (c) (d) None of these 3 20. The area bounded by the parabolas y2 = 4a (x + a) and y2 = – 4a (x – a) is

18. If the ordinate x 5 a divides the area bounded by the curve 8 y  1 2 and the ordinates into two equal parts, then a 5 x (a) 2 3

(b) 2 2

(c) 3

(d) None of these

(a)

16 2 a 3

(b)

8 2 a 3

(c)

4 2 a 3

(d) None of these

ANSwER KEy 1. (a) 2. (b) 3. (c)

4. (c) 5. (a) 6. (b, c)

7. (c) 8. (b) 9. (b)

10. (a) 11. (b) 12. (c)

13. (b) 14. (c) 15. (c)

16. (a) 17. (c) 18. (b)

19. (c) 20. (a)

SOLUTIONS 1. Since |sin x| is a periodic function of periiod π, 10 π

π

π

π

0

0

 lim x →0

∴ ∫ sin x dx  9.∫ sin x dx  9 ∫ sin x dx [ssin x  0in(0, π)] 9[cosx ]0π 9(11)  18. 1 h→ 0 h

2. lim

xh

∫ x

xh

= lim h→ 0

∫ x

x  h  ( x  h) 2  1 1

x  x2 1

. (1)

.

3. Let f ( x )  ax 2  bx  c ⇒ f '( x )  2 ax  b and f "( x )  2 a.

dz

We are given f (0)  c  1, f '(0)  b  2

z  z 2 1

f "(0)  2 a  6 ⇒ a  3, b  2 and c  1.

and

dz z  z 2 1 h



1

0   form  0  

 xh  dz → 0 as h → 0  ∴ ∫  x z  z 2 1    1 d   . ( x  h) 2 dh  z  z  1  zxh  liim x →0 1 [ Using L' Hospital's Rule]

2

∴ ∫ f ( x )dx = 1

2

∫ (3x

2

 2 x  1) dx

1

 [ x  x 2  x ]2− 1 3

 (8  4  2)  ( 1  1  1)  6  3  9. 4. For  2  x 1, we have 1 x  2 and 1 x  1 x ∴ max {(1 2 x), (1 1 x), 2} 5 1 2 x. For 2 1 2 7 ⇒ 3 < x or x < 7 ⇒ 3 < x < 7.

G

C

AB  AC ...(1) ∴ AD  2 E is middle point of CA BA  BC ∴ BE  ...( 2) 2 F is middle point of AB CB  CA CF  ......(3) 2 Adding (1), (2) and (3), we get AD 1 BE 1 CF 5 0 25. Take A as origin. Then position vectors of A, B C are given, 0, a and b resp. Then the centroid G has position vector 0 a  b a  b = , 3 3 ab ∴ AG  . 3 26. Given a 3 r 5 b, a ⋅ r 5 c and a ⋅ b 5 0 Since a 3 r 5 b ∴ a 3 (a 3 r) 5 a 3 b ⇒ (a ⋅ r) a 2(a ⋅ a) r 5 a 3 b ⇒ ca 2 a2 r 5 a 3 b or ca 2 r 5 a 3 b [ a is a unit vector ∴ a2 5 1] ∴ r 5 ca 2 (a 3 b). 27. As a, b, c are non-coplanar, b 3 c, c 3 a, a 3 b are also noncoplanar. So, any vector an expressed as a linear combination of these vectors. Let a  λ b  c  µ c × a  va × b ∴ a ⋅ a  λ [bca], a ⋅ b  µ[cab], a ⋅ c  v[ abc] ( a ⋅ a )b  c ( a ⋅ b)c  a ( a ⋅ c ) a  b ∴a    [bca] [cab] [ abc] ∴( a ⋅ a )b  c  ( a ⋅ b)c  a  ( a ⋅ c ) a  b  [bca]a.

1 . ( a ⋅ a)

or 5 2 x >2 2

n      n    n   2 30.  ∑ ai  . ∑ ai   ∑ |ai |  2 ∑ ∑ ai .a y  i 1   i 1  i 1 1≤ i j ≤ n       n ⇒ 0  n  2 ∑ ∑ ai .a y ⇒ ∑ ∑ ai .a y  2 1≤ i j ≤ n 1≤ i < j ≤ n

31. Let the position vectors B, C, D with A as origin be b,cc and b bc c  d d , d and hence those of L, M, N, P are , , 2 c 2 2  respectively LM  P.V of M  P.V. Of L bc b c    2 2 2  cd d c PN  P.V. of N  P.V of P    2 2 2 Dd

N

Cc

P M

Aa

L

b B

Above shows that ∠M parallel and equal to PN, similarly, we can shown MN is parallel and equal to ∠P. Hence, PLMN is a parallelogram. 32. It is perpendicular to the plane of a and b hence perpendicular to both a 3 b 5 7 (6i 1 2J 2 3K). Now verify by taking its dot product with both a and b it will be zero. 33. We have, |A 1 B| 5 |A| 1 |B| ⇒ A  B, A, B are Collinear. ⇒ i  λ j  k  t (i  j  k )

Vector Algebra ▌ 3.203 ⇒xb 1 yc 5 0 when b and c and non-parallel ⇒x 5 0, y 5 0.

⇒ t 5 1, t 5 λ Hence, λ 5 1

43. a 2  b2  c 2  1 | a  b  c |2  ( a  b  c )2

34. We have | a |  2,| b |  5,| a × b |  8

 a ⋅b b⋅c  c ⋅ a 0

Now 6 | a × b)  8 ⇒ ( a  b)2  64 ⇒ a b sin θ  64 ⇒ a b (1  cos θ)  64 2 2

2

2 2

2

⇒ a b  ( a  b)  64 ⇒ ( 4)( 25)  ( a ⋅ b)  64 2 2

2

2

⇒ ( a  b)2  36 ⇒ a ⋅ b  6    35. Find AB, BC and CA, and their module are 41, 6 , 35 ∴ AB  BC  CA ∴ ∠c  90 2

2

2

36. Let the position vectors of A, B, C be a, b, c relative to 0 as origin then those of D, E, F are respectively, a 1 c, b 1 c and c 1 a. The four diagonals are OP, CD, AE and BF

 ∑ a 2  2 ∑ a ⋅ b  3  0  3. 44. We have (a 1 b)2 5 c2 or a2 1 b2 1 2a⋅b 5 c2 or 9 1 25 1 2.35 cosθ 5 49 1 π ∴cos θ 5 so that θ 5 . 2 3 45. Squaring a 1 b 1 c 5 0 and noting that a2 5 |a|2 5 1 etc., we get a ⋅ b b⋅ c c ⋅ a 

3 2

46. Let P be along i so that resultant is Pbut along J. If the other force be θ then, Pi   Q  PJ  Pi  Pj ∴| Q |  P 2 Q   47. If A, B, C are collinear then AB = k AC   ∴ AB  AC  0 or ( b  a ) ( c  a )

   OP  OD  DP  a  b  c    CD  OD  OC  a  b  c    AE  OE  OA  b  c  a    BA  OF  OB  c  a  c Adding all we get ratio is 4. 37. (a 2 c) 3 b 5 0 ⇒a 2 c is parallel to b ∴ (a 2 c) 5 λb. 38. Area | a  b |  | 8( i  j  K ) |  8 3 1 1 39. Area  | d1  d2 |  10 3  5 3 2 2 12 0 λ 3 1 ∴546  12  44  6 λ 40. v  | abc |  0 2 1 15 ∴λ 5 2 3. 41. [ABC] 5 0 ⇒ Det 5 0 ⇒ 7λ 1 28 5 0 ∴λ 5 2 4. 42. (a. c) b 2 (a. b) c 5 0  1 ⇒ ( a ⋅ c )   b  ( a ⋅ b)c  0 2 

5 0 or b 3 c2 a 3 c 2 b 3 a 1 0 5 0 or b 3 c 1 c 1 a 1 a 3 b 5 0    48. Given : a 5 1, c 5 1 and b 5 4.   1 1 ⇒ a ⋅ c 5 1.1. 5 5 4 4        b  2c  λ a ⇒ a ⋅ b  2 a ⋅ c  λ a 2   1 ⇒ a ⋅ b  2.  λ 4   1 ⇒ a⋅bλ  2          Again, b  2c  λ a ⇒ b ⋅ b  2b ⋅ c  λ b ⋅ a    1 ⇒ 16  2b ⋅ c  λ  λ   2    λ2 λ ⇒ b⋅c 8  2 4        Alsoo, b  2c  λ a ⇒ b ⋅ c  2c  λ b ⋅ a ⇒ 8

 1 λ2 λ   2(1)  λ   2 4  4

⇒ λ 2  λ  12  0 ∴ λ  4, 3. 49. Any vector in the plane of b and c is r 5 b 1 λc 5 (1 1 2j 1 k) 1 λ(i 1 j 2 2k) 5 (1 1 λ) i 1 (2 1 λ) j 1 (2 1 2 2λ) k

3.204

▌ Mathematics

Projection of r on a is   ∴

r ⋅a a

2(1  λ )  ( 2  λ )  (1  2 λ ) 4  1 1 λ  1





2 3 0 v  | abc |  1 1 1  4 57. 3 0 1

λ  1 6

2 ⇒ λ  1  2; λ  3 or 1 3

6 Hence r  2i  j  5k or r  2i  3j  3k . 2

50. a  b is unit vector if |(a  b)|  1 or ( a  b)

58. By Definition 59. By Definition 60. By Definition

2

5 1 or 1 1 1 1 2.1.1 cos θ 5 1 as a and b unit vectors 1 ∴ cos θ   1cos( π 3 )  cos( π  π 3 )  cos ( 2 π 3 ) 2 ∴θ  2 π 3 51. If x be the position vector of B, then P(a) divides AB in the ratio 2:3 2 x  3( a  2b) ; 5a  3a  6b  2 x a 2 3 x  a  3b 52. We have, (a · j)3(2j 2 3k) 5 a. (j 3 (2j 2 3k)) 5 a. (2 3(j 3 k 5 2 3(a . i) [ j 3 k 5 i] 5 2 3 (4) 5 2 12

61. By Definition     62. a ⋅ b 5 ab cos θ 5 (1) (1) cos θ 5 cos θ 5 a, b       63. AC  3 AB ⇒ OC  OA  3 OB  OA      ⇒ c  a  3 b  a where c is P.V of point C .       ⇒ c  3b  3a  a  3b  2 a

(

)

)

(

  64. Since magnitude of a is zero∴ a is called zero vector.   65. Since magnitude of a is unit ∴ a is called unit vector.   66. b  4iˆ  6 ˆj  2( 2iˆ  3 ˆj )  2 a   a and b are paralllel.

53. a ⋅ b  ab cosθ  2 3(1 2 )  3

67. ∝⋅b 5 0 ⇒ 2 1 λ 5 0 ∴λ 5 2 2.

54. | ab sin θnˆ || ab cos θ | ∴ ab sin θ  ab cos θ or tan θ  1 θ  455

68. b, c, d are meaningless as there is neither dot nor cross between a scalar a.b. and vector c. Since a, b, c are coplanar [abc] 5 0 as a ⋅(b3 c)5 0.

55. Let r 5 r1i 1 r2 j 1 r3 k ∴ ∴ r�i 5 r1, r�j 5 r2, r�k 5 r3

 69. a  ( b  c )  ( a ⋅ c ) b  ( a ⋅ cb) c  0 as a ⋅ b  0 and a ⋅ c  0

and r  i  0  r2 ( j  i )  r3 ( k  i) r2 k + r3 j ∴( r ⋅ i ) ( r  i ) r1r2 k + r3r2 j Similarly ( r ⋅ i) ( r  j ) r2 r3i + r2 r1k And ( r ⋅ k ) ( r  k )  r3r1 j + r2 r3i ∴( r ⋅ i ) ( r ⋅ i )  ( r ⋅ j ) ( r  j )  ( r ⋅ k )( r  k )  0 15 56. Let b  6i  8 j  k 2 15 2 (6i  8 j  k ) 2 25 ∴a = a vector of length 50 along b 15  4(6i  8 j  k) 2 Since a makes obtuse angle with z-axis so we must have a⋅k 0 Thus a  24i  32j  30k. A unit vector along b is ±

70. a  b  PC

(1)

b  c  qa

( 2)

∴ a  b  c  ( P  1)c by

(1)

 ( q 1) a by

( 2)

This implies that a and c are collinear which is contrary to hypothes. 71. We have, p 5 AC 1 BD 5 AC 1 BC 1 CD 5 AC 1 λ AD 1 CD 5 (AC 1 CD) 1 λ AD 5 AD 1 λAD 5 (1 1 λ) AD Since p 5 µAD ∴µ 5 1 1 λ. 2 72. ( x  y )  1  1  2 x ⋅ y  2  2 cos φ 2  4 sin φ 1 ∴ ( x  y )  sin φ 2 2

( )

Vector Algebra ▌ 3.205 73. ( e  e )2  e ⋅ e  2e ⋅ e  e ⋅ e 1 2 1 1 1 2 2 2 ⇒ | e1  e2 |  | e |2  2 | e1 | | e2 | cos θ  | e2 |2 ⇒ | e1  e2 |2  1  2 ⋅ 1 ⋅ 1cos θ  1 [|| e1 |  | e2 |  1]  θ ⇒ | e1  e2 |2  2 (1  cos θ)  2  2 cos 2  2  ⇒ | e1  e2 |2  4 cos 2

θ θ 1 ⇒ cos  2 2 2

| e1  e2 |

77. Since EB is parallel to AB   1 2 ED 5 AB. ∴ AB 5 2 ED 2      78. OB  OA  AB  a  b [oc is parallel to AB and OC  2AB]         OC ≠ OB  BC , 2b  a  b = b  BC       ⇒ BC  2b  a  b = b  a 79. Clearly r. C 5 λ [a, b, c] 5

74. α , β  b2  0, 2α.β  a 2  0 Eliminate α , β,∴ a 2  2b2 or a  b 2 1   1 AB  AC  | 17 i  6 J  8k | 2 2  a   76. a 5 kaˆ ⇒ k 5 5 | a | aˆ 75. Area 

1 λ 8

1 r ⋅ a = µ[a bc]  µ 8 1 r ⋅ b = v[a bc]  v 8 ∴ r ⋅ ( a  b  c )  (λ  µ  v ) ∴λ  µ  v  8r ( a  b  c ) 80. (p, q ) 5 k (5, 1), k ≠ 0, a scalar ⇒ p 5 5k , q 5 k ⇒ p 5 5q

nDA bOX 1. What is the area of the triangle with vertices (0, 2, 2), (2, 0, −1) and (3, 4, 0)? 15 (a) sq unit (b) 15 sq unit 2 7 (c) sq unit (d) 7 sq unit 2   π 2. If the angle between the vectors a and b is 3   what is the angle between 25a and 6 b ? π 2π (a) (b) 6 3 3π 2π (c) (d) 5 7 3. Consider the following statements    1. For any three vectors a , b, c ;       a ⋅{( b  c )  (a  b  c )}  0 2. For any three coplanar unit vectorss      d, e f ; d  e ⋅ f  1

(

)

Which of the statements given above is/are correct? (a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2   4. Let a and b be two unit vectors and α be the angle between   them. If a 1 b is also the unit vectors, then what is the

(

value of α?

)

π 4 2π (c) 3 (a)

π 3 π (d) 2 (b)

5. What is the value of λ for which the vectors ˆi  ˆj  kˆ , 2ˆi  ˆj  kˆ , λˆi  ˆj  λkˆ are coplanar? (a) (b) (c) (d)

1 2 3 4

6. What is the geometric interpretation of the identity    (a  b)  (a  b)  2 (a  b) ? 1. If the diagonals of a given parallelogram are used as sides of a second parallelogram, then the area of the second parallelogram is twice that of the given parallelogram. 2. If the semi-diagonals of a given parallelogram are used as sides of a second parallelogram, then the area of the second parallelogram is half of the given parallelogram. Select the correct answer using the code given below (a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2  ˆ r  ˆi  (2  λ )jˆ  2kˆ are such that 7. If r1  λˆi  2ˆj  k, 2   |r1 |  |r2 |, then λ satisfies which one of the following? (a) λ  0 only

(b) λ  1

(c) λ  1

(d) λ 1

3.206

▌ Mathematics

 8. A vector b is collinear with the vector a  ( 2, 1, 1) and    satisties the condition a ⋅ b  3 what is b equal to? (b) ( 2 / 3, 1 / 3, 1 / 3) (a) (1, 1 / 2, 1 / 2) (c) (1 / 2, 1 / 4, 1 / 4)

(d) (1, 1, 0)

   9. The vectors a  xˆi + yˆj  zkˆ , b = kˆ , c are such that they  form a right handed system. Whatt is c equal to? (a) ˆj (b) yˆj  xkˆ (c) yˆi  xˆj

(d) xˆi  yˆj

   10. If a and bˆ are the unit vectors along a and b respectively,   them what is the projection of b on a?   (a) a ⋅ b (b) aˆ ⋅ bˆ    (c) aˆ ⋅ b (d) a 3 b 11. What are the unit vectors parallel to xy -plane and ˆ perpendicular to the vector 4iˆ  3jˆ  k?

( ) (c) ( 3ˆi  4ˆj) / 5 (a)  3ˆi  4jˆ / 5

( ) ˆ ˆ (d) ( 4 i  3j) / 5

(a) 2 aˆi 2 bˆj (c) 2 aˆi 2 bˆj

 13. Given a  2ˆi  3ˆj + 4kˆ and bˆ is a unit vector co-directional  ˆ If m is a scalar such that bˆ  m a, then what is the with a. value of m? (a) 1 / 5

(b) 1 / 5

(c) 1 / 29

(d) 1 / 29   14. The magnitude of the vector a and b are equal and the angle     between them is 60º. If the vector λ a 1 b and a 2 λ b are perpendicular to each other, then what is the value of λ? (a) 1 (b) 2 (c) 3 (d) 4     15. If a  3, b  4 and a  b )  7, then what is the value   of a  b ? (a) 3 (c) 1

(b)  4ˆi  3jˆ / 5

12. What is the vector in the xy-plane through origin and  perpendicular to the vector r  aˆi  bˆj and of the same length?

(b) aˆi 2 bˆj (d) bˆi 2 aˆj

(b) 2 (d) 0

16. Consider the diagonals of a quadrilateral formed by the vectors ˆ 3iˆ  6ˆj  2kˆ and 4iˆ  ˆj  3k. The quadrilateral must be a (a) Square (c) Rectangular

(b) Rhombus (d) None of these

AnSwER KEy 1. (a) 2. (b)

3. (a) 4. (c)

5. (a) 6. (c)

7. (d) 8. (d)

9. (b) 10. (a)

11. (a) 12. (d)

13. (d) 14. (a)

15. (c) 16. (b)

SOLUTIOnS 1. Let A 5 (0, 2, 2), B 5 (2, 0, −1) and C 5 (3, 4, 0)   Now AB  (2,  2,  3) and AC  (3,2,  2) 1   ∴ Area of triangle AB  AC 2 ˆi ˆj kˆ 1  2 2 3 2 3 2 2 1  [ˆi (4  6)  ˆj (  4  9)  kˆ (4  6)] 2 1  10ˆi  5ˆj  10kˆ 2 1 1  (10)2  (5)2  (10)2  225 2 2 15  2

2. From the figure it is clear that the angle between 6   2π b and  5a is 120 or 3

6b

120º

60º

b

α

5a  3. a

{( b  c) 3 (a  b c)} 0 









      a ⋅ ba  b b bc c a c  bc c

{

}

Vector Algebra ▌ 3.207       00a ⋅ bc 0a . c  b 0       a . bc a . bc 0    and for any three coplanar vectors d , e, f    d e ⋅f 0

)

(

(

(

)

)

(

)

)

(

Hence, statement (1) is correect.     4. ∵ a ⋅ b  a b cos α   ⇒ ⇒ cos α  a ⋅ b   Now, a  b  1 2 2   ⇒ a  b  2a ⋅ b  1 ⇒ 1 + 1  2cos α  1 2π 1 ⇒ cos α   cos 3 2 2π ⇒ α 3 5. Given vectors are coplanar, if 1 1 1 2 1 1  0 λ 1 λ ⇒ 1 ( λ  1)  1( 2λ  λ )  (2  λ )  0 ⇒ λ  1  3λ  2  λ  0 ⇒ 3λ  3 ⇒ λ  1 6. Both statements (1) and (2) are correct.

(

)

7. λ 2  4  1  1  ( 2  λ )2  4 ⇒ λ  4  1  1  4  λ 2  4λ  4 2

⇒ 4λ  4 ⇒ λ 1  8. Let b  (1, 1, 0)   a ⋅ b  (2,1, 1) ⋅ (1, 1, 0)  2 1 3  Hence, b  (1, 1, 0)    9. The vectors a = xiˆ  yjˆ + zkˆ, b  kˆ and c = yjˆ  xkˆ form a right handed system.

    a⋅b   10. The projection of b on a 5  5 a ⋅ b |a| 11. By taking option (a) 3ˆi  4jˆ 1  ⋅ 4ˆi  3jˆ  kˆ  (12  12) 5 5 Hence, option (a) true.

(

)

(

)

 12. Let r1  bˆi  ˆj   Now, r1 ⋅ r  ( b ˆi  a ˆj) ⋅ (a ˆi  b ˆj)  ab  ab  0 Hence, option (d) is the correct answer.  13. Given a  2iˆ  3jˆ  4kˆ  Also, bˆ  ma As bˆ is unit vector. Now, |2iˆ  3jˆ  4k66| = 4  9  16  29 Therefore. M should be

1 29

    14. Since, (λ a  b)2 ⋅ ( a  λb)  0     ⇒ λ | a |2  (1  λ )2 a ⋅ b  λ|b|2  0     ⇒ (1  λ )2 | a||b| cos 60  0 ∵ | a|  |b|

(

)

⇒ λ 1  2  2 2 2 15. Since, a  b  a  b  2  a  b     2 2 2 2 ⇒ a  b  7  2 [3  4 ]  2 ⇒ a  b 1   ⇒ a  b 1  16. Let d1  3iˆ  6ˆj  2kˆ  and d 2  4ˆi  ˆj  3kˆ   Now, d1 ⋅ d 2  3( 4)  6 (1)  2(3)  0  Hence, d1  33  62  22  7  d 2  42  12  33  26   ⇒ d1 ≠ d 2

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C

H

8

A

P

T

E

R

Statistics and Probability STATISTICS Mean

Weighted Mean

(i) For an ungrouped data If x1, x2 ... xn are n numbers, then the mean of numbers will be, x1  x2 x3  ...  xn n Σx . x n

x

(ii) For a grouped data If the frequencies of variables x1, x2, x3, ... xn be f1; f2, f3,....., fn, respectively, then the mean will be, x 

f1x1  f 2 x2  f3 x3  ...  f n xn f1  f 2  f3  ...  f n Σ fx . Σf

(iii) direct method If the mid-value of each class-interval is represented by x, then the mean is given by the formula Mean 5

Σ fx . Σf

(iv) Short-cut method d 5 x 2 A, then

If A is the assumed mean and

Σ fd Mean  A  . Σf (v) Step-deviation method

If A is the assumed mean and i x A , then is the class interval and u  i Σ fu Mean  A   i. Σf

If x1, x2, ... xn denote n values of a variable x and w1, w2, ... wn denote respectively their weights, then their weighted mean xw, is given by Σ wx xw 5 . Σw

Median If the n values in the raw data are arranged in ascending or descending order then the middle-most value is called the median. (i) If the number of values n in the raw data is odd, then the th  n 11 median will be the  value, when arranged in  2  order of magnitude. (ii) If the number of values n in the raw data is even, then th th n n  the median is taken to be the mean of   and  1 1  2 2  values, when arranged in order of magnitude.

Vital Statistics (i) Crude birth rate  It is defined as the number of birth  per 1000 of the population. CBR 

Number of births during a given year 1000 Mid- year population during the year

(ii) Crude death rate It is defined as the number of deaths  per 1000 of the population. CDR 

Number of death during a given year 1000 Mid- year population during year

3.210

▌ Mathematics

(iii) Specific death rate It is defined as the number of deaths  per 1000 population in a specified class in a given year.

where, S x 5 Strandardized population for group x and, D x 5 Specific death rate for group x.

Number of deaths in a age group in give year 1000 SDR  Mid- year population in given year

(vi) Price Index number

(iv) Infant mortality rate It is an important specific death  rate. It is the number of infants under one year of age dying in a year per 1000 live births of the same year.

(vii) Cost of living Index

IMR 

1 year in a given year 1000 Number of live births during the year

(v) Standardized death rate It is defined as follows: ST DR 5

Σ Sx D x Σ Sx

It is defined as follows: Current year's price 100 Price index number  Base yeaar's price

It is defined as follows: Σ p1i q0 i Cost of living index   100 Σ p0 i q0 i

where, p0i 5 Price of a commodity in the base year p1i 5 Price of the commodity in the current year q0i 5 Quantity of the commodity consumed in the base year.

CorrelATIon And regreSSIon Correlation

regression

Karl Pearson’s coefficient of correlation is denoted by r or p and covariance ( x , y )  σx  σ y 1 Σ(x  x ) ( y  y) n  1 1 Σ ( x  x )2 Σ ( y  y )2 n n 

Σ (u  u ) ( v  v ) Σ (u  u ) 2

( v  v )2

 Σu Σ v   Σ uv   n     (Σ v ) 2   (Σ u ) 2  2 Σu2     Σv    n   n  where, (x 2 a) 5 u and (y 2 b) 5 v for any a and b also 2 1 £ r £1

1. Regression line of y on x is ( y 2 y ) σy (x  x ) σx  byx ( x  x ) r

2. Regression line of x on y is ( x 2 x ) σx ( y  y) σy  bxy ( y  y ) r

Note that, (i) byx . bxy 5 r2 (ii) Both byx and bxy are either positive or negative. (iii) r is positive or negative according or byx or bxy are positive or negative, respectively.

ProbAbIlITy Random Experiment is an experiment which has the following two features:

(a) each element of S denotes an outcome of the experiment; and

A sample point or a case is an element of the sample space. An event is a subset of the sample space. A set of events is said to be mutually exclusive if occurrence of one of them precludes the occurrence of the remaining events from the set. A set of events is said to be equally likely if taking into consideration all the relevant evidences, there is no reason to expect one of them in preference to the others. A set of events is said to be exhaustive if the performance of the experiment results in the occurrence of at least one of them.

(b) any performance of the experiment results in an outcome that corresponds to exactly one point of S.

Classical Definition of Probability. If a random experiment results in N mutually exclusive, equally likely and exhaustive

(a) all possible outcomes of the experiment are known in advance. (b) the exact outcome of any specific performance of the  experiment is unpredictable (randomness). Sample Space associated with a random experiment is a set S of points such that

Statistics and Probability

(iii) P (exactly one of A, B, C occurs) 5 P(A) 1 P(B) 1 P(C) 2 2P(A Ç B) 2 2P(A Ç C) 2 2P(B Ç C) 1 3P(A Ç B Ç C), and (iv) P (exactly two of A, B, C occur)

cases, out of which n are favourable to the happening of an event A, then the probability of occurrence of A, usually denoted by P(A), is given by P(A) 5 n/N. Notation.

If A and B are two events, then

(i) A È B stands for the occurrence of at least one of A or B, A Ç B stands for the simultaneous occurrence of A and B, 2 A' or A or AC stands for non-occurrence or negation of A, A¢ Ç B¢ stands non-occurrence of both A and B, A Í B stands for ‘occurrence of A implies the occurrence of B’, and (vi) f stands for the impossible event.

(ii) (iii) (iv) (v)

Results 1.

If A is an event, then

(i) 0 £ P(A) £ 1 (ii) P(A) 1 P(A¢) 5 1. 2. If A and B are two events such that A Í B, then P(A) £ P(B). 3. If A, B are two events, then (i) P(A È B) 5 P(A) 1 P(B) 2 P(A Ç B) [In case A and B are mutually exclusive, P(A È B) 5 P(A) 1 P(B)], (ii) P [exactly one of A, B occurs] 5 P(A Ç B¢) 1 P(A¢ Ç B) 5 P(A È B) 2 P(A Ç B) 5 P(A) 1 P(B) 2 2P(A Ç B), (iii) P(A¢ Ç B¢) 5 1 2 P(A È B), and (iv) P(A¢ È B¢) 5 1 2 P(A Ç B). 4.

If A, B, C are three events, then (i) P(A È B È C) 5 P(A) 1 P(B) 1 P(C) 2 P(A Ç B) 2 P(A Ç C) 2 P(B Ç C) 1 P(A Ç B Ç C), (ii) P(at least two of A, B, C occur) 5 P(A Ç B) 1 P(A Ç C) 1 P(B Ç C) 2 2P(A Ç B Ç C),

▌ 3.211

5.

5 P(A Ç B) 1 P(A Ç C) 1 P(B Ç C) 2 3P(A Ç B Ç C). If A1, A2, ..., An are n events, then (i) P(A1 È A2 È ... È An) £ P(A1) 1 P(A2) 1 ... 1 P(An), and

6.

(ii) P(A1 Ç A2 Ç ... Ç An) ³ 1 2 P(A¢1) 2 P(A¢2) 2 ... 2 P(A¢n). (i) P(f) 5 0, and (ii) P(S) 5 1.

remark If x is the number of cases favourable to an event A, y the number of cases favourable to the event A¢, then odds in favour of A are x : y and odds against A are y : x. Conditional Probability The probability of occurrence of an event A given that B has already occurred is called the conditional probability of occurrence of A on the condition that B has already occurred. It is denoted by P(A/B). Compound Event When two or more events occur together, then a joint occurrence is termed as compound event. Theorem of Compound Probability If A and B are two events, then P(A Ç B) 5 P(A) P(B/A) if P(A) > 0 5 P(B) P(A/B) if P(B) > 0. If E1, E2, ..., En are n mutually exclusive and exhaustive events such that P(E6) > 0 for each i and A is an event, then P(A) 5 P(E1) P(A/E1) 1 P(E2) P(A/E2) 1 ... 1 P(En) P(A/En). bayes’ Theorem If E1, E2, ..., En (n > 1) are n mutually exclusive and exhaustive events such that P(Ei) > 0 (l £ i £ n) and A is an event, then for 1 £ i £ n, P(Ei/A) 5 P( E1 ) P( A / E1 ) P( E1) P( A / E2 ) 1 P( E2 ) P( A / E2 ) 1 ... 1 P( En ) P( A / En )

PrACTICe eXerCISe STATISTICS 1. The mean weight of 120 students in the second year class of a college is 56 kg. If the mean weights of the boys and that of the girls in the class are 60 kg and 50 kg respectively, then the number of boys and girls separately in the class are (a) 72, 64 (b) 38, 64 (c) 72, 48 (d) none of these 2. The mean of 100 items is 50 and their S.D. is 4. The sum of all the items and also the sum of the squares of the items is (a) 5000, 251600 (b) 4000, 251600 (c) 5000, 261600 (d) none of these

3. If the mean of the set of number x1, x2, ........, xn is , x then the mean of the numbers xi 1 2i, 1 ≤ i ≤ n is (a) x 1 2 n

(b) 1 n 1 1

(c) 1 2

(d) 1 n.

4. The weighted mean of first n natural numbers whose weights are equal to the squares of corresponding numbers is (a)

n 11 2

(b)

3n( n 11) 2 n( 2 n 11)

(c)

( n 11)( 2 n 11) 6

(d)

n( n 11) 2

3.212

▌ Mathematics

7 5 5. If a variable takes the discrete values x + 4,, x − , x − , 2 2 1 1 x − 3, x − 2, x + , x − , x + 5 ( x − g ), then the median is 2 2 5 1 (a) x  (b) x  4 2 5 (c) x  2 (d) x  4 6. If the s.d of a set of observations is 4 and if each observation is divided by 4, the s.d of the new set of observations will be (a) 4 (b) 3 (c) 2 (d) 1 7. If 25 % of the items are less than 20 and 25 % are more then 40, the quartile deviation is (a) 20 (b) 30 (c) 40 (d) 10 8. The A.M. of 2nC0, 2nC2, 2nC4,..., 2nC2n is (a)

2n ( n  1)

(b)

2 n 1

(c)

2 n 1

22 n n 1 n 1

(d)

2 ( n  1)

9. Median of 2nC0, 2nC1, 2nC2, 2nCn (when n is odd) is 1 2n ( c( n1)/ 2 2 n c( n1)/ 2 ) 2 (c) 2nCn (a)

(b)

2n

Cn

(d) none of these

10. In a factory, workers work in three shifts say shift 1, shift 2 and shift 3 and they get wages in the ratio 4 : 5 : 6 depending on the  shift 1, 2 and 3 respectively. Number of workers in the shifts are in the ratio 3 : 2 : 1. If total number of workers working is  1500 and wages per worker in Ist shift is Rs 400. Then mean wage of a worker is (a) Rs 467 (b) Rs 500 (c) Rs 600 (d) Rs 400 11. Mean of n items is x. If these n items are increased by 12, 22, 32, ..., n2 successively, then mean gets increased by ( n 1 1) ( 2 n 1 1) (a) 6 n2 (c) 2

n( n 1 1) ( 2 n 1 1) (b) 6 (d) remains same

12. The standard deviation of 25 numbers is 40. If each of the numbers is increased by 5, then the new standard deviation will be (a) 40 (b) 45 21 (c) 40 1 (d) none of these 25 13. If Q.D. is 16, the most likely value of S.D. will be (a) 24 (b) 42 (c) 10 (d) none of these

14. The sum of squares of deviations for 10 observations taken from mean 50 is 250. The coefficient of variation is (a) 50% (b) 10% (c) 40% (d) none of these 15. Let r be the range and S 2 

1 n ∑ ( x  x )2 be the S.D. of n  1 i 1 1

a set of observations x1, x2,..., xn, then (a) S  r

n n 1

(b) S  r

n n 1

(c) s  r

n n 1

(d) none of these

16. The S.D. of a variate x is σ. The S.D. of the variate where a, b, c are constants, is  a (a)   σ  c

(b)

 a2  (c)  2  σ c 

(d) none of these

ax 1 b c

a σ c

17. The standard deviation of a distribution is 30 and each item is raised by 3, then new S.D. is (a) 32 (b) 28 (c) 27 (d) none of these 18. Mean of the numbers 1, 2, 3,..., n with respective weights 12 1 1, 22 1 2, 32 1 3,..., n2 1 n is (a)

3n( n 1 1) 2( 2 n 1 1)

(b)

2n 11 3

(c)

3n 1 1 4

(d)

3n 1 1 2

19. The A.M. of a set of 50 numbers is 38. If two numbers of the set, namely 55 and 45 are discarded, the A.M. of the remaining set of numbers is (a) 38.5 (b) 37.5 (c) 36.5 (d) 36 20. The means of five observations is 4 and their variance is 5.2.  If three of these observations are 1, 2 and 6, then the other two are (a) 2 and 9 (b) 3 and 8 (c) 4 and 7 (d) 5 and 6 21. A sequence of odd positive integers is written as 1 3 5 7 9 11, 13, 15, 17, 19, 21, 23, 25, 27 The mean of the nth row is n3 ( 2 n2  1) 3 n( n  1) ( 2 n  1) (c) 6

(a)

n3 ( 4 n 2  2) 6 n( 2 n2  1) (d) 3 (b)

Statistics and Probability 22. The average salary of male employees in a firm was Rs 520 and that of females was Rs 420. The mean salary of all the employees was Rs 500. The percentage of male employees is (a) 80 (b) 60 (b) 40 (d) 20 23. An automobile driver travels from plane to a hill station 120 km distance at an average speed of 30 km per hour. He then makes the return trip at an average speed of 25 km per hour. He covers another 120 km distance on plane at an average speed of 50 km per hour. His average speed over the entire distance of 360 km will be 30 1 25 1 50 km / h 3 3 (c) km / h 1 1 1 1 1 30 25 50

(a)

(b) (30.25.50)

1 3

(d) none of these

24. The difference between the greatest and least value of observation is called (a) range (b) mean deviation (c) standard devition (d) variance 25. The variance of the first n natural numbers is n2  1 12 n2  1 (c) 6

(a)

n2  1 6 n2  1 (d) 12 (b)

26. The standard deviation for the set of number 1, 4, 5, 7, 8 is 24.45 nearly. If 10 are added to each number then the new standard deviation will be (a) 0.45 nearly (b) 24.45 nearly (c) 0.245 nearly (d) 12.45 nearly 27. The coefficient of variation is computed by Mean S.D S.D (c) 3100 Mean

(a)

S.D Mean Mean (d) 3100 S.D

(b)

28. The coefficient of correlation is independent of (a) origin but not scale (b) scale but not origin (c) both origin and scale (d) neither origin nor scale 29. If r is the correlation then, (a) r > 1 (c) | r| £ 1

(b) r £ 1 (d) |r|3 ³ 1

30. Quartile deviation for a frequency distribution is (a) Q  Q3  Q1 1 (c) Q  (Q3  Q1 ) 3

1 (b) Q  (Q3  Q1 ) 2 1 (d) Q  (Q3  Q1 ) 4

▌ 3.213

31. For a normal curve, the greatest ordinate is (a) 2πσ (c)

1 2πσ

(b) σ 2π (d)

1

σ 2π

32. If the standard deviation of 0,1, 2, 3, ........, 9 in K, then the standard deviation of 10, 11, 12, ...., 19 is (a) K (b) K 1 10 (c) K 1 10

(d) 10 K

33. The coefficient of correlation was defined by (a) Laplace (b) Pascal (c) De-Moivre (d) Karl-Pearson 34. For a given distribution of marks mean is 35.16 and its standard derviation is 19.76. The coefficient of variation is 35.16 19.76 (a) (b) 19.76 35.16 35.16 19.76 (c) 3 100 (d) 3 100 19.76 35.16 35. The average of n numbers X1, X2, X3, ......Xn is M. If Xn is replaced by x, then new average is nM  x n  x 1 (a) M  xn  x1 (b) n (c)

( n 1) M  x1 n

(d)

M  xn  x 1 n

36. The S.D. of the first n natural numbers (a) (c)

n 1 2 n2  1 12

(b)

n ( n 1) 2

(d) None of these

37. If m is the mean of a distribution, then åfi (yi 2 m) is equal to (a) M.D. (b) S.D. (c) 0 (d) None of these 38. The central value of the set of observations is called (a) mean (b) median (c) mode (d) G.M. 39. The mean of the first three term is 14 and means of next two  items is 18. The means of all the five terms is (a) 14.5 (b) 15.0 (c) 15.2 (d) 15.6 40. A group of 10 items has 6. If the mean of 4 of these is 7.5, then the mean of remaining terms is (a) 6.5 (b) 5.5 (c) 4.5 (d) 5.0 41. The arithmetic mean of an A.P. and the mean of first and last  term of an A.P. are (a) equal (b) unequal (c) square of each other (d) None of these

▌ Mathematics

3.214

42. The range of the following set of observations 2, 3, 5, 9, 7, 6, 5, 7, 4, 3 is (a) 11 (b) 7 (c) 5.5 (d) 6 43. The median of 10, 14, 11, 9, 18, 12, 6 is (a) 10 (b) 12 (c) 14 (d) 11 44. A firm of readymade garments make both men’s and women’s  shirts. Its profit average is 6 % of sales. Its profits in men’s shirts  average 8% of sales and women’s shirts comprise 60% of output. The average profit per sales rupee in women’s shirts is  (a) 0.0466 (b) 0.0166 (c) 0.0666 (d) none of these

46. Consider any set of observations x1, x2, x3,..., x101; it being given that x1 < x2 < x3 < ... < x100 < x101; then the mean deviation of this set of observations about a point k is minimum when k equals (a) x1 (b) x51 x1 1 x2 1 ... 1 x101 (d) x50 (c) 101 47. The mean weight of 9 items is 51. If one more item is added to the series the mean becomes 16. The value of the 10th item is (a) 35 (b) 30 (c) 25 (d) 20 48. If a variable takes values 1, 2, 3, ... , n with frequencies 12, 22 , ... , n2, then the mean is

45. The AM of n observations is M. If the sum of n 24 observations is a, then the mean of remaining 4 observations is nM  a 4 nM  a (c) 4

(a)

(b)

(a) Σn Σn3 Σn2 Σn3 (c) Σn (d) none of these (b)

nM  a 2

(d) nM  a.

AnSWer Key 1. 2. 3. 4. 5. 6.

7. 8. 9. 10. 11. 12.

(c) (a) (b) (b) (a) (d)

13. 14. 15. 16. 17. 18.

(d) (c) (a) (a) (a) (a)

(a) (b) (a) (b) (d) (c)

19. 20. 21. 22. 23. 24.

(b) (c) (d) (a) (c) (a)

25. 26. 27. 28. 29. 30.

31. 32. 33. 34. 35. 36.

(a) (b) (c) (c) (c) (b)

(c) (b) (d) (d) (b) (c)

37. 38. 39. 40. 41. 42.

(c) (b) (d) (d) (a) (b)

43. 44. 45. 46. 47. 48.

SoluTIonS 1. We know that the combined mean A

n1 A1  n2 A2 n1  n2

...(i)

Here A1 5 mean weight of boys 5 60 kg. A2 5 mean weight of girls 5 50 kg. A 5 combined mean 5 56 kg. and n1 1 n2 5 120.

...(ii)

So, from (I) and (II), 56 

2. Here n  100, A  50, σ  4. ∑ x ;∴Σx  nA  100  50  5,000. Now, A  n Again, from the formula, ∑ x 2 , we get Σx 2  n(σ2  A2 ). σ 2  A2  n ∴  n(σ2  A2 )  100(16  2500)  2,51,600.

n1 .60  n2 .50 ; 120

∴ 56 3 120 5 n1.60 1 (1202n1) 50 ; ∴ 120 (56250) 5 10n1 ; ∴ n1 5 72, n2 5 48. Thus, the number of boys 5 72 and the number of girls 5 48.

n

∑x

i

n

3. x  i 1  ∑ xi ⇒ nx n i 1 n

n

∴

∑ ( x  2i) ∑ x  2 (1  2  ...  n) i 1

i

n

 i 1

i

n n( n  1) nx 2 2   x  ( n  1) n

(a) (a) (a) (b) (c) (b)

Statistics and Probability 2 2 2 4. Weighted Mean  1.1  2.2  ....  n.n 2 2 2 1  2  ....  n n( n 1) n( n  1) Σn3 2 2  2 n( n 1) ( 2 n  1) Σn 6 3n( n  1)  2( 2 n  1)

12. If each item of a data is increased or decreased by the same constant, the standard deviation of the data remains unchanged. 3 13. We know that, S.D. Q.D. 2 3 ∴ S .D.   16  24. 2

7 5 5. Arrange the data as α  , α  3, α  , α  2, 2 2 1 1 α  , α  α  4, α 5. 2 2 Median  . 6. We know that if y 5 x/h when σy 5 σx /| h |. ∴ The s.d. of new set of observations will be 4/4 5 1. 40  20  10. 7. Q.D. 2

2n

C0 , 2nC2 , ... , 2nC2 n 

2n

2n

2n

Cn1/ 2 and Cn1/2 . So median is

Cn1/ 2  2nCn1/ 2 2

750  400  500  500  250  600 1500 Rs 467 per worker

∴ Mean 

11.

x

⇒ ⇒

σ  100 mean

5  100  10%. 50

15. We have r  max xi  x j i≠ j

1 n ∑ ( x  x )2 n  1 i 1 1

 x  x 2  ... x n  Now, ( x i  x )2   x i 1  n  

2

1 [(x  x1 )  (xi  x2 )  ...  (xi  xi  1) n2 i  (xi  xi  1)  ...  (xi  xn )]



1 [( n  1)r ]2 [ ∴ | xi  x j |  r ] n2 n

10. Workers in Ist shift 5 750, Wages in Ist shift 5 Rs 400 Workers is IInd shift 5 500 Wages in IInd shift 5 Rs 500 Workers in IIIrd shift 5 250, Wages in IIIrd shift 5 Rs 600





2n

2n 2n

Hence, coefficient of variation 

22 n 1 ( n  1)

9. C0, C1, C2, ... , Cn is even number of binomial coefficients  (when n is odd), and then middle terms are 2n

250  25  5 10

14. S.D. (σ ) 

and S 2 

8. (1 1 x)2n 5 2nC0 1 2nC1x 1 2nC2x2 1 2nC3x3 1 ...1 2nC2nx2n Put x 521 2n C022nC1 1 2nC222nC3 1 ... 1 2nC2n 5 0 ...(i) Now put x 5 1 2n C0 1 2nC1 1 2nC2 1 2nC3 1 ... 1 2nC2n 5 22n ...(ii) Adding (i) and (ii), we get 2n C0 1 2nC2 1 ... 1 2nC2n 5 22n1 A.M. of

▌ 3.215

x1  x2  x3  ... ...  xn n x1  12  x2  22  x3  32  ...  xn  n3 x n x1  x2  ...  xn 12  22  ...  n2 x n n ( n  1) ( 2 n  1) x x  6

⇒ ( x1  x )  r 2 ⇒ ∑ (x1  x )2  nr 2 i 1

⇒

1 n 1

n

∑ (x  x ) i 1

2

1



⇒ S r

nr 2 nr 2 ⇒S 2 ( n  1) ( n  1)

n n 1

ax  b a b i.e.,  y x  c c c a b i.e., y = Ax  B, where A  , B  c c ∴ y  Ax  B

16. Let y 

∴ y  y  A ( x  x ) ⇒ ( y  y ) 2  A2 ( x  x ) 2 ⇒ Σ ( y  y ) 2  A2  Σ ( x  x ) 2 ⇒ n.σ2y  A2 .σ2x ⇒ σ2y  A2 σ2x ⇒ σy  A σ x ⇒ σy  Thus, new S.D.

a σ. c

a σ c x

3.216

▌ Mathematics

17. S.D. of a series is unaltered if each item is raised (reduced) by the same scalar quantity, S.D. is independent of change of origin. Hence S.D. will be same as it was already. ∴ S.D. 5 30

∴ Tn of 1, 3, 11, 29, 61,...

element of nth row. Similarly, sequence of last terms of each row 5 1, 9, 27, 59,...

18. Here, for each xi 5 i, weight wi 5 i2 1 i n

Hence, the required mean 

Σwi xi Σwi

∑ i( i 

2

∴ tn 5

 i)

∑ (i

2

Hence, in the nth row elements can be written as

 i)

i 1

n

1 1 ( 2 n3  3n2  n  3), ... [2 n3  3n2  n  3] 3 3 (Note: adding 2 in the preceding number to get the succeeding  number) ∴ sum of the elements of nth row (using sum of n terms of A.P.)

n

2 2 ∑ i  ∑ i n ( n  1)  n( n  1) (2n  1) i 1 i 1 4 6  n  n n( n  1) ( 2 n  1) n( n 1) 2  ∑i ∑i 2 6 i 1 i 1 3





19. We have,

2

N n 2  4 n3  2 n  n3 2 ( A  L)     ( 2 n  1) 3 2 2  3  ∴ mean off the numbers in the nth row

n( n  1)  n( n  1) 2 n  1     2 3   2



 n( n  1)  2 n  1)  1  2 3  

n3 ( 2 n2  1) n ( 2 n2  1)  2 3 3 n

3n2  7 n  2 (3n  1) ( n  2) 3n  1   2(2 n  4) 4( n  2) 4

  N  n2 , A  Tn , L = tn   3 Here  4n  2n   ∴ A  L  Tn  tn    3

Σxi

 38,∴ Σxi  1900 50 New value of Σxi  1900  55  45  1800, and n  48

∴ New mean 

22. x1  520, x2  420 and x  500

1800 450 225    37.5 48 12 6

20. x  4, N  5 and

∑ (x  x )

Also, we know x 

 5.2

⇒ n1 : n2  4 : 1

∑ (x  x )2  (5.2)5

Hence, the percentage of male employees in the firm

∴∑ (x  x )2  26

 4    100  80%  4  1

∴ (1 4)  ( 2  4)  (6  4)  ( α  4)  (β  4)  26 2

2

2

2

4

where α, β are the other two observations. ∴ 9  4  4  ( α  4)2  (β  4)2  26

23. Average speed 

∴ ( α  4)4  (β  4)4  9 Also,

n1x1  n2 x2 n1  n2

⇒ 500 (n1  n2 )  520n1  420n2 ⇒ 20n1  80n2

2

N

1 [2n3 1 3n2 1 n23] 3

5 last element of the nth row.

i 1 n

1 5 (2n323n2 1 n 1 3) 5 first  3

1 2  6  α  β 4 5



∴ α  β  20  9  11 Clearly 4, 7 only satisfy the above eqaution in α, β

3 km / h 1 1 1   30 25 50

24. Range (by Def)

Hence reqd, number are 4, 7 21. The number of numbers in the nth row 5 n2   Sequence of first terms in different row is 1, 3, 11, 29, 61, ...

120  120  120 120 120 120   30 25 50

25.

n2  1 To get σ  12

n2  1 12 2 n −1 σ which is variance 12

Statistics and Probability 26. 24.45 S.D. remains same if each observation is increased by the same quantity, here each observation is increased by 10. 27.

S.D. 3 100. Formula of coefficient of variation. Mean

28. Both origin and scale.

37. zeros Σ fi ( y1 2 µ) 5 Σ fi yi 2 µ Σ fi 5 µ Σ fi 2 µ Σ fi 5 0. 38. Media is the central value. 39. 15.6 ∴ Mean 

29. Using Schwar’s inequality it can be proved that | r|  1 1 30. Q 5 (Q3 2 Q1) Def. of quartile deviation. 2 1

31.

7.5 

33. Karl Pearson. 19.76 34. 100 coefficient of variation. 35.16 S.D. 19.76   100  100 Mean 35.16 x1  x 2  x 3  ..... x n n nM  x1  x 2  x 3  ....  x 4



 

36. 

Σ yi 4

 ∑ x i  60

 ∑ yi  30

nM  x n  x1 n x1  x 2  x 3  ...  x n 1  x n  x n  x1

n



60  30  5.0. 60

a  ( a  d )  (a  2 d )  ... a  ( n  1) d

n     a n d 2 1 ( )   a  a  ( n  1) d n   n 2 2    Mean of first and least term or given A.P.

42. Range 5 Maximum observation 5 Minimum observation 5 9 2 2 5 7. 43. Arrange that data in ascending, i.e., 6, 8, 9, 10, 12, 14  7 1  ∴ Median   n observation  as n  7  2 

1

,∴ new avg

is odd. i.e., 4 th observation. ∴ Median  10.

nM  x n  x1 n

6

∴ A.M.

n

x1  x 2  .......  x n 1  x

Σ x i  Σ yi

41. Let the A.P. of ‘n’ terms be a, a 1 d, a 1 2d, a 1 (n 2 1)d

35. M 



10

3  14  2  18 78   15.6. 5 5

∴ Mean of remaining 6 items

32. K S.D. remain same if each observation is increased by same quantity, here each observation is increased by 10.



Σ xi

40. 5.06 

Property of normal distribution.

2Πσ

▌ 3.217

44. Here x 5 6, x1 5 8, n1 5 40, n2 5 60. Assuming that the total

n 1 S.D. of first n natural number 12

output is 100, we are required to find out x2 , we know that

2

2

n1x1  n2 x2 40  8  60  x2  40  60 n1  n2



 ∑x 1 ∑ x2   n  n 



n ( n  1) ( 2 n  1) n2 ( n  1)2  6n 4 n2



2 ( n  1) ( 2 n  1)  3( n  1)2 12

 



4 n  6 n  2  3n  6 n  3 12

45. Let the mean of the remaining 4 observations be X 1.



n 1 . 12

2

2

x

2

320  60 x2 100 600  320 280 14 ⇒ x2     4.66. 60 60 3 T   hus, the average profit in womens shirt is 4.66 per cent of sales  of Re 0.0466 per sale rupee. ⇒ 6

Then, M 

a  4 X1. nM  a ⇒ X1  . ( n  4)  4 4

3.218

▌ Mathematics

46. Mean deviation if minimum when it is considered about the item, equidistant from the beginning and the end i.e., the median. In this case, median is

101 1 1 th i.e., 51st item i.e., x51 2

47. Let the values of 9 items be x1, x2, …, x9 Therefore, mean of x1, x2, …, x9 is

⇒ x1, x2, …, x9 5 15 3 9 5 135 Let x10 be the 10th item. The mean of x1, x2, …, x9, x10 is 16. x1 ,  x 2  ...  x 9  x10  16 10 ⇒ x1, x2, …, x9, x10 5 160 ⇒ 135 1 x10 5 160 ⇒

∴ x10 5 1602135 5 25 3 48. 1.12  2.22  ...  n.n2 ∑ n  12  22  ...  n2 ∑ n2

x ,  x 2  ...  x 9 15  1 9

ProbAbIlITy 1. One of the events must happen. Given that the chance of one is two-third of the other, the odds in favour of the other are (a)  3 : 5 (b)  2 : 5 (c)  3 : 2 (d) none of these 2. For any two independent events E1 and E2, P{(E1 ∪ E2) ∩ (E1 ) ∩ (E2 )} is 1 4 1 (c)  2

(a) 

(b) 

1 4

(d) none of these

3. A determinant is chosen at random from the set of all determinants of order 2 with elements 0 or 1 only. The probability that value of the determinant chosen is positive is 5 16 3 (c) 16

(a)

(b)

7 16

(d) none of these

4. A man is known to speak truth 3 out of 4 times. He throws a dice and reports that it is six. The probability that it is actually six is 3 (a) 8 3 (c) 5

1 (b) 5 (d) none of these

5. Ram and Shyam throw with one dice for a prize of Rs 88 which is to be won by the player who throws 1 first. If Ram  starts, then mathematical expectation for Shyam is (a) Rs 32 (b) Rs 40 (c) Rs 48 (d) none of these

6. The probability that certain electronic conponent fails when first used is 0.10. If it does not fail immediately, the probability  that it lasts for one year is 0.99. The probability that a new component will last for one year is (a) 0.891 (b) 0.692 (c) 0.92 (d) none of these 7. A die is loaded so that the probability of face i is proportional to i, i 5 1, 2,...,6. The probability of an even number occuring when the die is rolled, is 2 7 3 (c) 7

(a)

(b)

4 7

(d) none of these

8. A student is given a true-false exam with 10 questions. If he gets 8 or more correct answers, he passes the exam. Given that he guesses at the answer to each question, the probability that he passes the exam, is 6 128 7 (c) 128

(a)

(b)

9 128

(d) none of these

9. The chance of an event happening is the square of the chance of a second event but the odds against the first are the cubes of  the odds against the second. The chances of happening of each event are 1 1 , 9 3 1 1 (c) , 3 6

(a)

(b)

1 1 , 6 9

(d) none of these

10. Four tickets marked 00, 01, 10 and 11 respectively are placed in a bag. A ticket is drawn at random five times, being replaced  each time. The probability that the sum of the numbers on the ticket is 15, is

Statistics and Probability

3 1024 7 (c) 1024

(a)

(b)

5 1024

(d) none of these

11. Numbers are selected at random one at a time, from the numbers 00, 01, 02,...,99 with replacement. An event E occurs if and only if the product of the two digits of a selected number is 18. If four numbers are selected, then the probability that E occurs at least 3 times, is 97 390625 72 (c) 390625

(a)

(b)

68 390625

(d) none of these

12. If A and B are such events that P(A) > 0 and P(B) ≠ 1, then P( A /B ) is equal to (a) 1 2 P(A / B )

(b) 1 2 ( P( A / B )

1 2 P(A ∪ B ) (c) P( B )

P( A) . (d) P( B )

13. An MBA applies for a job in two firms X and Y. The probability of his being selected in firim X is 0.7 and being rejected at Y is 0.5. The probability of at least one of his applications being rejected is 0.6. The probability that he will be selected in one of the firms, is (a) 0.6 (b) 0.4 (c) 0.8 (d) none of these 14. At the college entrance examination on, each candidate is admitted or rejected according to whether he has passed or failed the tests. Of the candidates who are really capable, 80% pass the test and of the incapable, 25% pass the test. Given that 40% of the candidates are really capable, then the proportion of capable college students is about (a) 68% (b) 70% (c) 73% (d) 75%. 15. The probability that a man aged x years will die in a year is p. The probability that out of n men A1, A2, A3,...An, each aged x, A1 will die and be first to die, is 1 n2 1 (c) 2 (1 2 (1 2 p ) n ) n

(a)

(b) 1 2 (1 2 p ) n (d)

1 (1 2 (1 2 p ) n ). n

16. Let 0 < P(A) < 1, 0 < P(B) < 1 and P(A ∪ B) 5 P(A) 1 P(B) 2P(A) P(B), then (a) P(B ∩ A’ ) 5 P(B) 2P(A) (b) P(A’ ∪ B’ ) 5 P(A’) 1 P(B’) (c) P(A ∪ B)’ 5 P(A’) P(B’) (d) P(A/B) 5 P(A).

▌ 3.219

17. 5 grils and 10 boys sit at random in a row having 15 chairs numbered as 1 to 15. The probability that end seats are occupied by the girls and between any two girls odd number of boys sit, is 20 3 10!3 5! 15! 20 3 5! (c) 15!

(a)

(b)

20 3 10! 15!

(d) none of these

18. A and B are two events. Odds against A are 2 : 1. Odds in favour  of A ∪ B are 3 : 1. If x ≤ P(B) ≤ y, then the ordered pair (x, y) is  5 3 (a)  ,   12 4  1 3 (c)  ,  3 4

2 3 (b)  ,  3 4 (d) none of these

19. Suppose X follows a binomial distribution with parameters n and p, where 0 < p < 1. If P(x 5 r) / P(X 5 n 2 r) is independent of n and r, then 1 2 1 (c) p 5 4

(a) p 5

(b) p 5

1 3

(d) none of these

20. A bag contains (2n 1 1) coins. It is known that n of these coins have a head on both sides, whereas the reamaining (n 1 1) coins are fair. A coin is picked up at random from the bag and tossed. If the probability that the toss results in a head is 31/42, then n is equal to (a) 10 (b) 11 (c) 12 (d) 13 21. If the mean of a binomial distribution is 25, then its standard deviation lies in the interval given below: (a) [0, 5) (b) (0, 5] (c) [0, 25) (d) (0, 25] 22. If the integers m and n are chosen at random between 1 and 100 then the probability that a number of the form 7m 1 7n is divisible by 5 is 1 1 (a) (b) 5 7 1 1 (c) (d) 49 4 23. If two events A and B are such that P(Ac) 5 0.3, P(B) 5 0.4 and P(ABc) 5 0.5, then P[B\ (A ∪ Bc)] 5 1 (a) (b) 3 1 (c) (d) none of these 4

3.220

▌ Mathematics

24. The probabilities of four cricketers A, B, C and D scoring 1 1 1 1 more than 50 runs in a match are , , and . It is known 2 3 4 10 that exactly two of the players scored more than 50 runs in a particular match. The probability that these players were A and B is 27 65 1 (c) 6

(a)

(b)

5 6

(d) none of these

25. The probabilty of sure event is (a) 1 (b) 2 1 (d) unlimited (c) 2 26. A dice is thrown once. Then the probability of getting a number greater than 3 is 1 2 (c) 6

(a)

2 3 (d) 0 (b)

27. The probabilty that a leap year selected at random will contain 53 Sundays is 1 7 6 (c) 7

(a)

2 7 6 (d) 14 (b)

28. Given two independent events A and B such that P(A) 5 0.30 and P(B) 5 0.60. Probability of not getting neither A or B is (a) 0.12 (b) 0.42 (c) 0.13 (d) 0.28 29. From a deck of 52 cards the probabilty of drawing a court card is 4 13 1 (c) 13

(a)

3 13 1 (d) 14 (b)

30. If three dice are thrown simultaneously the probabilty of getting a score of 5 is 5 216 1 (c) 36

(a)

1 6 1 (d) 72 (b)

31. A and B toss 3 coins. The probability that both obtain the same number of heads is 1 (a) 9 5 (c) 16

3 (b) 16 3 (d) 8

32. The probability that a card drawn at random from a pack of 52 cards is a king or heart is 1 13 1 (c) 4

(a)

1 52 16 (d) 52 (b)

33. From each of the four married couples, one of the partners is selected at random. The probabilty that those selected are of the same sex is 1 2 1 (c) 8

(a)

1 4 1 (d) 16 (b)

34. If A and B are two mutually exclusive events such that P(B) 5 2 P(A) and A È B 5 S, then P(B) is equal to 1 2 2 (c) 3

(a)

1 3 3 (d) 4 (b)

35. A coin is tossed 6 times. The probability of getting a head three times is 1 2 1 (c) 8

(a)

3 64 5 (d) 16 (b)

36. Two fair dice are tossed. The probability that both show even numbers is 1 4 1 (c) 6

(a)

1 3 5 (d) 36 (b)

37. The probability that a card drawn out of a packet of 52 is of diamond is 0 52 1 (c) 13

(a)

1 52 1 (d) 4 (b)

38. In a single throw of a pair of dice, p(a total > 1) is equal to (a) 0

(b) 1

1 (c) 2

(d)

1 8

39. In simultaneous tossing of two coins, the problem of getting 2 tails is (a) 0 (b) 1 (c)

1 2

(d)

1 4

▌ 3.221

Statistics and Probability 40. Vikas throws three coins. The probability of at least one head turning up is 1 1 (a) (b) 3 8 7 1 (c) (d) 8 4 41. A bag contains 7 white and 4 black balls, out of them one is drawn at random. The probability that the ball drawn will be red or black is 9 7 (a) (b) 20 20 13 11 (c) (d) 20 20 42. A problem in statistics is given to three students whose chance 1 1 1 of solving it are , , . The probability that the problem 2 3 4 solved is 1 1 (a) (b) 2 3 3 (c) (d) 1 4

1 8 1 (c) 81

2 11 20 (c) 121

3 11 42 (d) 121

4 , then value of P(x 5 0) is 3 1 64 (a) (b) 8 729 8 1 (c) (d) 729 729 47. The mean number of sixes in two tosses of a die is 1 2 1 (c) 4 (a)

(b)

1 3

(d) None of these

B (a) P(A)  P( B )  2 P( A) P    A  A (b) P(A)  P( B )  2 P( A) P   B (c) P(A)  P( B)  2 P( A ∪ B ) (d) None of these 49. A die is thrown three times getting a 5 or 6 considered as success. Then, the probability of three success will be

44. The letters are sent to different persons and addresses on the three envelopes are written at random. The probabilty that the letters go into the right envelopes is 1 27 1 (c) 9

(b)

48. P(A È B) 5 P(A Ç B) if the relation between P(A) and P(B) is

(b)

(a)

(b)

46. If the mean and variance of a binomial distribution are 2 and

43. One bag contains 6 blue balls and 6 green balls and another bag contains 7 blue and 4 green balls. Two balls are drawn, one from each bag. The probability of both being blue is (a)

1 27 1 (d) 16

(a)

1 6

(d) None of these

16 , then 45. If a binomial distribution n 5 4, P(x 5 0) 5 81 P(x 5 4) is

2 9 1 (c) 27

(a)

(b)

26 27

(d) None of these

50. If A and B are arbitary events, then (a) P(A Ç B) ³ P(A) 1 P(B) (b) P(A È B) ³ P(A) 1 P(B) (c) P(A Ç B) 5 P(A) 1 P(B) (d) None of these

AnSWer Key 1. 2. 3. 4. 5. 6. 7.

(c) (a) (c) (a) (b) (a) (b)

8. 9. 10. 11. 12. 13. 14.

(c) (a) (b) (a) (c) (c) (a)

15. 16. 17. 18. 19. 20. 21.

(d) (c, d) (a) (a) (a) (a) (a)

22. 23. 24. 25. 26. 27. 28.

(a) (c) (a) (a) (a) (b) (d)

29. 30. 31. 32. 33. 34. 35.

(b) (c) (c) (d) (c) (c) (d)

36. 37. 38. 39. 40. 41. 42.

(c) (d) (b) (c) (d) (d) (c)

43. 44. 45. 46. 47. 48. 49

(d) (b) (c) (b) (c) (a) (c)

50 (b)

3.222

▌ Mathematics

SoluTIonS 1. Let the given events be E1 and E2 2 Then P(E1 )  P(E2 ) 3 The events E1 and E2 are exhaustive ∴ p ( E1 ∪ E2 )  1 Since E1 and E2 are mutually exclusive p ( E1 )  p( E2 )  1 2 5 3 P( E2 )  P( E1 )  1 ⇒ P( E2 )  1 ⇒ P( E2 )  3 3 5 3 2 3 2 ∴ P( E2 )  and P( E1 )    5 3 5 5 3 Since P( E2 )  , odds in favour of E2 are 3: 5  3 i.e. 3: 2 5 ⇒

2. Since E1 ∩ E2  E1 ∪ E2

(

)

and ( E1 ∪ E2 ) ∩ E1 ∪ E2  φ 1 ∴ P [( E1 ∪ E2 ) ∩ ( E1 ∩ E2 )]  P (φ )  0  . 4 3. Since each element of the determinant can be placed in two ways 0 or 1, total number of ways 5 24 5 16. Since value of the determinant is 1ve, so we have only 3 cases: 1 0  1 1 1 0   ,  ,   0 1 0 1 1 1 Hence the reequired probability 5

3 . 16

4. Let A denotes the event that a six occurs and B the event that the man reports that it is a six. Then the probability that it is actually a six is given by P( A ∩ B ) P( A | B )  P(B ) 1 3 3 Now P( A ∩ B )    6 4 24 P( B )  P( A ∩ B )  P( ∩ B ) 1 3 5 1 8      6 4 6 4 24 3 / 24 3  . Hence P( A | B )  8 / 24 8 5. Probability of Shyam winning 5 1 5 5 5 1        ... 6 6 6 6 6 6 2  5  25  25   1      ... 36  36  36     1 5 36 5 5  .    36 1 25 / 36 36 11 11

Mathematical expectation for Shyam  Rs 5  88 11 5 Rs 40. 6. Given: probability, that electronic component fails when first  used 5 0.10 i.e., P(F) 5 0.10 ∴ P(F’ ) 5 1 2P(F) 5 0.90 and let P(Y) 5 Probability of new component to last for one year Obviously, the two events are mutually exclusive and exhaustive. ∴ P(Y/F) 5 0 and P(Y/F’) 5 0.99 Y  Y  ∴ P(Y )  P(F ) .P    P( F ') . P   F   F 5 0.10 3 0 1 0.90 3 0.99 5 0 1 (0.9) (0.99) 5 0.891. 7. Since the probability of the faces are proportional to the numbers on them, we can take the probabilities of faces, 1, 2,... 6 as k, 2k, ..., 6k, respectively. Since one of the faces must occur, we have k 1 2k 1 3k 1 4k 1 5k 1 6k 5 1 1 or k 5 . 21 ∴ The probability of an even number 1 4 5 . 5 2k 1 4k 1 6k 5 12k 5 12 3 21 7 8. n 5 total no. of ways 5 210 5 1024. Since each answer can be true or false. and m 5 favourable number of ways 5 10C8 1 10C9 1 10C10 5 45 1 10 1 1 5 56. since to pass the exam, he must give 8 or 9 or 10 true answers, m 56 7 Hence, p 5 5 5 . n 1024 128 9. Let the chance of the second event be p. Then the chance of the first event is p2. ∴ Odds against the first event are as 1 2p2 : p2 and odds against the second event are 12p : p Hence according to the condition given in the question, we have 3 1 p2  1 p    p  p2 or

(1  p )(1  p ) (1  p )3  p2 p3

or

p ( p  1)  (1  p )2

or

p 2  p  p 2  2 p  1 or 3 p  1

∴ p

1 and 3

p2 

1 9

1 9 1 and the probability of the second event  . 3 Hence the probability of the first event 

Statistics and Probability 10. Let S be the sample space and E be the required event. Now, n(S) 5 total number of cases 5 45 5 1024. and n(E) 5 coefficient of x15 in (x0 1 x 1 x10 1 x11)5 5 coefficient of x15 in [(1 1 x)5 (1 1 x10)5] 5 coefficient of x15 in (1 1 5x 1 10x2 1 10x3 1 5x4 1 x5) 3 (1 1 5x10 1 10x20 1 ...) 5 5 ∴ Required probability, P( E ) 5

n( E ) 5 5 . n( S ) 1024

11. Out of the numbers 00, 01, 02, ..., 99, those numbers the product of whose digits is 18 are 29, 36, 63, 92 i.e, only 4. 4 1 1 24  , q  P ( E ) 1  100 25 25 25 Let X be the random variable, showing the number of times E occurs in 4 selectios. Then P(E occurs at least 3 times) 5 P(X 5 3 or X 5 4) 5 P(X 5 3) 1 P(X 5 4) 5 4C3 p3 q1 1 4C4 p4 q0 5 4p3q 1 p4 p  P(E ) 

3

4

 1 24  1  97 4      25  25  390625  25  12. P( A / B )  

P( A ∩ B ) P( B )

1  P( A ∩ B ) P( B )



1  P( A ∪ B ) P( B )

13. Let A and B denote the events that the person is selected in firms X and Y respectively. Then in the usual notations, we are given: – P (A) 5 0.7 ⇒ P (A) 5 120.7 5 0.3 – P (B) 5 0.5 ⇒ P (B) 5 120.5 5 0.5 – – and P (A ∪ B) 5 0.6 The probability that the persons will be selected in one of the two firms X or Y is given by : P(A ∪ B) 5 12 P – – – – 1 2 [P (A) 1 P (B) 2 P (A ∪ B )] 5 12(0.3 1 0.520.6) 5 0.8. 14. Let A be the event that a really able candidate passes the test and let B be the event that any candidate passes this test. Then we have P(B/A) 5 0.8, P(B/Ac ) 5 0.25 P(A) 5 0.4, P(Ac ) 5 120.4 5 0.6 By Baye’s formula P ( A) P( B / A) P(A/B) 5 P( A) P( B / A)  p( Ac ) P( B / A)) 32 0.32   68%  0.32 + 0.15 47

▌ 3.223

15. The probability that a man does not die in a year 5 12p ∴ Probability that none of the n men dies in a year 5 (12p)n ∴ The probability that at least one man dies in a year 5 12(12p)n    Since every man can die first, the chance that A1, will die first  is . Hence the probability that A1, will die within a year and he 1 will be first to die   [1  (1  p) n ] n 16. P(A ∪ B) 5 P(A) 1 P(B)2P(A) P(B) ⇒ P(A) 1 P(B)2P(A ∩ B) 5 P(A) 1 P(B)2P(A) P(B) ⇒ P(A ∩ B) 5 P(A) P(B) ∴ A and B are independent events. ⇒ P(B ∩ A’ ) 5 P(B) P(A’ ) ≠ P(B)2P(A) and P(A’ ∪ B’ ) 5 P(A ∩ B)’ 5 12P(A ∩ B) 5 12P(A) P(B) ≠ 12P(A) 1 12P(B) 5 P(A’ ) 1 P(B’). Also P[(A ∪ B)’] 5 P(A’ ∩ B’ ) 5 P(A’ ) P(B’) Since A and B are independent P(A | B) 5 P(A). 17. There are four gaps in between the girls where the boys can sit. Let the number of boys in these gaps be 2a 1 1, 2b 1 1, 2c 1 1, 2d 1 1 5 0 , then 2a 1 1 1 2b 1 1 1 2c 1 1 1 2d 1 1 5 10 or a 1 b 1 c 1 d 5 3 The number of solutions of above equation 5 coefficient of x3 in (12x) 4 5 6C3 5 20 Thus boys and girls can sit in 20 3 10 ! 3 5 ! ways Total ways 5 15 ! Hence, the required probability 

20  10! 5! . 15!

1 3 18. We have, P(A)  and P(A ∩ B )  3 4 ∴ P(A ∩ B )  P( A)  P( B )  P(A ∩ B ) 3 1 ⇒  + P( B ) − P( A ∩ B ) 4 3 ⇒

5  P( B )  P(A ∩ B ) 12

P( B ) 

1 5 5 P ( A ∩ B ) ⇒ P( B )  12 2

Again, P(B ) 

5 + P( A ∩ B ) 12

⇒ P( B)  5 / 12  P( A)

[ P(A ∩ B)  P(A)]

5 1 3 ⇒ P( B )    2 3 4

...( ii )

From (i ) and (ii), we obtain 5/12  P(B )  3 / 4 Hence, x  5 / 12 and y  3/4.

▌ Mathematics

3.224

19. We have, n



n

P( X  r ) P( X  n  r )

Cr p r (1  p ) nr

Cn  r p

nr

(1  p )

r



n2 r

(1  p ) n2 r p n2 r n2 r

 1 p  1      1   p  p  1 and  1  0. ∴ the ratio will be independent of n p and r if

1 1  1  1 or p  . 2 p

20. Both heads appear on n coins and head and a tail appear on (n 1 1) coins so P(head) 

n

C1

2 n 1

C1

⋅ 1

n 1

C1

1 31 ⋅  2 n 1 C1 2 42

n 31 n   ⇒ 2 n 1 2( 2 n  1) 42 ⇒ 2n 1 n 1 1 5 (2n 1 1) (31/21) ⇒ 63n 1 21 5 62n 1 31 ⇒ n 5 10. 21. We have, np 5 25. Now, 0 ≤ p < 1 and 0 ≤ q ≤ 1 npq  np



⇒ 0 ≤ n pq ≤ np ⇒ 0 ≤



⇒ 0 ≤ S.D. ≤ 5. But p ≠ 0, therefore 0 ≤ S.D. < 5 ⇒ S.D. ∈ [0, 5).

22. We know 7k, k ∈ N, has 1, 3, 9, 7 at the units place for k 5 4p, 4p21, 4p22, 4p23 respectively, where p 5 1, 2, 3, ... Clearly, 7m 1 7n will be divisible by 5 if 7m has 3 or 7 in the unit place and 7n has 7 or 3 in the units place or 7m has 1 or 9 in the units place and 7n has 9 or 1 in the units place. ∴ For any choice of m, n the digit in the units place of 7m 1 7n is 2, 4, 6, 0 or 8. It is divisible by 5 only when this digit is 0. 1 ∴ the required probability 5 –. 5 23. P(A) 5 12P(Ac ) 5 120.3 5 0.7, P(B) 5 0.4 We know that P(A) 5 P(A ∩ B) 1 P(A ∩ Bc ) ⇒ 0.7 5 P(A ∩ B) 1 0.5 or P(A ∩ B) 5 0.720.5 5 0.2 P(A ∪ Bc ) 5 P(A) 1 P(Bc )2P(A ∩ Bc ) 5 0.7 1 (120.4)20.5 5 0.8 P [ B  ( A ∪ Bc ) Next P [ B | P( A ∪ B c )] 5 P[ A ∪ B c ] From set theory B  ( A ∪ B c ) 5 ( B  A) ∪ ( B  B c ) 5 ( B  A) ∪ φ 5 B  A P( A ∩ B ) 0.2 1 5 5 . ∴ Required Probability 5 P( A ∪ B c ) 0.8 4

24. Let E1 be the event that exactly two players scored more then 1 1 3 9 50 runs then P(E1)     2 3 4 10 1 2 1 9 1 2 3 1    ×     2 3 4 10 2 3 4 10 1 1 1 9 1 1 3 1         2 3 4 10 2 3 4 10 1 2 1 1 65      2 3 4 10 240 Let E2 be the event that A and B scored more than 50 runs, then 1 1 3 9 27 P(E1 ∩ E2 )      2 3 4 10 240 ∴ Required probability 5 P(E2/E1) 5

P( E1 ∩ E2 ) 27 5 P( E1 ) 65

25. Def. of probability. 26. Favourable cases are 4, 5, 6 3 1 ∴ Required probability 5 5 . 6 2 27. In a leap year, there are 52 weeks and two additional days which are Sunday and Monday, or Monday and Tuesday, or Tuesday and Wednesday, or Wednesday, Thursday or Thursday, Sunday and Sunday. 2 \ required probability 5 . 7 28. P(A  B) 5 P(A  B) 5 1 2 P(A È B) Þ 2P(P(A) 1 P(B) 2 P(A Ç B)) 5 1 2 P(A) 2 P(B) 1 P(A Ç B) 5 12 0.30 2 0.60 1 (0.30)(0.60) 5 0.28. 29. Total number of equally like cases 5 52. Number of court cards 5 12 12 3 \ Required probability 5 5 . 52 13 30. Total number of equally likely events are 6 3 6 3 6 5 216 and favourable outcomes are (2, 2, 1), (2, 1, 2), (1, 2, 2), (3, 1, 1), (1, 3, 1), (1, 1, 3), so required probability 5

6 1 5 . 216 36

31. P(both obtain some number of head) 5 P(no head or no head; head or two heads or three hands) 1 1 3 3 3 3 1 1 ⇒ ,  ,  ,  ⋅ 8 8 8 8 8 8 8 8 20 5   . 64 16

▌ 3.225

Statistics and Probability 32. If A be the event that there is a king and B be the event that there is a heart,

1 2 3 3 1 ⋅ ⋅  . 2 3 4 4

4 13 1 ; P( B )  ; P( A ∩ B )  52 52 52 P(A ∩ B )  P(A)  P(B )  P(A ∩ B )

P( A)  ∴

33. Required probability 5

2 1 1 5 35 . 4 8 2 2

34. P(A  B) 5 P(A) 1 P(B) Þ

P(A È B) 5 P(A) 1 2P(A) 5 3P(A)

Þ

P(S) 5 3P(A) Þ 1 5 3P(A) Þ P(A) 5

1 3

1 2 P(A) 1 P(B) 5 1 Þ P(B) 5 1 2 P(A) 5 1  . 3 3 35. Required probability 5

6

C3

26

5

20 5 5 . 64 16

36. Total number of equally likely cases 5 6 3 6 5 36. Favourable cases are 2, 4, 6, 2, 4, 6 ∴ Required probability 5

6 1 5 . 36 6

46. Mean  nP  2, variance  npq =

4 3

4  2 1 nPq 3 2  ⇒ q  ; ⇒ P = (1  q )   1    ⇒ 3 3 nP 2 3  6

1 2 64 ∴ n    2 ⇒ n  6; ∴ P( x  0)  q 6     . 729  3 3 1 6

∴ Mean 5 np 53

6 38. Required probability 5 5 1. 6

1 1 5 . 6 3

48. ⇒ P(A ∪ B) ⇒ P(A ∩ B) (given) ⇒ P(A) 1 P(B) 2 P(A ∩ B) 5 P(A ∩ B) 2P(A ∩ B) 5 P(A) 1 P(B) B P(A) 1 P(B) 5 2P(A ∩ B) 5 2P(A)P   .  A

26 1 5 . 52 2

1 1 1 1 40. Probability of one head (i.e., TTT) 5 , , 5 2 2 2 8 1 7 Required probability  1   . 8 8 41. P(red or white) 5 P(red) 1 P(white) 

1 1 ∴ P  1  q  ; ∴ P ( x  4 )  4 C4  P 4  . 81 3

47. n 5 2, P 5

13 1 37. Required probability 5 5 . 52 4

39. Required probability 5

6 7 42   . 11 11 121 44. There are three letters and three directed envelopes, therefore they can be put into envelope in 3p3 5 3 5 6 ways out of which only one is correct. 1 ∴ required probability 5 . 6 2 2 16 2 45. P ( x  0)  ⇒ q 4 =   ⇒ q  81 3  3 43. Required probability 

4 13 1 16   = . 52 52 52 52

⇒

 1  1  1 42. Required probability  1   1    1    1   2  3  4 

9 4 13   . 20 20 20

6

49. P 5 2 5 1 , q 5 2 ; and P ( x 5 3) 5 3C  2   1  3 6 3 3  3   3 1 5 . 27

50. P(A ∪ B )  P(A) + P(B )  P(A ∩ B )  P( A )  P( B ).

ndA boX 1. What is the standard deviation? 5 2 4 (c) 3

(a)

5 3 16 (d) 9 (b)

3

2. What is the coefficient of variance? (a) (c)

200 3 600 5

(b)

50 5 9

(d) 150

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▌ Mathematics

3. A coin is tossed 10 times. The number of heads minus the number of tails in 10 tosses is considered as the outcome of the experiment. What is the number of points in the sample space? (a) 10 (b) 11 (c) 21 (d) 99 4. In a study on the relationship between investment (X) and profit (Y) the following two regression equation were obtained based on the data on X and Y. 3X  Y  12 = 0 X  2Y  14 = 0

→

What is the mean X ? (a) 6 (c) 4

(b) 5 (d) 2

5. Following table gives the mean and variance of monthly demand for products A, B, C and D in a supermarket. Product

A

B

C

D

Mean demand

60

90

80

120

Variance

12

25

36

16

For which product, the demand is consistent? (a) Product A (b) Product B (c) Product C (d) Product D 6. What is the least value of the standard deviation of 5 integers, no two of which are equal? (a)

5

(b) 2 (c)

2

(d) No such lest value can be computed 7. What is the value of n for which the numbers 1, 2, 3 ……….n have variance 2? (a) 4 (b) 5 (c) 6 (d) 8 8. X 5 1 2 3 4 Frequency 2 3 f 5 The frequency distribution of a discrete variable X with one missing frequency f us given above. If the arithmetic mean of X is

23 , what is the value of the missing frequency? 8

(a) 5 (b)6 (c) 8 (d) 10

9. An observed event B can occur after one of the three events A1, A2, A3, if P(A1) = P(A2) = 0.4, P(A3) = 0.2 and P(B/A1) = 0.25, P(B/A2) = 0.4 P(B/A3) = 0.125, what is the probability of A1 after observing B? 1 3 20 (c) 57

(a)

6 19 2 (d) 5 (b)

10. Consider the following statements 1. The probability that there are 53 Sundays in a leap year is twice the probability that there are 53 Sundays in a non-leap year. 2. The probability that there are 5 Mondays in the month of March is thrice the probability that there are 5 Mondays in the month of April. Which of the statements given above is/are correct? (a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2 11. In tossing three coins at a time, what is the probability of getting at most one head? 3 7 (a) (b) 8 8 1 1 (c) (d) 2 8 12. Two balls are selected from a box containing 2 blue and 7 red balls. What is the probability that at least one ball is blue? 2 9 5 (c) 12

(a)

7 9 7 (d) 12 (b)

13. The probability of guessing a correct answer is

x . If the 12

2 probability of not guessing the correct answer is , then what 3 is x equal to? (a) 2 (b) 3 (c) 4 (d) 6 14. Three letters are randomly selected from the 26 capital letters of the English Alphabet. What is the probability that the letter ‘A’ will not be included in the choice? (a) 1/2 (b) 23/26 (c) 12/13 (d) 25/26

Statistics and Probability 15. Two numbers X and Y are simultaneously drawn from the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. What is the conditional probability of exactly one of the two numbers X and Y being even, given (X + Y) = 15? (a) 1 (b) 3/4 (c) 1/2 (d) 1/4

▌ 3.227

16. The chance of winning the race of the horse A is 1/5 and that of horse B is 1/6. What is the probability that the race will be won by A or B? (a) 1/30 (b) 1/3 (c) 11/30 (d) 1/15

AnSWer Key 1. (c) 2. (a)

3. (b) 4. (d)

5. (d) 6. (c)

7. (b) 8. (b)

9. (c) 10. (a)

11. (c) 12. (a)

13. (c) 14. (b)

15. (a) 16. (c)

SoluTIonS  1. ∵ x  2 ( given ) x

f

f (x  x )

0 2

4

3

12

1 0 1 2

1 0 1 4

4 4 4 3

4 0 4 12

1 2 3 4

xx

(x  x )

Hence, total number of points in the sample in the space is 11. 2

18 32 Now, SD =

Σ f (x  x ) N

16 4 32    9 3 18 2. Coefficient of variance

σ  100 x 4 1    100 3 2 200  3 

3. Head Tail 10 0 9 1 8 2 7 3 6 4 5 5 4 6 3 7 2 8 1 9 0 10

2

2

4. Since, lines of regression passes through ( X , Y ). ∴3 X  Y  12  0 .........(i ) and X  2Y  14  0 .........(ii ) On solving Eqs.(i) and (ii) we get X  2 and Y  6 5. Since, coefficient of variance 5

SD mean

Coefficient of variance A 5

12 3.46 5 5 0.057 61 61

Coefficient of variance B 5

25 5 5 5 0.055 90 90

Coefficient of variance C 5

36 6 5 5 0.075 80 80

16 4 5 5 0.033 120 120 Hence, we see that minimum coefficient of variancee is D, hence product is consistent.

Coefficient of variance D 5

6. Let us consider any five integers are 3, 4, 5, 6, 7 ∴ SD = 

(5  3)2  (5  4)2  (5  5)2  (5  6)2  (5  7 )2 5

4 1 0 1 4  2 5

7. For n  5, x = 3

∑ (x

 n )2 n 4 1 0  4 1 2 ⇒ 2 5 Variance  σ 2 

i

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▌ Mathematics 12. Required probability = P (one ball is blue) + P (both ball is blue)

23 8 1 2  2  3  3  f  4  5  2  3 f  5 23 2  6  3f  20 ⇒  8 10  f ⇒ 230  23f  224  24f ⇒ f  230  224  6

8. A.M 5

2 7 2 1     9 8 9 8 14 2 16 2     72 72 72 9

9. Required probability = P (A1/B) 

P(A1 ) P( B1 / A1 ) P(A1 ) P( B / A1 )  P( A2 ) P( B / A2 )  P( A3 ) P( B / A3 )

0.4  0.25 0.4  0.25  0.4  0.4  0.2  0.1225 0.1 0.1 20    0.1  0.16  0.025 0.285 57 

10. (1) ∴ Probability of 53 Sundays in a leap year = 1 2 2    4 7 28



13.  Probability of guessing a correct answer x 5 12 And probability of guessing a wrong answer. 2  3 x 2  1 12 3 x 8 1 12 ⇒ x  12  8  4 ⇒

2 7

14. ∴ Required probability

1 and probability of 53 Sundays in a non-leap year = 7



∴ Statements (1) is true. 3 (2) Probability of 5 Mondays in March 5 7 2 and probability of 5 Mondays in April 5 7 Statement II is wrong.



11. Possible samples are as follows {HHH , HTH , HHT , THH , TTH , THT , HTT , TTT } Required probability = Probability of getting one heat + Probability of getting no head =

3 1 4 1    8 8 8 2

25

C3

26

C3

25  24  23 23  26  25  24 26

15. Given, X + Y = 15 The total number of ordered pairs =(5,10) ,(6,9), (7,8), (8,7), (9,6), (10,5) ∴n(S) = 6 In each above pairs exactly one is even number. ∴n(E) = 6 6 ∴ Required probability = 51 6 16.

1 1 6  5 11    5 6 30 30

additional Support Material algebra 1. The set having only one subset is (a) {} (b) {0} (c) {1} (d) None of these 2. If A ∩ B 5 φ, then A ∪ B’, is (a) A (b) B (c) U2B (d) ϕ 3. If (y−2, x+1) = (x −1, 2), then values of x and y are respectively (a) 1, 2 (b) 2, 3 (c) 1, −2 (d) 1, −2 4. The number of relations on set A containing n elements is (a) 2 n (b) 22 n (c) 2 n

9. Vectors ((α1 , α2 ), (β1 , β 2 ) of V2 ( F) are linearly dependent if (a) α1β 2  α2β 2 ≠ 0 (b) α1β 2  α2β1  0 (c) α1β 2  α2β 2  1 (d) None of these 10. Which of the following statements is not true? (a) Addition is associative for natural numbers. (b) Subtraction is commutative for natural numbers. (c) Multiplication is associative for natural numbers. (d) Addition is commutative for natural numbers. 11. A non-zero integer p ≠ 1, is called a prime iff its only divisor (s) is /are (a) 1

(b) p

(c)  p

(d) 1,  p

2

(d) None of these 5. If f: R → R  , f(x)  x 2 then f is (a) injective but not surjective (b) surjective but not injective (c) bijective (d) None of these 6. Let a and b, be two elements of a group (G, *), then ∀a, b ∈ G, (a * b)21 equals (a) a 21 * b21 (b) a * b (c) b21 * a 21 (d) None of these 7. If R is a ring with zero element x, then ∀ a ∈ R, a.x field (a) a (b) x (c) 0 (d) None of these 8. Which of the following algebraic structures is not a field? (a) (Q, + , ×) (b) (R, + , ×) (c) (C, +, ×) (d) (I, +, ×)

12. The least positive integer modulo 5 to which 183 is congruent is (a) 3 (b) 5 (c) 36 (d) None of these 13. Between any two real numbers there lie (a) No irrational number (b) an infinite number of rational and irrational numbers (c) an infinite number of integers (d) an infinite numbers of complex numbers. z 1 is a purley imaginary number (z ≠ 1)), then | z | z 1 is equal to

14. If

(a) 1 (c) 3

(b) 2 (d) 5 3

3

 1 i   1 i  15. If      a  ib, then  1 i   1 i  (a) a = 0, b = 2 (b) a = 0, b = −2 (c) a = 2, b = 0 (d) a = 2, b = 2

3.230

▌ Mathematics

16. The value of ( 21)( 21) is (a) 1 (c) i

25. If x  1  1  1  1 then the value of x is

(b) −1 (d) −1

17. The value of

(a)

a 1 bω 1 cω 2 where ω is a complex cube c 1 aω 1 bω 2

root unity, is (a) ω 2 (c) 1

(c)  i 2

(d) (  2  2i)

27. If a + b + c = 0 then the quadratic equation 4ax 2  3abx  2c  0 has

3

of (x  1)  8  0 are (a) 1, 1  2ω , 1  2ω (c) 1, 1, 1

(b) 1, 1  2ω , 1  2ω (d) None of these

2

20. The complex cube roots of unity are in (a) A.P. (b) G.P. (c) H.P. (d) None of these 21. The expansion of cos5 θ is 1 (a) [cos 5 θ  5 cos 3 θ  10 cos θ] 16 1 (b) [sin 5 θ  5 sin 3 θ  10 sin θ] 16 1 (c) [cos 5 θ  5 sin 3 θ  10 cos θ] 16 1 (d) [cos 5 θ  5 sin 3 θ 10 cos θ] 16 22. The remainder when the expression x 2  2x 2  3x  4 is divided by 2 x −1, is 33 8 37 (c) 2 8 (a) 2

35 8 39 (d) 2 8 (b) 2

23. If the nth degree equation f (x) = 0, is satisfied by more than n values of x, then the coefficients of all powers of x in this equation (a) Must be different integers (b) must be different rational numbers (c) must be zero (d) may be any real numbers 24. The least number of imaginary roots of the equation 2x 7  x 4  x 3  5  0 is

(a) 2 (c) 6

(b) 4 (d) None of these

(d) 1

(a) a + b − 4 = 0 (b) a − b + 4 = 0 (c) a + b + 4 = 0 (d) None of these

19. If the cube roots of unity are 1, ω , ω 2 , then the roots 2

5 2

1 5 2

x 2  ax  b  0 and x 2  bx  a  0 is same; then

18. Value of i  1 is (b)  2

(c)

(b)

26. The difference between the corresponding roots of the equation

(b) ω (d) None of these

(a) 0

1 5 2

(a) roots of opposite signs (c) real roots

(b) imaginary roots (d) None of these

x 2 1 2x 1 1 28. The greatest and the least values of the expression 2 x 1 2x 1 7 for all real values of x, are respectively (a) 2, 0 (b) 1, 2 (c) 2, 3 (d) 1, 0

29. If the sum of the roots of the equation (a 1) x 2  ( 2a  3) x  (3a  4)  0 is  1, then the

product of the roots is (a) 0 (c) 2

(b) 1 (d) 3

30. If the 9th term of an A.P. is zero, then ratio of the 29th term to the 19th term is (a) 2 : 1 (b) 3 : 1 (c) 9 : 1 (d) None of these 31. The sum of the first m terms of an A.P is the same as the first n terms, where m ≠ n. Then the sum of the first (m + n) terms is (a) m (b) n (c) 0 (d) None of these 32. The sum of the series 1 5 1 (c) 11 (a)

1 1 1 1 1 1....∞, is 5.8 8.11 11.4

1 8 1 (d) 15 (b)

33. If x  1  y  y2  y3  ...∞, the value of y is x x (b) (a) 1 x x 1 1 x x 1 (c) (d) x x

Additional Support Material ▌ 3.231 34. If a, b, c are in A.P. as well as in G.P., then (a) a = b ≠ c (b) a ≠ b = c (c) a ≠ b ≠ c (d) a = b =c 35. If a, b, c are in A. P., b − a, c − b and a are in G.P., then a : b : c is (a) 1 : 2 : 3 (b) 1 : 3 : 5 (c) 2 : 3 : 4 (d) 1 : 2: 4 36. The sum of the 24 terms of the series

43. There are 10 true-false questions in an examination. Then these questions can be answered in (a) 20 ways (b) 100 ways (c) 512 ways (d) 1024 ways. 2 2 1/2 44. Coefficient of x in the expansion of (1 1 4x 1 x ) is (a) –3 (b) –1 (c) 2 (d) None of these

45. If (1  x  2x 2 )6  1  a 1 x  a 2 x 2  ...  a 12 x12 , then value

2 1 8 1 18 1 32 1 .... is

of a 2  a 4  a 6  ...  a 12 is equal to (a) 300 2

(b) 200 2

(a) 30

(b) 31

(c) 100 2

(d) None of these

(c) 32

(d) None of these

37. The third term of a G.P. is 4. The products of first five terms is (a) 43

(b) 45

(c) 44

(d) None of these

38. The number of numbers greater than a million that can be formed with the digits 2, 3, 0, 4, is (a) 380 (b) 420 (c) 360 (d) 960 39. The number of ways in which five identical balls can be distributed among ten identical boxes such that no box contains more than one ball, is (a) 10! (c)

10! (5!)2

(b)

10! (5!)

(d) None of these

40. A question paper is divided into two parts A and B. Each part contains 5 questions. The number of ways in which a candidate can answer 6 questions selecting at least two questions from each part, is (a) 50 (b) 100 (c) 200 (d) None of these 41. The total number of 3 digit even number that can be composed from the digits 1, 2, 3,….,9 When the repetition of digits is not allowed, is (a) 224 (b) 280 (c) 324 (d) 405

46. The value of ( 2  1)5  ( 2  1)5 is (a) 35 2

(b) 70 2

(c) 104 2

(d) 106 2

47. The value of

1 1 1 1    ....  n! 1!(n  1)! 3!(n  3)! 5!( n  5)!

if n is odd; is (a) 2 n  1 (c)

2 n 1 n!

31 41 51 None of these

2 n 1 ( n  1)!

(d) None of these

ex  B0  B1x  B2 x 2  ...  Bn x n  ..., then value 1 x of Bn  Bn 1

48. If

(a) 1 (c)

1 n!

(b)

1 n

(d) None of these

49. The sum of the series is 11 2 11 2 1 3 11 2 1 3 1 4 1 1 1 ...∞ 2! 3! 4! 3e (a) e (b) 2 5e (c) 2e (d) 2

42. If 56 pr 6 :54 pr 3  30800:1, then value of r is (a) (b) (c) (d)

(b)

50. The value 1 1 (a) e 3 (c) 3e

23 33 43 1 1 1 .... 1 ∞ is 0 2! 3! 4! (b) e 2 (d) 5e

3.232

▌ Mathematics

51. The sum of the series  n2  (a) log  2   n  1

58. Let ω be complex (but not real) cubic root of unity. Which one of the following is correct? (a ) ω is of absolute value greater than 1

1 1 1   ... is n 2 2 n 4 3n 6  n 2  1 (b) log    n2 

 n  (c) log  2   n  1

(b)  ω will be a cubic root of unity (c) The absolutee value of (  1  ω) is 1

2

52. Sum of the series (a) log e 2

(d) None of these

(d ) ( ω)6  1 but (ω)3 ≠ 1

1 1 1 1 1 1 ... ∞is 1.2 3.4 5.6 (b) log e 3

(c) 2 log e 2

2 59. If x  x  1  0, then what is the value of 2

(d) log e 5

53. Given positive integers r  1, n  2 and the coefficient of (3r )th and ( r  2) the terms in the binomial expansion of (1 x )2 n are equal. Then, (a) n  2r (b) n  2r  1 (c) n  3r (d) None of these

100

∑ ki

k 0

(a) (b) (c) (d)

50  50i 50  49i 49  50i 51  50i

k

1 55 1 (c) 110

1 550 1 (d) 1100 (b)

61. If z1, z2 and z3 are complex numbers which lie on a straight the values of a, b respectively? 2,25 3,27 2,4 None of these.

62. If z1, z2 and z3 are complex numbers which lie on a straight line L and if z4  az1  bz2  4 z3 lies on L, then what are the values of a, b respectively? (a) 2,  5 (b) 3,  7 (c) 2, 4

n

 1 i     1, is  1 i 

where i 2  1?

(b) 42 (d) 54

60. What is the value of the error when 2 is approximated by 11 0.18?

(a) (b) (c) (d)

(|z|2  |z  5|2  | z  zi |2 ) where z  x  iy , assumes minimum value? (a) x  5, y  3 (b) x  5, y  3 (c) x  3, y   5 (d) x  3, y  5 56. The smallest positive integer n for which (a) 8 (b) 16 (c) 12 (d) None of these.

2

line L and if z4  az1  bz2  4 z3 lies on L, then what are

 π (b) 0,   8  π (d) 0,   12 

55. For what values of x and y , the expression

57. What is the value of

(a) 27 (c) 45

(a)

 3 2 1 54. What is the image of 0,  under the inverse sine  2 2  function?

 π (a) 0,   10   π (c) 0,   15 

2

    1 1 1 1 x     x 2  2    x 3  3   ....  x 27   ? x x7      x  x 

(d) None of these

  63. What is the amplitude of sin π  i 1  cos π ?  5 5  π 5 π (c) 15

(a)

π 10 π (d) 2 5 (b)

64. What is the number of solution of the equatioon z2  | z |2  0, where z ≠ 0 ? (a) 1 (b) 2 (c) 3 (d) Infinitely many

Additional Support Material ▌ 3.233 65. If z satisfies | z  1 |  | z  3 |, then which one of the following is satisfied by ω  2 z  3  i? (a) | ω  5  i |  | ω  3  i | (b) Img (iω )  1 (c) Re (ω )  1 (d) | ω  5 |  | ω  3 | 66. Let h(x )  min {x , x 2 } for every real x. Then which one of the following is correct? (a) h is continuous but not differentiable for all x. (b) h is not differentiable at two values of x

67.

r 1 and S n  ∑ n where k ∈ z , n k C C ( ) ( )k ' r 0 r 0 r r S  then cos1  n  is  ntn  π π (a) ( b) 6 3 π π (c ) (d ) 4 2

68. A man has 7 relatives, 4 women and 3 men. His wife also has 7 relatives, 3 women and 4 men. What is the number of ways can they invite 3 women and 3 men so that 3 of them are the man’s relatives and 3 of them are his wife’s relatives? (a) 485 (b) 484 (c) 468 (d) 467 69. Ten different letters of an alphabet are given. Words with five letters are formed from these given letters. Then, the number of words which have at least one letter repeated, is (a) 69760 (b) 30240 (c) 99748 (d) None of these 70. What is the coefficient of x 5 in the expansion of (1  x )  (1  x )  (1  x )  ...  (1  x ) ? 21

22

75. x1 , x2 ,..................., x50 are 50 real num mbers such that x ,  xr 1 the prob bability that x20 is the middle number?

n

If tn  ∑

74. There are four bus routes between A and B and three bus routes between B and C. A man can travel round-trip in number of ways from A and C via B. If he does not want to use a bus route more than once, in how many ways can he make round-trip? (a) 14 (b) 19 (c) 72 (d) 144

for r  1, 2, .......... 49. Five numbers are picked up. What is

(cc ) h '( x )  1 for all x  1 (d ) h is differentiable for all x n

73. What is the sum of all numbers between 4000 and 4250 formed out of the digits 0,2,3 and 4 (no digit is repeated in the formation)? (a) 16488 (b) 8433 (c) 8405 (d) None of these

23

30

(a) C (51, 5)

(a) ( 20C2  30C2 ) / (50 C5 ) (c )

(19C2  31C3 ) (50 C5 )

( b)

( 30C2 ) / (19 C5 ) (50 C5 )

(d )

( 49C4 ) (50 C5 )

76. How many even numbers can be made with three digits such that if 3 is one of the digits, then 9 must be next digit? (a) 180 (b) 360 (c) 365 (d) None of these 77. What is the maximum number of points of intersection of 3 straight lines and 2 circles? (a) 12 (b) 14 (c) 17 (d) 21 78. How many four-letter words can be formed out of the letters of the word ‘TOMATO’ ? (a) 54 (b) 96 (c) 102 (d) 360 79. Let a  b  0 and c  0. Then, both the roots of the equation ax 2  bx  c  0 (a) are real and negative (c) have positive real parts

(b) have negative real parts (d) None of the above

80. The roots of the equation

( b) C (31, 5)  C ( 21, 5)

( a 4  b4 ) x 2  ( 4 abcd ) x  ( c 4  d 4 )  0 are given to be real.

(c ) C (31, 6)  C ( 21, 6)

Then which one of the following is correct? (a) The roots are always different. (b) The roots are always identical. (c) The roots may be different or identical. (d) No conclusion can be drawn.

(d ) C (31, 6)  C ( 21, 6) 71. In how many ways can the letters of the word ‘MACHINE’ be arranged so that the vowels may occupy only odd positions? (a) 5040 (b) 576 (c) 288 (d) 275 72. If the middle term is the greatest term in the expansion of (2 1 3x ) , then which one of the following is correct? 10

(a) 5/8 < x < 1 (c) 5/8 < x < 5/6

(b) 5/9 < x < 4/5 (d) 5/9 < x < 4/3

81. If x ∈ R, then what is the least value of the expression x2  6x  5 ? x2  2x 1 (a) 21 (c) 21/3

(b) 21/2 (d) 21/4

3.234

▌ Mathematics

82. If the roots of the cubic equation (   αβ)3  α ≠ 0 represent the vertices of a triangle, then what is the length of one of the sides of the triangle? (a )

3| α|

(c ) | α | / 3

88. If aN {ax : x ∈ N } and bN ∩ cN , where b, c ∈ N , b  2 are relatively prime, then which one of the following is correct? ( b) c  bd (a) b  cd

3 | β|

( b)

(d ) | αβ | / 3

83. If m ∈ Z and the equation mx 2  (2m  1) x  ( m  2)  0 has rational roots, then m is of the form (a ) n( n  2) (c ) n( n  2)

( b) n( n  1) (d ) n( n  3) where n ∈Z Z

Consider the following statements: 1. The number of binary operations that can be defined on B is 981.

2. The number of binary operations that can be defined on A 3 A is 816 . Which of the statements given above is/are correct? (a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2

2 P( A) 5 , then what is P( A ∩ B )? 3 1 (a ) ( b) 3 1 (d ) (c ) 2

1 and 2

2 3 1 6

86. Two numbers are randomly selected from the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and written in an ascending order ( x , y ); x  y. Consider the following statements I. The number of elementary events in the sample space is 45. II. The event that ( x 1 y ) is odd has 25 elementary events. Which of the following statements given above is/are correct? (a) I only (b) II only (c) Both I and II (d) Neither I nor II 87. A, B and C are events, A is the complement of A, A/B is the event that A will happen when B is known to have happened. Events A and B are independent. Consider the following statements: (1) A and B are independent (2) A/C and B /C are indepeendent (3) AC and BC are independent

(c ) d  bc

(d ) d 2 = bc

89. If Tr Vr 1  Vr  2, r  1, 2, 3, ..., where Vr is the sum of the first r terms of an AP whose first term is r and common difference is (2r  1), then Tr is always (a) prime and odd (c) composite and even

84. Let A 5{1, 2, 3}. Let P( A), the power set of A.

85. If A and B are two events such that P( A ∪ B ) 5

Which of the statements given above is correct? (a) 1 only (b) 2 only (c) 3 only (d) None of these

(b) composite and odd (d) odd but not prime

∞

∞

∞

n 0

n 0

n 0

90. If x  ∑ a n , y  ∑ bn , z  ∑ c n , y  wh and c are in AP, and |a|  1, | b |  1, | c |  1; then x, y and z are (a) AP (c) HP

(b) GP (d) AGP

91. If a1 , a2 , ..., a50 are in GP, then what is a1  a3  a5  ...  a49 equal to? a2  a4  a6  ...  a50 (b)) 1 a (d ) 25 a24

(a) 0 a (c) 1 a2

92. The third term of a geometric progression is 4. The product of the first five terms is (a ) 43 (c ) 4

( b ) 45

4

(d ) None of these

93. Which term of the sequence 9 2 8i, 8 2 6i, 7 2 4i is purely imaginary? (a) 5th (b) 9th (c) 10th (d) None of these n

94. If ∑ ti  i 1

n ( n  2)( n  1) n 1 then what is ∑ is equal to? 6 i 1 ti

2n n 1 n (c) n 1 (a)

2n n2 n (b) n2 (b)

95. If α is an nth root of unity, then what is 1  2 α  3 α α 2  ...nα n1 equal to? (a) n /(1  α ) (b)  n /(1  α ) (c) n /(1  α )2

(d)  n /(1  α )2

Additional Support Material ▌ 3.235 96. Consider the following statements: (I) There exits a natural number n such that 13  23  33  ...  n3  776 (II) For any real number r ≠ 0, r 21 , r 22 , r 23 is in GP and if r ≠ 1, then r 21 , r 22 , r 23 is not in HP. (a) I only (c) Both I and II

(b) II only (d) Neither I nor II

97. The sum of the squares of three distinct real numbers which are in GP is s2, if their sum is as, then which one of the following is correct? (a ) a 2 ∈ (1 / 3, 1)

( b) a 2 ∈ (1, 3)

(c ) a 2  3

(d ) None of these

98. M ( x1 , x2 , ..., X n ) defines a measure of central tendency based on n values x1 , x2 , ...xn . Consider the following measures central tendency: (I) Arithmetic mean (II) Median (III) Geometric mean Which of the above measures satisfy/satisfies the property x x M ( x1 , x2 , ....xn ) x  5 M  1 , 2 , ... , n  M ( y1 , y2 , .... yn ) yn   y1 y2

102. In a class consisting of 100 students, 20 know English and 20 do not know Hindi and 10 know neither English nor Hindi. The number of students knowning both Hindi and English is (a) 5 (b) 10 (c) 15 (d) 20 103. The function f : N → N : f ( x ) where N is the set of natural numbers is (a) one–one and onto (b) one–one but not onto. (c) onto but not one–one (d) neither one–one for onto 104. If f(x) = x2 and g(x) = x3 x being real, then (a) f is one–one but g is not one–one (b) f is not one–one and g is one–one (c) Neither f nor g is one–one (d) Both f and g are one–one 105. If x is the set of all integers and f is defined on x by f(x) = n2, then the image of set {–2, –1, 0, 1, 2} is (a) {–2, –1, 0, 1, 2} (b) {–2, 1, 0} (c) {0, 1, 2} (d) {0, 1, 4} 106. The range of the function f(x) = a cos (bx + c) + d, a > 0 is (a) [–a, a] (b) [–a, –d, a – d] (c) [–d – a, d + a] (d) [d – a, d + a] 107. Sum to n terms of the series

Select the correct answer using the code given below: (a) I only (b) II only (c) III only (d) I and III

(a)

2 1  5! ( n  1)!

99. If tan mθ  tan θ  0, then the values of θ form which series?

(c )

3!  1 1    4  3! ( n  2)! 

(a) HP (c) AP

(b) GP (d) None of these

100. What is the largest term in the sequence an  81 929 48 (c ) 543 (a )

( b)

n2 ? n  2000 3

8 89

(d ) None of these

101. If the sequence {a1 , a2 , a3 ......... an } forms an arithmetical progression then what is the value of ( a12  a32  a52  .....  a22n  1 )  ( a22  a42 )? n ( a 2 2 a22n ) 12 2n 1 n (c ) ( a 2 2 a22n ) 2 n 211 1 (a )

2n ( a 2 2 a22n ) n 21 1 2n (d ) ( a 2 2 a22n ) 2n 21 1 ( b)

1 1! 2! 3!     ... 5! 6! 7 ! 8! 1 1 n!  ( b)    4  4! ( n  4)!  (d ) none of these

a b aα  b 1 108. If b c bα  c  0 and α ≠ , then 2 2 1 0 (a) a, b, c are in A.P. (c) a, b, c are in H.P.

(b) a, b, c are in G.P. (d) none of these

109. If all permutations of the letters of the word AGAIN are arranged as in dictionary, the forty ninth word is (a) NAAGI (b) NAGAI (c) NAAIG (d) NAIAG 110. If 20% of three subsets (i.e., subsets containing exactly three elements) of the set A = {a1, a2,..., an} contain a1, then the value of n is (a) 15 (b) 16 (c) 17 (c) 18 111. A crocodile is known to have not more than 68 teeth. The total number of crocodiles with different set of teeth is (a) 68 (b) 68! (c) 1617 (d) 6868

3.236

▌ Mathematics

112. In a certain test, there are n questions. In this test 2k students gave wrong answers to at least (n - k) questions, where k = 0, 1, 2,..., n. If the total number of wrong answers is 4095, then value of n is (a) 11 (b) 12 (c) 13 (d) 15 113. The total number of terms in the expansion of (x + a)100 + (x 2 a)100 after simplification is (a) 202 (b) 51 (c) 50 (d) None of these 114. If the rth terms in the expansion of x4 then r is equal to (a) 2 (c) 4

(x3 2 2x )

10

2

contains

(b) 3 (d) 5

100

∑[

100

cx ( x  3)100  m 2 m ] is

(a )

100

C47

(d )

2 100

m0

C53

(a) 200 (c) 240

(b) 220 (d) 250

)

12

x3

117. ω is an imaginary cube root of unity. If (1 + ω2)m = (1 + ω4)m, then least positive integral value of m is (a) 6 (b) 5 (c) 4 (d) 3 2 118. The equation z 5 z has

(a) no solution (b) four solution (c) 2 solution (d) infinite number of solution n

  119. The smallest integer n for which 1  i  1 is  1  i  (a) 2 (c) 8

115. The coefficient of x5x in the expansion

(

116. Find the coefficient of x9 in the expansion of x 2 2 1

(b) 4 (d) 12

120 Given that the equation sin x + i cos 2x and cos x = i sin 2 x where P, q, r, s are non zero has real root, then C53

( b)

100

(d )

2 100

C2 100

(a ) Pqs  s 2  q 2 r

( b) Pqs  s 2  q 2 r

(c ) Pqs  r 2  P 2 s

(d ) Pqs  q 2  r 2 P

anSwer Key 1. (a)

16. (a)

31. (c)

46. (b)

61. (a)

76. (c)

91. (c)

106. (d)

2. (c)

17. (a)

32. (d)

47. (c)

62. (a)

77. (a)

92. (b)

107. (b)

3. (a)

18. (b)

33. (c)

48. (c)

63. (b)

78. (a)

93. (c)

108. (b)

4. (c)

19. (b)

34. (d)

49. (b)

64. (c)

79. (b)

94. (a)

109. (a)

5. (b)

20. (b)

35. (a)

50. (d)

65. (d)

80. (d)

95. (b)

110. (a)

6. (c)

21. (d)

36. (a)

51. (c)

66. (c)

81. (c)

96. (b)

111. (c)

7. (b)

22. (d)

37. (b)

52. (a)

67. (b)

82. (a)

97. (a)

112. (b)

8. (d)

23. (c)

38. (c)

53. (a)

68. (a)

83. (b)

98. (c)

113. (b)

9. (b)

24. (b)

39. (c)

54. (d)

69. (a)

84. (b)

99. (c)

114. (b)

10. (b)

25. (b)

40. (c)

55. (c)

70. (c)

85. (d)

100. (a)

115. (c)

11. (b)

26. (c)

41. (a)

56. (d)

71. (b)

86. (c)

101. (c)

116. (b)

12. (c)

27. (c)

42. (b)

57. (a)

72. (b)

87. (a)

102. (b)

117. (d)

13. (b)

28. (d)

43. (d)

58. (c)

73. (a)

88. (c)

103. (b)

118. (b)

14. (a)

29. (c)

44. (b)

59. (d)

74. (c)

89. (c)

104. (b)

119. (b)

15. (b)

30. (a)

45. (b)

60. (b)

75. (b)

90. (c)

105. (d)

120. (a)

Additional Support Material ▌ 3.237

SolutionS 1. The null set is a subset of every set.

β a  2 [from ( ii)] b α2

2. ∴ A ∪ B′  U  B 3. y  2  x  1 And x + 1 = 2 ∴x=1 And y – 2 = 1 – 1= 0 Or y = 2. 4. We know, if a set a contains n elements, then the numbers of ordered pairs of A × A is n2. Number of relations from A to B = Number of subsets of A × B = 22 Hence the correct altenative is .(c)

∴

β1 α1



β2 α2

⇒ α1 β 2  α2 β1  0

Thus, given vectors are linearly dependent if α1β 2  α1β1  0 10.

If a , b ∈ N then a 2 b ≠ b 2 a Thus, subtraction is not commutative.

11. An integer which is divisible by itself or 61 is called a prime number.

5. f ( x )  x 2 , for x 1, ie. image of x  f ( x )  x 2 , for x  1,,

12. 183 = 5 × 36 + 3 Thus 183 = 3 (mod 5). Hence the least integer is 36.

is the same element 1 in B; so the function is surjective but not injective.

13. Between any two real numbers, there are an infinite number of rational and irrational numbers.

6. Let e be the identity element of the group (G, *) Now b1 * a 1 ) * (a * b)  b1 *[a 1 *(a * b)] ( Associative low )

14. Let z  x  iy z  1 x  iy  1  ∴ z  1 x  iy  1  ( x  1)  iy   ( x  1)  iy      ( x  1)  iy   ( x  1)  iy  ( x 2  y2  1)  2iy  ( x  1)2  y2

 b1[(a 1 * a ) * b] ( Associative low ) 1

1

 b * [(e * b) ( a * a  e )  b1 * b ( Identity law )  e ( b1 * b  e ) ∴( b1 * a 1 ) * (a * b)  e

Now ,

By definition of inverse

x 2  y2 − 1  0

(a * b)1  b1 * a 1

Or x 2  y2  1

7. Since x is zero element a  x  a ; ∀a ∈ x Also a .a  a (a  x ) ( a  a  x ) Or a. a + x = a. a + a. x ( x is zero element) Using cancellation law, X = a. x Or a. x = x

z −1 is purely imaginary, if z 1

2

Or z  1 or z  1 15.

1  i (1  i)(1  i) (1  1)2   1  i (1  i)(1  i) 1 i 

1  i2 + 2i 1  1  2i  i 2 2

8. The ring Q, R, C are fields and the only structure not constituting a field is (I, +, ×)

1  i (1  i)(1  i) (1  i) 2   1  i (1  i)(1 − i) 11

9. Let a .b ∈ F, then these vectors are linearly deppendent, if a ( α1  α2 )  b (β1 , β 2 )  (0, 0)



(a α1  b β1 , a α2 )  (0, 0) a α1  b β1  0 ........( i) a α2  b β 2  0 .......( ii) a β1  [from ( i)] b α1

1  i 2  2i 1  1  2i   i 2 2 3

3

 1 i   1 i  3 3 3 3 ∴     ( i)  ( i)  i  i  1 i   1 i   i  i  2i ∴a  ib  2i Hence, a  0, b  2

3.238

▌ Mathematics

16. (  1   1)   1   1)  i  i  i2  1  i2  1 [ ( a )( b)   (  a )  (  b)]

23. An equation of nth degree has more than n root of it is an identity i.e., the equation is satisfied for all real values of x. In this case coefficients of all powers of x must vanish. 24. Let f ( x )  2 x 7  x 4  x 3  5 f ( x ) has three changes of signs. Thus the equation f (x) = 0, can have a maximum of three

17.

18.

a  bω  cω2 a  bω  cω2  2 c  c  aω  bω ω   a  bω  ω   a  bω  cω2  1 2  ∴  ω  ω(cω2  a  bω)  ω  a  bω  cω2 1   ω2 ω(cω2  a  bω2 ) ω 1 1 (1  i)2  (1  i) 2 2 1 1 (1  i)  1  (1  i)2  2 2

f(-x) has no change of sign, so f(x) = 0 has no negative real roots. The equation f (x) = 0 is of seventh degree, so the least number of complex roots is = 7−3 = 4 25. x  1  1  1  1  ... Or , x  1  x Or , x 2  1  x

i

i  1 

2

 2

2 3 19. ( x  1)  8  8 ( 1) Or x  1  2( 1)1/ 3

Or x  1  2( 1)1/ 3 ∴ x  1,  1  2ω ,  1  2ω 2 ( ( 1)1/ 3  1,  ω ,  ω 2 ) 20. Roots are 1, ω , ω 2 , hence in G .P. 21. We Know cos n θ 

1

[cos nθ  nC1 cos( n  2)θ n1

2 + nC2 cos( n  4)θ  ...]

∴cos5 θ  

1 2

51

[cos 5θ  5C1 cos 3θ  5C2 cos θ]

1 [cos 5θ  5 cos 3θ  10 cos θ] 16

22. Let f ( x )  x 3  2 x 2  3x  4 Putting, 2 x  1  0 ⇒ x  1 / 2

positive real roots. Also, f ( x )  2 x 7  x 4  x 3  5 Thus,

emainder of f (x), when it is divided by 2x−1, is given R by f (1/2) ∴ f (1 / 2)  (1 / 2)3  2(1 / 2) 2  3(1 / 2)  4



= 1/8 + 1/2 − 3/2 − 4



= (1+ 4 − 12 − 32) /8 = − 39/8



Thus, the remainder is −39/8.

Or x 2  x  1  0 1  1  4 1  5 Or x   2 2 1 5 (as x  0) Or x  2 26. Let α , β be the roots of the first equation and y, δ that of the second equation. ∴α  β  a , αβ  b, γ  δ  b, γδ  a Now , α  β  γ  δ Or ( α  β)2  ( γ  δ)2 Or ( α  β )2  4αβ  ( y  δ)2  4 γδ Or a 2  4 b  b2  4a Or a 2  b2  4(a  b)  0 Or (a  b) (a  b  4)  0 Or a  b  4  0 (  a ≠ b) 27. D  9 b2  32ac  9(a  c )2  32ac [ b  a (a  c )]  9a 2  9c 2  18ac  32ac  9a 2  9c 2  14ac  (3a  3c )2  4ac  0 ∴ Both roots are real. 2 28. Let , x  2 x  1  k (say) 2 x  2x  7

(1  k ) x 2  2(1  k ) x  (1  7 k )  0 This equation has real roots if 4(1  k )2  4(11  k )(1  7 k ) Or (1  k ) (1  k 1  7 k )  0 Or (1  k ) (6k )  0 ∴ k  0, k  1 Hence least value is zero and the greatest value is 1.

Additional Support Material ▌ 3.239 Adding, we get

29. Sum of roots = 21



 2a  3 ∴   1  a  1 

1 1 1 s∞ t1  t 2  t 3  ...    3 5 15

Or 2a + 3 = a + 1 Or a = −2

(all other term cancel)

Now, product of roots 

3a  4  6  4  2 a 1  2 1

33. x 

1 1 y

Or 1  y  30. t 9  a  (9  1) d  a  8d  0 ∴ a  8d

1 x 1 Or y  1  x x

1 x

34. Since a, b, c are in A.P., 2b = a + c

t 29  a  ( 29  1) d  a  28d

Also, a, b, c are in G.P.

 8d  28d  20d ( a  8d )

∴ b2  ac

t19  a  (19  1) d  a  18d

 a c   a c  Or    ac  b  2 2    

  8d  18  10d ∴

t 29 20d 2   t 29 10d 1

∴ t 29 : t19  2 : 1

m [2a  ( m  1)d ] 2 n s n  [2a  ( n  1)d ] 2 Now , s m  s n m n ∴ [2a  m  1)d ]  [2a  n  1)d ] 2 2 Or 2a ( m  n )  ( m  n ) ( m  n  1) d  0 Or 2a  ( m  n  1)d  0 ...( i) ( m ≠ n ) mn [2a  ( m  n  1)d ]  0 [ by( i))] ∴s m  n  2

31. s m 

32. n th term of the A.P., 5, 8, 11,….. = 5 + (n −1) × 3 = 3n + 2 nth term of the A.P., 8, 11, 14,… = 8 + (n − 1) × 3 = 3n + 5 t n  n th term of the given series 

1 (3n  2) (3n  5)

1 1 1   tn    ( partial fractions) 3  3n  2 3n  5  1 1 1  t1     3 5 8  1 1 1  t2     3  8 11  1 1 1 t 3     and so on. 3 11 14 

2

Or (a  c )2  4ac  0 or (a  c )2  0 ⇒ a  c From (i); 2b = 2a or b = a ∴a=b=c 35. a, b, c are in A.P., ∴ 2b = a + c Now, b − a, c −b, a are in G.P., ∴ (c−b)2 = ( b − a). a 2

 a c   a c   a c  Or  c  a  . a  b     2 2 2       Or

( 2c  a  c )2 (a  c  2a ) . a  4 2

Or (c  a )2  2(c  a ) . a Or c  a  2a

( c ≠ a )

Or b  3a Also, 2 b  a  c  a  3a  4a Or b  2a ∴ a : b : c  a : 2a : 3a  1 : 2 : 3 36. S24  2  2 2 + 3 3  4 2  .... upto 24 terms  2 (1  2  3  4 ......  24)  2

24  ( 24  24  1)  300 2 2

37. ar2 = 4 P5 5 product of first five terms 5 a.Ar. Ar2. Ar3 ar4 = a55. R10 = (ar2)5 = 45

3.240

▌ Mathematics

38. There are four different digits, namely, 0, 2, 3, 4. A number greater than a million has 7 digits. No. of 7 digit numbers 

7! 6!   420  60  360 3!2! 3!2!

Total number of seven digit numbers = 360

Putting x  1, we have 0  1  a 1  a 2  ...  a 12

10! 10!  5!(10  5)! (5!)2

40. The candidate can select 6 questions as under. (i) 2 questions from part A + 4 questions from part B (ii) 3 questions from part A + 3 question from part B (iii) 4 questions from part A + 2 question from part B ∴ Total number of ways  5 C2  5C4  5c 3  5c 3  5C4  5c 2

Putting x 1, we get

Adding (i) and (ii) 64  2(1  a 2  a 4  ...  a 12 ) Or a 2  a 4  ....a 12 

 2 ( 2 )5  6c 3  ( 2 )3  6c5 ( 2)1 ]  2 [4 2  15  2 2  6 2 ]  2[35 2]  70 2 47. The given expression

41. The number is even if the unit digit is either of 2, 4, 6, 8. Thus the unit place can be filled in 4 ways. Once the unit’s place is filled, the remaining places can be filled in p2 ways. 8

Total number of numbers  4  8 p2  4  56  224. 56 54

p r 6 p r 3



30800 1

(54  r  3)! 56!   30800 54! (56  r  6)!



 1  n! n! n! n!      n! 1!(n  1)! 3!(n  3)! 5!(n  5)! n!(n  n)!

1 n [ C  nC3  nc5  ...n c n ] n! 1 2 n 1 n [  C1  nC3  nC5  ...  2 n 1 ]  n! 

48. We have e x  (1  x) (B0  B1 x  B2 x 2  Bn 1 x n 1  Bn x n  ...) Writing expansion of e x , We get x1  x 

56  55  (51  r)  30800 51  r 

30800  100 56  55

R  41. 43. Each question can be answered as true or false i.e., in 2 ways. Total number of ways = 2 ×2 × 2 × 2 × 2 × 2 ×2 × 2 × 2 × 2 = 210 = 1024 ways 1 44. Now, (1  4x  2x 2 )1/ 2  1  ( 4 x  2x 2 ) 2

62  31 2

46. The given expansion

= 10 × 5 + 10 × 5 × 10 = 200

42.

...( i)

64  2(1  a 1  a 2  ...  a 12 ) ...( ii)

39. The question reduces to selecting five boxes out of 10. This can be done in  10 C5 ways 

45. (1  x  2x 2 )6  1  a 1 x  a 2 x 2  ...  a 12 x12

x2 x3 xn   ...   ... n! 2!! 3!

 (1  x ) ( Bn 1 x n 1  Bn x n  ...) Equating coefficient of x n on both sides, we have Bn  Bn 1 

49. Tn  

1 n!

1 + 2  3  ...  n n(n  1)  2(n!) n!

n(n  1) 2n(n  1)!

 11   1 22  .( 4 x  2 x 2 )2  ...  1.2

1  n 1 2     2  (n  1)! 

1 ∴ Coefficient of x 2  1   16  1 8

Tn =

1  n 1 2     2  (n  1)! ( n  1)!!

Additional Support Material ▌ 3.241

1  n 1 2     2  (n  2)! ( n  1)! 1 1 2 ∴ T1     2  0! 0! Or Tn 

1  1 2    2  0! 1! 1 1 2  Ta     2 1! 2! T2 

T4 

1 2

1 2    and so on  2! 3!

Adding and using (1)! As ∞, We get T1  T2  T3  T4  ... 

 1 1 0  1  1  !.... ∞  2 2 

 2 1 1  1  1    ....∞  2 2! 3!  1 3  e e  e 2 2 n 3 n.n 2 n2 50. T    n! (n  1)! n! n

( n 2  1)  1 (n  1)! ( n  1) ( n  1) 1   ( n  1)! (n  1)! n 1 1   (n  2)! ( n  1)! n 23 1   (n  2)! ( n  1)! 1 3 1    (n  3) ( n  2)! ( n  1)!! ∞ ∞ 1 1 ∴∑ Tn  ∑ 3∑  ( ) n 3 ( n 2)!   n 0 n 0 ∞ 1  e  3e  e  5e ∑ n 0 ( n  1) Or Tn 

51.

52. Here, T1  1 

1 2

1 1 T2   3 4 1 1 T3   and so on 5 6 ∴ T1  T2  T ... ∞   1 1 1 1 1   1       ...  2 3 4 5 6    log e (1  1)  log e 2 2n 2n 3r 1 53. In the expansion (1  x ) , t3r  C3r 1 ( x )

and tr 2  2 n Cr 1 ( x )r 1 Since, binomial coefficients of t3r and tr 2 are equal. ⇒

2n

C3r 1  2 n Cr 1

⇒ 3r  1  r  1 or 2 n  (3r  1)  ( r  1) ⇒

2r  2 or

⇒

r  1 or

But

r  1,

2 n  4r n  2r

∴ We take n  2r  3 21  54. Given, interval 0,   2 2    1 3 1 1  0,  ⇒ . 2 .  2 2   2 2  ⇒ ⇒

  π π π π  0,  sin .cos  cos .sin  3 4 3 4      π π  0, sin     3 4    π  0, sin  12  

Given series

Now w, the image of Eq. ( i )

  1  1  1 2 1  1 3     2    2    2   ...   n  2  n  3n    

Under the inverse sin e function is

 n2 1   1   log  1  2   log  2  n    n 

  π   sin1 (0),sin1  sin   12     π   sin1 (sin 0),sin1  sin  12   

 n2   log  2   n 1 

 π  0,   12 

....( i )

▌ Mathematics

3.242

{

55. Given, | z |2  | z  5 |2  | z  zi |2 where

z  x  iy

59. Since, x 2  x  1  0 ⇒ x  ω , ω 2

}

2

2

2

 2 1  3 1  27  1 1   x  x    x  x 2    x  x 3   ....   x  x 27 

⇒ iz  ix  i 2 y  y  ix z  zi  ( x  iy )  ( y  ix )

2

2

z  5  x  iy  5  (x  5)  iy

(

(

) (

)

 | x  iy |2  | ( x  5)  iy |2  | ( x  y )  i( y  x )2 |

 (1)2  (1)2  22  ...  ( 2)2

 ( x 2  y 2 )  ( x  5)2  y 2  ( x  y )2  ( y  x )2

(1  ω  ω 2  0)

⇒ x 2  y 2  x 2  25  10 x  y 2  x 2  y 2  2 xy  y 2  x 2 Let y  f ( x ) is minimum at x  3 and y  5

= 18(1) + 9(4) = 18 + 36 = 54

n

n

 1 i   1 i 1 i     1 ⇒   1 1  i    1 i 1 i  n

 2i  n  2  5 1 ⇒ i 5 1

⇒

n

The smallest positive inteeger n for which i 51 is 4. 100

∑ ki

k

k 0

⇒ 0 + i + 2i 2  3i 3  4i 4  ... 100i100 ⇒ ( i  2  3i  4)  (5i  6  7 i  8)  ...( i  i, i 2  1, i 3  i, 14  i(1  5  9  ...  97 ) 2(1  2  5  ...  49) i(3  7  11  ...  99) 4(1  2  3  ...  25)   25    25  25  i  (1  97 )    ( 2  98)   i  (3  99)   2   2 2 25  {4  10 4 25  {98i  100  102i  104} 4 25  {4i  4}  25(2i  2} 2  50i  50 58. ω  cube root of unity. (a ) ϖ is absolute value but equal to one.

60. Let error be x. 2 ∴ 0.18   x 11 2 18  11 100 200  198 2 1    1100 1100 550 ⇒ x

61. For 2  x  3, 0  f ( x )  1 ∴ [ f ( x )]  0 ∴

∫

3 2

0 dx  0

62. Condition for straight line, 1  a  b  4  0 ⇒ a  b  3 Hence, option (a) is correct. 63. sin

 π π  i  1  cos  5 5 

 2 sin

 π π π cos  i  2 sin 2  10 10 10  

 2 sin

π  π π π i10π cos  i sin   2 sin e  10  10 10  100

Hence, amplitude is

π . 10

64. z 2  | z |2  0 ⇒ z 2  z z  0 ⇒ z( z  z )  0 Given,

1  i 3 , which is absolute value and equal to 2 (d ) If ( ϖ)6  1then (ϖ)3 1

⇒ z  z 0

So, option ( c ) is correct.

2

2 Similarly, for x 5 ω , we get the same value.

(b) ω will not be a cubic root of unity . (c) 1  ω 

)

2

2 2  1  ω  ω 2  ω 2  ω  (1  1)2  ....   1   (ω 31 ) 1 

Then, | z |2  | z  5 |2  | z  iz |2

56. Since,

2

2    1 1  1    ω     ω 2  2   ω 3  ω 3  .....   ω 27  27  ω    ω  ω 

⇒ ( x  y )  i( y  x )

57.

x  ω,

When

( i 2  1)

( z ≠ 0 given)

⇒ 2 Re( z )  0 Hence, z is only imaginary.

Additional Support Material ▌ 3.243 65. | z  1 |  | z  3 | ω i3 ω i3 1  3 2 2

⇒ ⇒

ω  i  3 2 ω  i  3 6  2 2

⇒ ⇒



( 21 C5  22C5 x  23C5 x 2  ...30 C5 )



( 30 1)

{ C  r

r

r 1

n

Cr  ... Cr 



31

C6 

( 20 1)

ω  i 5 ω i3  2 2



31

C6 

21



C (31, 6)  C ( 21, 6)

ω  i 5  ω  i 3 ⇒ ω 5  ω 3

So, option ( c ) is correct .

66. Here, h is continuous for all values of x Hence, h'(x )  1 ∀x  1 67. S n 

0 1 2 n  n k n  ...  n k k ( C0 ) ( C1 ) ( C2 ) ( Cn ) k

Sn 

n n 1 n2 n  n k n  ...  n k k ( C1 ) ( C2 ) ( C0 ) ( C0 ) k

n

n

⇒

Sn 1  ntn 2

Cr 1

}

C( 51)

C6

71. In the word ‘MACHINE’, vowels are (A,E,I) and consonants are (C,H,M,N). Vowels occupy the odd places, the number of ways is 4 P3 . Rest of the four position, four consonants can be arranged in 4!. ∴ Required number of ways  4 P3  4! 4! 4!

then

 S  1 π cos  n   cos1    n n  2 3  

72. Since middle term is the greatest term.

1

T6 T  1 and 7  1 T5 T6 10

⇒

10

4 3 3 4 68. ( P1  P2 )  ( P2  P1 )

C5 ( 2)5 (3x )5 1 C4 ( 2)6 (3x )4 C6 ( 2)4 (3x )6 1 10 C5 ( 2)5 (3x )5

10

 ( 4 P2 3 P1 )  ( 3 P1 4 P2 )  ( 4 P3 3 P0 )  ( 3 P0 4 P3 )  ( 4 P0 4 P3)  ( 3 P3 4 P0 )  ( 4  3)  (3  4)  (6  3)  (3  6)  ( 4  1)  (1  4)  (1  1)  (1  1))  144  324  16  1  485 69. Total number of five letters words formed from ten different letters  10  10  10  10  10  10

}

n 1

 24  24  576

2 S n  ntn



{

C( 51)  5 C5  6C5  ...20 C5

5

Number of five letters words having no repetition  10  9  8  7  6  30240 ∴ Number of words which have at least one letter repeated

and

⇒

6 3 5 3x ⇒  x  1 and   1 5 2 6 2 5 4 5 4 ⇒ x  andx  ⇒  x  9 5 9 5 73. The number between 4000 and 4250 are 4023, 4032, 4203, 4230 ∴ Sum of all numbers = 4023+4032+4203+4230=16488 74. Number of ways a man travels from A to C  4C1 3 C1  12 Number of ways a man travels from C to A 2

C1 3 C1  6

∴ Total number of ways = 12 × 6 = 72

5

 10  30240  69760 70. (1 1 x )21 1 (1 1 x )22 1 ... 1 (1 1 x )30 ⇒ ( 21 C0 1 21C1x 1 21C2 x 2 1 ...121 C21x 21 ) 1 ( 22 C0 1 22C1x 1 22C2 x 2 1 ...122 C22 x 22 ) 1 ... 1 ( 30 C0 1 30C1x 1 30C2 x 2 1 ...130 C30 x 30 ) Now, equating the coefficient of x 5

30

75. Required probability 

C2  19C2 50

C2

76. If 3 is one of the digits then total number of even numbers =1×1×5=5 If 9 must be next digit = 8 × 9 – 9 × 5 = 360 ∴ Total number = 360 + 5 = 365 77. Required number of ways = 2! 3 3! = 2 3 3 3 2 = 12

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▌ Mathematics

78. Required number of possible ways

3 3 82. Given, ( z  αβ )  a Its roots are

z  αβ  α, αω, aω2

4!  3! 2! 1 2  3  4 1 2  3  4  2 1 1 2  3 1 2  4 p4 + 2C1 + 2C2 

∴ z1  α  αβ, z2  αω  aβ, z3  αω2  αβ

 24  24  6  54

∴| z1  a 2 |  | α  αω |

79. Since, a, b, c  0 and ax 2  bx  c  0 ⇒

x

 1 3  ∴| α | 1     i  2 2  

b2  4 ac b  2a 2a

| α |

Caase I when b2  4 ac  0 ⇒

x

b2  4 ac b2  4 ac b b and   both roots. 2a 2a 2a 2a

Case II when b2  4 ac  0 ⇒

x

b i.e., both roots are equal and negative. 2a

Case III wheen b2  4 ac  0 ⇒

x

4 ac  b b i have negative reall part. 2a 2a

 1  4m For perfect square m is of the foorm n(n  1) [ P( A)  power set of A]

⇒ (8)(8) or 864 2

( 4 abcd )  4( a  b ) ( c  d )  0 4

4

4

4 4

4 4

4

4

4

4

⇒

( 4 a 2 b2 c 2 d 2  a 4 c 4  b4 c 4  a 4 d 4  b4 d 4 )  0

⇒

[( a 4 c 4  b4 d 4  2 a 2 b2 c 2 d 2 )  ( a 4 d 4  b4 c 4  2 a 2

⇒

( a 2 c 2  b2 d 2 )  ( a 2 d 2  b2 c 2 )2  0

⇒

( a 2 c 2  b2 d 2 )  ( a 2 d 2  b2 c 2 )2  0

Which is not posssible. So, "No conclusion can be drawn" for given equation. x2  6x  5 x2  2x 1

⇒ x 2 ( y  1)  x ( 2 y  6)  ( y  5)  0

Defined on ( A  A)is ⇒ (9)( 9 ) ⇒ 981. 2

16 a b c d  4( a c  b c  a d  b d )  0

81. Let y 

 (2m  1)2  4 m( m  2)

So, the number of binary operations that can be deefined on B is 4

2

83. Now, discriminate

A  A  3 3 ⇒ 9

B 2  4 AC  0 2 2 2

9 3 |a| 2 3  3| α |   2 4 4

84. B  P( A) ⇒ 8

4 4 2 4 4 80. ( a  b ) x  ( 4 abcd ) x  ( c  d )  0 Since, the roots arre real, then D  0

2

| α |

 4 m 2  1  4m  4 m 2  8m

2

∴ From above discussion, both roots have negative real parts.

⇒

3 3  i 2 2

 The number of binary operations in any set containing n elements ⇒ So, option ( b).

85. P(A) 

2 3

⇒ P(A) 

1 3

⇒ P( A ∩ B )  P( A ∪ B )  P( A) 1 1 1    2 3 6

Now, D  ( 2 y  6)2  4( y  1) ( y  5)  0

86. S 5{1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.

⇒ 4 y 2  36  24 y  4 ( y 2  6 y  5)  0

87. A, B and C are events, A 5 complement of A A/B is the event that A will happen when B is known to have happened. Here event A and B are indeppendent so, A and B will be independent.

⇒ 48 y  16  0 1 ⇒ y  3

Additional Support Material ▌ 3.245 88. aN {ax : x ∈ N} bN ∩ cN , b,c ∈ N, b  2, c  2 are relatively prime bN {bx / x ∈ N}  the set is of  ve integral multiplies of b. cN {cx / x ∈ N}  the set is of  ve integral multiplies of c. Hence, bN ∩ cN is the set is of ve integral multiplies of bc.  bcN[  b and c are prime numbers] again bN ∩ cN  dN ⇒ d  bc 89. Tr  Vr 1  Vr  2, r  1, 2, 3, ... vr  Sum of the first r terms of an AP. r Vr  {2r  ( r  1)( 2r  1)} 2 r  {2r  2r 2  2r  r  1} 2 r  {22r 2  r  1} 2 ( r  1) Vr 1  [2r  ( r  1  1)( 2r  1)] 2 ( r  1)  [2r  2r 2  r ] 2 r  ( r  1) ( 2r  1) 2 r r Now, Tr  ( r  1) ( 2r  1)  ( 2r 2  r  1]  2 2 2 r  {2r 2  2r  r  1  2r 2  r  1})  2 2 r  ( 4r )  2 2  2r 2  2

x 1 x  | a |1 | x 1 | 1 ⇒ |x| | x 1 |  | x | ⇒ 1 1 Similarly , y  and z  1 b 1 c y 1 and b y z 1 Similarly , c z  a, b, c are AP. Then, b a c b y 1 x 1 z 1 y 1 ⇒    y x z y xy  x  xy  y zy  y  yz  z ⇒  zy xy zx  zy  xy  xz ⇒ ⇒ xy  zy  2 zx Divided by xyz on both sides,  1 1 2  z  x  y  which represents x , y , z are in HP. 91. Given, a1 , a2 ,..., a50 are in GP. Let a, ar , ar 2 , ... ,ar 49 are in GP. where, ( a1  a;a2  ar 2 ;...; a50  ar 49 ) Then, 

a  ar 2  ar 4  ar 6  ...  ar 48 ar  ar 3  ar 5  ...  ar 49 1  r 2  r 4  r 6  ...  ar 48  r  r 3  a5  ...  r 49 a 1 1  1   r  a2  a2    a1 

[tr  2( r  1) ( r  1)]

T2  2.3.1 ⇒ 6 T3  2.4.2 ⇒ 16 T4  2.5.3 ⇒ 30

∞

∞

∞

n 50

n 50

n 50

90. x 5 ∑ a n , y 5 ∑ bn , z 5 ∑ c n ⇒ ⇒ ⇒ ⇒

x  1  a  a 2 ...∞ 1 x 1 a 1 1 a  x 1 a 1 x

a1  a3  a5  ...  a49 a2  a4  a6  ...  a50



T1  2.3.0 ⇒ 0

i.e., Tr  compoosite and even.

a

⇒

92. Here,

t3 5 4 ⇒ ar 2 5 4

∴ Product of first five terms 5 a.ar.ar 2 .ar 3 .ar 4 5 a5 r 10 5 ( ar 2 )5 5 45 93. The term (9  8i ) (8  6i ), (7  4i ) are in AP. First term  9  8i  a Comman difference, d  (8  6i)  (9  8i )  ( 1  2i)

3.246

▌ Mathematics

Then, nth term Tn  a  ( n  1) d

 (9  8i )  ( n  1)( 2i  1)  9  8i  2 in  n  2i  1  (10  10i )  n( 2i  1)  10(1  i )  n( 2i  1) Here, T5  10(1  i )  5( 2i  1) 10  10i  10i  5 55th ( purely real ) t9  10(1  i )  9( 2i  1)  10  10i  18i  9 (1  8i) ( complex number ) T10  10(1  i )  10( 2i  1)

 10  10i  20i  10  10i ( purely imaginary ) So, opption ( c ) is correct . n( n  1)( n  2) ⇒ sn 6 i 1 n( n  1)( n  1) S n1  6 Now, tn  S n  S n1

94. Given

n

∑t  i

n( n  1)( n  2) n( n  1)( n  1)  6 6 n( n  1)  {n( n  2)  n( n  1)} 6 n( n  1) n( n  1)  tn  (3) 6 2



⇒

1 2  tn n( n  1) n 1 1 2 ∑ ∑ n( n  1) i 1 ti 1 1   2∑     n n  1  1 1 1 1 1 1   2  1    ...       ...   n n 2 3 2 3  1      1  2n  2 1    n  1 n 1  

95. Let S  1  2α + 3a 2  ...  na n1 αS  α  2α2  ...  (n  1) α n1 + nα n ??? (1  α) S  1  α  α2  ...  α n1  nα n 1 ( α n  1)  nα n ⇒ (1  α) S  α 1 1  1) ⇒ (1  α) S   n(1) [ α n  1] α 1 n ⇒ S  1 α

n

3 96. I. ∑ r 5 776 r 51

2

 n( n + 1)  ⇒  5 776 2   Hence no value of n exist. II. Statement II is correct. 97. Let three distinct real number in GP be According to the given condition, a  a  ar  ar , s ≠  1 r 1 Let r  y , then y ≠  2 r a2 Now, 2  a 2  a 2 r 2  s 2 r  1  2  2 ⇒ a  r  2   1  s 2 r    2    1 ⇒ a 2  r 2    1  s 2 r     2 2 2 a ( y  1)  s ...(i)

a , a, ar . r

a2  a 2  a 2 r 2  s2 r2

and

and

a  ar  a  as r

⇒ a( y  1)  α s ⇒ a 2 ( y  1)2  α2 s 2

...(ii)

Dividing Eq. (i) by Eq. (ii), we get s2 a 2 ( y 2  1)  a 2 ( y 2  1)2 a 2 s 2 ⇒

y 1 1  y  1 α2

⇒ y

a2 1 a2 1

...(iii)

1 a2 1 ⇒r  2 r a 1 ⇒ r 2 ( α2  1)  ( α2  1)r  ( a 2  1)  0 Since, r is real, ∴ Discriminant  0 ⇒ ( a 2  1)2  4( a 2  1))2  0 ⇒ ( a 2  1)2  [2( a 2  1)2  0

{

}{

}

⇒ ( a 2  1)  2( a 2  1) ( a 2  1)  2( a 2  1)  0

Additional Support Material ▌ 3.247 102. Let A and B be the sets of students knowing English and Hindi, respectively. Then,

⇒ (3  a 2 )(3α 2  1)  0 ⇒ ( a  3)(3α  1)  0 2

2

∴ n(A ∩ B )  (100  10)  90, n( A)  20 and

1 ⇒  a2  3 3

n(B )  (100  20)  80.

98. Only geometric mean satisfy the given property. 99. tan mθ  tan nθ  0 sin mθ sin nθ  0 cos mθ cos nθ ⇒

sin mθ cos nθ  sin nθ cos mθ 0 cos mθ cos nθ

⇒

sin( mθ  nθ) 0 cos mθ cos nθ

1 1 Then, x 5 y. If y = 1 then x = ∉ N So, f is not onto. 2 2 104. f is not one–one Since 1 ≠ –1 and f(1) = f(–1) = 1 g(x1) = g(x2) ⇒ x13 5 x23 ⇒ x1 5 x2 . So, q is one–one.

⇒ ( m  n )θ  πx

105. f (–2) = (–2)2 = 4, f (–1) = (–1)2 = 1, f (0) = 02 = 0 f (1) = 12 = 1 and f (2) = 22 = 4, So, image of given set = {0, 1, 4}

πx ⇒ θ ∀x 1, 2, ... m−x

106.  1∉ cos( bx  c ) ⇒  a  a cos( bx  c ) ∉ a ⇒ a  d  a cos( bx  c )  d  a  d

Hence, these values are in A.P. n2 n  200

∴Range ( f )  [ d  a, d  a].

... ( i )

have n2 real number hence it should be square member . From option ( a ) n2  81 ⇒ n9 By putting n = 9 in equation (1) we get

107. We have, tr 

r! ( r  1)! and tr  1  ( r  4)! ( r  5)!

Now , rtr  ( r  5)tr  1 

r! r!  0 ( r  4)! ( r  4)!

⇒ rtr  ( r  1)tr  1  4tr  1

(9) 2 3 (9  200) 81  (729  200) 81  929 an 

n 1

n 1

r 1

r 1

⇒ 4 ∑ tr  1  ∑ [rtr  ( r  1)tr  1 ] ⇒ 4(t2  t3  ...  tn )  1t1  ntn ⇒ 4(t1  t2  ... tn )

101. ( a12  a32  a52  ...  a22n  1 )  ( a22  a42  ....  a22n )  ( a12  a22 )  ( a32  a42 )  ( a32  a42 )  ( a52  a62 )  ....  ( a22n  1  a22n )  ( a1  a2 )( a1  a2 )  ( a3  a4 )( a3  a4 )  ( a5  a6 )( a5  a6 )  ....  ( a2 n  1  a2 n )( a2 n  1  a2 n )  d ( a1  a2 )  d ( a3  a4 )  d ( a5  a6 ) ....  d ( a2 n  1  a2 n )



 0! n( n  1)!  5t1  ntn  5     5! ( n  4)!

1 n!  4! ( n  4)!

11 n!  ⇒ t1  t2  ...  tn    4  4! ( n  4)!  108. Expanding along R3, we get 2

b aα  b a aα  b  00 c b α  c b bα  c

⇒ 2 (b2a 2 bc 2 aca + bc) 2 (aba 2 ac 2 aba + b2) = 0

 d  a1  a2  a3  .... an 

 2n   d  {2 a1  ( 2 n  1) d } 2    a  a4 ( a2 n  a1 )     n 2 a  ( 2 n  1)  2 n 2 x  1   1  2n 1

 (20  80  90)  10. 103. b f(x 1) = f(x2) ⇔ 2x1 = 2x2 = x1 = x2. So; f is one–one. Let y = 2x.

⇒ sin( mθ  nθ)  sin πx

100. an 

∴ n(A ∩ B )  n( A)  n( B )  n ( A ∪ B )

⇒ 2a (b2 2 ac) 2 (b2 2 ac) = 0 or (b2 2 ac) (2a 2 1) = 0   

or b2 2 ac = 0 ∴ b = ac 2

∴ a, b, c are in G.P.

[ (2α 2 1) ≠ 0]

3.248

▌ Mathematics

109. Starting with the letter A and arranging the other four letters, there are 4! = 24 words. These are the first 24 words. Then, starting with G and arranging A, A, I and N in different ways, there are 4! 5 12 words, the 37th word start with I and there are 2!1!1! 12 words starting with I. Total goes up to 80 words. Now, 49th word is NAAGI. 110. The number of subsets of A containing exactly three elements is nC3 whereas the number of three subsets of A that contain a1 is n 2 1C2. We are given, 20 n ( C3 ) 100 ( n  1)( n  2) 1 n( n  1)( n  2)  ⇒ 6 2 5 ⇒ n 15 n 1

C2 

111. The required number is equal to the number of all the subsets of a 68 elements set i.e., = 268 = 1617 112. The number of students answering at least r questions incorrectly is 2n - r. ∴ The number of students answering exactly r (1 ≤ r ≤ n 2 1) questions incorrectly is 2n 2 r 2 2n 2 (r + 1). 113. There will be 101 terms in each 50. Even terms will cancel and odd will be 51 in number which will be added. 11 r

x 114. Tr  10 cr 1   3

∴ (11 2 r)

22

 2   2   x 

r 1

(r 2 1) = 4; ∴ 9 = 3r; ∴ r = 3.

115. The given sigma is expansion of [( x  3)  2]100  ( x  1)100  (1  x )100. ∴ x 53 will occurr is T54 T54 = 100c53 (- x)53 ∴ coefficient is - 100c53

116. The general term r

 1 Cr ( x 2 )12r  3   12Cr x 245r x  Now , ( 24  5r )  9, then r  3.

12

Thus, the required term  12C3 

12.11.10  220. 1.2.3

117. (1  ω2 ) m  (1  ω4 ) m ∴ ( ω) m  ( ω2 ) m or ωm  ω2 m Which is only satisfied for m  3, or ω3  ω6  1 2 2 118. z  z ⇒ ( x  iy )  x  iy 2 2 ⇒ x  y  x  i( 2 xy  y )  0 ⇒ x 2  y 2  x  0 And 2xy  y  0 1 Now , 2 xy  y  0 gives y  0 or x  2 When y  0, x 2  y 2  x  0 gives x 2  x  0 or x  0, 1 1 1 1 2 , x  y 2  x  0 gives  y 2   0 When x  4 2 2 3 or, y  2

Hence, there are four solutions. 119. Consider 

1 i 1 i 1 i   1 i 1 i 1 i

(1  i )2 1  2i  i 2 1  2i  1   r 11 1 i2 1 i2 n

 1 i  ∴   1 ⇒ n  4, 8, 12, 16 ...  1 i  Hence, the smallest integer , n  4. ( i 2 )2 x 1  i 4 n i 2  1 120. Since the equation has real solution ∴ z = z + iy = x Putting z = x and equating real and imaginary parts We get, x 2  Px  r  0 and qx  s  0 or x  s / q Eliminating x , we get or Pqs  s 2  q 2 r

 s s2  P    r  0 q2  q

Additional Support Material ▌ 3.249

MatriCeS anD DeterMinatS 43 1 6    1. The value of the determinant 35 7 4  is 17 3 2    (a) 0 (c) 15

(b) 10 (d) None of these

 x 11 x 1 2 x 1 4    2. The value of the determinant  x 1 3 x 1 5 x 1 8  is  x 1 7 x 1 10 x 1 14    (a) 22

(b) x 2 1 2

(c) 2

(d) None of these

a b c 3. If a, b, c > 0 then the value of the determinant b c a is c a b (a) always positive (c) always zero

(b) always negative (d) None of these

a b c bc c a a  b 4. Let ∆1  c c b and ∆ 2  a  b b  c c  a then b a a ca a  b bc (a) ∆1  ∆ 2

(b) ∆1  2∆ 2

(c) ∆ 2  2∆1

(d) None of these

2 2ω 2ω2 5. The value of the determinant 1 1 1 Where ω, ω2 are 1 21 0 complex cube roots of the unity, is (a) 2 (b) 4 (c) ω (d) 0 1 i 1 i i i 1  i is 6. The value of 1  i i 1 i 1 i (a) 4 2 7i (c) 24 1 7i

(b) 4 1 7i (d) None of these

1 2 3 3 2 4 A 5 2 3 4 , B 5 2 1 3 then A and B are 7. If 1 5 6 5 1 6 (a) (b) (c) (d)

Equal equal but opposite in sign A5B None of these

8. If capita letters, denote the cofactors of the corresponding elements , and if a1

b1

c1

A1

B1

C1

∆1 5 a 2

b2

c2 , ∆ 2 5 A 2

B2

C2 ,

a3

b3

c3

B3

C3

A3

(a) ∆1 5 ∆ 2

(b) ∆1 5 ∆ 22

(c) ∆ 2 5 ∆12

(d) None of these

9. The value of λ for which the system of equations 3x  y  4z  3 x  2y  3z  2 has infinite number of solutions, is 6x  5y  λz  3 (a) 5 (c) 0

(b) 25 (d) None of these

10. If the equations x  au  z, y  z  ax, az  x  y are consistent having a non-zero solution, then value of a2 is (a) 1 (c) 3

(b) 2 (d) None of these

11. The value of the determinant is 0 a 2 b a 2c b 2a 0 b 2c c 2a c 2 b 0 (a) abc (c) ab 1 bc 1 ca

(b) a 1 b 1 c (d) 0

 cos θ sin θ   12. Consider G =   : θ ∈ R  sin θ cos θ   For any θ ∈ R, donate the matrix  cos θ sin θ    by Aθ .  sin θ cos θ  Consider the following statemennts. 1. Adj. (Aθ )  Aθ for some θ ∈ R, then θ 

(2r  1) π for 2

some integer r. 2. Adj (Aθ )  AθT holds for only finitely many θ∈ R, Which of the statements given above is/are correct? (a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2

3.250

▌ Mathematics 18. What is the identify element under the matrix multiplication in the group

1 0 0    13. Let A  2 1 0   3 2 1  

  x G 5    x 1 (a )  0 2 ( b)  2

If U1 , U 2 and U 3 are column matrices satisfying 2 2 1        AU1  0  , AU 2   3 AU 3   3 1 0  0        and U 3  3 matrixx whose columns

(c )

are U1 , U 2 and U 3 , then what is U is equal to ? ( b)

(c ) 2 14. If A   12

(d )

3 2 (d ) 3

(a ) 3

3 1

(a) 25 (c) 8

 x  : x ∈ R and x ≠ 0  ? x  0  1 2  2

1 1   2 2 1 1  2 2  1 1   1 1

19. If A be the square matrix of order n, then what is the value of

 then what is A1003 2 5 A1002 is equal to ?  (b) 1 (d) 21

1 1 1    15. Let A  2 1 3 and 1 1 1   4 2 2   (10) B  5 0 α  1 2 3   

Adj (Adj A) ? (a )

A

(c )

A

n ( n21)

n ( n22 )

( b)

A

(d )

A

n ( n22 )

n ( n21)2

20. Consider the set A of all determinants of order 3 with entries 0 or 1 only. Let B be the subset of A consisting of all determinants with value 1. Let C be the subset of A consisting of all determinants with value -1. Then, (a) C is empty. (b) B has as many elements as C

If B is the inverse of matrix A, then what is the value of a ? (a) 22 (b) 21 (c) 2 (d) 5 16. If A is a square matrix, then what is adj (AT) 2 (adj A)T equal to?

(c) A 5 B ∪ C (d) B has twice as many elements as C 1  a 2 x (1  b2 ) x (1  c 2 ) x 21. If f ( x )  (1  a 2 ) x 1  b2 x (1  c 2 ) x (1  a 2 ) x (1  b2 ) x 1  c 2 x

(a ) 2 A

( b) 2 A I

And a 2  b2  c 2  2, then what is the degree of the

(c ) Null matrix

(d ) Unit matrix

polynomial f (x)? (a) 0 (c) 2

17. Which one of the following is correct?

22. An equilateral triangle has each side equal to a. If the coordinates of its vertices are ( x1 , y1 ), ( x1 , x2 ), ( x3 , y3 ). then

1 0  0 1 If I    and E    , the 0 1   1 0  n the valuess of A and B, such that

x1

(AI  BE )2  E are respectively (a)



3

,

1

2 (c)



2 3

(b)

3 1  , 2 2

(d)



2 ,

1 2

(b) 1 (d) 3

3 1 , 2 2

what is the square of the determinant x2 x3 3 2 4 (a ) 3a ( b) a 4 (c ) 3a 4 / 2

(d ) 2 a 4

y1 1 y2 1 equal to? y3 1

Additional Support Material ▌ 3.251 23. If the value of third order determinant be 11, then what is the value of the square of determinant formed by its cofactors? (a) 11 (b) 121 (c) 1331 (d) 14641 an 24. If bn cn

an 1 2 bn 1 2 cn 1 2

an 13 bn 1 3 cn 1 3

 ( a  b)( b  c )( c  a )( a1  b1  c1 ), what is the value of n? (a) 1 (c) 21

(b) 2 (d) 22

25. If α, β, γ are the rots of x 3  px  q  0 then what is the value of the determinant

(b) 4 (d) -2

26. Which one of the following is correct? The determinant a b aα  b b c bα  c is equal to zero for all value aα  b bα  c 0 (a ) a, b, c are in AP

( b) a, b, c are in GP

(c ) a, b, c aree in HP

(d ) a  b  c  0

27. If A and B are two invertible matrices of order 3, then (AB)21 is equal to (a ) A2 B21

( b) B21 A21

(c ) AB

(d ) A21 B

21

28. If A and B are symmetric matrices of order n( A ≠ B ) then (a) A 1 B is known symmetric (b) A 1 B is symmetric (c) A 1 B is a diagond matrix (d) A 1 B is a zero matrix 29. Matrices A and B will be inverse of each other, only if (a) AB 5 0, BA 5 I (b) AB 5 BA (c) AB 5 BA 5 0 (d) AB 5 BA 5 I 1 0 0    30. If A 5 0 1 0  then 0 0 1    (a) (b) (c) (d)

2   1 2   44 

0 1  32. The matrix   is the matrix of refection in the line 1 0  (a ) x  1 ( b) y  1 (c ) x  y  1 (d ) x  y

α β γ β γ α? ? γ α β (a) 0 (c) 2

1 2  31. If A    then Adj. A is equal to 3 4  4 3 4 (a )  ( b)   2 1   3 1  4 2  '(c)  (d )    2 1 3  

Adj. A 5 A Adj. A 5 A Adj. A 5 A21 All the statements are correct

33. The rotation through 180º is identical to (a) the reflection in x-axis (b) the refection in y-axis (c) a point refection (d) identity transformation 34. If A is a square matrix of order 3 and entries of A are positive integers, then | A | is (a) different from zero (b) positive (c) 0 (d) an arbitrary integer 35. In a lower triangular matrix A  [ aij ]m n the element for aij  0 (a) 1 < j (c) i < j a  36. If A 5  f g 

(b) i > j (d) i 5 j f b h

g  h  , then A is c 

(a) diagonal matrix (c) symmetric matrix

(b) scalar matrix (d) skew-summetric matrix

 3 4 2 2  1 37. If A   , B    , then ( A  B)  2 3  0 1 (a) A1  B1 (b) does not exist (c) is a skew symmetric matrix (d) None of these 38. If A 1 B are two matrices such that A 1 B and A 1 B and AB are both defined then A and B are (a) both null matrices (b) both identity matrices (c) both square matrices of same order (d) None of these

3.252

▌ Mathematics 48. Consider the set A of all determinants of order 3 with entries o or 1 only. Let B be the subset of A consisting of the determinant with value 1, then (a) C is empty (b) B has as many elements as C (c) A 5 B ∪ C (d) B has twice as many elements as C

5 8 0    5 0  is 39. The matrix, A   3 1 2 1  (a) orthogonal (c) nilpotent

(b) involutary (d) orthogonal

40. If A is skew-symmetric then A3 is (a) symmetric (b) skew-symmetric (c) Hermitina (d) None of these 41. If A is an idempotent non-zero matrix other than the unit matrix then A is a (a) triangular matrix (b) diagonal matrix (c) singular matrix (d) non-singular 42. If A is a 3-rowed square matrix such that | A | then adj(adj A) is equal to (a) 3 A (b) 9 A (c) 27 A (d) None of these 43. In matrices (a ) ( A  B )  A  2 AB  B 2

2

(d ) ( A  B )2  A2  2 AB  B 2 44. With 1ω, ω2 as cube roots of unity, inverse of which of the following matrices exists?

(a) (b) (c) (d)

mn

ω 2 ( b)  1

1  ω

(d ) None of these

is a matrix then

(a ) | A |  o

( b) | A | 1

(c ) | A |  2

(d ) None of these

47. If S  6r  1 4r 3  2 nr 2

is independent of (a) x only (c) x, y, z, n

x

1 x yz    zx  and β 5 1 y 1 z xy  

x2   y 2  then z 2 

(a ) α ≠ β

( b) α  β

(c ) α ≠ 2β

(d ) α  β

n( n  1)

y n2 ( 2 n  3) then the value of z n3 ( n  1) (b) y only (d) n only

(a) 1 (c) 0

(b) 2 1 (d) None of these

1 1 1   52. The value of  a b c  is  a 3 b3 c 3    (a) a 1 b 1 c 2 3abc (b) 3(a 1 b)(b 1 c)(c 1 a) (c) (a 2 b)(b 2 c)(c a) (d) (a 2 b)(b 2 c)(c 2 a)(a 1 b 1 c)

1  54.  4 42 

46. If A is an orthogonal matrix, then

2r

1 x  50. If α 5 1 y 1 z 

53. If A and B are square matrices of order 3 such that | A | 5 1, | B | 5 3, then the determined of 3AB is equal to (a) 2 9 (b) 2 27 (c) 2 81 (d) 81

r 5 min(m, n) r > min(m, n) r ≤ min(m, n) None of these

r

(b) x2 1 2 (d) None of these

2 2ω 2ω2    1  is 51. If ω is an imaginary cube root of unity, then 1 1 1 21 0   

(c ) ( A  B )2 ≠ 2 AB  B 2

45. If A   aij 

(a) 2 2 (c) 2

2

( b) ( A  B )2  A2  B 2

1 ω  (a )  2 ω ω   ω ω2  (c )  2  1 ω

 x 11 x 1 2 x 1 4    49. The values of the determinant  x 1 3 x 1 5 x 1 8  is  x 1 7 x 1 10 x 1 14   

1 3 32

1  2  equals 22 

(a) 2 (c) 1 n

∑S

r 51

r

0 a b   55. If ∆  a 0 c  then b c 0    (a) ∆ 5 0 (c) ∆ 5 2 abc

(b) 2 2 (d) 0

(b) ∆ 5 2abc (d) ∆ 5 a2 1 b2 1 c2

Additional Support Material ▌ 3.253

anSwer Key 1. 2. 3. 4. 5. 6. 7.

(a) (a) (b) (c) (d) (b) (c)

8. 9. 10. 11. 12. 13. 14.

(c) (b) (a) (d) (b) (d) (d)

15. 16. 17. 18. 19. 20. 21.

(d) (c) (b) (c) (d) (b) (c)

22. 23. 24. 25. 26. 27. 28.

(b) (d) (c) (a) (b) (b) (b)

29. 30. 31. 32. 33. 34. 35.

36. 37. 38. 39. 40. 41. 42.

(b) (a) (b) (d) (c) (d) (c)

43. 44. 45. 46. 47. 48. 49.

(c) (d) (c) (b) (b) (c) (a)

50. 51. 52. 53. 54. 55.

(c) (c) (c) (b) (c) (b)

(b) (c) (d) (c) (b) (a)

(a)

SolutionS 7  6 1 1 6 1. ∆  7  4  7 7 4 7 23 3 2 7 6 1 6 1 1 6  74 7 4  7 7 4 72 3 2 3 3 2 6 1 6 1 1 6 7 4 7 4  7 7 4 2 3 2 3 3 2 7 000 (∴ In first determinant c1  c 3 and in the second determinant c1  c 2 ) x 1 x  2 x  4 2. ∆  x  3 x  5 x  8 x  7 x  10 x  14 Applying, C2 → C2  C1 , C3 → C3 → C1 x 1 1 3 ∆  x 3 2 5 x 7 3 7 Applying, R 2 → R 2  2R 1 , R 3 → R 3 → 3R 1 x 1 1 3 ∆  x  1 0 1 2x  4 0 2

a b c 3. ∆  b c a c a b Applying, R 1 → R 1  R 2  R 3 a  bc b c ∆  bc a c a c a  b a

1 b c  (a  b  c ) 1 c a 1 a b Applying, R 2 → R 2  R 1 , R a → R 3  R 1 1 b c ∆  (a  b  c ) 0 c  b a  c 0 a  b bc

 (a  b  c ) {(c  b) (b  c)  (a  c) (a  b)}

{

 (a  b  c ) a 2  b2  c 2  ab  bc  ca

5 2 1 (2x 2 222x 1 4) 5 22

}

{(a  b  c )}{a 2  b2  c 2  ab  bc  ca} 1  (a  b  c ) {2a 2  2 b2  2c 2  2ab  2 bc  2ca} 2 1  (a  b  c ) {(a  b)2  ( b  c)2  (c  a)2 } 2 Since a, b, c are positive: ∴∆  0 4. Applying, C1 → C1  C2 ,  C3 ∆2 

2(a  b  c) 2(a  b  c) 2(a  b  c) ab bc c a c a ab bc

Taking 2 common and appllying R 1 → R 1  R 2 c

Expanding by second column, we get

b

a

b

∆2  2 a  b b  c c  a c a a  b bc

3.254

▌ Mathematics

Applying, R 3 → R 3  R 1 c a b ∆2  2 a  b b  c c  a a b c

7. Applying, R 2 → R 2  2 R 1 , R 3 → R 3  R 1 1 2 3 A  0 1 2  3  6  3 0 3 3

Applying, R 2 → R 2  R 3 c a b ∆2  2 b c a a b c Applying, R 1 ↔ R 3 and R 2 ↔ R 3 a b c ∆ 2  2 (1)2 c a b  2∆1 b c a

R1 → R 2  R 2 , 1 1 1 Applying, B  2 1 3 5 1 6 Applying, C2 → C2  C1 , C3 → C3 → C1 , 1 0 0 B  2 1 1 1  4  3 5 4 1 ∴ A  B.

1 1 1 2 5. ∆ 2  2 2ω ω 1 1 0

( R1 ↔ R 2 )

C2 → C2  C1 , C3 → C3 → C3  C1

Applying,

1 0 0 ∆  2 2ω  2 ω2  2 1 2 1 1 [1( 2ω  2)  2 ( ω2  2)]  [2ω  2  2ω2  4]  [( 2ω  2ω  2)  ( ω  ω  1)  0] 2

2

6. Applying, R 1 → R 1  R 2  R 3 2i 2i 2i ∆  1 i i 1 i i 1 i 1 i 1  ( 2  i) 1  i i

a1

b2

c2

A1

B1

C1

a3

b2

c2

A3

B3

C3

8. ∆1 ∆ 2  a 2 b2 c 2  A 2 B2 C2

1 1 i 1 i 1 i 1 i

Applying, C2 → C2  C1 , C3 → C3  C1

a 1A1  b1B1  c1C1

a 1A 2  B2  c1C2

a 2 A 3  b2 B3  c 3C3

a 3A1  b2 B1  c 3C1

a 3A 3  b3B3  c3C3

a 3A 2  b3B2  c 3C2

∆ 2 0 0    ∆1 ∆ 2   0 ∆1 0   ( ∆1 ) 3 0 0 ∆  1  2 ∴ ∆ 2  ( ∆1 ) 9. The given system of equation can have infinite solutions, if  3 1 4    ∆  1 2 3  0 6 5 λ   Applying, R 1 → R 1  3R 2 , R 3 → R 3  6R 2 0 7 13    ∆  1 2 3   0 0 7 λ  18  

1 0 0 ∆  ( 2  i) 1  i 1  2i 2i i 1 1  2i

Or  [  7λ  126  91]  0

 ( 2  i) [(1  2i) (1  2i)  2i]

Or  7λ  35  0

 ( 2  i) [(1  2i)2  2i]  ( 2  i) [3  2i]  ( 2  i) (3  2i)  4  7i

a 1A 3  b1B3  c1C3

a 2 A1  b2 B1  c 2C1 a 2 A 2  b2 B2  c 2C2

λ  5. 10. The system of equations x  ay  z  0, ax  y  z  0 x  y  az  0,

Additional Support Material ▌ 3.255 has a non-trivial solution if 1 a 1    ∆  a 1 1   0 1 1 a    Applying, C2 → C2  aC1 , C3 → C3  C1

On comparing, ( x1  1), ( y1  2), ( z1  1)  1 0 0   x2     AU 2  2 1 0   y2   3 2 1  Z    2 

 2   x3      2 x2  y2   3 3x  2 y  z  0  2 2    2

1 0 0    ∆  a 1  a 2 1  a   0 1 1  a 1 a   Simplifying, we get a  or a 2  1.  0 a  b a  c   11. ∆   b  a 0 b  c a  c c  b 0   On interchanging rows and columns,  0 ba c a   ∆  a  b 0 c  b a  c b  c 0   Taking ( 1) common from each column,  0 a  b a  c   ∆  (1) (1) (1)  b  a 0 b  c  ∆ c  a c  b 0   ∴ 2∆  0 ⇒ ∆  0.

θ  , A T   cos θ sin θ  12. Aθ   cos θ sin  sin θ cos θ  sin θ cos θ  θ Cofactors of Aθ a11  cos θ, a12  sinθ, a21  sinθ, a22  cosθ a a12   cos θ sin θ  Let B =  11    a21 a22  sin θ cos θ cos θ sin θ  Adj. Aθ  B '     sin θ cos θ Adj Aθ  AT ; θ ∈ R ⇒ But Adj Aθ ≠  Aθ So,, option (b) is correct 13. Let U1  [ x1 y1 z1 ]T , U z  [ x2 y2 z2 ]T , [ x3 y3 z3 ]T 1 0 0   x1     Then AU1  2 1 0   y1   3 2 1  Z    1   1   x1     2 x1  y1  0  3x  2 y  z  0  1 1    1

On comparing, ( x2  2), ( y2 1), ( z2  4) 1 0 0   x3     and AU 3  2 1 0   y3   3 2 1  Z    3   2   x3     2 x3  y3   3 3x  2 y  z  1  3 3    3 On comparing, ( x3  2), ( y3 1), ( z3  4)  x1 x2 X 3   1 1 2      So U 3   y1 y2 y3   2 1 1 Z Z Z   1 4 3  2  1 U  1 (3 4)  2 (6  1)  2 (8 1)  114 18  3 2 14. A   1 1 A2   1

3  1 3 3 1 0   A    I 2 2   0 1 2 3 A4     A 1 2  A1003  ( A3 )334 . A  (I 2 )334 . A  I 2 . A  A A1002  ( A3 )334 . A  (I 2 )334  I 2 A1003  5 A1002  A  5 I 2

( 2  5) 3 3 3  1 (1  5) 1 6)  18  3  21 

1 1 1    15. A  2 1 3 1 1 1   A  1 (1 3)  1 ( 2  3)  1 ( 2  1)  4  5  1  10

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▌ Mathematics

A1 

Adj ( A) Adj ( A)  10 A

(1  a 2 x ) (1  b2 ) x (1  c 2 ) x 21. f ( x )  (1  a 2 ) x (1  b2 x ) (1  c 2 ) x (1  a 2 ) x (1  b2 ) x (1  c 2 x )

10 A1  Adj ( A) Given, B = A1

Apply C1 → C1  C2  C3

4 2 2   1 ⇒ (10)B  10 A  Adj ( A)  5 0 α  1 2 3    α  ( a32 of A)

f ( x) 1  ( a 2  b2  c 2  2) x (1  b2 ) x (1  c 2 ) x  1  ( a 2  b2  c 2  2) x (1  b2 ) x (1  c 2 ) x 1  ( a 2  b2  c 2  2) x (1  b2 ) x (1  c 2 ) x

where a32 → cofactor of A.

Given a 2  b2  c 2  2

1 1 2 3 α  (3  2) α  (5) α 5 α 

⇒ a 2  b2  c 2  2 = 0 1 (1  b2 ) x (1  c 2 ) x Then, f ( x )  1 (1  b2 ) x (1  c 2 ) x 1 (1  b2 ) x (1  c 2 ) x

T T T T 16. adj ( A )  ( adjA)  ( adjA)  ( adjA)  0  Null matrix

17.

( AI  BE )3  E 3 2 3 2 ⇒  3AA2B3ABB2 BA3  33 AABB2    10 01    ⇒ A3  3 AB 2  0 and B 3  3 A2 B 1 We are givenn ⇒ A( A2  3B )2  0 ( i ) and B ( B 2  3 A2 ) 1 solving Eqs. ( i ) and ( ii ), we get A 

3 1 and B  2 2

G ( a ).G ( x ) 5 G ( x ) ⇒  aa aa  5  xx xx  ax 2a  5  xx xx  ⇒  22 ax ax 2 ax    ⇒ 2 ax 5 x

∴ G5

x1 3 2 1 ∴ a 5 x2 4 2 x3 

{( ) : x ∈ R and x ≠ 0} 5  x x

Expand with respect to C1 f ( x )  (1  x ) 2 f ( x)  x2  2x 1 ⇒ Second degree polyno omial. y1 1 y2 1 y3 1

Also, triangle is an equilateral, then area 5

1 2 x x

R2 → R2  R1 R3 → R3  R1 1 (1  b2 ) x (1  c 2 ) x f ( x)  0 1 x 0 0 0 1 x

x1 1 22. Since area of triangle 5 x2 2 x3

18. for matrix multiplication

⇒ a5

Apply ,



1 2 1 2

1 2 1 2

  

19. We know the property of matrix of adj (adj A)  A

x1 ⇒ ∴ x2 ( n  1)2

Where A is a nonsingular squar matrix. 20. Since, A is the determinant of order 3 with entries 0 or 1 only. Also, B is the subset of A consisting of all determinants with value 1. (We know by interchanging any two rows or columns among themselves sign changes) Given that, C is the subset having determinant with value-1. ∴B has as many elements as C

y3 1

y1 1 y3 1

x1

y1 1

x3

y2 1

y2 1 5

x3 ⇒ ∴ x2

y1 1

3 2 a 4

3 y2 1 5 a 2 4 y3 1

23. Given ∆  11 Determinant formed by cofactors ∆′  ∆ 3 1  (11)2  121 ∴( ∆ ′)2  (121)2  14641

3 2 a 4

Additional Support Material ▌ 3.257

ax x 24. b x c

ax 2 bx  2 cx  2

ax x 3 bx  3  cx 3

( a  b)( b  c )( c  a )( a1  b1  c1 ) Here the degree of determinant is n  n  2  n  3  3n  5 and the degree of expression on right side is 2. ⇒ ⇒ ⇒

1 2  31. | A |    3 4   4 3  4 2  ∴ Adj A     ( find it ) 2 1  3 1 

3x  5  2 3x 3 x = 1

α β y 25. β y α y α β α β  y β y β  y  α y α (C1  C2  C3 ) y  α β α β 1 β y  (α  β  γ ) 1 y α 1 α β

...( i)

We are given α, β and γ are thee roots of x 3  px  q  0 ⇒ α β  γ 0 ...( ii) From Eqs.( i ) and ( ii ), we get α β y 1 β y β y α  0 ⇒ 1 y α (α  β  γ )  0 y α β 1 α β

26.

1 0 0    | A | 5 30. 0 1 0  0 0 1    1 0 0    ∴ Adj A 5 0 1 0  5 A ( find it ) 0 0 1   

a b aα  b b c bα  c aα  b bα  c 0 aα  b b aα  b bα  c c bα  c 2 aα  βα  bα  c bα  c 0 C1α → C1α  C2 a b 0 b c 0 2 aα  b ba  c ( aα  bα  ba  c ) Now using C3 → C3  C1  (aα2  2bα  c )( ac  b2 ) ⇒ b2  ac Hence, a, b, c are in GP. =

27. (AB )1  B1 A1 ( Theorem on inverse of matrices) 28. ∴ (A  B )1  B1  A1  A  B [ A` A, B` B]

 x  0 1   x   y  32. Here,     5   y  1 0   y   x  ⇒ x 5 y, y 5 x 33. Rotation through 180º gives x 5 2 x, y 5 y. Hence, this is a point reflection. 1 1 1 1 1 1 34. If A  1 1 1 | A |  0 If A  2 2 4 1 1 1 3 4 4 1 0 0 A  2 0 2  2 ≠ 0 3 1 1 Also, | A | is negative. Hence, | A | can be arbitrary integer.  3 4  2 2  1 2  37. ( A  B )     2 3  0 1 2 2  ∴ det (A 1 B) 5 2 4 5 2 2 ≠ 0 ∴ (A 1 B)21 exists Adj (A 1 B)21 exists  2 2  Adj (A  B)   2 1  1  2 2  1 2  ∴( A  B)1     1 2 1   1 1 / 2  1  3 4   3 4  1 1 1 2  A1 =  ,B =    1 2 3  2 3  2  0 2  1 / 2 1    1  0 5 / 2 3 1 1 1 A1  B1    , ∴( A  B) ≠    2 2   38. Since A 1 B in defined, A and B are matrices of the same type, say m × n; Also AB is defined. So the number of columns in A must be equal to the number of rows in B i.e., n 5 m, Hence, A and B are square matrices of the same order.

▌ Mathematics

3.258

5 8 0  5 8 0  1 0 0       39. A   3 5 0  3 5 0   0 1 0   I 3 Hence, 1 2 1  1 2 1 0 0 1  A is involuntary. 2

40. Let A be skew symmetric. Then, At  A ∴( A3 )t  ( AAA)t  At . At . At  ( A).( A).( A)  A3 Hence, A3 is skew symmetric.

x 1 x  2 x  4 49. x  3 x  5 x  8 operate x  7 x  10 x  14 R2 → R2 → R1 R3 → R3 → R1 

41. Since A is idempotent, we have A2 5 A. Now, A2  A ⇒ A( A  I )  0 ⇒ | A|  0 and |A  I |  0 [∴ A ≠ 0 & A  I ≠ 0] ⇒ A is singular and (A  I ) is singular. n2 42. adj (adj A)  | A | A Hence n  3 and | A |  3 So, adj (adj A)

 3( 3 2 ) A  3 A. 44. 

1 ω ω2 5 0, 2 ω ω 1

1 ω

ω ω 5 0 Hence, inverse does not exist.  ω

x

n( n  1)

r 1

47.

n

n

∑ S ∑ (6

r 1

r

r 1

r

r 2  1)

r

r  2 nr ) z

n

∑ (4

r 1

y n2 ( 2 n  3)

2

n( n  1) n( n 1)( 2 n  1)  n

x 1 1 3 1 3 2 1 2 .2 1 2 2 0 0

Operate R3 → R3 → 2 R2 

1 x 50. α 5 1 y 1 z x xyz y 5 xyzz z

46. AA` 1 ∴| AA`|  | 1 | ⇒ | A || A |  1 ∴| A |  1

∑ 2r

x 1 1 3 1 2 Operate  2 C2 → C2 → C1 6 0 0 C3 → C3 → C2

 2( 2  3)  2. 50

2

n

x 1 x 1 x  4 x  2 x  4 x  6 2 3 4 . 2 3 4 6 8 10 6 8 10

x

n ( n  1) 3

n( n  1)

y n2 ( 2 n  3)  0 n2 ( n  1)2  n2 ( n 1) z n3 ( n  1) [ ∴ C1 and C2 are identical] Which is independent of x y z and n 1 0 1 48. Let A  1 1 1 As B ⊂ A 1 1 0 Let B 

1 0  1  0  1, As C ⊂ A 1 1

Let C 

1 0 1 1 1

B has many elements as C.

yz x 1 zx 5 y xyz xy z x2 1 1 x y2 1 5 1 y z2 1 1 z

x2 y2 z2

xyz xyz xyz

x2 y2 α 5 β z2

51. 2 2ω ω2 1 1 1 Operate C2 → C2  C1 1 −1 0 2 2(1  ω) ω2 1 2 1 1

0

0

 2 (1  ω  ω2 )  2(0)  0 52. 1 a a3 1  a

1 b b3

1 C → C2  C1 c Operate 2 C3 → C3  C1 c3 0 b a

0

ca a 3 b2  a 3 c 3  a 3 Expand by R1  ( b  a )( c  a ) 1 1 2 2 b  ba  a c  ca  a 2

 ( b  c )( c  a )( c 2  b2  a( c  b)  ( b  a )(cc  a ) ( c  b)( c  b  a )  ( b  c )( c  a )( a  b)( a  b  c )

Additional Support Material ▌ 3.259

trigonoMetry 1. The equation sin 2 θ 

x2  y2 , is possible if 2 xy

(a) x = y (c) 2x = y

(b) x = − y (d) None of these

2. The greatest value of sin x cos x is (a) 2 (b) 1 1 (c) −1 (d) 2 3. The value of tan 75º + cos 75º is equal to (a ) 2 3

( b) 2  3

(c ) 2  3

(d ) None of these 



4. If tan θ  cos 9  sin 9 , then θ is cos 9  sin 9 (a) 9º (b) 54º (c) 18º (d) None of these 5. The value of (a )

sin 70 1 cos 40 , is cos 70 1 sin 40

1

( b) 1

3 (c )

11. If a cos 2

A  5A  3 6. If cos A 5 , then sin   Sin  5 4 2  2  (a) 7 (c) 11

(b) 8 (d) None of these

7. If tan 2 θ  2 tan 2 ϕ  1 then cos 2θ  sin 2 ϕ equals (a) −1 (c) 1

(b) 0 (d) None of these

8. The general solution of the equation sin x + cos x = 1, is (a ) x  2 nπ

(c ) x  nπ  (1) n

π π  4 4

(d ) None of these 9. The equation (a  b)2 sec 2 θ  4ab is possible only when (a) a = 0 (b) a + b = 0 (c) a = b (d) b = 0

(b) 2b = b + c (d) None of these

12. In the triangle ABC, a = 2b, and ∠A 5 3 ∠B, then angle A is (a) 30º (c) 90º 13. The value of

(b) 60º (d) None of these 1 1 1 1 1 1 1 equals r 2 r 21 r 2 2 r 2 3

(a ) 0 ( b)

a 2 1 b2 1 c 2 ∆

(c )

∆2 a 1 b2 1 c 2

(d )

a 2 1 b2 1 c 2 ∆2

2

14. If r1 , r2 , r3 are A.P then (a 2 b)(s 2 c ) equals (a) (b−c) (s−a)

(b) (c−a) (s−b)

(c) (a − b) (s − c)

(d) None of these

15. The sides of an equilateral triangle, a square and a regular hexagon circumscribed in a circle are in (a) AP (b) GP (c) HP (d) None of these 16. If

7π 6

C A 3b  c cos 2  , then 2 2 2

(a) 2b = a + c (c) 2c = a + b

(d ) None of these

3

( b) 2 nπ 

10. The angle of elevations of the tops of two vertical towers as seen from the middle point of the line joining the foot of the towers are 60º and 30ºrespectively. The ratio of the heights of the towers is (a) 2 : 1 (b) 3 : 1 (c) 3 : 2 (d) 3 : 1

2 cos A  cos B  cos3 B, 2 sin A  sin B  sin 3 B. Then, | sin( A  B ) | is equal to

(a) 1/2 (c) 2/3 4 17. If tan θ  , then sin θ is 3 4 4 (a)  but not 5 5 4 4 (c) but not 5 5

(b) 1/3 (d) 1/5

4 4 (b)  or 5 5 (d) None of these

3.260

▌ Mathematics

18. Let π be a positive integer, cos n  sin n 1  n  32. If  tan 50 cos n  sin n then which one of the following is correct? (a ) n must be a prime of the form 8m  7 for some non-negative integer m. (b) n must be composite with p 2 , divisor of n for some prime number p (c) n must be a prime number and is of the form 4t 1 for some positive integer t . (d) n is a product of 3 distinct primes

Then (tan A – 1) (tan B + 1) = 2 tan A Which of the above is/are correct? (a) I only (b) II only (c) Both I and II (d) Neither I nor II x2  y2 , x, y ∈ R 2 xy

Then which one of the following is correct? (a) x ≠ y (b) x  y ≠ 0 (d) x  0,y may be zer

21. If cos ( θ  α)  a and cos (θ  β )  b, then what is the value of sin 2 ( α  β )  2 ab cos( α  β )? (a ) a  b 2

( b) a  b

2

2

( c )  a 2  b2

2

( c )  a 2  b2

22. If sin x  sin y  3 (cos y  cos x ) then what is the value of sin 3x cosec 3y? (a) 21 (b) 0 (c) 1 (d) None of these 23. The real value of x such that esin x  esin x  4 3π π (a ) is ( b) is 4 3 π (c ) is (d ) Does not exist 6 24. Let A, B be the acute angles such that cos A  and cos B 

(d ) 30  ( A  B )  60

π 25. If A  0, B  0 and A  B  , then what is the maximum 3 value of tan A tan B ? 1 (a ) 3 ( b) 3 1 (c ) 3 (d ) 3

(a) The above trigonometric equation has no solution. (b) The solution to the above trigonometric equation is

1 1 II. If A  47 , B  77  2 2

(c) x ≠ y ≠ 0

(c ) 15  ( A  B )  30

x cos θ 5 23/ 4 Which one of the following is correct?

1 1 1 I. cos 2 52   sin 2 7   2 2 2 2

If sin 2 θ 

( b) 60  ( A  B )  90

26. Let x  2  1 Consider the trigonometric equation

19. Consider the following expressions

20.

(a ) ( A  B )  90

n 1 ( n  1)2  n2 2n 1 ( 2 n  1)2  1

For some positive integer n, which one of the following is correct?

tan1 ( x  2)  k π , where k ranges through integers (c) The solution to the above trigonometric equation is tan1 ( x  2)  mπ , where m ranges through odd integers (d) None of these  π sin (α 1β β1γ) is 27. If α , β,γ ∈ 0,  , then the value of sin α 1 sin β 1 sin γ  2 (a) less than 1 (c) 1

(b) greater than 1 (d) greater than or equal to 1

28. What is the number of integral values of k for which the equation 7 cos x  5 sin x  2 k  1 has a solution? (a) 4 (c) 10

(b) 8 (d) 12

29. Which one of the following is correct? For real values of , the expression (3+cos) cosec can have no values between (a) 2 2 2 and 2 2 ( b) 2 2 2 and 4 (c ) 2 4 and 4 ( d ) 2 4 and 2 2 2   4  2  30. The value of tan cos1    tan1    is  5  3   6 17 (a) (b) 17 6 16 (c) (d) None of these 7 31. What is the value of tan cos1 ( 4 / 5)  tan1 ( 2 / 3)  ?   (a) 6/17 (c) 16/7

(b) 7/18 (d) 17/6

Additional Support Material ▌ 3.261 32. Which one of the following is correct? The equation sin

1

{

}

2x (1  x )  2cos 2

1

x

38. If sin (A  B  C )  1, tan ( A  B ) 

(b) [1, 0]

(a) A  120, B  60, C  0

 1  (c) 0,  2 

 1  ,1 (d)   2 

(b) A  60, B  30, C  0

33. AB is a vertical pole. The end A is on the level ground. C is the middle point of AB.Pis a point on the level ground. The portion BC subtends an angle batp. If AP= n(AB), then what is tan b equal to? (a )

n 2n 1

( b)

(c )

n n 1

(d ) None of these

2

n n 1 2

34. T F is a tower (T denotes the top and F denotes the bottom) of height 30(3 1 3 ). AB is a building (A denotes the top and B denotes the bottom) of height 60 m. It is given that the angle of elevation of T as seen from is 60º and the angle of elevation of T as seen from A is α. If ∠A T B 5 θ? 3 21

(a) (c)

2 2 1

(b)

3 21 3 2

(d) None of these

3 π π  θ  , then 2 2 (a) cos log θ  log cos θ

35. If e 

(b) cos logg θ  log cos θ (c) cos log θ  log cos θ 2 (d) cos log θ  log cos θ 3 1 36. If sin A  , then 4 cos3 A  3 cos A is equal (0  A  90) 2 (a) 1 (b) 0 (c) (d)

3 2 1 2

1 1 tan φ 5 , then cos 2θ equals 7 3 (a) 4 sin φ

37. If tan θ 5

(b) 2 sin 2 φ (c) sin 3 φ (d) sin 4 φ

3

and sec ( A  C )

 2, then

(a) [0,1]

2

1

(c) A  90, B  60, C  30 (d) None of these 39. If A and B are acute positive angles and satisfies the equalities 3sin 2 Asin 2 B1 and 3sin 2 A2 sin 2 B 0, then A2 B  π (a) π (b) 2 π π (c) (d) 4 6 40. log (cot 1 ) (cot 2 ). ... (cot 89 ) is equal to (a) ∝ (c) 0

(b) 1 (d) None of these

41. 3 sin x  4 cos x  1 lies in the interval (a) [5, 5] (b) [4, 6] (c) [6, 4]

(d) None of these

 7π  π  3π  5π  42. 11cos 11cos 11cos 11cos  is equal to 8 8 8 8      1 1 (a) (b) 8 4 1 (c) (d) None of these 6 2π 2  3π  2  5π  2  13π  43. sin    sin    sin    sin   is equal to 8 8 8        8 

(a) 1 (c) 2

(b) 0 (d) None of these

44. If 3cos x  2 cos 3x  cos y , 3sin x  2 sin 3x  sin y , then the value of cos 2x is: (a) -1 (c) 0

(b) 1 (d) None of these

45. If sec θ  tan θ  k , then sin θ is equal to: (a)

k 2 1 k 2 1

(b)

k 1 k 1

(c)

k 1 k 1

(d)

k 2 1 k 2 1

46. The number of solutions of the equation n sin x = 7x2 + 2x + 4, 0 ≤ x ≤ 2 is/are (a) 2 (b) 0 (c) 4 (d) 3

3.262

▌ Mathematics

47. The number of solutions of the equation | cos x | = 2 [x], where [ . ] is the greatest integer, is: (a) one (c) infinite

(b) two (d) nil

48. If θ lies in the first quadrant and cos θ 8 /177, then the value of cos (30  θ )  cos ( 45  θ )  cos (120  θ ) is  3 1 1  23  (a)    2 2  17 

 3 1 1  23  (b)    2 2  17 

 3 1 1  23  (c)    2 2  17 

 3 1 1  23  (d)    2 2  17 

cos(A  C ) , then tan A, tan B, tan C are in 49. If cos 2B = cos(A  C ) (a) A.P. (c) H.P.

(b) G.P. (d) none of these

π 50. If n 5 , then tan α . tan 2α . tan 3α ... tan (2n 2 1) α is equal 4 α to (a) 1 (c) ∞

(b) 2 1 (d) none of these

51. The ratio of the greatest value of 2 2 cos x + sin2x to its least value is 1 4 13 (c) 4 (a)

(b)

9 4

(d) none of these

54. cos

(c)

(b) 2 2 2 k 2

22 k2

56. If x 

(d) 2 k 2 2 k 2

2sinα 1  cos α  sin α  y , then y  1  cos α  sin α 1  sin α

1 x (c) 1  x

(b) x

(a)

(d) 1  x

1 57. If sin x  cos x  y  , x ∈ [0, π ], then y π (a) x 5 , y 5 1 4

(b) y 5 0

(c) y 5 2

(d) x 5

(b)

3π 4

58. The value of sin8x 1 7 sin 6 x 1 18 sin 4 x 1 12 sin 2 x is equal to sin 1 7 x 1 6sin 5x 1 12 sin 3x (a) 2 cos x (c) 2 sin x

(b) cos x (d) sin x

59. If the sides a, b, c of a triangle ABC are roots of the equation x3 – 15x2 + 47x – 82 = 0, then the value of equal to 225 164 131 (c) 82

131 164 169 (d) 82 (b)

60. The value of equals of

10



p 1



∏  sin

(a) 1 (c) 21

2 pπ 2 pπ   i cos  equals 11 11 

(b) 2i (d) i

1 2 1

(d)

2 2

53. If y = 4 sin2θ 2 cos 2θ, then y lies in the interval (a) ( 21, 5) (c) (2∞, 21) ∪ (5, ∞)

(a) k 2 2 k 2

(a)

52. If sin (π cos θ )  cos (π sin θ ), then the value of  π cos  θ   is 4  2 (a) 2 1 (c) 2 2

55. If cos 25° + sin 25° = k, then cos 50° is equal to

(b) [21, 5] (d) none of these

π 2π 4π cos cos  7 7 7

(a) 0

1 (b) 2

1 (c) 4

1 (d)  8

61. If cosec θ 

π θ pq , then cot      pq 4 2 q p

(a)

p q

(b)

(c)

pq

(d) pq

62. cos2θ  2 cosθ is always 3 (a) Greater than  2 (b) Less than or equal to

3 2

3 (c) Greater than or equal to  and less than or equal to 3 2 (d) none of these.

Additional Support Material ▌ 3.263

63. The value of expression cos  sin

π 3π 5π sin sin 14 14 14

(a) 7/24 (c) 7/50

1 4 1 (d)  8 (b) 

(a) 0 (c)

65. The tangents of two acute angles are 3 and 2. The sine of twice their difference is

π 2π 10 π cos cos 7 7 7

1 4

66. The general value of y satisfying the equation 122x2x25tan2(x + y)1cot2 (x1y) is

64. General solution of the equation log2 sin x – log2 cos x – log2 (12 tan x) – log2 (1 + tan x) + 1 = 0 is (a) ( 2 n 1 1) / π8 (c) ( 2 n 1 1) π 4

(b) 7/48 (d) 7/25

(a) 2 nπ 6

(b) (16 n 1 1) π8

π 4

(b) nπ 6

nπ π 6 4 2

(c)

(d) none of these

π 4

(d) none of these

anSwer Key 1. 2. 3. 4. 5. 6. 7. 8. 9.

10. 11. 12. 13. 14. 15. 16. 17. 18.

(a) (d) (d) (b) (c) (c) (b) (c) (c)

(d) (a) (c) (d) (a) (b) (b) (b) (c)

19. 20. 21. 22. 23. 24. 25. 26. 27.

(a) (b) (a) (a) (d) (d) (b) (d) (a)

28. 29. 30. 31. 32. 33. 34. 35. 36.

(b) (c) (b) (d) (d) (a) (a) (a) (b)

37. 38. 39. 40. 41. 42. 43. 44. 45.

46. 47. 48. 49. 50. 51. 52. 53. 54.

(d) (b) (b) (c) (c) (a) (a) (a) (d)

55. 56. 57. 58. 59. 60. 61. 62. 63.

(b) (d) (b) (b) (a) (c) (d) (b) (d)

(a) (b) (a) (a) (b) (c) (b) (c) (b)

SolutionS 1. sin 2 θ  1 

x 2  y2 ∴ 1 2 xy

3 1



( 3  1)2 ( 3  1)( 3 )



42 3 2

2 3

Or x 2  y2  2 xy 2

3 1

2

Or x  2 xy  y  0

cos 75  cos(( 45  30 )

Orr ( x  y)  0 ⇒ x  y

 cos 45 cos 30  sin 45 sin 30

2

1 1 2. sin x cos x 5 ( 2 sin x cos x ) 5 (sin 2 x ) 2 2 Now the greatest value of 2x is 1. 1 1 The greatest value of sin x cos x   1  2 2 3. tan 75º = tan( 45  30 ) 1  tan 30   1  tan 30

1 1



2 2

3

1



2 2



3 1



2 2

6 2 4

∴ tan 75  cos 75  2  3 

4. tan tan θ 

1 3 1

3



6 2 4

cos 9  sin9 1  tan 9  cos 9  sin 9 1  tan 9

tan 45  tan 9  tan( 45  9 ) 1  tan 45 tan 9

∴ tan θ  tan 54 ⇒ θ  54

64. (b) 65. (d) 66. (d)

3.264

▌ Mathematics

5. sin 70  cos(90 50 ) sin 70  sin(90 50 ) sin 70  sin 50  cos 70  cos 50  70  50   70  50  2 sin    cos  2 2       70  50   70  50  2 cos   cos   2 2     

sin 60 cos10 cos 60 cos10

2 9. sec θ 

 16(cos 2A  cos 3A )

π π ∴ x  nπ  (1) n .  4 4

or

(a  b)2  4ab

or

(a  b)2  4ab  0

or

(a  b) 2  0 ∴(a  b) 2  0 ⇒ a  b

From ∆ ABM h1 1  tan 60  h1  3x ∴ x 3 From ∆ CDM h2 x 1  tan 30  or h 2  x 3x 3

5A A sin 2 2  A 5A   16 2 sin sin  2 2  

8. sin x  Cos x  1  1  1 2 sin x  cos x   1 2  2  ( Multiplying and dividing L.H.S by 2 ) π  1 π  sin or sin   x   4 4 2   π n π ∴  x  nπ  (1) . 4 4

cos 2 θ 

10. Let AC= 2x and M the mid-point of AC.

6. 32 sin

7. tan 2 θ  2 tan 2 ϕ 1  1  cos 2θ  1  cos 2θ  2.  1 1  cos 2θ  1  cos 2θ   3  cos 2ϕ  1  cos 2θ  2.  (On simplification ) 1  cos 2θ  1  cos 2ϕ  1  cos 2ϕ  cos 2θ  cos 2θ cos 2ϕ  3  3cos 2θ  cos 2ϕ  cos 2θ cos 2ϕ Or 4 cos 2θ  2 cos 2ϕ  1  3  2 or 2 cos 2θ  2 cos 2ϕ  1 or 2 cos 2θ  (1  2 sin 2 ϕ )  1 or 2 cos 2θ  2 sin 2 ϕ  0 cos 2θ  sin 2 ϕ  0 Or

(a  b) 2 1 4ab

or

(a  b)2  / 0;

 tan 60  3

 16[2 cos 2 A  1( 4 cos3 A  3cos A )  16[3cos A 2 cos 2 A  4 cos3 A  1]  3  9 27  16 3.  2.  4 ⋅  1 16 64  4   36  18  27  16   16    11 16  

4ab (a  b) 2

∴

h1 3 ∴ h1 : h 2  3 : 1  3x  h2 x

11. a cos 2

C A 3b  c cos 2  2 2 2

 1  cosC   1  cosA  3b Or a   c  2 2     2  a c  1 3b or    (a cos C  c cos A)  2  2  2 Using, a cos C  c cos A  b, we get  a  c  b 3b a  c 3b b       or 2 2 2  2  2 2 or 2b  a  c 12.

sin A sin B  a b ⇒

sin 3B sin B  2b b (  ∠A  3∠B and a  2b) sin 3B  2sin B

or

3sin B  4 sin 3 B  2 sin B or 4sin 3 B  Sin B 4sin 2 B  1 2

sin B  1/4 1 sin B  ⇒ B  30 2 ∠A  3∠B  3  30  90

( sin B ≠ 0)

Additional Support Material ▌ 3.265

13.

16. sin( A  B )  sin A cos B  cos A sin B

1 1 1 1    r 2 r 21 r 2 2 r 2 3 s 2 (s  a )2 (s  b)2 (s  c )2  2   ∆2 ∆ ∆2 ∆2 1  2 {4s 2  2s (a  b  c)  a 2  b2  c 2 } ∆ 1  2 {4s 2  2s)(2s)  a 2  b2  c 2 } ∆ 2



2

a  b c ∆2

Substituting the values of cos A and sin A from the given Eq. (i), we have sin( A  B ) 

∴ r2  r1  r3  r2 or

∆ ∆ ∆ ∆   − s b sa sc s b

or

1 1 1 1    s b sa sc s b

ba c a  (s  a) (s  b) (s  a) (s  b) ∴( b  a ) (s  c )  (c  a ) (s  a ) or (a  b) (s  c)  ( b  c ) (s  a ) 15. Let r be the radius of the circle and n the number of the sides of π a the polygon and a the side of the polygon. Then r 5 cot   2 n (Standard result) π or a 5 2r tan   n

...( i)

Let a1, a2, a3 respectively be the sides of the triangle, square and a hexagon respectively. (i) For an equilateral triangle, n = 3; from (i) we get

On squaring and adding given equation, we get

∴ Angle θ is either in the second or fourth quadrant . Then, sin θ  0 or  0 4 4 ∴ sin θ may be or  5 5 18. cos n  sin n  tan 50 cos n  sin n ⇒ cos n (1  tan n )  tan 50 cos n 1  tan n ) tan 45  tan n ⇒ 1  tan 45 tan n   tan 50 ⇒ tan( 45  n )  tan 50 (1) 45  n  50 ⇒ n  5 Hence n must be a prime number and is of the from (4t  1) for some  v integer t. 1 1 19. cos 2 52  sin 2 7 2 2  (0.6087 )2  (.1305)2  0.3534  0.3534 1  2 2

π a 1 5 2r tan   2 3r 3 (ii) For a square, n = 4 π a 4 5 2r tan   5 2r 4 (iii) For a hexagon, n = 6  π  2r a 3 5 2r tan   5 3 6

1 3

2r 3

.

…(ii)

2

17. Since, tan θ, 0.

14. r1, r2 , r3 are in A.P.

ratio as

sin B cos B

cos 2 B  1 / 3 sin ( A  B ) 1/ 3

2

Clearly, 2 3r , 2r,

…(i)

are in G.P with common

x2  y2 2 xy 1  sin θ  1

20. sin 2 θ  

⇒ 0  sin 2 θ  1 ⇒ 0

x2  y2 1 2 xy

⇒ 0  x 2  y 2  2 xy ⇒

x 2  y 2  2 xy  0

⇒ ( x  y )2  0 ⇒

x y

where ( y ≠ 0)

▌ Mathematics

3.266

1 21. cos ( θ  α)  a ⇒ θ  α  cos a and cos (θ  β)  b ⇒ θ  β  cos1 a

24. cos A 

⇒ (θ  β )  ( θ  α)  cos1 b  cos1 a

{

(α  β )  cos1 ab  1  a 2 cos (α  β )  ab  1  a

1  b2

1 b

2

}

2

2 2

2 ab 1  a

1 b

2

2

( n  1)2  n2

Sin2 A  1  cos 2 A  1  cos B 

and

cos (α  β )  a b  (1  a ) (1  b )  2

n 1

2

( 2 n  1) ( 2 n  1)2  1

sin 2 B  1  cos 2 B

2

1

cos 2 (α  β)  1  a 2  b2  2 a 2 b2  2 ab 1  a 2

n 1 ( n  1) 2  n2

1  b2

sin B 

( 2 n  1)2 ( 2 n  1)2  1

1 ( 2 n  1)2  1

sin 2 (α  β)  a 2  b2  2 a 2 b2  25. A  0, B  0 and A  B 

2 ab 1  a 2 1  b2 Now, sin 2 (α  β )  2 ab cos (α  β )  a  b  2 ab b  2 ab 1  a 2

2

{

2 2

2aab ab  1  a

2

1 b

2

2

}

1 b  2

 a  b  2 a b  2 ab 1  a 2 1  b2  2

2

2 2

2 a 2 b2  2 ab 1  a 2 . 1  b2  ( a 2  b2 ) 22. sin x  sin y  3 (cos y  cos x ) sin x  3cos x  3 cos y  sin y r cos( x  α)  r cos( y  α) 1 3 x  α   ( y  α)

where r  10 , tan α  ⇒

⇒ x  y or x  y  2α Thus, ( x  y ) satisfies relation.  sin 3x.cos ec 3 y sin 3x sin 3x 1   sin 3 y sin 3x sin x sin x 4 23. e  e 1 sin x e  sin x  4 e ( esin x )2  4.( esin x )  1  0 Let ( y  esin x ) y 2  4 y 1 0 ⇒ 42 2 y ⇒ 2 ⇒ ( y  2  5) 2  5  3 ⇒ esin x  2  5   2  5  2  2  esin x  3 i.e., no solution ( does not exist ).

π 3

We know that , AM  GM tan A  tan B ⇒  (tan A. tan B )1/2 2 ⇒ (tan A  tan B )2  4(tan A. tan B) ( i) tan A  tan B Also, tan ( A  B)  1  tan A. tan B π tan A  tan B tan      3 3 1  tan A. tan B   Squaring on booth sides (tan A  tan B ) 2  3 (1  tan A. tan B ) 2 From Eq. ( i ) ⇒ 3(1 − tan A. tan B )2  4(tan A. tan B ) ⇒ 3  3(tan A. tan B )2  6(tan A. tan B )  4(tan A. tan B) ⇒ 3 (tan A. tan B ) 2  10(tan A. tan B )  3  0 ⇒ 3 (tan A. tan B  1)  10(tan A. tan B  3)  0  tan A. tan B ≠ 3 1 ⇒ 0  tan A. tan B  3 1 So, maximum value of tan A. tan B  3 26. Now, ( x )2  ( x  2)2  2 x  2  2 2  2  2  23/ 4 x2 sin θ  1 23/4 2 ⇒ cos α cos θ  sin α sin θ  1

∴

x

3/ 4

cos θ 

when tan α  

x2 x



x2 x x2 x2



x2 x2  2x

Additional Support Material ▌ 3.267



hence, for real value of θ the expression can have no values between (3  1,  3, 1)  (4,  4)

x2 2 1 2 2  2 2  2  x  2

∴cos( θ  α)  2 mπ ⇒ θ  2 mπ  α Wherre α  tan1 ( x  2) Hence, option (d) is correct. 27. Now, sin α  sin β  sin γ  sin ( α  β  γ )  sin α  sin β  sinn γ  sin α cos β cos γ  cos α sin β cos γ  cos α cos β sin γ  sin α sin β sin γ  sin α (1  cos β cos γ )  sin β (1 cos α cos γ ) sin γ (1 cos α cos β )  sin α sin β sin γ > 0 ∴sin α sin β sin γ  sin ( a  β  γ ) ⇒

sin ( α  β  γ ) 1 sin α  sin β  sin γ

28. Given, 7 cos x  5 sin x  2 k  1 ⇒



7 2

2

7 5

cos x 

5

sin x

2

7  52

2k  1 2

72 5

⇒ cos a cos x + sin α sin x  tan α  ⇒

2k  1

whhere

7 2  52

5 7 cos( a  x ) 

2k 1 74

−1  cos ( a  x )  1 ⇒ −1 

2k  1

1

74 Since,

⇒  74  2 k + 1  74 ⇒  8.6  1  2 k  8.6  1 ⇒  9.6  2 k  7.6 ⇒  4.8  k  3.8

 4  2  30. tan cos1    tan1   5  3    3  2  tan  tan1    tan1   4    3     1  3  1  4   cos    tan   4 5        3 2       tan  tan1  4 3    1  3 . 2     4 3     17  17  tan cos1     16  6   4  2  31. tan cos1    tan1   5  3    1  3   2   tan  tan    tan1    4  3     3 2      1 tan  tan  4 3    1  3  2      4 3    17  17 tan  tan1     16  6 

{

If we denote x cos1 x by y, then Since, 0  cos1 x  π ⇒

1 sin θ  3 cos ec θ  cot θ

...( i)

π π  sin1 ( 2 x 1  x 2 )  2 2 π π 1 ⇒  sin sin( 2 y)  2 2 π π  2y  ⇒ 2 2 U sin g Eqs. (i) and (ii), we get π 2 π 0 y 4 0  cos1 x  π / 4

0  2y 

Hence, integral value of k is 8.

 (3 + cos θ)

0  2y  2 π

also since

∴ k  4,  3,  2, 1, 0,11,2,3

29. (3  cos θ) cos ec θ

}

1 2 1 32. sin 2x (1  x )  2cos x

⇒ ⇒

which holds if

1 2

 x 1

3.268

▌ Mathematics

33. ln ∆APC , tan θ 5

AC AB / 2 5 AP AP B

30(1  3 ) AT

⇒

sin 45 

⇒

AT  30 2 (1  3 )

Applying sine ruule in ∆ABT , C

⇒

1 sin θ  60 2  30 2 (1  3 )

⇒

sin θ =

β P

A AB / 2 1  n ( AB ) 2 n





AB AP AB tan β  tan θ  ⇒ 1  tan β tan θ AP 1 tan β  2 n 1 ⇒ 1 n 1  tan β  2n ln ∆APB, tan(β  θ) 

tan β 

⇒

1 2 (1  3 )

α

35. Since e  π /2  θ  π / 2, ∴ log e − π / 2  log θ  log π / 2 π i.e.,   log e  log θ  log π / 2 2 π i.e.,   log θ  log π / 2  1  π / 2 2 ( because π / 2  e ∴ log π / 2  log e  1) Also, 1  π /2 ∴  π / 2  log θ  π / 2 ∴ cos( log θ)  0 But 0 cos θ  1;

∴ log ( cos θ)  log1  0

T

36. 4 cos3 A  3cos A  cos 3A  cos 90º  0 1 [because sin A = ⇒ A = 30º ] 2

D 30(3+√3)

1  tan 2 θ  37. Cos 2θ  1  tan 2 θ

60m 30º 60º

F

Here TD  30(3  3 )  60  30 (1  3 ) ln ∆TAD, 30(1  3 ) 30(1  3 )  tan α  AD 30(3  3 )



1 ⇒ α 

3  3) 30(1  3 ) Again,, sin α  AT

2  tan 2 φ  Sin 2φ  2  tan 2 φ

1  tan 2 φ  Cos 2φ  1  tan 2 φ

1 49  48  24 1 50 25 1 49 1

1 2( ) 3 2 9 3 1 3 10 5 1 9 1 −9  8  4 1 10 5 1 9

1

∴ Sin 4φ 2 Sin 2φ Cos 2φ  2 (3 / 5) ( 4 / 5) =

3 3(1  3 )

3 1

3 1

Q

B

3 1

Hence, cos (log θ) > log ( cos θ)

34. ∆BTF , tan 60  30(3  3 ) BF 30(3  3 ) ln BF  BF

A



2 2

1 1 tan β   2n n 2 n2 n 1 1 ⇒ tan β  2 tan β (1  2 ) 2n 2n 2n 1

⇒

sin θ sin 30  AB AT

π 4

24 25

38. A  B  C  90   A  B  30  ⇒ B  30, A  60, C  0. A  C  60 

Additional Support Material ▌ 3.269 46. From the graph, there is no point of intersection

39. 3sin 2 A  2 sin 2 B  1 ⇒ 3 sin 2 A  cos 2 B 3 Also, 3sin 2 A  2 sin 2 B  0 ⇒ sin 2 B  sin 2 A. 2 Now, cos( A  2 B )  cos A cos 2 B  sin A sin 2 B

y 0.7

3  cos A.3sin A  sin A. sin 2 A  3 sin 2 A cos 2 π A  3sin 2 A cos A  0; ∴ A  2 B  . 2 2

(– 1 , 48 ) 7 7 2 1

40. A  log (cot 1 cot 2 .... cot 45 .... cot 89 ) cot 89  tan 1, cot 88  tan 2; and cot 1 tan 1  1, cot 2 tan 2  1,cot 45  1, and A  log (1)  0.

47. From the graph, | cos x | and 2 [x] do not cut each other for any real value of x. Hence number of solutions is nil. y 4

 π 3π   5π   7π  42.  1  cos   1  cos   1  cos   1  cos  8 8 8 8     

3

 3π   π π 3π     1  cos   1  cos   1  cos   1  cos  8  8 8 8  

2 1

 π 3π    1  cos 2   1  cos 2  8  8 

0

 π 3π  π 2 3π 2 π sin sin cos 2    8  8 8 8 2

π 1 1  sin 2    . 4 4 8

3

x

(sin 30 sin 45  sin 120 ) 8  3 1 1  15  1 1 3         17  2 2  2 2  17  2 2  3 1 1  23  .    2 2  17 

π  5π   5π  π  sin 2    cos 2    sin 2    sin 2    1. 8 8 8      8   



44. Squaring and adding given relation, we get 9 cos 2 x  4 cos 2 3x  12 cos x cos 3x  9 sin 2 x  4 sin 2 3x  12 sin x sin 3x  1

49.

9  4  12 (cos 2 x )  1 ⇒ cos 2 x 1.

⇒

2

45 sin θ  cos 120 cos θ  sin 120 sin θ  cos θ (cos 30  cos 45  cos 120 )  sin θ

 5π   π  π 3π  5π   sin    cos 2     sin 2    sin 2  π   8 8  8 2 8       

1 1 k 2 1 sec θ  k   2 k 2k 2k k 2 1 ∴sin θ 2 . cos θ 2 k 1 k 1

 4

(17 )2  82 15 8 ⇒ sin θ   17 17 17 Now, the given expression is equal to cos 30 cos θ  sin 30 sin θ  cos 45 cos θ  sin

2

⇒

1

48. cos θ 

2 π 2  3π  2  5π  2  13π  43. sin    sin    sin    sin   8 8 8        8 

45. sec θ  tan θ  k , sec 2 θ  tan 2 θ  1; So sec θ  tan θ 

x

0 1 2

2 2 2 2 41. y  a sin x  b cos x  c lies in c  a  b , c  a  b    So, 3 sin x  4 cos x  1 lies in [6, 4].

 sin 2

y= n y= n sin x

1 k

We have,

cos 2B cos ( A  C )  . 1 cos ( A  C )

Applying componendo and dividendo, we get 1  cos 2 B cos ( A  C )  cos ( A  C )  1  cos 2 B cos ( A  C )  cos ( A  C ) ⇒

2 sin 2 B 2 sin A sin C  2 cos 2 B 2 cos A cos C

 tan 2 B  tan A tan C ⇒ tan A, tan B, tan C are in G.P

3.270

▌ Mathematics

 π  1 α 50. We have, tan α ⋅ tan ( 2 n  1) α  tan α ⋅ tan   2α  π   tan α ⋅ tan   α  2   tan α cot α  1. ∴ The given expression 1.

55. We have, cos 50° = cos225°2sin225° = (cos 25° + sin 25°) (cos25°2sin 25°) = k(cos25°2sin 25°). But (cos 25° + sin 25°)2 + (cos25° 2sin 25°)2 = 2 ⇒ cos 25  sin 25  2  k 2 ( As cos 25  sin 25 is  ve )

51. We have, 2  cos x  sin 2 x  2  cos x  1  cos 2 x (cos 2 x  cos x )  3 2  1 1    cos x      3 2 4    2



13  1   cos x   . 4  2

∴ Maximum value occurs at cos x 

1 2

13 and minimum value occurs at cos 4 x 1 and it is 1 13 ∴ The required ratio is. 4 52. We have,  π sin (π cos θ )  cos (π sin θ )  sin   π sin θ   2 π π ⇒ π cos θ   π sin θ ⇒ π (cos θ  sin θ )  2 2 1 ⇒ cos θ  siin θ  2 π π 1 1 ⇒ cos θ ⋅ cos  sin θ ⋅ sin  . 4 4 2 2  1  π π ∴ cos  sin   4 4 2   1 π ⇒ cos  θ    . 4 2 2   53. We have, y = 4 sin2θ – cos 2θ = 4 sin2θ – (1 – 2 sin2θ) = 6 sin2θ – 1 y 1  sin 2 θ 6 But 0  sin 2 θ  1. y 1 ∴ 0 1 ⇒ 0  y 1  6 6 ⇒ 1  y  5 ⇒ y ∈[1, 5]. ∴

π 2π 4π .cos . . 7 7 7

2 sin α 1  cos α  sin α ⇒ x (1  cos α  sin α )  2 sin α

56. Given, x 

⇒ x (1  cos α )  ( 2  x )sin α Let y 

and it is

54. cos

cos 50  k 2  k 2

∴

x 2 sin α (1  sin α ) 1  cos α  sin α . Now  y (1  sin α )2  cos 2 α 1  sin α

⇒

x 2 sin α (1  sin α )  1 y 1  sin 2 α  2 sin α  1  sin 2 α

⇒

x y

57. Since

y

1  2 y

and |sin x  cos x|  2 , therefore 1 Hence y   2 and sin x  cos x , y which is possible for y  1, x  58. Numerator of given expression = (sin 8x + sin 6x) + 6(sin 6x + sin 4x) + 12 (sin4x + sin 2x) = 2 sin 7x cos x + 12 sin 5x cos x + 24 sin 3x cos x = 2 cos x (sin 7x + 6 sin 5x + 12 sin 3x) Hence, given expression 

2cos x (sin 7 x  6 sin 5x  12 sin 3x )  2 cos x. sin 7 x  6 sin 5x  12 sin 3x

59. As a, b, c are roots of x3 2 15x2 + 47x 2 82 = 0 ∴ a + b + c = 15 ab + bc + ca = 47 abc = 82 Now cos A cos B cos C a 2  b2  c 2    2abc a b c 2 2 2   a c b , so on  ∴cos B  2abc  



( a  b  c )2  2( ab  bc  ca ) 2abc 225  2( 47 ) 131   2(82) 164



Additional Support Material ▌ 3.271 10

60.  ∏ (sin 2 p π  i cos 2 p π)

Also, cos

p 1

10

 ( i )10 ∏ (cos 2 p π  i sin 2 p π)

so, cos π / 7 cos 2 π / 7 cos10 π / 7  sin

p 1

( 7 ) cos (2π 7 ) cos (4π 7 ) sin (8π ) 7  1  2.  4 π 2 .sin ( ) 7

2 pπ 2π (1  2 ... 10)i  ∏ 11  e 11 p 1  e10π i 1 (cos 10 π  i sin 10 π ) 1

3

pq 1 pq ⇒  pq sin θ p  q Apply componendo and dividendo, we get 1  sin θ p  q  p  q ⇒  1  sin θ p  q  p  q

61. Given, cosec θ 

2

  θ θ θ  cos 2  sin 2  p 1  tan 2  p ⇒   ⇒    θ θ θ q q  cos  sin  1  tan    2 2  2      q π θ π θ p ⇒ tan 2     ⇒ cot 2     4 2 4 2 q     p 62. We have, cos 2 θ  2 cos θ  2cos 2θ  1  2 cos θ 2

 3 1  2  cos θ    2 2   2

 1 Now 2  cos θ    0 for all θ 2  2

 1 3 ∴ ∴ 2  cos θ     0 for all θ 2 2   3 ⇒ cos 2 θ  2cosθ  for all θ 2 Also, maximum value of this exprression is 3. 5π π 3π sin sin 14 14 14  π 3π   π 5π  π π   cos    cos    cos    14 2 14 2    2 14   

 3π   2π π π  cos   cos   cos    7   7  7  4π π  cos ( π / 7 ) cos ( 2 π / 7 ) cos    7 

π 3π 5π sin sin 14 14 14

 cos π

10

63. sin

10 π 4π  cos 7 7

64. If sin x  0, cos x  0 and tan x  1, the given equationn can be written as tan x 1 2 tan x log 2 1 ⇒ log 2 . 1 2 1  tan 2 x 1  tan 2 x 1 ⇒ 1 tan 2 x  21  ⇒ tan 2 x  1 2 2 ⇒ x  2 nπ  π  (16 n  1) π 8 8

( )

65. Given tan α  3 and tan β  2 3 2 1 tan α  tan β   ∴ tan (α  β)  1  tan α tan β 1  3  2 7 7 1 ⇒ sin (α  β )  and cos( α  β )  50 50 ∴sin 2 (α  β )  2 sin (α  β ) cos (α  β) 2

1 50



7



50

7 . 25

66. The given equation can be written as 3 – 2x 2 x2 = 1 + tan2 (x + y) + 1 + cot2 (x + y) ⇒ 4 – (x + 1)2 = sec2 (x + y) + cosec2 (x + y) ⇒ cos2 (x + y) sin2 (x + y) [4 (x + 1)2] = 1 ⇒ sin2 (2x + 2y) [42(x + 1)2] = 4

...(1)

Since sin (2x + 2y)  1 and 42(x + 1)  4 2

2

∴ (1) holds only if sin2 (2x + 2y) = 1

...(2)

and 42 (x + 1) = 4

...(3)

2

From (3), we get x =1 Putting in (2), we get sin (2y 2 2) = ± 1 ⇒ 2y22 = , n ∈ Z ⇒ y = , n ∈ Z ∴ x 1, y  1  ( 2 n  1)

π , n∈ Z 4

3.272

▌ Mathematics

CoorDinate geoMetry 1. If the vertices of a triangle have integral coordinates, then the triangle is (a) Equilateral (b) Never equilateral (c) Always isosceles (d) None of these 2. The area of the quadrilateral with vertices at (221), (423), (1, 2) and (−3,−2) is (a) 18 (b) 36 (c) 54 (d) None of these 3. Locus of the point, the sum of squares of whose distances from the points (a,0) and (2a,0) is equal to 2c2, is (a) x 2  y2  c 2  a 2 2

2

2

(c) x  y  c  a

2

(b) x 2  y2  c 2  a 2 (d) None of these

4. The equation of the right circular cylinder of radius a with its axis parallel to the x-axis is (a) x 2  y2  a 2 2

2

(c) z  x  a

2

(b) x 2  z 2  a 2 (d) None of these

5. The distance between the lines 3x + 4y = 9 and 6x +8y = 15, is 3 (a) 2 (c) 6

3 (b) 10 (d) None of these

6. The lines represented by ax 2  2hxy  ax 2  0 are (a) coincident (b) perpendicular (c) parallel (d) None of these 7. The tangents to the circle x 2  y2  169 at the points (5, 12), (12, −5) are (a) parallel (b) perpendicular (c) coincident (d) None of these 8. The length of the intercept cut off by the circle x 2  y2  4 x  8 y  12  0 on the x-axis (a) 4 (c) 8

(b) 6 (d) None of these

9. The distance of the point (1,2) from the radical axes of the circles x 2  y2  6 x  16  0 and x 2  y2  2 x  6y  6  0 is (a) 1 (c) 3

(b) 2 (d) 4

10. The angel between the tangents from the point (4,3) to the circle x2 + y2 − 2x − 4y = 0 is (a) 30º (b) 45º (c) 60º (d) 90º

11. The point on the curve y2 = x, the tangent at which makes an angle of 45º with the x-axis is given by  1 1 (a)  ,   2 4

 1 1 (b)  ,   2 2

(c) ( 2, 4)

 1 1 (d)  ,   4 2

12. If (x1 , y1 ) (x 2 y2 ) and ( x 3 , y3 ) be three points on the parabola y2 5 4ax, the normals at which meet in a point, then value of y1 + y2 + y3 is (a) 0 (b) 1 (c) −1 (d) None of these 13. The eccentricity of the ellipse 16x 2  7 y2  112 1 7 (b) 4 16 3 3 (c) (d) 7 4 14. The eccentricity of the ellipse represented by the equation (a)

25x 2  16 y 2  150 x 175  0,is 2 5 4 (c) 5 (a)

(b)

3 5

(d) None of these

15. The coordinates of the foci of the ellipse represented by the equation in question 74 are (a) (3, 6 3)

(b) ( 2, 6 2)

(c) (1, 61)

(d) None of these

x 2 y2   1, the equation of the diameter a 2 b2 b conjugate to the diameter y 5 .x , is a

16. In the ellipse

b (a) y  x x b (c) x  y a

a (b) y  x b (d) None of these

17. If the eccentricity of ellipse becomes zero, then it takes the form of (a) a circle (b) a parabola (c) a straight line (d) None of these 18. When m varies, the locus of the point of intersection of the straight lines

x y x y 1   m and   is a b a b m

(a) a parabola (c) a hyperbola

(b) an ellipse (d) a circle

Additional Support Material ▌ 3.273 19. The eccentricity of a rectangular hyperbola is always (a) 2 (c)

(b) 2

3

(d) None of these

x 2 y2   1 and a 2 b2 P (x, p) any point on it, the value of S’P − SP is

20. If S and S’ are the foci of the hyperbola (a) a (c) 2a

(b) b (d) 2b

21. The distance of the points P (x, y, z) from the x-axis

29. The axis of the right circular cylinder x 2  y2  a 2 is parallel to the (a) x-axis (b) y-axis (c) z-axis (d) None of these 30. Tw o c i r c l e s x 2  y 2  6 a n d x 2  y 2  6 x  8  0 a r e given. Then the equation of the circle through their points of intersection and the point (1,1) is (a) x 2  y 2  6 x  4  0

(b) x 2  y 2  3x  1  0

(c) x 2  y 2  4 y  2  0

(d) None of the above

(a)

y2 1 z 2

31. All points whose distance from the nearest point on the circle

(b)

z2 1 x2

( x  1)2  y 2  1 is half the distance from the line x = 5 lie on

(c)

x 2 1 y2

(d)

x 2 1 y2 + z 2

(a) an ellipse (c) a parabola

22. The projection of a line on the coordinate axes are 2, 3, 6. The length of the line is (a) 2 (b) 3 (c) 6 (d) 7 23. The angle between the two diagonals of a cube is (a) 30º (b) 45º 1 (c) cos21   (d) None of these  3 x y z 24. The plane   1 meets the coordinate axes at A, B, a b c C; then coordinates of the centroid of the ∆ ABC are (a) (a, b, c) (c) (ab, bc, ca )

a b c (b)  , ,   3 3 3 (d) None of these

25. The three planes x + y + z − 6 = 0, 2x + 3y + 4z − 20 = 0, x − y +z − 2 = 0 (a) intersect in a point (b) intersect in common line (c) form a triangular prism (d) None of these 26. The number of spheres of radius a touching the coordinate axes is (a) 4 (b) 6 (c) 8 (d) None of these

32. AB and CD are the diameters of a circle of radius 1 unit with centreO. PB and PD are tangents at B and D which meet at the point P. If BC = 1 unit, then what is the area of the DPBD ? (a) 3 3 sq units

(b) 3 3 / 2 sq units

(c) 3 3 / 4 sq units

(d) 3 3 / 2 untis

33. What is the area bounded by the circle x 2 + y2 = r 2 , r =1,2 and the rays given by 2x 2  3xy  2 y2  0, y  0? 3π 3π (a) sq unit (b) sq unit 2 4 3π (d) 3π sq unit (c) sq unit 8 34. What is the locus of z = x + iy on Argand plane such that Arg z 2 π   ? z 2 2 (a) A straight line (b) A circle with centre origin and radius 2 (c) A circle with centre origin and radius (d) An ellipse

2

35. What is the equation of the circle which passes through the points (2,0), (0,2) and is orthogonal to the circle 2 x 2  2 y 2  5x  6 y  4  0 ? (a) 7x 2  7y 2  8x  8y  12  0

27. The radius of the circle x 2  y2  z 2  x  z  2  0; x  2y  z  4, is (a) 1 (c) 3

(b) a pair of straight lines (d) a circle

(b) 2 (d) 4

(b) x 2  y 2  8x  8y  6  0 (c) x 2  y 2  8x  8y  0 (d) 7x 2  7y 2  8x  8y  0 36. The curve described parametrically by

28. The spheres x 2  y2  z 2  x 2  y2  z 2  24x  40y  18z  225  0

x  t 2  t  1, y  t 2  t  1 represents

(a) touch internally (c) intersect orthogonally

(a) a pair of straight lines (c) a parabola

(b) touch externally (d) None of these

(b) an ellipse (d) a hyperbola

3.274

▌ Mathematics

37. One end of a focal chord of the parabola y 2 5 8ax is the point ( 2 at 2 , 4 at ). What is the other end of the chord?  a 2 2 3a   (a)  2 , t  t    2a 2 4a  (c)  2 ,  t  t

 a 2 2a  (b)  2 ,  t   2t  2a 4a  (d)  2 ,  t  t

38. If P  ( x , y ), F1  (3, 0), F2  (  3, 0) and 16x 2  25 y 2  4000, then PF1  PF2 is equal to (a) 8 (c) 10

(b) 6 (d) 12

39. The equation

x y   1, r  1 represents 1 r 1 r

(a) an ellipse (c) a circle

(b) a hyperbola (d) None of these

2

2

x2 y2   1. Let c1 and a 2 b2 c2 respectively be the y-axis with the tangents at P and Q. If (xl, y1) and (x2, y2) are the coordinates of the points P and Q, then which one of the following is correct? (a) cl y1 2 c2 y2 = 0 (b) cl y2 2 c2 y1 = 0

40. Let P and Q be points on an ellipse

(c) cl y1 2 c2 y2 = 1

(d) cl y22 c2 y2 = 1

41. AB is the chord of the parabola y2 = 4ax, a > 0 with vertex at A. BC is drawn perpendicular to AB meeting the axis of the parabola at C.What is the projection of BC on the axis of the parabola? (a) a (b) 2 a (c) 4 a (d) 8 a 42. An ellipse

x 2 y2   1 slides between two perpendicular a 2 b2

straight lines. What is the locus of its centre? (a) Circle (b) Parabola (c) Ellipse (d) Hyperbola 43. What is the length of latusrectum of the elliipse 5x 2  9 y2  45? (a) 5/3 (c)

2 5 3

y  tx  2at  at 3 (b) y  tx  2at  at 3 (c) y  tx  2at  at 3 (d) None of the above

(a) tan21 ( t 2 )

(b) cot21 (t 2 )

(c) tan21 (t)

(d) cot21 (t)

46. If the parametric equation of a parabola is x  t 2  1, y  2t  1, then what is the Cartesian equation of its directrix? (a) x = 0 (c) x + y = 0

10 3

(d)

5 3

(b) y = 0 (d) x 2 1 = 0

47. The points ((2a, 2 b), (0, 0), ( a, b) and are (a) (b) (c) (d)

collinear vertices of a rectangle vertices of a parallelogram None of the above

48. It is given that three distinct points (x 1, y1), (x2, y2) and (x3, y3)are collinear. Then a necessary and sufficient condition for (x 2, y 2) to lie on the line segment joining (x 3, y 3) to (x1, y1) is (a) either (x1  y1 )  ( x2  y2 )  ( x3  y3 ) or (x1  y1 )  ( x2  y2 )  ( x3  y3 ) (b) either (x1  y1 )  ( x2  y2 )  ( x3  y3 ) or (x1  y1 )  ( x2  y2 )  ( x3  y3 )  x  x3   y2  y3  (c) either 0   2   1 or 0   y  y   1  x1  x3   1 3 (d) None of these 49. A point P lies on the line passing through the points (12, 3, 3) and (– 3, 0, 9). If the z-coordinate of P is 1, what are the x and y coordinates? (a) x 5 4, y 5 17

(b) x 5 7, y 5 2

(c) x 5 2, y 5 7

(d) x 5 17, y 5 4

x4 y2 z k   lies in 1 1 2 the plane 2x  4 y  z  7, is (b)  7 (a) 7

50. The value of k such that

(c) No real value (b)

44. What is the equation of normal at P? (a) y  xt  at

45. What is the angel at which the tangent at P to the parabola is inclined to the tangent at P to the cirle through T, P, G?

(d) 4

51. The number of values of C such that the straight line y = 4x + c touches the curve (a) 0 (c) 1

x2  y 2  1 is 4 (b) 2 (d) ∞

52. Consider two points A(a,b)and B(c,d), where c > a > 0 and d > b > 0 then abscissa of a point on x-axis such that sum of its distances from A and B is minimum, is (a) ( bc  ad ) / ( b  d )

(b) ( ac  bd ) / ( b  d )

(c) ( ac  bd ) / ( b  a )

(d) ( ad  bc ) / ( b  d )

Additional Support Material ▌ 3.275

53. Consider the equation L  a sin x  a 3 cos x (a  0, x is real). For any given a > 0, the value of L is (a) 23 (b) 4 (c) 2 (d) 24 54. Consider the equation L  a sin x  a

3 cos x

(a  0, x is real). For

any given a >0, the value of L is (a) greater than 2 (b) greater than or equal to 2 /a (c) less than 2 /a (d) less than 2 55. What is the set of points (x, y)on the curve that lie on the coordinate axes? (a) {(22, 0), ( 2, 0)} (b) {(0, 0), (0, 2)} (c) {( 2, 0), (0, 4)}

(d) {(0, 0), ( 4, 0)}

56. A point (4,1) undergoes the following three transformations? (1) Reflection about the line y = x (2) Translation through a distance 2 unit along the positive direction of x- axis π (3) Rotation through an angle about the origin in the 4 anticlockwise direction Which one of the following points gives the final location of the original points?  7  (b)  22,  2 

7   2 7   2

 1 (a)  ,  2  21 (c)  ,  2

(d) ( 2 , 2 )

57. What is the distance of the point (a, b)from the line Ix + my + n = 0 measured parallel to a line making an angle q with the x - axis? (a) Ia 1 m β 1 n Ia 1 m β 1 n (c) I cos θ 1 m sinn θ

(b) I cos θ 1 m sin θ I cos θ 1 m sin θ (d) Ia 1 mβ 1 n

58. If the sum of the squares of the distance of the point (x, y) from the points (a, 0) and (-a, 0) is 2b2, then (a) x 2  a 2  b2  y 2

(b) x 2  a 2  2b2  y 2

(c) x  a  b  y

(d) x 2  a 2  b2  y 2

2

2

2

2

59. The coordinates of the point P which divides the join of A (5, –2) and B(9, 6) in the ratio 3:1 are (a) (4, – 7) (c) (8, 4)

(b) (7/2, 4) (d) None of these

60. The area of the region enclosed by 4 | x | + 5 | y | ≤ 20 is (a) 10 (b) 20 (c) 40 (d) none of these 61. The vertices of a DOBC are O (0, 0), B (23,21) and C 21,23). The equation of a line parallel to BC and intersecting sides OB 1 and OC whose distance from the origin is , is 2

(a) x  y 

1

0

2

1 (c) x  y   0 2

(b) x  y  0 1 (d) x  y   0 2

62. If two vertices of an equilateral triangle have integral coordinates then the third vertex will have (a) coordinates which are irrational (b) atleast one coordinate which is irrational (c) coordinates which are rational (d) coordinates which are integers 63. The distance of the point (2, 3) from the line 3x + 2y = 17, measured parallel to the line x y = 4 is (a) 4 (b) 5 (c)

2

(d) none of these

64. Through the point (1, 1), a straight line is drawn so as to form with coordinate axes a triangle of area S. The intercepts made by the line on the coordinate axes are the roots of the equation (a) x2 2 | S | x + 2 | S | = 0 (b) x2 + | S | x + 2 | S | = 0 (c) x2 2 2 | S | x + 2 | S | = 0 (d) none of these 65. The base of a triangle lies along the line x = a and is of length a. The area of the triangle is a2, if the vertex lies on the line (a) x = 0 (b) x = 2 a (c) x = 3a (d) x = 2 3a 66. If a, b, c are three terms of an A.P., then the line ax + by + c = 0 (a) has a fixed direction (b) always passes through a fixed point (c) always cuts intercepts on the axes such that their sum is zero (d) forms a triangle with the axes whose area is constant. 67. If the straight line drawn through the point P( 3, 2) and making π an angle with the x-axis meets the line 3 x–4y + 8 = 0 at Q, 6 then the length of PQ is (a) 4 (b) 5 (c) 6 (d) none of these 68. The equation of the common tangent to the curves y2 = 8x and xy = 2 1 is (a) 3y = 9x + 2 (b) y = 2x + 1 (c) 2y = x + 8 (d) y = x + 2 x2 y2   1 through 69. The number of tangents to the hyperbola 4 3 (4, 1) is (a) 1 (b) 2 (c) 0 (d) 3 70. If a, b, c form a G.P., then twice the sum of the ordinates of the points of intersection of the line ax + by + c = 0 and the curve x + 2y2 = 0 is (a) b/a (b) c/a (c) a/c (d) a/b

3.276

▌ Mathematics

71. If the vertices P, Q, R of a DPQR are rational points, which of the following points of the DPQR is (are) always rational point(s)? (a) centroid (b) incentre (c) circumcentre (d) orthocentre (A rational point is a point both of whose coordinates are rational numbers) 72. If the distance of any point P(x, y) from the origin is defined as d(x, y) = Max.{|x|, |y|} and d(x, y) = k (non-zero constant), then the locus of the point P is (a) a straight line (b) a circle (c) a parabola (d) none of these 73. The image of the point P(a, b) with respect to the line y = 2 x is the point Q and the image of Q with respect to the line y = x is R, then the mid point of R is (a) (α + β, β + α) (b) (0, 0) (c) (α 2 β, β 2 α) (d) none of these 74. The least value of |k| so that the lines x = k + m, y = 2 2 and y = mx are concurrent, is (a) 2 2

(b)

(c) 4 2

(d) none of these

2

75. A straight line through the origin O meets the parallel lines 4x + 2y = 9 and 2x + y + 6 = 0 at points P and Q respectively. The point O divides the segment PQ in the ratio (a) 1 : 2 (b) 3 : 4 (c) 2 : 1 (d) 4 : 3 76. Two points (a, 3) and (5, b) are the opposite vertices of a rectangle. If the other two vertices lie on the line y = 2x + c which passes through the point (a, b) then the value of c is (a) 2 7 (b) 2 4 (c) 0 (d) 7 77. If algebraic sum of distances of a variable line from points (2, 0), (0, 2) and (2 2, 2 2) is zero, then the line passes through the fixed point (a) (2 1, 2 1) (b) (0, 0) (c) (1, 1) (d) (2, 2) 78. Line 2x + 3y = 12 meets x-axis at A and y-axis at B. The line passing through (5, 5) is perpendicular to AB and it meets x-axis, y-axis and AB at points C, D and E respectively. If O is the origin, then the area of OCEB is (a) 23 sq. unit (c)

23 sq. unit 3

23 sq. unit 2 23 (d) sq. unit 4 (b)

79. The straight lines of the family x(a + b) + y(a 2 b) = 2a (a and b being parameters) are (a) not concurrent (b) concurrent in (1, 2 1) (c) concurrent at (1, 1) (d) none of these

80. The locus of the mid-point of the portion intercepted between the axes by the line x cos a + y sin a = p passes through the point (p + 1, p 2 1), then p4 2 5p2 is equal to (a) 2 2 (b) 2 1 (c) 0 (d) 2 81. The straight line y = x 2 2 rotates about a point where it cuts the x-axis and becomes perpendicular to the straight line ax + by + c = 0. Then its equation is (a) ax + by + 2a = 0 (b) ax 2 by 2 2a = 0 (c) by + ay 2 2b = 0 (d) ay 2 bx + 2b = 0 82. The algebraic sum of the perpendicular distances from the angular points (xi, yi) where i = 1, 2, 3 of a triangle to a variable straight line ax + by + c = 0 is zero. The line always passes through the (a) incentre of the triangle (b) circumcentre of the triangle (c) centriod of the triangle (d) none of these 83. The slopes of the lines joining the origin to the points of trisection of the portion of the line 3x + y = 12 intercepted between the axes are the roots of the equation (a) 2x2 2 15x + 18 = 0 (b) 3x2 2 20x + 12 = 0 2 (c) 12x 2 20x + 3 = 0 (d) 18x2 2 15x + 2 = 0 84. If the point (1, a) lies in between the lines x + y = 1 and 2(x + y) = 3, then a lies in (a) (2∞, 0) ∪ (1, ∞ )

 1 (b)  0,   2

1 (c) (2∞, 0) ∪ ( , ∞ ) 2

(d) None of these

π in the 4 clockwise direction about the point (0, 2). The equation of the line in its new position is (a) 7y + x 2 14 = 0 (b) 7y 2 x 2 14 = 0 (c) 7y + x 2 2 = 0 (d) 7y 2 x = 0

85. The line 3x 2 4y + 8 = 0 is rotated through an angle

86. The larger of the two angles made with the x axis of a straight line drawn through (1, 2) so that it intersects x + y = 4 at a point distant (a) 105º (c) 60º

6 from (1, 2) is 3 (b) 75º (d) 25º

87. Area of the parallelogram formed by the lines y = mx, y = mx + 1, y = nx and y = nx + 1 equals (a) (c)

mn ( m  n) 2 1 mn

(b)

2 mn

(d)

2 mn

Additional Support Material ▌ 3.277 88. The point whose coordinates are x = x1 + t(x2 2 x1), y = y1 + t(y22 y1) divides the join of (x1, y1) and (x2, y2) in the ratio (a)

t (1  t )

(b)

(1  t ) t

(c)

t (1  t )

(d)

(1  t ) t

89. The equation of a circle with origin as centre and passing through the vertices of an equilateral triangle whose median is of length 3a is (a) x2 + y2 = 9a2 (b) x2 + y2 = 16a2 2 2 2 (c) x + y = 4a (d) x2 + y2 = a2 90. If a straight line through C(2 8 , 8 ) making an angle of 135º with the x-axis cuts the circle x = 5 cos q, y = 5 sin q at points A and B, then the length of AB is (a) 3 (b) 7 (c) 10 (d) none of these 91. An equation of the circle in which the chord joining the points (1, 2) and (2, 2 1) subtends an angle of π/4 at any point on the circumference is (a) x2 + y2 2 15 = 0 (b) x2 + y2 2 6x 2 2y + 5 = 0 (c) x2 + y2 + 6x + 2y 2 15 = 0 (d) x2 + y2 2 2x 2 4y + 4 = 0 92. An isosceles DABC is inscribed in a circle x2 + y2 = a2 with the vertex A at (a, 0) and the base angles B and C each equal to 75º then length of the base BC is (a) a / 2 (b) a (c) 2 a / 3

(d)

3a / 2

93. The point (1, 4) lies inside the circle x2 + y2 2 6x 2 10y + p = 0 which does not touch or intersect the coordinate axes, then (a) 0 < p < 29 (b) 25 < p < 29 (c) 9 < p < 25 (d) 9 < p < 29 94. If the circle x2 + y2 + 4x + 22y + c = 0 bisects the circumference of the circle x2 + y2 2 2x + 8y 2 d = 0, then c + d is equal to (a) 60 (b) 50 (c) 40 (d) 56 95. The equation of the circle which touches the circle x2 + y2 2 6x + 6y + 17 = 0 externally and to which the lines x2 2 3xy 2 3x + 9y = 0 are normals, is (a) x2 + y2 2 6x 2 2y 2 1 = 0 (b) x2 + y2 + 6x + 2y + 1 = 0 (c) x2 + y2 2 6x 2 6y + 1 = 0 (d) x2 + y2 2 6x 2 2y + 1 = 0 96. The equation of the circle passing through the point a (2a, 0) and whose radical axis is x 5 with respect to the 2 circle x2 + y2 = a2, will be (a) (b) (c) (d)

x2 + y2 2 2ax = 0 x2 + y2 + 2ax = 0 x2 + y2 + 2ay = 0 x2 + y2 2 2ay = 0

97. The two points A and B in a plane are such that all points p PA satisfying 5 k , then k will not be equal to PB (a) 0 (b) 1 (c) 2 (d) none of these 98. The locus of the centre of the circle which touches the x-axis and also the circle x2 + y2 2 4x 2 4y + 4 = 0 externally is (a) x2 + y2 2 4x 2 4y 2 8 = 0 (b) x2 2 4x 2 8y + 4 = 0 (c) x2 + y2 2 4x 2 5y + 6 = 0 (d) x2 2 4x + 4 = 0 99. If q1, q2 be the inclination of tangents drawn from the point P to the circle x2 + y2 = a2 with x-axis, then the locus of P, if given that cot q1 + cot q2 = c, is (a) c(x2 2 a2) = 2xy (b) c(x2 2 a2) = y2 2 a2 (c) c(y2 2 a2) = 2xy (d) none of these 100. The chords of contact of the pair of tangents drawn from each point on the line 2x + y = 4 to the circle x2 + y2 = 1 pass through a fixed point (a) ( 2, 4)

 1 1 (b)  2 , 2  4  2

1 1 (c)  ,  2 4

(d) (22, 2 4)

101. If the equation of a given circle is x2 + y2 = 36, then the length of the chord which lies along the line 3x + 4y 2 15 = 0 is (a) 3 6

(b) 2 3

(c) 6 3

(d) none of these

102. The locus of a point which divides the join of A(2 1, 1) and a variable point P on the circle x2 + y2 = 4 in the ratio 3 : 2 is (a) 25(x2 + y2) + 20(x 2 y) + 28 = 0 (b) 25(x2 + y2) + 20(x 2 y) 2 28 = 0 (c) 20(x2 + y2) + 25(x 2 y) + 28 = 0 (d) none of these 103. The equation of the line through the point (1,2,3) parallel to the line

x  4 y  1 z  10 are   2 3 8

x 1 y  2 z  3   2 8 3 x  4 y  1 z  10 (c)   1 2 3 (a)

(b)

x 1 y  2 z  3   1 2 2

(d) None of these

104. If P1 = 0 and P2 = 0 be two non-parallel planes, then the equation P1 + λP2 = 0, λER represents the family of all planes through the line of section of the plane P1 = 0 and P2 = 0 except the plane (a) P1 = 0 (b) P2 = 0 (c) P1 + P2 = 0 (d) P1 – P2 = 0.

3.278

▌ Mathematics

105. A (3, 2, 0), B (5, 3, 2), C (–9, 6, –3) are three points forming a triangle. If AD, the bisector of ∠BAC meets BC in D, then co-ordinates of D are  219 57 17  (a)  , ,   8 16 16   19 57 217  (c)  , ,   8 16 16 

 19 257 17  (b)  , ,   8 16 16   19 57 17  (d)  , ,   8 16 16 

1 1 1 106. If the direction cosines of a line are  , ,  then c c c (a) 0 < c < 1 (b) c > 2 (c) c > 0 (d) c  3 107. The centre of the sphere which passes through (a, 0, 01), (0, b, 01), (0, 0, c) and (0, 0, c) is a  (a)  , 0, 0  2    c (c)  0, 0,  2 

 b  (b)  0, , 0   2  a b c (d)  , ,  2 2 2

108. The equation of the sphere passing through (0, 0, 0), (a, 0, 0), (0, b, 0), (0, 0, c) is (a) x2 + y2 + z2 + 2ax + 2by + 2cz = 0 (b) x2 + y2 + z2 – 2ax – 2by – 2cz = 0 (c) x2 + y2 + z2 – ax – by – cz = 0 (d) x2 + y2 + z2 + ax + by + cz = 0 109. The equation of the sphere on the join of (2, 3, 5) (4, 9, –3) as diameter is (a) x2 + y2 + z2 – 6x – 12y – 2z + 20 = 0 (b) (x – 2)(x – 4)(y – 3)(y – 9) = 0 (c) (x – 2)(x – 4) + (z – 5)(z + 3) = 0 (d) (y – 3)(y – 9) + (z – 5)(z + 3) = 0 110. The equation of the sphere concentric with the sphere x2 + y2 + z2 – 4x – 6y – 8z = 0 and which passes through the origin is (a) x2 + y2 + z2 – 4x – 6y – 8z = 0 (b) x2 + y2 + z2 – 6y – 8z = 0 (c) x2 + y2 + z2 = 0 (d) None of these 111. Centre of the sphere (x – x1)(x – x2) + (y – y1)(y – y2) + (z – z1) (z – z2) = 0 is (a) ( x2 , y2 , z2 )  x  x2 y1  y2 z1  z2  (b)  1 , ,  2 2   2  x  x2 y1  y2 z1  z2  , , (c)  1  2 2   2 112. The latus rectum of the parabola y = 5x + 4y + 1 is (b) 10 (d)

115. The eccentricity of the conic 9x2 + 25y2 = 225 is 2 5 1 (c) 3

4 5 1 (d) 5

(a)

(b)

116. The slope of the tangent to the hyperbola 2x2 – 3y2 = 6 at (3, 2) is (a) –1 (b) 1 (c) 0 (d) 2 117. The radius of the director circle of the hyperbola x2 y2   1 is a 2 b2 (a) a  b (c)

a b 2

2

(b)

ab

(d)

a 2  b2

118. Equation of the parabola whose vertex is (2 3, 2 2), axis is horizontal and which passes through the point (1, 2) is (a) y2 + 4y + 4x 2 8 = 0 (b) y2 + 4y 2 4x + 8 = 0 (c) y2 + 4y 2 4x 2 8 = 0 (d) none of these 119. The mirror image of the directrix of the parabola y2 = 4 (x + 1) in the line mirror x + 2y = 3 is (a) x = 2 2 (b) 4y 2 3x = 16 (c) 3x 2 4y + 16 = 0 (d) none of these 120. Eccentric angle of a point on the ellipse x2 + 3y2 = 6 at a distance 2 units from the centre of the ellipse is

π 4 3π (c) 4 (a)

π 3 2π (d) 3 (b)

x y x2 y2   2  touches the ellipse 2  2  1, then its a b a b eccentric angle q is equal to (a) 0 (b) 90º (c) 45º (d) 60º

2

5 4 (c) 5

114. The locus of the point of intersection of tangents to the ellipse x2 y2   1 which meet at right angles is a 2 b2 (a) a circle (b) a parabola (c) an ellipse (d) a hyperbola

121. If

(d) ( x1 , y1 , z1 )

(a)

113 . Two perpendicular tangents to y2 = 4ax (a) x + 4a = 0 (b) x + 2a = 0 (c) x = a (d) x + a = 0

x2 y2   1  1 intercepts equal 16 9 lengths l on the axes, then l = (a) 3 (b) 5 (c) 5 (d) none of these

122. If any tangent to the ellipse

Additional Support Material ▌ 3.279 123. The product of length of perpendiculars drawn from the two foci to the tangent at any point on the ellipse 25x2 + 4y2 = 100 is (a) 30 5

(b) 25 2

(c) 152

(d) none of these

124. The locus of the foot of the perpendicular from the centre of 2 2 the hyperbola x  y  1 on a variable tangent is 2 2 a b (a) (x2 + y2)2 = a2x2 + b2y2 (b) (x2 + y2)2 = a2x2 2 b2y2 2 2 2 2 2 2 2 (c) (x + y ) = b x 2 a y (d) none of these 125. If the circle x2 + y2 = a2 intersects the hyperbola xy = c2 in four points P(x1, y1), Q(x2, y2), R(x3, y3), and S(x4, y4), then (a) x1 + x2 + x3 + x4 = 0 (b) y1 + y2 + y3 + y4 = 2 (c) x1x2x3x4 = 2c4 (d) y1y2y3y4 = 2c4 126. Equation of the straight line, passing through the point (3, 4) and farthest from the circle x2 + y2 + 8x + 6y + 16 = 0, is (a) x 2 y + 1 = 0 (b) 3x + 4y = 25 (c) x + y 2 7 = 0 (d) none of these 127. The tangents to the hyperbola x 2 y = 3 are parallel to the st. line 2x + y + 8 = 0 at the following points (a) (2, 1) or (1, 2) (b) (2, 2 1) or (2 2, 1) (c) (2 1, 2 2) (d) (2 2, 2 1) 2

2

128. The equation of the common tangent touching the circle (x 2 3)2 + y2 = 9 and the parabola y2 = 4x above the x-axis, is (a)

3 y  3x  1

(c)  x  3

(b) ( x  3) (d) (3x  1)

129. On the parabola y = x2, the point at a least distance from the straight line y = 2x 2 4 is (a) (1, 1) (b) (1, 0) (c) (1, 2 1) (d) (0, 0) 130. The number of points with integral coordinates that lie in the interior of the region common to the circle x2 + y2 = 16 and the parabola y2 = 4x is (a) 8 (b) 10 (c) 16 (d) none of these 131. The length of the chord intercepted by the parabola y2 = 4x on the straight line x + y = 1 is (a) 4

(b) 4 2

(c) 8

(d) 8 2

132. The equation of the common tangent touching the circle (x 2 3)2 + y2 = 9 and the parabola y2 = 4x above the x-axis is: (a)

3 y  3x  1

(b)

3 y ( x  3)

(c)

3y  x  3

(d) (3x  1)

133. The locus of the centre of the circle described on any focal chord of a parabola y2 = 4ax as diameter is (a) x2 = 2a(y 2 a) (b) x2 = 2 2a(y 2 a) (c) y2 = 2a(x 2 a) (d) y2 = 2 2a(x 2 a)

134. The shortest distance between the parabolas y2 = 4x and y2 = 2x 2 6 is (a) 2

(b)

5

(c) 3

(d) none of these

135. If p is the length of the perpendicular from a focus upon the tangent at any point P of the ellipse x2/a2 + y2/b2 = 1 and r is the 2a b2 distance of P from the focus, then  2 r p (a) 2 1 (b) 0 (c) 1 (d) 2 x2  y 2 1 at (3 3 cosθ , sinθ ) 27 where q ∈ (0, π/2). Then the value of q such that sum of intercepts on axes made by this tangent is minimum, is (b) π/6 (a) π/3 (c) π/8 (d) π/4

136. Tangent is drawn to ellipse

137. Line 2x + y = 3 cuts the ellipse 4x2 + y2 = 5 at P and Q. If q is the angle between normals at P and Q, tan q = 1 2 3 (c) 5 (a)

(b)

3 4

(d) 5

138. In ellipse having foci (3, 1), (1, 1) and passing through the point (1, 3) has the eccentricity (a)

2 1

(c) 1  2

(b)

2 1

(d) none of these

139. If x = 9 is the chord of contact of the hyperbola x2 2 y2 = 9, then the equation of the corresponding pair of tangents is (a) 9x2 2 8y2 + 18x 2 9 = 0 (b) 9x2 2 8y2 2 18x + 9 = 0 (c) 9x2 2 8y2 2 18x 2 9 = 0 (d) 9x2 2 8y2 + 18x + 9 = 0 140. If x = 9 is the chord of contact of the hyperbola x2 2 y2 = 9, then the equation of the corresponding pair of tangents is (a) 9x2 2 8y2 + 18x 2 9 = 0 (b) 9x2 2 8y2 2 18x + 9 = 0 (c) 9x2 2 8y2 2 18x 2 9 = 0 (d) 9x2 2 8y2 + 18x + 9 = 0 141. If (5, 12) and (24, 7) are the foci of a hyperbola passing through the origin then the eccentricity of the hyperbola is (a)

386  12

(b)

386 / 13

(c)

386 / 25

(d)

386 / 38

142. Consider the set of hyperbolas xy = K, K ∈ R. Let e1 be the eccentricity when K = 4 and e2 be the eccentricity when K = 9, then e1 2 e2 is equal to (a) 1 (b) 2 3 (c) 0 (d) . 2

3.280

▌ Mathematics

143. If chords of the hyperbola x2 2 y2 = a2 touch the parabola y2 = 4ax, then the locus of the middle points of these chords is the curve (a) y2(x 2 a) = x2 (b) y2(x 2 a) = x3 (c) y2 = x3 (d) none of these

144. The locus of point of intersection of tangents at the end of normal chord of hyperbola x2 2 y2 = a2 is (a) a2(y2 2 x2) = 4x2y2 (b) a2(y2 + x2) = 4x2y2 (c) y2 + x2 = 4a2x2 (d) none of these

anSwer Key 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34.

(b) (a) (a) (b) (b) (b) (b) (c) (a) (d) (d) (a) (d) (b) (a) (a)

(c) (a) (a) (d) (c) (b) (a) (c) (a) (b) (c) (b) (d) (b) (b) (b)

(a)

35. (a)

(c)

36. (c)

37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54.

(c) (c) (b) (a) (c) (c) (b) (a) (c) (a) (a) (c) (d) (a) (b) (a) (a) (b)

55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72.

(c) (c) (c) (d) (c) (c) (a) (b) (c) (c) (b, c) (b) (c) (d) (b) (b) (a) (a)

73. 74. 75. 76. 77. 78. 79. 80. 81. 82. 83. 84. 85. 86. 87. 88. 89. 90.

(a) (b) (a) (b) (a) (b) (c) (c) (a) (d) (c) (a) (b) (a) (b) (c) (c) (c)

91. 92. 93. 94. 95. 96. 97. 98. 99. 100. 101. 102. 103. 104. 105. 106. 107. 108.

(b) (b) (b) (b) (d) (a) (b) (b) (c) (c) (c) (b) (a) (b) (d) (d) (d) (c)

109. 110. 111. 112. 113. 114. 115. 116. 117. 118. 119. 120. 121. 122. 123. 124. 125. 126.

(a) (a) (c) (c) (d) (a) (b) (b) (c) (c) (c) (a, c) (c) (b) (d) (b) (a) (c)

127. 128. 129. 130. 131. 132. 133. 134. 135. 136. 137. 138. 139. 140. 141. 142. 143. 144.

(b) (c) (a) (a) (c) (c) (c) (b) (c) (b) (c) (a) (b) (b) (a) (c) (b) (a)

SolutionS 3 2 1. Area of the equilateral triangle with side a 5 a 4 (an irrational number). Let ((x1 , y1 ), (x 2 , y2 ) and ( x 3 , y3 ) be vertices of an equilateral triangle. Area of the equilateral triangle x1 1 5 x2 2 x3

y1 1 y2 1 5 a rational number, y3 1

12[2(3  2)  1(4  1)  1(8  3)] 

1 2

[4(2  2)  3( 1  3) 11(2  6)] 1 1  ( 2  5  11)  (16  6  8) 2 2  9  9  18 3. Let the moving point be P (x,y) and A(a,0) and B(-a, 0) be the fixed points.

If x1 , y1; x 2 , y2 and x 3 , y3 are integers.

Now, PA 2  PB2  2c 2

An irrational number can’t be equal to a rational number. Hence a triangle with coordinates of the vertices as integers can’t be an equilateral triangle.

or [(x  a)2  (y  0)2 ]  [(x  a)2  (y  0)2 ]  2c 2

2. Area 

2 1 1 4 3 1 1 1 4 3 1  1 2 1 2 2 1 2 1 3 2 1

Or 2x 2  2y2  2a 2  2c 2 Or x 2  y2  c 2  a 2 4. The equation of the right circular cylinder in which coefficient of y2 is zero, has axis as x 2  z 2  a 2 .

Additional Support Material ▌ 3.281 5. Any point on line 3x + 4y = 9 is obtained by putting y = 0, which gives the point as (3,0). Required distance = ⊥ distance of (3, 0) from 6x = 8y = 15 is 6  3  8  0  15 2

6 8



2

11. Equation of parabola is y2 5 x, ∴a 5

1 4

M  tan 45   1  a 2a   1 1   2 ,  i.e  ,  m m  4 2

3 10

6. Since, coefficient of x2 + coefficient of y2 = 0, the lines are perpendicular. 7. Equation of tangent to given circle at the points (5,12) and (12, -5) are respectively x.5  y.12  169 (  xx1  yy1  a 2 )

12. Let the equation of the normal be y  mx  2am  am 3 Or am 3  m(2a  x)  y  0

...(i)

It is cubic in m, so it has three roots, say m1 , m 2 , m 3 . Then y1  2am1 , y2 2am 2 y3  2am 3

and 12x  5y  169

∴ y1  y2  y3  2a (m1  m 2  m 3 ) ...(ii)

or 5x  12y  169 12x  5y  169

From (i) Sum of roots  m1  m 2  m 3

which are obviously at right angles. 8. The given circle meets the x I axis, where y = 0 i.e., at points given by



coeff.of m 2 0   0 a coeff.of m 3

...(iii)

Hence, from (ii) and (ii) x 2  4x  12  0 ⇒ ( x  6) ( x  2)  0 ⇒ x  6, x  2

y1  y2  y3  0 13. Equation of the ellipse is

Length of intercepts = 6-(-2) = 8 Alternatively, intercept on the x-axis  2 g 2  c  2 (2)2  (12)  2 16  8 9. Equation of the radical axis is

x 2 y2  1 7 16 a 2  16, b2  7 Now, b2  a 2 (1  e 2 ) ⇒

(x 2  y2  6x  16)  (x 2  y2  2x  6y  6)  0 Or

Or 8x  6y  10  0 Or 4x  3y  5  0 4.1  3.2  5 2

2

4 3

7 1 e2 16

...(i)

Distance of the Point (1, 2) from (i) 

b2 1 e2 a2

or e 2  1 

5  1 5

7 3  9 / 16 ⇒ e  16 4

14. The given equation is

10. Combined equation of the two tangents is givnen by SS1 = T2 Where,

(25x 2  150 x)  16y2  175 25(( x 2  6 x  9)  16y2  175  225 (Adding 225 on both sides)

S  x 2  y2  2x  4y

25(x  3)2  16y2  400

S1  42  32  2 xy  4  3  5 T  4x  3y  ( x  y)  2(y  3)  (3x  y  10)

Let x  3X, y  y; the above equation becomes

i.e 5(x 2  y2  2x  4y)  (3x  y  10)2

Here, a 2  25, b2  16 ( ∴a  b)

X 2 y2  1 16 25

Or 4x 2  6xy  4y2  50 x  100  0 ..( i) 2

b2  a 2 (1  e 2 )

2

Since, Coefficient of x  coefficient of y  0 Thus, the lines represented by (i) are perpendicular.

Or e 2  1 

b2 16 9 3 1  ⇒ e  25 25 5 a2

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▌ Mathematics

15. The given equation reduces to X 2 y2  1 16 25 Coordinates of foci are X  0, Y  ae x  3  0,y 3 ( e  3/5)  5. 5 Or x  3, y  3 ⇒ (3,  3) Or

b 16. Here m1 5 . Let slope of the second conjugate diameter a be m, Then

Figure PL  y2  z 2 22. Let AB be the required line and its direction cosines be cos a, cosb, cosg. Then AB cos α  2. AB cos β  3, AB cos γ  6. Squaring and adding (AB)2 (cos 2 α  cos 2 β  cos 2 y)  22  32  62  49 Or (AB)2  49 ⇒ AB  7

2

m1.m 

b a2

b b2 Or . m  2 a a b ⇒ m a The equation of the other conjugate diameter is b y  x (  y  mx) a 17. Equation of ellipse is

2

2

2

When e  0, b  a Equation (i) becomes 2

x  y  a , Which is a circle. 18. Eliminating m, (by multiplying the two equations) the equation obtained is x 2 y2  1; a 2 b2 Which is a hyperbola. 19. A hyperbola is said to be rectangular if a = b. Thus equation of the rectangular hyperbola is x 2  y2  a 2 Now a 2  a 2 (e 2  1) Or e 2  2 or e  2 20. Sp = focal distance = ex −a S’P = ex + a S’P − SP = 2a

23. Let each edge of the cube be a. Then OP and QA are two diagonals of the cube, where P (a, a, a) and A (o, a, a ). D.rs. of OP: a-0, a-0, a – 0 i.e., a, a, a Figure D. rs. Of QA: 0 −a, a − 0, a −0 i.e., −a, a, a If q be the angle between OP and QA, then a.(a)  a.a  a.a

Cos θ 

2

a a a

Now, b  a (1  e )

2

(  cos 2 α  cos 2 β  cos 2 y  1)

2

x 2 y2   1; x 2 b2

2

21. PL  (x  x)2  ( y  0)2  ( z  0)2

2

2

2

a a a

 2

1 3

1 or θ = cos1    3 24. The given plane meets the x-axis where y = 0, z = 0. ∴A(a, 0, 0) Centroid of ∆ABC is  a 1010 01 b10 0101c  a b c , ,   i.e.  , ,  3 3 3    3 3 3 1 1 1 25. D 4  2 3 4  4 ≠ 0 1 1 1 Hence, the given planes intersect in a point, as no two planes are parallel. 26. Let equation of one of the spheres be x 2  y2  z 2  2ux  2wz  d  ...(i) The sphere (i) meets the x-axis, where y  0 and z  0. Substituing y  0 and z  0 in equation (i), We get x 2  2ux  d  0.

..(ii)

Since the sphere (i) touches the x-axis, roots of equation (ii) are coincident. Thus

Additional Support Material ▌ 3.283

30. The required equation of circle is, S1  λ ( S2  S1 )  0.

4u 2  4d,

∴

or u 2  d

( x 2  y 2  6)  λ (6 x  14)  0

Also, pas sin g through (1,1).

Similarly, since the sphere touches

⇒

y-axis and the z-axis,

1 2 ∴ Required equation of cirrcle is,

2

⇒

2

u  d, w  d. Also, the radius of the sphere  a,

or

λ

x 2  y 2  6  3x  7  0 or

u 2  v 2 w 2  d  a,

i.e

4  λ (8)  0

x 2  y 2  3x  1  0

31. Let P( x1 , y1 ) be the nearest point from the circle.

u2  v2  w 2  d  a2

Substituting the values of u,v,w

Equation of circle

and d in equation (i), we obtain

( x  1)2  y 2  1

2(xx 2  y2  z 2 )  2 2 a(  x  y  z) a 2  0 which are the required equations of the spheres. Considering the possible sign combinations, there are eight spheres in all, one in each of the eight octants. 27. The equation of the given sphere is x 2  y2  z 2  x  z  2  0

Now, distance between P and C is d1  (xx1  1)2  ( y1  0)2

...(i)

Also, thhe equation of line is

...(ii)

Now, the distance between P and line;

1 1 (i) are ( , 0  ) and its radius is 2 2 2

and Redius, r → 1

d1  x12  y12  2 x1  1

and the plane is x  2y  z  4

Centre, C → (1, 0)

2

1  1 5 2  R (say).    0     2  2 2 2     P  perpeendicular distance from C on the plane (ii)) 1 1 0 4 3 1 2 2    √6 6 2 12  22  (1)2 Radius of the circle  R 2  p2 5 6    1. 2 4 28. c1 (0, 0, 0) and r1  5 C2 (12,20,9) and r2  122  202  92  225  20

x 5 ie, Q(5, 0) d2  ( x1  5)2  ( y1  0)2 d2  x12  y12  10 x1  25

 144  400  81  25 Since C1C2  r1  r2 Hence, the given spheres touch externally. 29. The equation of the right circular cylinder in which coefficient of z2 is zero, has axis as z-axis.

...( ii )

From the question, 1 d1  d2 2 Squaring on both sides 4 d12  d22 ⇒

4( x12  y12  2 x1 1)  ( x12  y12  10 x1  25)

⇒ ⇒ ⇒

3x12  3 y12  2 x1  25  2  3 x12  x1   3 y12  25 3    2 2 x1 1  1 3 x1      3 y12  25 3 9 3  2

C1C2  (12  0)2  ( 20  0)2  (9  0)2

...( i )

⇒

 1 76 3 x1    3 y12  3 3   2

 1  x1   y2 3  ⇒  1 1  76   76      9    9  Here, a  b and h  0 and ∆ ≠ 0

▌ Mathematics

3.284

32. Since, all sides of a ∆ OBC are equal, then all angles are 60. ------------------ ODB  180 0  60  120 ⇒ ODB  OBD  30 Applying sin rule inn ∆ OBD,

  π 4y ⇒ tan1  2  2   x y 4   2



l

 (x  2)  iy (x  2)  iy  π ⇒ Arg     (x  2)  iy (x  2)  iy  2  (x 2  4  y2 )  i{y(x  2)  y(x  2)}  π ⇒ Arg   (x  2) 2  y 2   2

sin 30 sin120 l  l 1 ⇒

 3 2  π  x  iy  2  π 34. Arg    ⇒ Arg   3  2 2    x  iy  2  2

cos 30  3 cos 30

⇒

Also,

π 4y  tan  ∞ 2 x 2  y2  4

 D  90, B  90

⇒ x 2  y2  4 = 0

Then,  BDP  DBP  60

⇒ x 2  y2  4

⇒

which represents a circle with centre origin and radiius 2.

 DBP  60

Hence, ∆PDB is an equilaeral triangle. ∴

Area of ∆ PDB 

x 2  y 2  2 gx  2 fy  c  0

3 2 l 4

2 x 2  2 y 2  5x  6y  4  0 ⇒

2x 2  3xy  2 y 2  0 ⇒ (2x  y ) ( x  2 y )  0

∴

2( g1g 2  f1 f 2 )  c1  c2

⇒

 5 3 2  g.  f .   c  2 2  4

⇒

5g  6 f  2c  4

Also (1) passes through ( 2, 0) and (0, 2)

90  π(2)2 = π 360

Area of sector OAB, A2 

90 π  π(1)2  360 4

∴

A 2 90º

5g  6 g  2c  4

1 0

x’

 g  2c  4 x

Solving Eqs. (iv) and (v), we gett g

∴Required area  A1  A2  π  3π sq unit 4

4  12 y , c and f  7 7 7

∴ Eq. (i) reduces to

y’



g f

put in Eq. ( iii ) we get ,

C B

4  4g  c  0 44 f c 0

⇒

y D

5 x 2  y 2  x  3y  2 = 0 2

...( ii)

Since both circles are perpendicular

Here, a  2, b  2 ∴a  b  2  2  0 Now, Hence, lines are perpendicular. 0  πr 2 Arrea of sector ODC , A1  360 

...( i)

and given circle be

3 3 3 ( 3)2  sq units 4 4 33.

35. Let the required eq be

π 4

8 8 12 x2  y2  x  y   0 7 7 7 ⇒ 7 x 2  7 y 2  8 x  8 y  12 0

...( iii )

Additional Support Material ▌ 3.285 36. Given curves are x  t 2  t  1 and y  t 2  t 1 On subtracting Eq. (ii) from Eq. (i), x  y  2t Thus x  t 2  t  1  x y   x y  x  ⇒   1  2   2  ⇒ 4 x  ( x  y )2  2 x  2 y  4 ⇒ ( x  y ) 2  2 ( x  y  2) ⇒ x 2  y 2  2 xy  2 x  2 y  4  0 Now, ∆  1.1.4  2.(1)(1)(1) 1  (1)2  1  (1)2  4(1)2  4  2  1  1  4  4  ∆≠0 and ab  h2  1.1  (1)2  1 1  0, Hence, it represents a equation of parabola. 2 37. Given, parabola y  8ax

The parametric coordinate of one end of focal chord of parabola is A{2at 2 , 4at}. We know that the focal coordinates of any parabola  a 2 a  like y 2  4 ax is A{2at 2 , at} and  2 ,  B. t  t If wer compare the equation of both parabola we a  2 a in y 2  4 ax , we get

 2  5  10 Thereforee, (c) is the answer. 39. Given equation is x2 y2   1, where r  1. 1 r 1 r 1  r is positive and 1 r is positive.

⇒

x2 y2  1 a 2 b2 Hence, it represent as a hyperbola when r 1.

∴

Given equation is of form

40. Equation of tangent at P(x1 , y1 ) is It intercept y  axis, then x  0 yy ∴ 21  1 b b2 ⇒ y   c1 ( given ) y1 ⇒b

2  y2 e1

xx1 a

2



yy1 b2

1

...( i)

Similarly, at point Q( x2 , y2 ) is b2  y2 c2

...( ii)

∴ y1c1  y2 c2 41. Let equation of chord AB is y  mx y 2  4 ax

Also,

observe that

--------

y 2  8ax So, we put a  2a in second coordinate of focal chord of parabola y 2  4ax and we get the second coordinate of foccal chord parabola y 2  8ax.  2a 2( 2 a )   2 a 4 a ).  which is B →  2 ,  ,  2 ,   t t  t t    38. Given, 16 x 2  25y 2

Here,

x2 y2  1 25 16 2 a  25, b2  16

But

b2  a 2 (1  e2 )

⇒

a 5 PF1  PF2  major axis  2 a

⇒ Now,

16  1  e2 25 16 9 3 ⇒ e2  1   ⇒ e 25 25 5 Now, foci of the ellipse are ( ae, 0)  ( 3, 0). ⇒

16  25(1  e2 ) ⇒

We have,

3  a.

3 5

 4a 4 a  On solving we get, B  2 ,  m m  Equation of perpendicular to BC is my  x  c  0  4a 4 a  It passes through B  2 ,  m m  ∴ 4a 

4a C0 m2

 m2  1  C  4a   2  m   m2  1  ∴ Equation is my  x  4a   = 0 2  m   ( m2  1)  Its interesection on x  axis is c  4a , 0  . m2   4 a (1  m2 ) 4a  2  4a ∴ DC  AC  AD  m2 m ⇒

42. Hence, the locus of the centre is an ellipse. 43. Given equation can be rewritten as

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▌ Mathematics

x2 y2  1 9 5 Here, a 2  9, b2  5

46. x  1  t 2 ,

2

2b 2 Length of latusrectum  a 2  5 10   3 3

⇒

 y 1     x 1  2 

⇒

( y  1)2  4( x  1)

y 2  4 x where Y  y  1

44. Now

X  x 1

y 2  4ax dy 2y  4a dx  dy  2a 1 ⇒     dx ( at 2 ,2 at ) 2 at t

⇒

x  1  1

Equation of normal at P ( at 2 , 2 at ) is

⇒

x  0.

Eq. of direction x 1

y  2at tt ( x  at 2 ) or

47. The point O (0, 0)is the mid point of A(– a, –b)and B( a, b).

y  tx  2 at  at 3

Therefore, A, O, B are collinear and equation of line A O B is

45. Equation of tangents and normals meet the axis at

b y5 x a

T (at 2 , 0) and G ( 2 a  at 2 , 0).

Since, the fourth point D(a2, ab)satisfies the above equation. Hence, the four points are collinear.

y 2

P(at ,2at) θ 90° x G (2aat2,0)

(at2,0)T



y 1 t 2

It implies that T and G are the end points of a cirle. ∴ Equation of circle is (x  x1 ) (x  x2 )  (y  y1 ) (y  y2 )  0 ⇒

( x  at 2 ) ( x  at 2  2 a )  ( y  0) ( y  0)  0

⇒

x 2  a 2t 4  2 a( x  at 2 )  y 2  0

⇒

x 2  2 ax  y 2  a 2t 4  2 a 2t 2  0

48.

 A( x1 , y1 )  Let  B( x2 , y2 )   → A B C   and C ( x3 , y3 ) By definition of line segment. If B lies on AC , then B  Aλ  (1  λ ) C , where 0  λ  1 ( x2 y2 )  ( x1 y1 ) λ  (1  λ ) ( x3 , y3 ) ⇒

x2  x1λ  (1  λ ) x3

⇒

x2  x1λ  x3  λx3

⇒

( x2  x3 )  ( x1  x3 )λ

⇒

λ

x2  x3 x1  x3

Tangent at P(at 2 , 2 at )

and

y2  y1λ  (1  λ ) y3

x (at 2  a )  3a 2t 2  y 2 at  a 2t 4  0

⇒

y2  y1λ  y3  λy3

⇒

( y2  y3 )  ( y1  y3 )λ

Slope of this equation, m1  And slope of PT is, m2 

a  at 2 (1  t 2 )  2 at 2t

1 t

1 (1  t 2 )  m2  m1 2t ∴tan 0  t 1  m1m2 1 (1  t 2 )  t 2t 2 2 (1  t ) / 2t  t ⇒ 0  tan1 (t )  (1  t 2 ) / 2t 2

λ

y2  y3 y1  y3

...( i)

...( ii)

But we know that 0  λ 1 m Eqs. (i) and (ii), So, from x2  x3 y  y3  1 or 2 1 Either 0  3 y1  y3 x1  x3 option (c) is correct.

Additional Support Material ▌ 3.287 49. Let the coordinates of P → ( x , y , z ) given P lies on the line

⇒

passing through the point (12, 3, 3) and (–3, 0, 9), then the equation of line is

⇒

x  x1 y  y1 z  z1   x2  x1 y2  y1 z2  z1

⇒ ⇒ ⇒

x  12 y 3 z 3   3  12 0  3 9  3 x  12 y  3 z  3   15 3 6

⇒ ...( i )

( x  a) ( x  a)  b ( x  a )2 2

= 1, then from Eq.(i)

54.

x  12 y  3 1  3   6 3 15 x  12 y  3  2   15 3 6

L  a sin x  a 3cos x Using AM  GM

a sin x 

3 cos x

3 cos x

x4 y2 z k   1 1 2 Since, the line lies in the pllane 2x  4 y  z  7 ∴ Point (4, 2, k ) must satisfy the plane. 88 k 7 k 7

51. For ellipse, condition of tangency is c2 = a2m2 + b2

c 2  4  42 1  65

⇒

c  65

x2  y2 1 4

52. Let P(x , 0) Then, D  AP  BP ∴ D  ( x  a ) 2  b2  ( x  c ) 2  d 2 On differentiating w.r.t x, we get dD ( x  a) ( x  c)   2 2 dx ( x  a)  b ( x  c)2  d 2 Put

dD 0 dx

 a sin x a

 a

3 cos x

 π 2 sin  x   3 

 π Since, minnimum values of sin  x   is  1. 3  ∴

a sin x  a 2

3 cos x

⇒

a sin x  a

3 cos x

50. Given equation of straight line

⇒

( x  c)2  d 2 ( x  c)2

 ( x  a ) 2  b2 ( x  c)2  d 2 2 2 2 d ( x  a)  b ( x  c)2 d ( x  a )  b( x  c ) x ( d  b)  ad  bc ad  bc bc  ad  x d b b d

a sin x  a 2

x  12 y 3 1 1 ⇒  and  15 3 3 3 ⇒ x  12  5 ⇒ x  17 3 and y  3 3 ⇒ y  3 1 ⇒ y4 So, option (d) is correct.

Given line is y  4 x  c and curve

( x  c)

53. Condition for straight line, 1  a  b  4  0 ⇒ a  b 3 Hence, option (a) is correct.

Given the z 5 coordinate of P is z

⇒ ⇒

 2

 a2 

2 a

55. The set of points on the curve which lie on the coordinate axes is {( 2, 0), (0, 4)}. 56. Given that the point (4,1) undergoes three transformation. 1. Reflection about the line y = x ∴ (4, 1)will be transformation to (1,4) 2. Translation through a distance 2 unit along the positive direction of x-axis. ∴(1, 4) will be transformed to (1 + 2, 4) = (3, 4) π 3. Rotation through an angle about the origin is the anti4 clockwise direction. If (x′, y′)be the final location then x'  x cos θ  y sin θ 3 4 1    (∵ θ  45 ) 2 2 2 and y'  x sin θ  y cos θ 3 4 7    2 2 2 Hence, the finall location of original point is  1 7  ,  .  2 2

3.288

▌ Mathematics

57. Distance of the point (a, b) from the line Ix  my  n  0 measured parallel to a line making an angle θ with x-axis = 58.

I α 1 mβ 1 n I cos θ 1 m sin θ

Let the given point be P(x, y), A(a, 0) and B(- a, 0). Then, PA2 + PB2 = 2b2 ⇒ [(x – a)2 + ( y – o)2] + [(x + a)2 + (y + 0)2] = 2b ⇒ x2 + a2 = b2 – y2.

62. Let the vertices of the equilateral triangle be (x1, y1), (x2, y2) and (x3, y3). If none of xi and yi (i = 1, 2, 3) are irrational, then x1 y1 1 1 area of ∆ 5 x2 y2 1 5 rational. 2 x3 y3 1 But the area of an equilateral triangle 5

59. 60.

 3  9  1  5 3  6  1  (2)  Required point is  ,   (8, 4). 3 1 3 1   The four lines enclosing the given region are 4x + 5y = 20, 4x 2 5y = 20, 2 4x + 5y = 20 and 2 4x 2 5y = 20. Clearly, the four lines form a rhombus having diagonals of length 10 and 8.

3 (side ) 2 5 irrational. 4

Thus, the two statements are contradictory. Therefore, both the coordinates of the third vertex cannot be rational. 63. Coordinates of any point on the line through (2, 3) and parallel to the line x 2 y = 4, at a distance r, are (2 + r cos θ, 3 + r sinθ), where tan θ = 1. A(2,3)

Y



4x  5y

5y





20

4x

r



5

O

5

x y 4

20

4

X

 

5y

 5y 

4x

P

20

If this point lies on the line 3x + 2y = 17,

4x



20

3x2y17 4

3 (2 + r cosθ) + 2 (3 + r sinθ) = 17

then ⇒

1 ∴ Required area   10  8  40. 2

⇒

61. The equation of line BC is x + y + 4 = 0. ∴ Equation of a line parallel to BC is x + y + k = 0. 1 This is at a distance from the origin. 2

⇒

2

1

1 2

2

2

.  6  2r .

1

 17

2

r 5

r  2.

1 1 Since it passes through (1, 1), ∴   1 a b .

2 Since BC and the required line are on the same side of the origin ∴k=

5

1

64. If a, b are the intercepts made by the line, then the equation of x 1 the line is   1. a b

∴ k 51 . 2 2 ∴k=±

6  3r

.

Hence, the required line is x + y +

1 2

= 0.



⇒

a+b 51. ab

...(1)

Also, area of the triangle made by the straight line on the coordinate axes is S 1 ab = | S | i.e., ab = 2 | S | ...(2) ∴ 2 So, by (1), a + b = 2 | S | ...(3) From (2) and (3), the intercepts a and b are the roots f the equation x2 2 2 | S | x + 2 | S | = 0.

Additional Support Material ▌ 3.289 65. Let h be the height of the triangle. Since, the area of the triangle is a2 ∴

1 × a × h = a2 ⇒ h = 2a. 2

Clearly, (d) satisfies these equations. Hence common choice is (d).. 69. Since 

Since the base lies along the line x = a, the vertex lies on the line parallel to the base at a distance 2a from it. So, the required lines are x = a ± 2a i.e., x = 2a or x = 3a. 66. Let a, b, c be pth, qth and rth terms of an A.P. whose first term is A and common difference is d. The given line is ax + by + c = 0 ⇒ [A + (p 2 1) d] x + [A + (q– 1) d] y + [A + (r 2 1) d] = 0 ⇒ A (x + y + 1) + d ((p 2 1) x + (q 2 1) y + r 2 1) = 0 ⇒ The given line passes through the point of intersection of lines x + y + 1 = 0 and (p 2 1) x + (q 2 1) y + r 2 1 = 0, which is a fixed point. 67. The given line is P( 3,2

∴ the point (4, 1) lies outside the hyperbola, hence the number of tangents through (4, 1) is two. 70. Since the distance between the parallel lines lx + my + n = 0 and lx + my + n’ = 0 is same as the distance between the parallel lines mx + ly + n = 0 and mx + ly + n’ = 0. Therefore, the parallelogram is a rhombus. Since the diagonals of a rhombus π are at right angles, therefore the required angle is . 2 71. Let a, b, c be in G.P. with common ratio r. Then, b = ar and c = ar2. So, the equation of the line is ax + by + c = 0 ⇒ ax + ary + ar2 = 0 ⇒ x + ry + r2 = 0 This line cuts the curve x + 2y2 = 0 Eliminating x, we get 2y2 2 ry + r2 = 0 If the roots of the quadratic equation are y1 and y2, then r b c y1  y2  ⇒ 2 ( y1  y2 )  r   . 2 a b

r

Q

3x4y80

72. Let P ≡ (x1, y1), Q ≡ (x2, y2); R ≡ (x3, y3), where xi, yi (i = 1, 2, 3) are rational numbers. Now, the centroid of DPQR is  x1 1 x2 1 x3 y1 1 y2 1 y3  ,   3 3  

3 x 2 4y + 8 = 0 ...(1) Let PQ = r. Then, the coordinates of Q are

which is rational point. Incentre, circumcentre and orthocentre depend on sides of the triangle which may not be rational even if vertices are so. For example, for P (0, 1) and Q (1, 0);

 π π  3 r r, 2 1 .  3 1 t cos , 2 1 r sin  or  3 1 6 6  2 2   Since the point Q lies on the given line, ∴

 16 1 x2 y2   1    1  0, 4 3  4 3

  3 6 3 3  r  4  2   80  2 2  

Hence, PQ = 6.

PQ =

73. We have d(x,y) = Max. {| x |, | y |} = k If |x| > | y |, then k = | x | ⇒ x = ± k If | y | > | x |, then k = | y | ⇒ y = ± k Hence, the locus represents a straight line. 74.

Y

2 68. Any tangent to the parabola y2 = 8x is y = mx 1 . Clearly, m (a), (b) and (d) satisfy the equation (by m = 3, 2, 1). For xy = 1 Equatioin of tangent with slope m, to xy = 1 are  1  y  m  m  x   m   dy  1 1 , y  m    2  m ∴ x  6 m  dx x 

2.

Q y



P(α,β)

x

R X

The image of the point P(a, b) w.r.t. the line y = 2x is Q(−b, 2a) (interchange and change signs). The image of Q(2 b, 2 a) w.r.t. the line y = x is R(2 a, 2b) (merely interchange). ∴ The mid-point of PR is (0, 0).

3.290

▌ Mathematics

75. Since the given lines are concurrent, therefore 1 0 k  m 0 1 2 = 0 ⇒ m2  km  2  0 m 1 0

...(1)

Since m is reaal, the discriminant of (1) is  0

81. The line x cos a + y sin a = p meets the coordinate axes at (p/cos a, 0) and (0, p/sin a) whose mid point is (h, k) where

⇒ k 2  8 or |k |  2 2 Thus, the least value of |k| is 2 2 .

r1 9 3 5 5 and the required ratio is thus 3 : 4. r1 12 4 77. Mid-point of the line joining the given points lie on the given line 3  b 2(a  b)  c 2 2 ⇒ 2a + 2c 2 b + 7 = 0 ...(i) Also, since the given line passes through (a, b) b = 2a + c ...(ii) Solving (i) and (ii), we get c = 2 7 ∴

78. Let the equation of the variable line be ax + by + c = 0 then according to the given condition a 2  b2



2b  c a 2  b2



2 a  2b  c a 2  b2

p p ,k5 2 cos α 2 sin α Eliminating α , we get h5

76. It is clear that the lines lie on opposite sides of the origin O. x y 5 5 If Let the equation of any line through O be cos θ sin θ. OP = r1 and OQ = r2 then the coordinates of P are (r1 cos q, r1 sinq) and that of Q are (2 r2 cos q, 2 r2 sinq). Since P lies on 4x + 2y = 9, 2r1 (2 cos q + sinq) = 9 and Q lies on 2x + y + 6 = 0, 2 r2(2 cos q + sinq) = 2 6 so that

2a  c

80. The given equation can be written as a(x + y 2 2) + b(x 2 y) = 0 or (x + y 2 2) + (b/a)(x 2 y) = 0 This is a family of lines concurrent at x 2 y = 0 and x + y 2 2 = 0. On solving these two equations, we get (1, 1).



1 1 4   2 h2 k 2 p

1 1 4   2 x2 t 2 p Since it passes through the point (p  1, p  1)

∴ Locus of (h, k ) is

1 1 4   2 (p  1)2 (p  1)2 p ⇒

(p  1)2  (p  1)2 4  2 (p 2  1)2 p

⇒ 2(p 2  1)p 2  4(p 2  1)2 ⇒

p4  5 p2  2  0

b , since it is ⊥ to the a line ax + by + c = 0 and it cuts the x-axis at (2, 0). Hence, b the required line passes through (2, 0) and its slope is . a Therefore, its equation is

82. Slope of the line in the new position is

Y axbyc0

0

⇒ c 0 yx2

which shows that the line passes through (0, 0) for all values of a, b. 79. Line 2x + 3y = 12 meets x-axis at A(6, 0) and y-axis at B(0, 4) The equation of a line passing through (5, 5) and perpendicular 3 to 2x+3y=12 is y  5  ( x  5) ⇒ 3x  2 y = 5 2 This cuts x-axis, y-axis and AB at points 5   25  c  , 0  , D  0,  , E ( 3,2 ) 3    2 

Respectively. OCEB Now, ar = ar(∆OCE) + ar(∆OEB) ar (OCEB) 5

23 sq. unit 3

 

X

A(2, 0)

b y  0  (x  2) a ⇒ ay = bx 2 2b ⇒ ay 2 bx + 2b = 0 3

∑ ax  by  c

83. Given: i 1

i

i

a 2  b2

0

a ( x1  x2 + x3 )  b ( y1  y2 + y3 )  3c  0

Additional Support Material ▌ 3.291 84. The given line meets x-axiz at A(4,0) and y-axis at B(0,12). Let P divides AB in the ratio 1: 2 and Q divides AB in the ratio 2: 1 then coordinates of P and Q are (8/3, 4) and respectively. So the equations of OP and OQ are respectively

3 3π tanθ  , φ  θ 4 4  3π  m  tan φ  tan  θ   4 

4 8 x and y 5 x 8/3 4/3

y5

3 1 tan θ  1 4  1    3 7 1  tan θ 1 4

or 2y = 3x and y = 6x whose slopes 3/2 and 6 are the roots of the equation

1 Equation of AC is y  2  ( x  0) 7 i.e., x  7 y  14  0

3  3 x 2    6  x   6  0 ⇒ 2 x 2  15x  18  0 2 2  85.

AB  PC 

2  3 2 2

AP 5 r 5

2 2

2a  1 2 2

A

87.

1



, PD 

2a



a

2 2

6 3 x 1 y  2  r cos θ sinθ

Equation to AP:

2 Y

2x2y20 D P(1,a)

4

2

2x2y30 C

B

A

P θ (1,2)

r

Clearly, PC  AB 2a 1

⇒

1



2 2

2 2

⇒ | 2a 1 |  1

...( A )

|a| 2



1 2 2

1 1 ∴  a 2 2

⇒ | a |

X xy4

 1  1 m  n  A , ,  , B (0, 1) and C  . 2 n 2 m m 2 n m 2 n n m    .

Also, PD  AB ⇒

4

1

∴ P ≡ (1 + r cos q, 2 + r sin θ) ‘P’ will satisfy x + y = 4 88. The vertices of the parallelogram are O(0, 0), , B(0, 1) and .

⇒ 1  2 a  1  1 ∴ 0  a 1

O

1 2

Y B

...( B)

1 From (A) and (B), 0  a  2 86.

x

y

m

1

y nx



1

I

C

II

x

x m y

A(0,2)

φ

θ B

A

y m

π 4

O(0,0) O

C

X

3.292

▌ Mathematics

89. We have, x  x1  t ( x2  x1 )  x1  tx2  tx1

⇒ ⇒ ⇒

(1  t ) x1  tx2 (1  t )  t (1  t ) y1  ty2 Similarly, y  (1  t )  t ⇒ x  (1  t ) x1  tx2 

⇒ The point (x, y) divides the join of (x1, y1) and (x2, y2) in the ratio of t : (1 2 t). Since the triangle is equilateral

(given)

BOC is an equilateral triangle BC = OB = the radius of the circles BC = a.

93. Since the circle does not touch or intersect the coordinates axes, the absolute values of x and y coordinates of the centre are greater than the radius of the circle. Coordinates of the centre of the circle are (3, 5) and the radius is 9  25  p

∴ centroid of the triangle is same as the circumcentre and hence, radius of the circumcircle is

so that 3  9  25  p ⇒ p  25

2 2 r 5 ( median ) 5 (3a ) 5 2 a 3 3 ∴ The equation of the circumcircle whose centre is at (0, 0) and radius 2a is x2 + y2 = (2a)2

and the point (1, 4) lies inside the circle ⇒ 1 + 16 2 6 2 10 × 4 + p < 0 ⇒ p < 29

90. Line AB is x + y = 0, which is diameter of the circle x2 + y2 = 25. Its length = 2r = 10. 91. Let P(x, y) be any point on the circle passing through A(1, 2) and B(2, 2 1) The slope of AP 

y2 y 1 and slope of BP  . x 1 x2

Since they include an angle of π/4, so

⇒ (y + 1)(x 2 1) 2 (x 2 2)(y 2 2) = ± [(x 2 1)(x 2 2) + (y 2 2)(y + 1)] ⇒ 3x + y 2 5 = ± [x2 + y2 2 3x 2 y] ⇒ x2 + y2 2 5 = 0 or x2 + y2 2 6x 2 2y + 5 = 0 92. ∠B = ∠C = 75º ⇒ ∠BAC = 30º ⇒ ∠BOC = 60º

...(3)

From (1), (2), (3) we get 25