CamIGCSE Maths Core & Extended Study & Revision Guide 3rd edition 1510421718, 9781510421714

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Cambridge IGCSE™

Mathematics Core and Extended Third edition John Jeskins, Jean Matthews, Mike Handbury, Eddie Wilde Brian Seager (series editor)

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Acknowledgements Every effort has been made to trace all copyright holders, but if any have been inadvertently overlooked, the Publishers will be pleased to make the necessary arrangements at the first opportunity. Although every effort has been made to ensure that website addresses are correct at time of going to press, Hodder Education cannot be held responsible for the content of any website mentioned in this book. It is sometimes possible to find a relocated web page by typing in the address of the home page for a website in the URL window of your browser. This text has not been through the Cambridge endorsement process. All questions, sample answers and marks awarded in this title were written by the authors. In examinations, the way marks are awarded may be different. Hachette UK’s policy is to use papers that are natural, renewable and recyclable products and made from wood grown in well-managed forests and other controlled sources. The logging and manufacturing processes are expected to conform to the environmental regulations of the country of origin. Orders: please contact Bookpoint Ltd, 130 Park Drive, Milton Park, Abingdon, Oxon OX14 4SE. Telephone: (44) 01235 827827. Fax: (44) 01235 400401. Email [email protected] Lines are open from 9 a.m. to 5 p.m., Monday to Saturday, with a 24-hour message answering service. You can also order through our website: www.hoddereducation.com ISBN: 978 1 5104 2171 4 © Brian Seager, John Jeskins, Jean Matthews, Mike Handbury and Eddie Wilde 2019 First published in 2004 Second edition published in 2016 This edition published in 2019 by Hodder Education, An Hachette UK Company Carmelite House 50 Victoria Embankment London EC4Y 0DZ www.hoddereducation.com Impression number 10 9 8 7 6 5 4 3 2 1 Year

2023 2022 2021 2020 2019

All rights reserved. Apart from any use permitted under UK copyright law, no part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying and recording, or held within any information storage and retrieval system, without permission in writing from the publisher or under licence from the Copyright Licensing Agency Limited. Further details of such licences (for reprographic reproduction) may be obtained from the Copyright Licensing Agency Limited, www.cla.co.uk Cover photo © Shutterstock/ju.grozyan Illustrations by Integra Software Services Typeset by Integra Software Services Pvt. Ltd., Pondicherry, India Printed in Spain A catalogue record for this title is available from the British Library.

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Contents





Introduction

v

Topic 1 Number hapter 1 C Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10

Number and language Accuracy Calculations and order Integers, fractions, decimals and percentages Further percentages Ratio and proportion Indices and standard form Money and finance Time Set notation and Venn diagrams

1 6 9 12 16 19 23 26 29 30





33 40 41 49 54 58 60 64 72 79





83





90 96 101 105





112 114





120 122 126





129 134





Topic 2 Algebra and graphs hapter 11 C Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18 Chapter 19 Chapter 20

Algebraic representation and manipulation Algebraic indices Equations and inequalities Linear programming Sequences Proportion Graphs in practical situations Graphs of functions Differentiation and the gradient function Functions

Topic 3 Coordinate geometry Chapter 21

Straight-line graphs

Topic 4 Geometry hapter 22 C Chapter 23 Chapter 24 Chapter 25

Geometrical vocabulary and construction Similarity and congruence Symmetry Angle properties

Topic 5 Mensuration hapter 26 C Chapter 27

Measures Perimeter, area and volume

Topic 6 Trigonometry Bearings Trigonometry Further trigonometry

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hapter 28 C Chapter 29 Chapter 30

Topic 7 Vectors and transformations hapter 31 C Chapter 32

Vectors Transformations

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Topic 8 Probability hapter 33 C Chapter 34

Probability Further probability

138 142





hapter 35 C Chapter 36 Chapter 37

Mean, median, mode and range Collecting, displaying and interpreting data Cumulative frequency

147 152 164







Answers Index

170 193

Topic 9 Statistics







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iv

Cambridge IGCSE Mathematics Core and Extended Study and Revision Guide

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Introduction Welcome to the Cambridge IGCSE™ Mathematics Core and Extended Third Edition Study and Revision Guide. This book is intended to help you prepare for the Cambridge examinations in IGCSE Mathematics, Core and Extended Tiers. The book covers all you need to know for the examination. If you are preparing for the Core, you need only look at the sections shown with blue dots. For the Extended syllabus, you need also to cover the sections shown by purple dots.

How to use this book

Core material is indicated by blue dots on the side of the page. Extended material is indicated by purple dots on the side of the page.

Key objectives The key points covered in the chapter.



23 Similarity and congruence

22

Geometrical vocabulary and construction

These tests are: l l

Key objectives

l

● To use and interpret the geometrical terms:

point, line, parallel, bearing, right angle, acute, obtuse and reflex angles, perpendicular, similarity and congruence. ● To use and interpret the vocabulary of triangles, quadrilaterals, circles, polygons and simple solid figures including nets.

l

● To measure and draw lines and angles. ● To construct a triangle given the three sides

using only a ruler and compasses.

All the corresponding sides are equal (SSS). Two corresponding sides are equal and the angle between them is the same (SAS). The angle between the sides is often called the ‘included angle’. Two angles and a corresponding side are equal (AAS). The triangles are right angled and have their hypotenuses and a corresponding side equal (RHS).

Sample question and answer

● To read and make scale drawings.

TA and TB are tangents to the circle centre O.

Angles and lines ● ● ● ● ● ●

● ● ● ●

O

T

PAGE 282

Therefore triangle TOA is congruent to triangle TOB (RHS).

B

Acute angles are angles between 0° and 90°. Right angles are 90°. Obtuse angles are between 90° and 180°. Reflex angles are between 180° and 360°. Perpendicular lines meet at a right angle. Parallel lines never meet and are marked with arrows.

Triangles

In the triangles TOA and TOB: 1 TO = TO (common side) 2 AO = BO (radii) 3 Angle OAT = angle OBT = 90° (Angle between tangent and radius = 90°)

A

Show that triangle TOA is congruent to triangle TOB.

Test yourself 8 Which of these show congruent triangles? If they are congruent, state the abbreviation for the appropriate test. 5 cm a 5 cm

6.5 cm

PAGES 283–285

6.5 cm

7.5 cm

Isosceles triangles have two equal sides and two equal angles. Equilateral triangles have three equal sides and three equal angles. Scalene triangles have all three sides with different lengths. In a right-angled triangle the longest side is called the hypotenuse.

7.5 cm

b

Tip When asked to show two triangles are congruent, list the three equal sides or angles with the reasons. Then state the conclusion with the correct abbreviation for the test for congruence.

4 cm

63°

6 cm

4 cm

6 cm

63° PAGE 285

c

Centre

Radius

gm

er

The segment in the right-hand diagram is called a minor segment. A segment that is more than half the circle is called a major segment.

Se

The sector in the right-hand diagram is called a minor sector. A sector that is more than half the circle is called a major sector.

Quadrilaterals

Chord

Arc

Diamet

The names connected with circles are shown in these two diagrams.

Centre

3.8 cm

en

71°

6.1 cm

3.8 cm

62°

t

Circumference

6.1 cm 62°

Sector

9 BDEF is a parallelogram. B is the midpoint of AF. Show that triangle ABC is congruent to triangle EDC.

Tangent

47°

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Circles

A

B

C

D

PAGES 285–286 E

F

The table on the next page summarises the special quadrilaterals: Answers on page 181

90

Cambridge IGCSE Mathematics Core and Extended Study and Revision Guide

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Test yourself

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Page references to the Student Book are shown by this icon:

99

Tip Advice to help you give the perfect answer.

Questions for you to check your understanding and progress.

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Sample questions and answers

Upper and lower bounds

Sample questions and answers

These questions help explain the concepts being considered.

1 a Minimum perimeter = 2 × 3.45 + 2 × 4.55 = 16 cm. Maximum perimeter = 2 × 3.55 + 2 × 4.65 = 16.4 cm. b Minimum area = 3.45 × 4.55 = 15.6975 = 15.7 cm2 to 3 s.f. Maximum area = 3.55 × 4.65 = 16.5075 = 16.5 cm2 to 3 s.f. 2 Lower bound of A = 3 × 3.615 = 2.0028 = 2.00 to 3 s.f. 5.415 Upper bound of A = 3 × 3.625 = 2.0120 = 2.01 to 3 s.f. 5.405

1 A rectangle has sides 3.5 cm and 4.6 cm measured to 2 s.f. Find the minimum and maximum value of: a the perimeter b the area.

Sample answers that show how the questions might be answered are in the blue box.

3b

2 A =  c , b = 3.62, c = 5.41 to 2 d.p. Find the lower and upper bounds of A.

Test yourself

Sample exam question and answer

6  P = ab − 2c, a = 2.1, b = 5.4 and c = 3.6 correct to 2 s.f. Find the lower  and upper bounds of the value of P. 7  A rectangle has an area of 14.5 cm2 and a length of 4.6 cm, both  measured to the nearest 0.1 units. Work out the limits between which  the width must lie. Answers on page 170

Model exam questions and answers show you how to earn the marks.

Sample exam question and answer 1 When a ball is thrown upwards, the maximum

Exam-style questions

height h it reaches is given by h = 

It is given that U = 4.2 and g = 9.8, both correct to 2 s.f. Calculate the upper and lower bounds of h. [4]

Examination-style questions for you to try to see what you have learned.

1 Upper bound = 0.926. Lower bound = 0.874. To find the upper bound of h, use the upper bound of U and the lower bound of g. To find the lower bound of h, use the opposite. 2 Upper bound of h = 4.25 = 0.926 282 2 × 9.75 = 0.926 to 3 s.f. 2 Lower bound of h = 4.15 = 0.874 238 5 2 × 9.85 = 0.874 to 3 s.f.

Exam-style questions

Exam tips

1  A pane of glass is 3.5 mm thick and measures 1.4 m by 0.9 m.   Work out the volume of the glass: a in mm3, b in m3.  [2] 2  The volume of a solid metal cylinder is 600 cm3. Each cm3 of the metal  has mass 15 g. Calculate the mass of the metal cylinder in kilograms. [2] v−u . u = 17.4, v = 30.3 and t = 2.6,  3  A formula used in science is a =  t all measured correct to the nearest 0.1. Find the maximum  possible value of a.  [3] 4  The population of Kenya is 2.6 × 107, correct to 2 s.f. The area of   Kenya is 5.8 × 105 km2, correct to 2 s.f. Calculate the lower and upper  bounds for the number of people per square kilometre in Kenya.  [4] 5  In 2018 a firm produced 1.2 billion litres of fizzy drink (correct to 2 s.f.).  The volume of a standard swimming pool is 390 m3 (correct to 2 s.f.).  What is the greatest number of these swimming pools that could   possibly be filled with this amount of fizzy drink?  [3]

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The examination papers will not usually tell you the formulae you need to use. You will have to learn and remember them. All those you need to know are in the book. In some cases, you may have learned a formula in a different form. If so, continue to use it.

U2 . 2g

Answers on page 170

Cambridge IGCSE Mathematics Core and Extended Study and Revision Guide Make sure you read the instructions carefully, both those on the front of the paper and those in each question. Here are the meanings of some of the words used: ●  Write, write down, state – little working will be needed and no explanation is required. ●  Calculate, find – something is to be worked out, with a calculator if appropriate. It is a good idea to show the steps of your working, as this may earn marks, even if your answer is wrong. ●  Solve – show all the steps in solving the equation. ●  Prove, show – all the steps needed, including reasons, must be shown in a logical way. ● Deduce, hence – use a previous result to help you find the answer. ● Draw – draw as accurately and as carefully as you can. ● Sketch – need not be accurate but should show the essential features. ●  Explain – give (a) brief reason(s). The number of features you need to mention can be judged by looking at the marks. One mark means that probably only one reason is required.

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1





Number and language

Key objectives l

To identify and use natural numbers, integers, prime numbers, square and cube numbers, common factors and common multiples, rational and irrational numbers, real numbers and reciprocals.

l

To calculate with squares, square roots, cubes and cube roots and other powers and roots of numbers. l To use directed numbers in practical situations.

Natural numbers and integers

PAGE 4

Natural numbers are the counting numbers 1, 2, 3, 4, … and zero. They are positive whole numbers and zero. Integers are the positive and negative whole numbers and zero. So they are: …, −4, −3, −2, −1, 0, 1, 2, 3, 4, …

Multiples and factors

PAGES 6–8

The multiples of a number are the numbers in its times table. The factors of a number are the numbers that divide into it exactly.

Sample questions and answers 1 6, 12, 18, 24, 30 2 1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6; the factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18 and 36.

1 Find the multiples of 6 up to 30. 2 Find the factors of 36.

Test yourself

Tip

1 Find the first six multiples of the following: a 4 b 9 2 Find the factors of the following: a 24 b 40

It is easier to find factors by looking for pairs of numbers that multiply to give the number.

c 15

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c 32

Answers on page 170

Prime numbers and prime factors

PAGES 5–7

Prime numbers are numbers that have only two factors: 1 and the number itself. Prime factors are factors that are also prime numbers.

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Highest common factor and lowest common multiple

Sample questions and answers 1 Find the first five prime numbers. 2 Write 48 as a product of prime factors.

1 2, 3, 5, 7, 11 2 48 = 2 × 2 × 2 × 2 × 3 or 24 × 3 48 6 3

8 2

2

4 2

2

Test yourself

Tip

3 Write the following as a product of prime factors: a 36 b 140

Keep breaking the numbers into factors until you end up with prime numbers.

c 84

Answers on page 170

Highest common factor and lowest common multiple PAGES 7–8 The highest common factor (HCF) is the largest number that will divide into each of the given numbers. The lowest common multiple (LCM) is the smallest number that can be divided exactly by each of the given numbers.

Sample questions and answers

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1 Find the HCF of 120, 36 and 84. 2 Find the LCM of 120, 36 and 84.

1

120 =  2  ×  2  × 2 ×  3  × 5 Express each as a product of prime factors. 36 =  2  ×  2  × 3 ×  3 84 =  2   ×   2   ×   3  × 7 HCF = 2 × 2 × 3 = 12 Choose the factors that are common to all three numbers.

2

120  =  2 ×  2 ×  2  × 3  × 5 Express each as a product of prime factors. 36 = 2 × 2 ×  3  ×  3 84 = 2 × 2 × 3 ×  7 LCM = 2 × 2 × 2 × 3 × 3 × 5 × 7  For each prime, choose the largest number of times it appears.  = 2520 

Test yourself 4 Find the HCF and the LCM of the following: a 15 and 24 b 12 and 40

c 90, 54 and 36

Answers on page 170

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1 Number and language

Directed numbers

PAGE 11

Negative numbers are used in practical situations such as temperatures.

Sample question and answer 1 8 °C − 12 °C = −4 °C

1 The temperature at midday was 8 °C. At midnight it was 12 °C colder. What was the temperature at midnight?

Test yourself 5 a The balance in Aisha’s bank account was −$20. She paid $50 into her account. How much was in her account then? b The car park of a hotel was below ground on floor −2. Vivek went from the car park to his room on floor 4. How many floors did he go up? Answers on page 170

Squares, cubes and roots

PAGES 5–6, 9–10

To find the square of a number, multiply it by itself: 52 = 5 × 5 = 25. Here are the squares of the integers 1 to 15. Integer

1

2

3

4

5

6

7

8

9

10

11

13

14

15

Square number

1

4

9

16

25

36

49

64

81

100

121 144 169

196

225

12

As 64 is the square of 8, the positive square root of 64 is 8. This is written as 64 = 8. Note that (−8)2 is also 64. To find the cube of a number, multiply it by itself and then by itself again: 53 = 5 × 5 × 5 = 125. Here are the cubes of the integers 1 to 10. Integer

1

2

3

4

Cube number

1

8

27

64

5

6

125 216

7

8

9

10

343

512

729

1000

You should memorise these square and cube numbers. You will also need them for finding square and cube roots without a calculator.

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If you know that 53 = 125, then you also know that the cube root of 125 is 5. This is written as 3 125 = 5. 

Tip

Sample questions and answers Calculate the following without the use of a calculator: 1 702 2 144 3 3 512

1 702 = 70 × 70 = 4900 2 144  = 12, as 122 = 144 3 3 512 = 8, as 83 = 512

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Rational and irrational numbers

Test yourself 6 Without a calculator, find the following: a the square of 40 b 81

c 6 3

d

3

343

Answers on page 170

Other powers and roots

PAGES 10–11

25 = 2 × 2 × 2 × 2 × 2 = 32   We say that 2 to the power of 5 equals 32.

Tip

Using roots, we say that the fifth root of 32 is 2. This is written as 5 32  = 2.

Make sure you practise using your calculator to find powers and roots, so that you are sure that you know how to use the special buttons on your calculator.

There are special buttons on scientific calculators to help you find powers and roots. They have labels such as x y and y x . Find the buttons on your calculator and learn how to use them.

Sample questions and answers Use a calculator to find: 4

1 7 

2

6

1 2401

2 3

729

Test yourself 7 Use a calculator to find: a 85 b 3.5 4

c

4

1296

d 5 161051

Answers on page 170

Rational and irrational numbers

PAGES 8–9

m

A rational number is a number that can be written as an ordinary fraction, n , where m and n are integers. These are terminating or recurring decimals. An irrational number cannot be written in this form. Decimals that neither terminate nor recur are irrational numbers. For example, π and 6 are irrational. It is illegal to photocopy this page

Real numbers are the set of all rational and irrational numbers. Any number on a number line is a real number.

Test yourself

Tip

8 Here are some numbers. Decide which of these is irrational. π a 126 b 121 c 6 . d 3 17 e 1.36 f ( 3)2

Recurring decimal notation: 0.3˙ means 0.333..., ˙ ˙3 means 0.030303..., 0.0 ˙ 03 ˙ means 0.3 0.303303303...

Answers on page 170

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1 Number and language

Reciprocals

PAGE 5

The reciprocal of any number is the result of dividing 1 by the number. This means that the reciprocal of a fraction

b a is . a b

Sample questions and answers 1 1 ÷ 0.4 = 2.5 4 2 5 41 = 21 4 . So its reciprocal is 21

Find the reciprocal of: 1 0.4

2 5 14

Test yourself 9 Find the reciprocal of: a 5

b 0.05

c 3 31

Answers on page 170

Sample exam questions and answers 1 Which of these numbers is irrational? [1] 3  27 , 48, 0.12, 2π, 0.3 2 Write 30 as a product of prime factors. [1]

1 Irrational: 48 (as 48 is not a square number) and 2π (rational × irrational = irrational). 3 27 = 3 = 31 , 12 and 0.3  = 1 , so these are all rational. 0.12 = 100 3 2 30 = 2 × 3 × 5 30 5

6 2

3

Exam-style questions [1] [1] [1] [2] [2] [3]

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1 From the numbers in the range 10 to 20, choose: a a multiple of 7 b a factor of 36 c a prime number. 2 Write 264 as a product of prime factors. 3 Find the HCF and LCM of 12 and 16. 4 Find the HCF and LCM of 10, 12 and 20. 5 Find the value of 4 multiplied by the reciprocal of 4. Show clear working to explain your answer. 6 Find the value of (-5)2 + 4 × -3.

[2] [2]

Answers on page 170

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2





Accuracy

Key objectives ●

To be able to make estimates of numbers, quantities and lengths. ● To be able to give approximations to specified numbers of significant figures and decimal places. ● To be able to round off answers to reasonable accuracy in the context of a given problem.

Accuracy of answers

●

To be able to give appropriate upper and lower bounds for data given to a specified accuracy. ● To be able to obtain appropriate upper and lower bounds to solutions of simple problems given data to a specified accuracy.

PAGES 13–17

Sometimes you are asked to round to a given accuracy – either to a given number of decimal places or to a given number of significant figures. Sometimes you need to work to a reasonable degree of accuracy.

Checking your work Use the following methods to check your work: l

Common sense – is the answer reasonable? l Inverse operations – work backwards to check. l Estimates – one significant figure is often easiest.

Tips l

If the value of the first digit being ignored is 5 or more, round up. l Look at the context and the accuracy of the information you have been given.

Sample questions and answers

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1 Pat calculated 67.4  ×  504.9 and got 12.3 the answer 418 572.2. Use estimates to show her answer is wrong. 2 Estimate the result of this calculation: 34 ×  3 60 .

1 70 × 500  = 3500 10 Her answer is far too large. (In fact, Pat had multiplied by mistake instead of dividing.) 2 34 = 81   We know that 43 = 64, so 3 64 = 4 34 x 3 60  ≈ 80 x 4 ≈ 300

Tip ≈ means ‘is approximately equal to’.

Test yourself 1 Round these to two decimal places (d.p.): a 30.972 b 4.1387 c 24.596 2 Round these to three significant figures (s.f.): a 5347 b 61.42 c 3049.6 3 Use 1 s.f. estimates to find approximate answers, showing your estimates: 5857 3.13 × 4 20 a $792 ÷ 19 b c 3 130  × 3 3 d 62 × 20.3 6.05 Answers on page 170

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2 Accuracy

Bounds of measurement

PAGES 17–21

A length measured as 45 cm to the nearest centimetre lies between 44.5 and 45.5 cm. So the bounds are 44.5 and 45.5 cm. A mass measured as 1.38 kg correct to 3 s.f. has bounds 1.375 and 1.385 kg. As 1.385 rounds up to 1.39 correct to 3 s.f., if we use the bounds to write an inequality for the mass, we have 1.375 kg  mass ,