Allen PRMO/IOQM MOCK TEST 1 [1 ed.]

Citation preview

Time : 3 Hours

ALLEN

CAREER INSTITUTE Path to Success KOTA (RAJASTHAN)

Max. Marks : 102 TM

FORM NUMBER

(ACADEMIC SESSION 2019-2020)

Pre Nurture and Career Foundation Division MAJOR TEST -1

PRE - RMO

DATE : 21-07-19

INSTRUCTIONS 1. Use of mobile phones, smartphones, ipads, calculators, programmable wrist watches is STRICTLY PROHIBITED. Only ordinary pens and pencils are allowed inside the examination hall. 2. The correction is done by machines through scanning. On the OMR Sheet, darken bubbles completely with a black pencil or a black or blue ball pen. Darken the bubbles completely only after you are sure of your answer; else, erasing may lead to the OMR sheet getting damaged and the machine may not be able to read the answer. 3. Incomplete/Incorrectly and carelessly fillled information may disqualify your candidature. 4. Each question has a one or two digit number as answer. The first diagram below shows improper and proper way of darkening the bubbles with detailed instructions. The second diagram shows how to mark a 2-digit number and a 1-digit number. Q. 1

INSTRUCTIONS

Q. 2

1. "Think before your ink".

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2. Marking should be done with Blue/Black Ball Point Pen only.

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3. Darken only one circle for each question as shown in

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5. Make the marks only in the spaces provided.

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6. Please do not make any stray marks on the answer sheet.

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Example Below. WRONG METHODS A

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CORRECT METHODS D

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4. If more than one circle is darkened or if the response is marked in any other way as shown "WRONG" above, it shall be treated as wrong way of marking.

5. The answer you write on OMR sheet is irrelevant. The darkened bubble will be considered as your final answer. 6. Questions 1 to 6 carry 2 marks each; questions 7 to 21 cany 3 marks each; questions 22 to 30 carry 5 marks each. 7. All questions are compulsory. 8. There are no negative marks. 9. Do all rough work in the space provided below for it. You also have blank pages at the end of the question paper to continue with rough work. 10. You may take away the question paper after the examination.

PRE REGIONAL MATHEMATICS OLYMPIAD 1.

2.

a, b, c and d are 4 whole numbers from 0 to 9

7.

Let abcdef

MAJOR TEST-1

be a 6-digit integer such that

(a ¹ 0). If abcd + abc + ab + a = 2671 . Find

defabc is 6 times the value of abcdef . Find

2a + 3b + 4c + 5d.

the value of a + b + c + d + e + f. 8.

Define na! for n and a positive to be na! = n(n – a) (n – 2a)(n – 3a)...(n – ka) where k is the greatest integer for which n >

In which base is the number 221 a factor of 1215 ?

9.

ka.

There are a few integers n such that n2 + n + 1 divides n2013 + 61. Find the sum of the squares

If the quotient 728! / 182! is equal to 2 . Find x. x

3.

A line segment is divided so that the lesser part is to the greater part as the greater part is to the

of these integers. 10.

BC at D, BE ^ AC at E, AD intersects BE at H.

whole. If R is the ratio of the lesser part to the

Find ÐCHD in degrees.

greater part, then find the value of

R 4.

[R ( R

2 + R -1 )

+ R -1 ]

+R

11.

-1

12.

value of m. 13.

Let S = êë 1 úû + êë 2 úû + ... + êë 2018 úû . Find sum of the digits of êë S úû , where êë x úû is

pq + p + 1 0 and pq ¹ 1. Find the value of . q Convex quadrilateral ABCD has AB = 9 and CD

where êë x úû is greatest

integer function of x.

If p, q are two real numbers satisfying the relationship 2p2 – 3p – 1 = 0 and q2 + 3q – 2 =

Determine the number of real solutions of

é x ù é 2x ù ê2ú + ê 3 ú = x ë û ë û

some two digit natural number m. Find the

6.

Find the last non-zero digit of 50! = 1 × 2 × 3 × ... × 50.

It is known that 2726,4472,5054,6412 have the same remainder when they are divided by

5.

In DABC, ÐACB = 60°, ÐBAC = 75°, AD ^

greatest integer function of x. 14.

Find the square root of the sum of all positive

= 12. Diagonals AC and BD intersect at E, AC

rational numbers that are less than 10 and that

= 14, and DAED and DBEC have equal areas.

have denominator 30 when written in lowest

What is AE?

terms. Space for Rough Work

Your Hard Work Leads to Strong Foundation

1/4

PRE REGIONAL MATHEMATICS OLYMPIAD 15.

Given a set S = {1, 2, 3,...,20}. The subset A = {a, b, c} is said to be “nice”, if a, b, c are in

20.

A.P. then how many “nice” subsets does S have? 16.

by the total number of matches she has played.

21.

four matches, winning three and losing one. At

If (l + tan1º)(1 + tan2º)(1 + tan3º).....(1 + tan44º)(1 + tan45º) = 2n , find n.

At the start of a weekend, her win ratio is exactly 0.500. During the weekend she plays

2p ö 4p ö æ æ = csin ç q + If a sinq = b sin ç q + , then ÷ 3 ø 3 ÷ø è è the value of ab + bc + ca

A tennis player computes her "win ratio" by dividing the number of matches she has won

MAJOR TEST-1

22.

In DABC, AB = AC =

3 and D is a point on

the end of the weekend her win ratio is greater

BC such that AD = 1. Find the value of

than 0.503. If the largest number of matches

BD × DC.

that she could have won before the weekend began is k. Find 17.

k . 4

23.

The 12 number a 1 , a 2 , ....., a 12 are in numbes is 354. let P = a2 + a4 + ..... + a12 and Q = a1 + a3 + ..... + a11. If the ratio P : Q is 32

24.

PQR is a triangle with PQ = 15, QR = 25,

: 27, the common difference of the progression

RP = 30. A, B are points on PQ and PR

is

respectively such that ÐPBA = ÐPQR. The

Find the number of rectangles that can be

perimeter of the triangle PAB is 28, then the

obtained by joining four of the twelve vertices

length of AB is________.

of a 12-sided regular polygon 19.

x+y+z , then the 2

value of x + y + z is.

arithematical progression. The sum of all these

18.

x +1 + y + z - 4 =

If

25.

a, b are distinct natural numbers such that

Let ‘n’ be the number of pairs of consecutive integers in the set. {1000, 1001, 1002,..., 2000} Such that, when two integers are added, then no carrying is required. Find sum of the digits of ‘n’.

1 1 2 + = . a b 5 If

a + b = k 2 , the value of k is_____

Space for Rough Work

2/4

Your Hard Work Leads to Strong Foundation

PRE REGIONAL MATHEMATICS OLYMPIAD 26.

Suppose that n is a positive integer and a, b are

29.

positive real number with a + b = 2. Find the smallest possible value of 27.

1 1 + . n 1+ a 1 + bn

28.

Consider a cube of edge 1 cm. then the sum of area of all equilateral triangle, isosceles right angled triangle, scalene right angled triangle, by joining any three of the vertices of the cube can be represented as p + q r + s t . where, r and t are prime number. Find p + q + r + s + t.

The least area of the right angled triangle whose inradius is 1 unit can be expressed as m + n ,

MAJOR TEST-1

30.

Let CH be an altitude of DABC. Let R and S be the points where the circles inscribed in

find m + n

triangles ACH and BCH are tangent to CH . If

The positive integers a, b, c are connected by

AB = 2019, AC = 2018, and BC = 2017, then m RS can be expressed as , where m and n are n relatively prime positive integers. Find

the inequality a2 + b2 + c2 + 3 < ab + 3b + 2c then the value of a + b + c is_________

ém + nù ê 11 ú , where [x] represent greatest integer ë û function of x.

Space for Rough Work

Space for Rough Work

Your Hard Work Leads to Strong Foundation

3/4

PRE REGIONAL MATHEMATICS OLYMPIAD SPACE FOR ROUGH WORK

Space for Rough Work

4/4

Your Hard Work Leads to Strong Foundation

MAJOR TEST-1