Sample-paper-7.pdf

PRMO Practice Test-7 Time : 3 Hrs. General Instruction: 1. All the answers are single digit or double digit. 2. Each question carries 1 mark. 3. No negative marking.

M.M. : 30

8n an integer? 9999  n

1.

For how many natural numbers n between 1 and 2014 (both inclusive) is

2.

In a farm, both men and women were working. Exactly one–third of the staff brought one child each. One day, each male employee planted 13 trees and each women employee planted 10 trees and each child planted 6 trees. A total of 159 trees were planted on that day. How many women employees were there in that form?

3.

The sum of all angles except one of a convex polygon is 2190° (where the angles are less than 180°). Find the possible number of sides of the polygon.

4.

A lady takes her triplets and a younger niece to a restaurant on the birthday of the triplets. The restaurant charges Rs. 75 for the mother and Rs. 5 for each completed year of a child's age. If the total bill is Rs. 160 and if her niece is younger than her triplets, than find the age of the triplets.

5.

Find the number of prime numbers less than 100 which can he expressed as the sum of the squares of two natural numbers.

6.

Given x and y are two positive numbers satisfying the system 1– find

12 2 12 6  and 1 +  y  3x y  3x x y

y . x 9y 2 3 9x 2 3 9z 2 3  y,  z,  x. 2 2 2 1  9x 2 1  9y 2 1  9z 2

7.

Find number of triplets (x, y, z) satisfying the system

8.

It is known that there is such a number S such that if real numbers a, b, c, d are all neither 0 nor 1, 1 1 1 1 1 1 1 1 satisfying a + b + c + d = S and    = S, then     S Find S. a b c d 1 a 1 b 1 c 1 d

9.

Find the sum of squares of all real solutions of the equation | x2 – 4 | x | + 5 | = 16  8x  x 2  1

10.

Find number of ordered pains (x,, y) of real numbers satisfying the equation (16x200 + 1) (y200 + 1) = 16(xy)100.

11.

Given that the lengths of three sides of ABC are 3, 4 and 5. P is a variable point in the interior of ABC (not on its boundary). If the maximum value of product of distances from P to the three sides m mn AB, BC, CA is where m, n are relatively prime positive integers, then find . n mn

12.

Given a, b are real numbers such that a + b = 17. If the minimum value of 2a + 4b is N 3 4 then sum of





all digits of N is equal to. H.O. 92, Rajeev Gandhi Nagar, Kota (Raj.) Mob. 97831-97831, 70732-22177, Ph. 0744-2423333 www.nucleuseducation.in

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PCCP 13.

In the triangle ABC, the angle bisector BK of B intersect AC at K. If BC = 2, CK = 1, BK = area of ABC is

3 2

and

m 7 where m, n are relatively prime positive integers find (m + n). n

14.

Given AB is a diameter of a circle with centre O and C is a point on the tangent line to the circle at A. The line segment OC intersects the circle at D and the extension of BD intersects AC at E, as shown in the given diagram. If AC = AB = 2 and length of AE = n  1. Find n.

15.

A circle of radius 2 and a circle of radius 3 are tangent externally at T. The line MN is an external MT m common tangent, where M, N are the two tangent points. If  where m, n are relatively prime NT n positive integers. Find (m + n)

16.

A big circle with centre O and radius R, a circle with centre B and radius 2r and two circles with centers r 4 2 m A, A and radius r are tangent pairwise as shown in the given figure. If the ratio  where R n m, n are relatively prime positive integers, then (m2 – n2) has the value equal to A

A O B

17.

ABCD is a square of side 1 unit, M is the center of the circle taking AD as diameter, E is a point on the m side AB such that CE is tangent to the circle. If the area of BCE is (where m, n are relatively n prime positive integers). Find (m + n)

18.

In the figure below, ABC is an isosceles triangle inscribed in a circle with centre O and diameter AD, with AB = AC. AD intersects BC at E and F is the midpoint of OE. Given that BD is parallel to FC and BC = 2 5 cm. If CD = n cm. Find n. B OF

A

E

D

C 99

 10  n

19.

Given

n 1 99

 a  b where a, b are relatively prime positive integers. Find (a + b).

 10  n n 1

20.

Let Sn be the sum of first n terms of an AP {an}. If S15 > 0 and S 16 < 0 and if among

Si is the maximum ratio ai

S1 S2 _____ S15 , . Find i. a1 a 2 a15

H.O. 92, Rajeev Gandhi Nagar, Kota (Raj.) Mob. 97831-97831, 70732-22177, Ph. 0744-2423333 www.nucleuseducation.in

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2

2

PCCP 2

3

100

21.

Find the remainder if the sum 910  910  910      910

22.

If n has four zeroes at the end and (n  1) has six zeroes at the end. Find the value of n.

23.

In the sequence 1, 23, 45, 67, 89, 1011, 1213, ……….. 9899, 100101, 102103 ……….. (from the second term on wards, two consecutive natural numbers written together in the natural order, form the terms). Find the number of terms having four digits.

24.

The sides of a right angled triangle are a, a + d and a + 2d with a and d both positive, then

is divided by 7.

a has the d

value equal to 25.

Find the number of ordered pairs (x, y) of integers which satisfy the equation x2 – y2 = 1.

26.

Find the number of non-negative integral solution of the equation x + y +z = 15 where x  3 and y  4.

27.

Let N denotes the positive integral solutions of the equation x1x2x3x4x5 = 1050, then last two digits of N is equal to

28.

If number of five digit numbers which can be formed by using the digit 1, 2, 3, 4, ……. 9 such that no two consecutive digits are identical is 9 × 2n, then n is equal to

29.

Out of 10 people, 5 are to be seated around a round table and 5 are to be seated across a rectangular table. If then number of was to do so is (2 n ). Find n.

30.

If N denotes the total number of 6–digit numbers x1x2x3x4x5x6 having the property x1 < x2  x3 < x4 < x5  x6, then sum of digits of N is equal to

H.O. 92, Rajeev Gandhi Nagar, Kota (Raj.) Mob. 97831-97831, 70732-22177, Ph. 0744-2423333 www.nucleuseducation.in

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