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MATHMAA

# KVPY2013PartII

KVPY2013PartII contains the questions of part II. To go back to Part I questions please click Back link and to go to the solutions click NEXT link which are at the bottom of the page.

## PART II

61) Let a, b, c, d, e be natural numbers in an arithmetic progression such that a+b+c+d+e is the cube of an integer and b+c+d is square of an integer. The least possible value of the number of digits of c is

A) 2   B) 3  C) 4   D)5

62) On each face of a cuboid, the sum of its perimeter and its area is written. Among the six numbers so written, there are three distinct numbers and they are 16, 24 and 31. The volume of cuboid lies between

A) 7 and 14    B) 14 and 21  C) 21 and 28   D) 28 and 35.

63) Let ABCD be a square and let P be a point on the segment CD such that DP:PC = 1:2 . Let Q be a point on segment AP such that $$\angle BQP =90^{\circ}$$. Then the ratio of the area of the quadrilateral PQBC to the area of ABCD is

A) $$\frac{31}{60}$$    B) $$\frac{37}{60}$$  C) $$\frac{39}{60}\ D)\(\frac{41}{60}$$

64) Suppose the height of the pyramid with a square base is decreased by p% and the lengths of the sides of the square base are increased by p% (where p > 0 ). If the volume remains the same , then

A) 50 < p < 55   B) 55 < p < 60  C) 60 < p < 65  D) 65 < p < 70.

65)  There are three kinds of liquids X, Y, Z . Three jars J1, J2, J3 contains 100 ml of liquids X, Y, Z respectively . By an operation we mean three steps in the following order :

- Stir the liquid in J1 and transfer 10 ml from J1 into J2 ;

- Stir the liquid in J2 and transfer 10 ml from J2 into J3 ;

- Stir the liquid in J3 and transfer 10 ml from J3 into J1 ;

After perfoming the operation four times, let x,y, z be the amounts of X, Y, Z respectively, in J1 . Then

A) x > y > z   B) x > z > y  C) y > x > z  D) z > x > y.

This completes PartII  KVPY 2013 questions which carry two marks each.

To go to Part I questions click Back and solutions of Part I click Next.