Math KVPY 2017

1. It is given that number 43361 can be written as product of two distinct primes p1 and p2. Assume that there are 42900 numbers less than 43361 and coprime to it. Then value of  is

A. 462
B. 464         
C. 400          
D. 402

2. Let S be a circle in plane which touches x-axis at point A and y-axis at point B. Unit circle  touches circle S at point C externally. If O denotes the origin then angle OCA is

A.
B.
C.
D.

3. Let  be vertices of nine sided polygon with each side equal to 2 units. Then difference between diagonals A1A5 and  A2A4 is

A.
B.
C. 6
D. 2

4. Let a1a2 … an be n-nonzero real numbers of which 'p' are positive and remaining (n-p) are negative. Number of ordered pairs (j,k), j < k, for which ajak is positive, is 55. Number of ordered pairs (j,k), j < k for which ajak is negative, is 50. The value of  is 

A. 629             
B. 325     
C. 125                
D. 221

5. In isosceles trapezium, length of one of the parallel sides and length of non-parallel side is 30 units. If area of trapezium is maximum then smallest angle of trapezium is:

A.
B.
C.
D.

6. Let ABCD be a square and E be a point outside the square such that E, A, C are collinear points in that order. If area of triangle EAB is equal to area of square ABCD and  then area of square ABCD is

A. 8
B. 10
C.
D.

7. Let A={1,2,3....,30} be a set. Then number of ways to select three distinct integers in set A so that their product is divisible by 9 is:

A. 1590           
B. 1505      
C. 1110      
D. 1025

8. Let  be real numbers such that  . Then smallest value of    lies the interval

A. (0, 1.5)     
B. (1.5, 2.5)                 
C. (2.5, 3)            
D. (3, 3.5)

9. Let S be set of all ordered pairs  of integers which satisfy . Then

A. S is an infinite set
B. S is empty set
C. S contains exactly one element
D. S is finite set with at least two elements

10. Let A, G, H be A.M., G.M, and H.M. between two distinct positive numbers and a be the smallest of two real roots of equation:. Then

A.
B.
C.
D.

11. Let  then value of x+y is:

A. 3
B.
C. - 3
D.

12. Let r(x) be the remainder on dividing polynomial  by polynomial x- x then

A. r(x) = 0
B. r(x) is non-zero constant
C. degree of r(x) is 1
D. degree of r(x) is 2

13. The sum of all non-integer roots of equation:  is

A. –6             
B. –11
C. –5
D. 3

14. Let a,b,c,d be distinct elements of the set {1, 2, 3, …, 9}. The minimum value of  is

A.
B.
C.
D.

15. Lengths of sides of quadrilateral are distinct integers and the length of second largest side is 10 units. Then maximum possible length of largest side is

A. 25
B. 26
C. 28
D. 27

16. Let ABCD be a unit square. A circle is drawn with centre O, on extended line CD, passing through point A. If diagonal AC is tangent to circle then area of shaded region is:

A.
B.
C.
D.

17. Let AB be line segment of length 2 units. Construct a semicircle S with AB as diameter and let C be mid point of arc AB. Construct another semicircle T, external to triangle ABC with chord AC as diameter. Then area of region inside semicircle T but outside semicircle S is

A.
B.
C.
D.

18. The exponent of 2 in 

A. 98
B. 99
C. 100
D. 101

19. Given a right angled triangle ABC with angle ACB equal to 90o, P is a point on AB such that CP is perpendicular to AB. Let M and N be points on lines CB and CA such that MP is parallel to CA and NP is parallel to CB. If MP = 5 and NP = 4 then length of sides CB and CA are:

A.
B.
C.
D.