1

KVPY PAPER KVPY PAPER KVPY PAPER KVPY PAPER ––––IIIIIIIIIIII

1.1.1.1. The product of the first 100 positive integers ends withThe product of the first 100 positive integers ends withThe product of the first 100 positive integers ends withThe product of the first 100 positive integers ends with (a) 21 zeroes(a) 21 zeroes(a) 21 zeroes(a) 21 zeroes (b) 22 zeroes(b) 22 zeroes(b) 22 zeroes(b) 22 zeroes (c) 23 zeroes (c) 23 zeroes (c) 23 zeroes (c) 23 zeroes (d) 24 zeroes(d) 24 zeroes(d) 24 zeroes(d) 24 zeroes

2.2.2.2. How many zeroes will be there in the end of the product How many zeroes will be there in the end of the product How many zeroes will be there in the end of the product How many zeroes will be there in the end of the product 2! 4! 6! 8! 10!2! .4! .6! .8! .10!

(a)(a)(a)(a) 10! 6!+ (b)(b)(b)(b) 10! 8! 6!+ + (c)(c)(c)(c) ( )2 10! 8! 6!+ + (d)(d)(d)(d) ( )2 10! 3.3.3.3. Given four 1 gm stones, four 5gm stones, four 25 gm stones, four 125 gm stones Given four 1 gm stones, four 5gm stones, four 25 gm stones, four 125 gm stones Given four 1 gm stones, four 5gm stones, four 25 gm stones, four 125 gm stones Given four 1 gm stones, four 5gm stones, four 25 gm stones, four 125 gm stones

each, it is possible to weigh material of any integral each, it is possible to weigh material of any integral each, it is possible to weigh material of any integral each, it is possible to weigh material of any integral weight up toweight up toweight up toweight up to (a) 600 gms (a) 600 gms (a) 600 gms (a) 600 gms (b) 625 gms(b) 625 gms(b) 625 gms(b) 625 gms (c) 624 gms(c) 624 gms(c) 624 gms(c) 624 gms (d) 524gms(d) 524gms(d) 524gms(d) 524gms

4.4.4.4. The sum of the first n terms of an A.P. where 1The sum of the first n terms of an A.P. where 1The sum of the first n terms of an A.P. where 1The sum of the first n terms of an A.P. where 1stststst term is a (not necessarily term is a (not necessarily term is a (not necessarily term is a (not necessarily positive) integer and common difference is 2, is known to be 153. If n > 1, then positive) integer and common difference is 2, is known to be 153. If n > 1, then positive) integer and common difference is 2, is known to be 153. If n > 1, then positive) integer and common difference is 2, is known to be 153. If n > 1, then the number of possible values of n is the number of possible values of n is the number of possible values of n is the number of possible values of n is (a) 2(a) 2(a) 2(a) 2 (b) 3 (b) 3 (b) 3 (b) 3 ((((c) 4c) 4c) 4c) 4 (d) 5(d) 5(d) 5(d) 5

5.5.5.5. Consider a cubical box of 1m side which has one corner at (0,0,0) and the opposite Consider a cubical box of 1m side which has one corner at (0,0,0) and the opposite Consider a cubical box of 1m side which has one corner at (0,0,0) and the opposite Consider a cubical box of 1m side which has one corner at (0,0,0) and the opposite corner placed at (1,1,1). The least possible distance that an ant crawling fromcorner placed at (1,1,1). The least possible distance that an ant crawling fromcorner placed at (1,1,1). The least possible distance that an ant crawling fromcorner placed at (1,1,1). The least possible distance that an ant crawling from P to P to P to P to Q must travel isQ must travel isQ must travel isQ must travel is

(a)(a)(a)(a) 6 mmmm (b)(b)(b)(b) 5 mmmm (c)(c)(c)(c) 2 3 mmmm (d)(d)(d)(d) 1 3+ mmmm

6.6.6.6. letletletlet 1 1x <− andandandand 11

nn

n

xx

x+ =

+for allfor allfor allfor all 1n≥ ,then,then,then,then

(a)(a)(a)(a){ } 1nx →− as as as as n→∞ (b)(b)(b)(b){ } 0nx → as as as asn→∞

(c) (c) (c) (c) { } 1nx → asasasas n→∞ (d)(d)(d)(d) { }nx divergesdivergesdivergesdiverges 7.7.7.7. The number of perfect cubes among the first 4000 positive integers isThe number of perfect cubes among the first 4000 positive integers isThe number of perfect cubes among the first 4000 positive integers isThe number of perfect cubes among the first 4000 positive integers is

(a) 16(a) 16(a) 16(a) 16 (b) 15(b) 15(b) 15(b) 15 (c) 1(c) 1(c) 1(c) 14444 (d) 13(d) 13(d) 13(d) 13

8.8.8.8. The roots of the equation The roots of the equation The roots of the equation The roots of the equation 4 2 1x x+ = areareareare (a) all real & positive(a) all real & positive(a) all real & positive(a) all real & positive (b) never real (b) never real (b) never real (b) never real (c)2 positive & 2 negative (c)2 positive & 2 negative (c)2 positive & 2 negative (c)2 positive & 2 negative (d) one positive, one negative and two non(d) one positive, one negative and two non(d) one positive, one negative and two non(d) one positive, one negative and two non----realrealrealreal

9.9.9.9. Prove that the equation Prove that the equation Prove that the equation Prove that the equation 3 22 5 0x x x+ + + = has ohas ohas ohas only one real root nly one real root nly one real root nly one real root α such thatsuch thatsuch thatsuch that

[ ] 3α =− where where where where [ ]x → the G.I.F. of x. the G.I.F. of x. the G.I.F. of x. the G.I.F. of x.

10.10.10.10. If the equation If the equation If the equation If the equation 3 22 4 0x ax bx+ + + = has three real roots where a, b > 0, has three real roots where a, b > 0, has three real roots where a, b > 0, has three real roots where a, b > 0, show that a + b > show that a + b > show that a + b > show that a + b > ---- 6. 6. 6. 6.

11.11.11.11. Show that the foShow that the foShow that the foShow that the following equation can have at most one real root llowing equation can have at most one real root llowing equation can have at most one real root llowing equation can have at most one real root 5 33 5 21 3sin 4cos 5 0x x x x x− + + + + = ....

12.12.12.12. If the positive real number x be such thatIf the positive real number x be such thatIf the positive real number x be such thatIf the positive real number x be such that [ ] [ ], x x and x x− are in G.P., where are in G.P., where are in G.P., where are in G.P., where [ ]x → the G.I.F. of x.the G.I.F. of x.the G.I.F. of x.the G.I.F. of x. Find x. Find x. Find x. Find x.

13.13.13.13. Consider two quadratic expressionsConsider two quadratic expressionsConsider two quadratic expressionsConsider two quadratic expressions ( ) 2f x ax bx c= + + & & & & ( ) 2g x ax px q= + + , (a, b, , (a, b, , (a, b, , (a, b,

c, p, q c, p, q c, p, q c, p, q R∈ , b, b, b, b ≠ q, aq, aq, aq, a ≠ 0)such that their discriminants are equal. If f(x) = g(x) 0)such that their discriminants are equal. If f(x) = g(x) 0)such that their discriminants are equal. If f(x) = g(x) 0)such that their discriminants are equal. If f(x) = g(x) has a root x=has a root x=has a root x=has a root x= α , then, then, then, then (a) (a) (a) (a) α will be A.M. of the roots of f(x)=0 & g(x) =0;will be A.M. of the roots of f(x)=0 & g(x) =0;will be A.M. of the roots of f(x)=0 & g(x) =0;will be A.M. of the roots of f(x)=0 & g(x) =0; (b(b(b(b)))) α will be A.M. of the roots of f(x)will be A.M. of the roots of f(x)will be A.M. of the roots of f(x)will be A.M. of the roots of f(x)=0 =0 =0 =0 ;;;; (c)(c)(c)(c) α will be A.M. of the roots of f(x)=0 or g(x) =0will be A.M. of the roots of f(x)=0 or g(x) =0will be A.M. of the roots of f(x)=0 or g(x) =0will be A.M. of the roots of f(x)=0 or g(x) =0;;;; (d)(d)(d)(d) α will be A.M. of the roots of g(x) =0;will be A.M. of the roots of g(x) =0;will be A.M. of the roots of g(x) =0;will be A.M. of the roots of g(x) =0;

14.14.14.14. If the graph ofIf the graph ofIf the graph ofIf the graph of ( )y f x= ,,,, where where where where ( ) 2 ,f x ax bx c= + + a ,a ,a ,a , b, c b, c b, c b, c R∈ ,,,, a a a a ≠ 0,0,0,0, has the has the has the has the

maximum vertical heightmaximum vertical heightmaximum vertical heightmaximum vertical height 4 , then , then , then , then

2

(A) a > 0(A) a > 0(A) a > 0(A) a > 0 (B) a < 0(B) a < 0(B) a < 0(B) a < 0

(C)(C)(C)(C) ( )2 4b ac− is negative is negative is negative is negative (D) nothing can be said(D) nothing can be said(D) nothing can be said(D) nothing can be said

15.15.15.15. The number of quadratic equations which are unchanged by their roots, isThe number of quadratic equations which are unchanged by their roots, isThe number of quadratic equations which are unchanged by their roots, isThe number of quadratic equations which are unchanged by their roots, is (A) 2 (A) 2 (A) 2 (A) 2 (B) 4(B) 4(B) 4(B) 4 (C) 6(C) 6(C) 6(C) 6 (D) none of these(D) none of these(D) none of these(D) none of these

16.16.16.16. If p, q,If p, q,If p, q,If p, q, r, s r, s r, s r, s R∈ , the equation, the equation, the equation, the equation ( )( )( )2 2 23 2 0x px q x rx q x sx q+ + − + + − + − = hashashashas

(a) six real roots(a) six real roots(a) six real roots(a) six real roots (b) at least two real roots(b) at least two real roots(b) at least two real roots(b) at least two real roots (c) two real & four imaginary roots(c) two real & four imaginary roots(c) two real & four imaginary roots(c) two real & four imaginary roots (d) (d) (d) (d) 4 real & real & real & real & 2 imaginary roots imaginary roots imaginary roots imaginary roots

17.17.17.17. If a, If a, If a, If a, b, c are positive integers forming an increasing G.P. whose common ratio is a b, c are positive integers forming an increasing G.P. whose common ratio is a b, c are positive integers forming an increasing G.P. whose common ratio is a b, c are positive integers forming an increasing G.P. whose common ratio is a natural number, (b natural number, (b natural number, (b natural number, (b –––– a) is a cube of a natural number and a) is a cube of a natural number and a) is a cube of a natural number and a) is a cube of a natural number and 6 6 6log log log 6a b c+ + = then (a + b + c) =then (a + b + c) =then (a + b + c) =then (a + b + c) = (a) 100(a) 100(a) 100(a) 100 (b) 111(b) 111(b) 111(b) 111 (c) 122(c) 122(c) 122(c) 122 (d) 189(d) 189(d) 189(d) 189

18.18.18.18. If a, b, c, dIf a, b, c, dIf a, b, c, dIf a, b, c, d R+∈ such that (abcd) = 1, then the minimum value of such that (abcd) = 1, then the minimum value of such that (abcd) = 1, then the minimum value of such that (abcd) = 1, then the minimum value of

( )( )( )( ) ( ), , ,

1 1 1 1 1a b c d

a b c d a+ + + + = +∏ isisisis

(a) 4(a) 4(a) 4(a) 4 (b) 1(b) 1(b) 1(b) 1 (c) 16(c) 16(c) 16(c) 16 (d) 18(d) 18(d) 18(d) 18

19.19.19.19. For the series For the series For the series For the series ( ) ( ) ( )2 2 21 1 11 1 2 1 2 3 1 2 3 4 ....

1 3 1 3 5 1 3 5 7S = + + + + + + + + + +

+ + + + + +

(a)(a)(a)(a) 7 16t = (b)(b)(b)(b) 7 18t = (c)(c)(c)(c) 10505

4S = (d)(d)(d)(d) 10

405

4S =

20.20.20.20. IfIfIfIf ( )( )( )( )

1

13

n

r

n a n b n cr r

=

+ + ++ =∑ , where a < b < c , where a < b < c , where a < b < c , where a < b < c, then , then , then , then

(A) 2b = c(A) 2b = c(A) 2b = c(A) 2b = c (B)(B)(B)(B) 3 3 38 8a b c abc− + =

(C) c is a prime number(C) c is a prime number(C) c is a prime number(C) c is a prime number (D)(D)(D)(D) ( )2

0a b+ =

21.21.21.21. let let let let( )1111...111

n times

na = , then , then , then , then

(a)(a)(a)(a) 912a is not prime is not prime is not prime is not prime (b)(b)(b)(b) 951a is not prime is not prime is not prime is not prime

(c)(c)(c)(c) 480a is not prime is not prime is not prime is not prime (d)(d)(d)(d) 91a is not prime is not prime is not prime is not prime

22.22.22.22. L L L Letetetet log 4a b = ,,,, log 2c d = , where a, b, where a, b, where a, b, where a, b, c, d, c, d, c, d, c, d N∈ . Given that (b . Given that (b . Given that (b . Given that (b –––– d) = 7, then the value d) = 7, then the value d) = 7, then the value d) = 7, then the value of (c of (c of (c of (c ––––a ) isa ) isa ) isa ) is (a) 1(a) 1(a) 1(a) 1 (b) (b) (b) (b) ----1111 (c) 2(c) 2(c) 2(c) 2 (d) (d) (d) (d) ----2222

23.23.23.23. If a, bIf a, bIf a, bIf a, b ≠ 1, ab > 0 &1, ab > 0 &1, ab > 0 &1, ab > 0 & log logb aa b= , then ab =, then ab =, then ab =, then ab = (a) ½(a) ½(a) ½(a) ½ (b) 1(b) 1(b) 1(b) 1 (c) 2(c) 2(c) 2(c) 2 (d) 10(d) 10(d) 10(d) 10

24.24.24.24. The numThe numThe numThe number of solutions for the equation in (x ber of solutions for the equation in (x ber of solutions for the equation in (x ber of solutions for the equation in (x –––– 1 + ln x) = 0 is 1 + ln x) = 0 is 1 + ln x) = 0 is 1 + ln x) = 0 is (a) 1(a) 1(a) 1(a) 1 (b) 2(b) 2(b) 2(b) 2 (c) 4(c) 4(c) 4(c) 4 (d) none of these(d) none of these(d) none of these(d) none of these

25.25.25.25. The number of multiples of 4 among all 10 digits number isThe number of multiples of 4 among all 10 digits number isThe number of multiples of 4 among all 10 digits number isThe number of multiples of 4 among all 10 digits number is

(a)(a)(a)(a) 825 10× (b)(b)(b)(b) 725 10× (c)(c)(c)(c) 7225 10× (d)(d)(d)(d) 7234 10× 26.26.26.26. The larger diagonal of a parallelogram of area 8 s.u. must have lengthThe larger diagonal of a parallelogram of area 8 s.u. must have lengthThe larger diagonal of a parallelogram of area 8 s.u. must have lengthThe larger diagonal of a parallelogram of area 8 s.u. must have length

(a) at least 4(a) at least 4(a) at least 4(a) at least 4 (b) equal to 8(b) equal to 8(b) equal to 8(b) equal to 8 (c) at most 4(c) at most 4(c) at most 4(c) at most 4 (d) equal to (d) equal to (d) equal to (d) equal to 8

27.27.27.27. Let Let Let Let { }na be a sequence of positive re be a sequence of positive re be a sequence of positive re be a sequence of positive real numbers such thatal numbers such thatal numbers such thatal numbers such that ( )1 2nn nLt a a

n++

→∞ is is is is

finite.finite.finite.finite. Then Then Then Then

3

(a)(a)(a)(a) { }na converges to 1converges to 1converges to 1converges to 1 (b)(b)(b)(b) { }na converges to 0converges to 0converges to 0converges to 0

(c) (c) (c) (c) { }na converges to ½ converges to ½ converges to ½ converges to ½ (d)(d)(d)(d) { }na convergconvergconvergconverges to es to es to es to 1

2

28.28.28.28. Consider the following two statements about a positive integer n and choose the Consider the following two statements about a positive integer n and choose the Consider the following two statements about a positive integer n and choose the Consider the following two statements about a positive integer n and choose the correct option below correct option below correct option below correct option below (I): n is a perfect square(I): n is a perfect square(I): n is a perfect square(I): n is a perfect square (II): The number of positive integer divisors of n is odd.(II): The number of positive integer divisors of n is odd.(II): The number of positive integer divisors of n is odd.(II): The number of positive integer divisors of n is odd. (a) (I) & (II) a(a) (I) & (II) a(a) (I) & (II) a(a) (I) & (II) are Equivalent;re Equivalent;re Equivalent;re Equivalent; (b) (I) implies (II) but not conversely(b) (I) implies (II) but not conversely(b) (I) implies (II) but not conversely(b) (I) implies (II) but not conversely (c) (II) implies (I) but not conversely(c) (II) implies (I) but not conversely(c) (II) implies (I) but not conversely(c) (II) implies (I) but not conversely (d) neither statement implies the other.(d) neither statement implies the other.(d) neither statement implies the other.(d) neither statement implies the other.

29.29.29.29. A particle starts at the origin and travels along the X A particle starts at the origin and travels along the X A particle starts at the origin and travels along the X A particle starts at the origin and travels along the X----axis. For the first one axis. For the first one axis. For the first one axis. For the first one second, its speed second, its speed second, its speed second, its speed is 1 is 1 is 1 is 1 m/sec. There after its speed at any time t is at mostm/sec. There after its speed at any time t is at mostm/sec. There after its speed at any time t is at mostm/sec. There after its speed at any time t is at most

( ).

9/10ths of its speed at (t of its speed at (t of its speed at (t of its speed at (t –––– 1). Then 1). Then 1). Then 1). Then

(a) the particle reaches any point x > 0 at some time(a) the particle reaches any point x > 0 at some time(a) the particle reaches any point x > 0 at some time(a) the particle reaches any point x > 0 at some time (b) the particle must reach x = 10(b) the particle must reach x = 10(b) the particle must reach x = 10(b) the particle must reach x = 10 (c) the particle may or may not reach x (c) the particle may or may not reach x (c) the particle may or may not reach x (c) the particle may or may not reach x = 9 but it will never reach= 9 but it will never reach= 9 but it will never reach= 9 but it will never reach x = 10x = 10x = 10x = 10 (d) (d) (d) (d) nothing of the above nature can be predicted without knowing the exact speednothing of the above nature can be predicted without knowing the exact speednothing of the above nature can be predicted without knowing the exact speednothing of the above nature can be predicted without knowing the exact speed

30.30.30.30. Let Let Let Let ( ) ( )min ,x xf x e e−= for any real number xfor any real number xfor any real number xfor any real number x, then , then , then , then

(a) f has no maximum(a) f has no maximum(a) f has no maximum(a) f has no maximum

(b) f attains its maximum at a point where(b) f attains its maximum at a point where(b) f attains its maximum at a point where(b) f attains its maximum at a point where ( )/ 0f x =

(c) (c) (c) (c) f attains its maximum at a point where it’s not differentiablef attains its maximum at a point where it’s not differentiablef attains its maximum at a point where it’s not differentiablef attains its maximum at a point where it’s not differentiable

(d)(d)(d)(d) ( )( ): max :M f x x R= ∈ <∞ but there is no numberbut there is no numberbut there is no numberbut there is no number ox such thatsuch thatsuch thatsuch that ( )of x M=

31.31.31.31. Suppose x is an irrational number Suppose x is an irrational number Suppose x is an irrational number Suppose x is an irrational number and a and a and a and a, b, c, d are non, b, c, d are non, b, c, d are non, b, c, d are non----zero rational numbers. If zero rational numbers. If zero rational numbers. If zero rational numbers. If

ax b

cx d

+

+ is rational, then we must have is rational, then we must have is rational, then we must have is rational, then we must have

(A) a = c = 0(A) a = c = 0(A) a = c = 0(A) a = c = 0 (B) a = c; b = d(B) a = c; b = d(B) a = c; b = d(B) a = c; b = d (C) ad = bc(C) ad = bc(C) ad = bc(C) ad = bc (D) a + d = b + c(D) a + d = b + c(D) a + d = b + c(D) a + d = b + c

32.32.32.32. Let f(x) be a polynomial of degree 3 such that f(0) = 1, f(1) = 2, x = 0 is a Let f(x) be a polynomial of degree 3 such that f(0) = 1, f(1) = 2, x = 0 is a Let f(x) be a polynomial of degree 3 such that f(0) = 1, f(1) = 2, x = 0 is a Let f(x) be a polynomial of degree 3 such that f(0) = 1, f(1) = 2, x = 0 is a critical point but f(x) doesn’t have local extremum at x = 0, then prove that the critical point but f(x) doesn’t have local extremum at x = 0, then prove that the critical point but f(x) doesn’t have local extremum at x = 0, then prove that the critical point but f(x) doesn’t have local extremum at x = 0, then prove that the

value ofvalue ofvalue ofvalue of ( )2

2 2

2

( ) 714 ln 7

37

f x xdx x x x c

x

+= − + + + +

+∫

33.33.33.33. Evaluate : Evaluate : Evaluate : Evaluate : 4 2

sin cos

sin cos

x xdx

x x

+

+∫

34.34.34.34. The set of all real numbers which satisfyThe set of all real numbers which satisfyThe set of all real numbers which satisfyThe set of all real numbers which satisfy 2

2

2 32

2 2

x x

x x

− +≥

− + is is is is

(a) the set of al(a) the set of al(a) the set of al(a) the set of all integers;l integers;l integers;l integers; (b) the set of all rational numbers; (b) the set of all rational numbers; (b) the set of all rational numbers; (b) the set of all rational numbers; (c) the set of all positive real numbers(c) the set of all positive real numbers(c) the set of all positive real numbers(c) the set of all positive real numbers (d) the set of all real numbers(d) the set of all real numbers(d) the set of all real numbers(d) the set of all real numbers

35.35.35.35. If x > y are positive integers such that ( 3x + 11y ) leaves a remainder 2 when If x > y are positive integers such that ( 3x + 11y ) leaves a remainder 2 when If x > y are positive integers such that ( 3x + 11y ) leaves a remainder 2 when If x > y are positive integers such that ( 3x + 11y ) leaves a remainder 2 when divided by 7 and ( 9x + 5ydivided by 7 and ( 9x + 5ydivided by 7 and ( 9x + 5ydivided by 7 and ( 9x + 5y ) ) ) ) leaves a remainder 3 wleaves a remainder 3 wleaves a remainder 3 wleaves a remainder 3 when divided by 7, then the hen divided by 7, then the hen divided by 7, then the hen divided by 7, then the remainder ( x remainder ( x remainder ( x remainder ( x –––– y ) divided by 7, equals y ) divided by 7, equals y ) divided by 7, equals y ) divided by 7, equals (a) 3(a) 3(a) 3(a) 3 (b) 4(b) 4(b) 4(b) 4 (c) 5(c) 5(c) 5(c) 5 (d) 6(d) 6(d) 6(d) 6

36.36.36.36. The average of scores of 12 students in a test is 74. The highest score is 79 The average of scores of 12 students in a test is 74. The highest score is 79 The average of scores of 12 students in a test is 74. The highest score is 79 The average of scores of 12 students in a test is 74. The highest score is 79. Then . Then . Then . Then the minimum possible lowest score must the minimum possible lowest score must the minimum possible lowest score must the minimum possible lowest score must bebebebe (a) 25(a) 25(a) 25(a) 25 (b) 12(b) 12(b) 12(b) 12 (c) 19(c) 19(c) 19(c) 19 (d) 28(d) 28(d) 28(d) 28

4

37.37.37.37. If a, b, c are real numbers so that If a, b, c are real numbers so that If a, b, c are real numbers so that If a, b, c are real numbers so that ( ) ( )3 2 2 1x ax bx c x g x+ + + = + for some for some for some for some

polypolypolypolynomial g(x), then nomial g(x), then nomial g(x), then nomial g(x), then (A) b = 1, a = c(A) b = 1, a = c(A) b = 1, a = c(A) b = 1, a = c (B) b = 0 = c(B) b = 0 = c(B) b = 0 = c(B) b = 0 = c (C) a = 0(C) a = 0(C) a = 0(C) a = 0 (D) none of the above(D) none of the above(D) none of the above(D) none of the above

38.38.38.38. Let ABC be a triangle such that the three medians divided it into six pa Let ABC be a triangle such that the three medians divided it into six pa Let ABC be a triangle such that the three medians divided it into six pa Let ABC be a triangle such that the three medians divided it into six parts of rts of rts of rts of equal area. Then, the triangle equal area. Then, the triangle equal area. Then, the triangle equal area. Then, the triangle (a) can’(a) can’(a) can’(a) can’t existt existt existt exist (b) can be any triangle(b) can be any triangle(b) can be any triangle(b) can be any triangle (c) must be equilateral(c) must be equilateral(c) must be equilateral(c) must be equilateral (d) need not be equilateral but must isosceles(d) need not be equilateral but must isosceles(d) need not be equilateral but must isosceles(d) need not be equilateral but must isosceles

39.39.39.39. From a bag containing 10 distinct objects, the number of ways one can select an From a bag containing 10 distinct objects, the number of ways one can select an From a bag containing 10 distinct objects, the number of ways one can select an From a bag containing 10 distinct objects, the number of ways one can select an odd number of obodd number of obodd number of obodd number of objects isjects isjects isjects is

(a)(a)(a)(a) 102 (b)(b)(b)(b) 92 (c) 10!(c) 10!(c) 10!(c) 10! (d) 5(d) 5(d) 5(d) 5 40.40.40.40. Consider the two statements: Consider the two statements: Consider the two statements: Consider the two statements:

(I) between any two rational numbers, there is an irrational number(I) between any two rational numbers, there is an irrational number(I) between any two rational numbers, there is an irrational number(I) between any two rational numbers, there is an irrational number (II)(II)(II)(II) between any two irrational numbers, there is a rbetween any two irrational numbers, there is a rbetween any two irrational numbers, there is a rbetween any two irrational numbers, there is a rational number; Thenational number; Thenational number; Thenational number; Then (a) both (I) & (II) are true(a) both (I) & (II) are true(a) both (I) & (II) are true(a) both (I) & (II) are true (b) (I) is true but (II) is not(b) (I) is true but (II) is not(b) (I) is true but (II) is not(b) (I) is true but (II) is not (c) (I) is false but (II) is not(c) (I) is false but (II) is not(c) (I) is false but (II) is not(c) (I) is false but (II) is not (d)(d)(d)(d) both (I) & (II) are falseboth (I) & (II) are falseboth (I) & (II) are falseboth (I) & (II) are false

41.41.41.41. Let f(x) =a Let f(x) =a Let f(x) =a Let f(x) =a 2x + bx + c, where a, b, c + bx + c, where a, b, c + bx + c, where a, b, c + bx + c, where a, b, c R∈ . Suppose. Suppose. Suppose. Suppose f(x) f(x) f(x) f(x)≠ x for any real number x for any real number x for any real number x for any real number x. Then the number of solutions of f(f(x)) = x in real numbers x isx. Then the number of solutions of f(f(x)) = x in real numbers x isx. Then the number of solutions of f(f(x)) = x in real numbers x isx. Then the number of solutions of f(f(x)) = x in real numbers x is (a) 4(a) 4(a) 4(a) 4 (b) 2(b) 2(b) 2(b) 2 (c) 0(c) 0(c) 0(c) 0 (d) can’t be determined(d) can’t be determined(d) can’t be determined(d) can’t be determined

42.42.42.42. Let Let Let Let1

nn

S ne∞ −

==∑ . Then. Then. Then. Then

(a)(a)(a)(a) 1S≤ (b)(b)(b)(b)1 S< <∞ (c)(c)(c)(c) S is infiniteis infiniteis infiniteis infinite (d)(d)(d)(d) 0S =

43.43.43.43. Let f(x) =aLet f(x) =aLet f(x) =aLet f(x) =a 3x + b+ b+ b+ b 2x + cx + d be a polynomial of degree 3 where a, b, c, d+ cx + d be a polynomial of degree 3 where a, b, c, d+ cx + d be a polynomial of degree 3 where a, b, c, d+ cx + d be a polynomial of degree 3 where a, b, c, d R∈ . . . . ThenThenThenThen (a)f(x)(a)f(x)(a)f(x)(a)f(x)→∞ asasasas x x x x→∞ (b)(b)(b)(b) f is 1 f is 1 f is 1 f is 1----1 as well as ONTO1 as well as ONTO1 as well as ONTO1 as well as ONTO (c)(c)(c)(c) the graph of f(x) meets the x the graph of f(x) meets the x the graph of f(x) meets the x the graph of f(x) meets the x----axis in one or three points;axis in one or three points;axis in one or three points;axis in one or three points; (d)(d)(d)(d) f must be ONTO but need not be 1 f must be ONTO but need not be 1 f must be ONTO but need not be 1 f must be ONTO but need not be 1----1.1.1.1.

44.44.44.44. LetLetLetLet 1 2, ,...., na a a be arbitrary integers and suppose be arbitrary integers and suppose be arbitrary integers and suppose be arbitrary integers and suppose 1 2, ,...., nb b b is a is a is a is a

permutation of thepermutation of thepermutation of thepermutation of the ia ‘s. Then‘s. Then‘s. Then‘s. Then 1 1 2 2 ............. n na b a b a b− + − + + − (a)(a)(a)(a) is is is is ≤ nnnn (b)(b)(b)(b) can be an arbitrary positive integer can be an arbitrary positive integer can be an arbitrary positive integer can be an arbitrary positive integer (c) can be any even nonnegative integer(c) can be any even nonnegative integer(c) can be any even nonnegative integer(c) can be any even nonnegative integer (d) must be zero(d) must be zero(d) must be zero(d) must be zero