Maths Paper - i _question Paper

MATHEMATICS PAPER – I (ADVANCED)

CENTERS: MUMBAI / DELHI / AKOLA / KOLKATA / LUCKNOW / NASHIK / GOA # 14

PART III : MATHEMATICS

SECTION I: (SINGLE CHOICE QUESTIONS)

This section contains 10 multiple choice questions. Each question has four choices (A), (B), (C) or (D) out of which ONLY ONE is correct.

41. Let f x be a continuous function which takes positive values for 0x and 0

xf t dt x f x

with 11 .2

f Then the value of 2 1f is

(a) 1 (b) 2 (c) 4 (d) 14

42. The number of points ,b c lying on the circle 22 3 8x y such that the equation

2 0x b x c has real roots is b,c R (a) 1 (b) 2 (c) 3 (d) 4

43. If tan xf xx

then 1

20

lim f xx

f x x

(where [.] denotes the greatest integer function and

{.} denotes fractional part). (a) 3 (b) log 3 (c) 3e (d) Does not exist 44. The number of possible triplets , ,x y z of positive integers, satisfying 2 2 2 2336yx z is (a) 72 (b) 6 (c) 3 (d) 18

45. Let f x be continuous function on 0,1 and if 1 1

1 2

00 0

1, 2 3f x dx xf x dx and x f x dx .

Then the number of roots of 0f x in 0,1 is _____ (a) exactly one (b) atleast one (c) atmost one (d) zero

SPACE FOR ROUGH WORK



46. If f x be positive, continuous and differentiable on the interval ,a b . If lim 1f xx a

and

1

4lim 3f xx b

also 3 1'f x f x

f x then

(a) 24

b a (b)

24b a (c)

12b a (d)

24b a

47. Consider 2

0

8 13 sinx

at t dt xx

and , 0a x R x takes the values for which the equation

has a real solution, then the number of values of a 0,100 is ___ (a) 1 (b) 2 (c) 3 (d) 4 48. If 2sgn sin sin 1f x x x has exactly four points of discontinuity for 0,x n n N then n

can be (a) only 4 (b) 4 or 5 (c) only 5 (d) 5 or 6 49. All the digits 1 to 9 are permutated for any permutation, the nine digits occupy positions 1 to 9 in

some order, what is the probability of choosing a nine digit number such that the product of the digits of any six consecutive positions is divisible by 35.

(a) 112

(b) 512

(c) 712

(d) 14

50. If ‘t’ is real and 2

2

3 43 4

t tt t

then the equations 3 4 3x y z ,

2 3 2, 6 5 3 x y z x y z has _________ real solutions. (a) one for any possible (b) two for any possible

(c) infinitely many for some (d) no solution for some possible




SECTION II: (MULTIPLE CHOICE QUESTIONS) This section contains 5 multiple choice questions. Each question has four choices (A), (B), (C) or (D) out of which ONE or MORE is/are correct. 51. Which of the following statement (s) is/are true? (a) maximum value of P such that 3P divides 100! is 48 (b) maximum value of P such that 3P divides 50! is 22 (c) maximum value of P such that 3P divides 99 97 95 ......... 51 is 14 (d) maximum value of P such that 3P divides 25! is 10 52. Which of the following is/are true?

(a) 6 66 6 65 5 5 5 61 2 3 4 25 4 3 2 1 5. . . . . C C C C C

(b) 5 5 55 5 56 6 6 6 61 2 3 4 16 5 4 3 2 1 0. . . . . C C C C C

(c) 6 6 66 6 66 6 6 6 651 2 3 46 5 4 3 2 1 720. . . . . C C C C C

(d) 5 5 55 5 56 6 6 6 6 551 2 3 4 26 5 4 3 2 1 6. . . . . . C C C C C C

53. Let , ,x y z be positive reals. Then

(a) 4 9 16 81x y z if 1x y z (b)

32

x y zy z z x x y

(c) If 1,x y z then 1 1 1 0 x y z (d) If 1,x y z then 1 1 1 9x y z

54. Let nA be a n n matrix in which diagonal elements are 1,2,3,.....,n

11 22 33. ., 1, 2, 3,....., ,...... nniii e a a a a i a n and all other elements are equal to ' 'n then

(a) nA is singular for all ' 'n (b) nA is nonsingular for all ' 'n (c) 5det 120.A (d) det 0. nA 55. Let ,: f R R such that 2 2" ' xf x f x f x e and 0, ,' f x x R then which of the

following can be correct (a) , f x f x x R (b) , f x f x x R

(c) 3 5 f (d) 3 7 f




SECTION III: INTEGER VALUE CORRECT TYPE This section contains 5 questions. The answer to each question is a single digit integer, ranging from 0 to 9 (both inclusive) 56. Suppose a cubic polynomial f(x) = x3 + px2 + qx + 72 is divisible by both x2 + ax + b and x2 + bx + a

(where a, b, p, q are constants and a b ), then the value of p is

57. Consider a triangle having vertices at the points i /22A e3

, i /62B e3

, i5 /62C e3

. Let P

be any point on its incircle, then the value of AP2 + BP2 + CP2 is

58. If

e

2

sgn x 2 log x , 1 x 3f x

x , 3 x 3.5

where [.] denotes the greatest integer function and {.} represents the fractional part function, then the number of integral points of discontinuity is

59. If the length of the shortest distance between the lines 2 12 3 4

x y z ; 2x + 3y – 5z – 6 = 0 =

3x – 2y – z + 3 is K, then 3 K is equal to (where [.] denotes greatest integer function)

60. ABCD and PQRS are two variable rectangles, such that A, B, C and D lie on PQ, QR, RS and SP

respectively and perimeter ‘x’ of ABCD is constant. If the maximum area of PQRS is 32, then 4

x


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Maths Paper - i _question Paper