Maths Paper - i _question Paper

MATHEMATICS PAPER – I (ADVANCED)

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. x

41.

Let f  x  be a continuous function which takes positive values for x  0 and with

1 f 1  . Then the value of f 2

(a) 1 42.



0 f  t  dt  x



2  1 is

(b) 2

(c) 4

(d)

1 4

2

The number of points

the circle x 2   y  3  8 such that the equation

x2  b x  c  0 (a) 1

(c) 3

 b, c  lying on has real roots is  b, c  R  (b) 2

(d) 4

1 tan x then lim  f  x    x 2  f  x   (where [.] denotes the greatest integer function and x 0 x {.} denotes fractional part). (a) 3 (b) log 3 (c) e3 (d) Does not exist





43.

If f  x  

44.

The number of possible triplets  x, y, z  of positive integers, satisfying 2 x  2 y  2 z  2336 is (a) 72 (b) 6 (c) 3 (d) 18 1

45.

f  x

Let f  x  be continuous function on 0,1 and if

 f  x  dx  1, 

1

0

0

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

1

xf  x  dx  2 and  x 2 f  x  dx  3 . 0

(d) zero

SPACE FOR ROUGH WORK

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MATHEMATICS PAPER – I (ADVANCED) 46.

If f  x  be positive, continuous and differentiable on the interval  a, b  . If

lim f  x   1 and

xa

1 3 1 lim f  x   3 4 also f '  x    f  x    then f  x xb

(a) b  a 

 24

(b) b  a 

 24

(c) b  a 

 12

(d) b  a 

 24

x

47.

 a Consider   t 2  8t  13 dt  x sin   and  a, x  R  0 x takes the values for which the equation   x 0

has a real solution, then the number of values of a   0,100 is ___ (a) 1 (b) 2 (c) 3 (d) 4 48.

If f  x   sgn  sin 2 x  sin x  1 has exactly four points of discontinuity for x   0, 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. 1 5 7 1 (a) (b) (c) (d) 12 12 12 4

50.

If ‘t’ is real and  

t 2  3t  4 then the equations 3 x  y  4 z  3 , t 2  3t  4 x  2 y  3z  2, 6 x  5 y   z  3 has _________ real solutions. (a) one for any possible  (b) two for any possible  (c) infinitely many for some  (d) no solution for some possible  SPACE FOR ROUGH WORK

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MATHEMATICS PAPER – I (ADVANCED) 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) 56  5C1.46  5C2 .36  5C3 .26  5C4 .16  6C2 . 5 (b) 65  6C1.55  6C2 .45  6C3 .35  6C4 .25  6C1.15  0 (c) 66  6C1.56  6C2 .46  6C3 .36  6C4 .26  6C5 .16  720 (d) 65  6C1.55  6C2 .45  6C3.35  6C4 .25  6C5 .15  5C2 . 6

53.

Let x, y, z be positive reals. Then 4 9 16 (a)    81 if x  y  z  1 x y z

x y z 3    yz zx x y 2 1 1 1 (c) If x y z  1, then 1  x 1  y 1  z   0 (d) If x  y  z  1, then    9 x y z 54.

(b)

Let An be a n  n matrix in which diagonal elements are 1, 2,3,....., n

 i.e., a11  1, a22  2, a33  3,....., a ii  i,......a nn  n  and all other elements are equal to ' n ' then

55.

(a) A n is singular for all ' n '

(b) A n is nonsingular for all ' n '

(c) det .A5  120

(d) det . An  0

Let f : R  R, such that f "  x   2 f '  x   f  x   2 e x and f '  x   0,  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) f  3   5

(d) f  3  7 SPACE FOR ROUGH WORK

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MATHEMATICS PAPER – I (ADVANCED) 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.

57.

58.

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  2 i /2   2  i/ 6   2  i5 / 6  Consider a triangle having vertices at the points A  e  , B e e  , C  . Let P  3   3   3  be any point on its incircle, then the value of AP2 + BP2 + CP2 is

sgn  x  2    log e x  , 1  x  3 If f  x    2 3  x  3.5 x  , where [.] denotes the greatest integer function and {.} represents the fractional part function, then the number of integral points of discontinuity is

x  2 y 1 z   ; 2x + 3y – 5z – 6 = 0 = 2 3 4 3x – 2y – z + 3 is K, then  3 K  is equal to (where [.] denotes greatest integer function)

59.

If the length of the shortest distance between the lines

60.

ABCD and PQRS are two variable rectangles, such that A, B, C and D lie on PQ, QR, RS and SP x respectively and perimeter ‘x’ of ABCD is constant. If the maximum area of PQRS is 32, then  4 SPACE FOR ROUGH WORK

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