AMU –PAST PAPERSMATHEMATICS - UNSOLVED PAPER - 2004
SECTION – I   CRITICAL REASONING SKILLS
01   Problem     The units place digit in the number 1325 + 1125 – 325 is :     a. 0     b. 1     c. 2     d. 3
02   Problem     The angle of intersection of the curves y = x2, 6y = 7 – x3 at (1, 1) is :     a.      /4     b.      /...
03   Problem     the value of x for which the equation 1 + r + r2 + …+ rx (1 + r2) (1 + r2) (1 + r4) (1 +     r8) holds is...
04   Problem                 2     If f (x)  x  1     for energy real number x, then minimum value of f(x) :            ...
05   Problem     The value of a for which the sum of the squares of the roots of the equation     x2 – (a - 2) x – a – 1 =...
06    Problem     Suppose A1, A2….,A30 are thirty sets each having 5 elements and B1, B2, ….., Bn are                     ...
07   Problem     The number of onto mappings from the set A = {1, 2……., 100}to set B = {1, 2} is :     a. 2100 – 2     b. ...
08   Problem     Which of the following functions is inverse of itself ?                 1 x     a. f(x) =   1 x     b. ...
09   Problem     If f(x) = log (x + x 2  1 ), then f(x) is :     a. Even function     b. Odd function     c. Periodic fun...
10   Problem     The solution of log99(log2 (log3x)) = 0 is :     a. 4     b. 9     c. 44     d. 99
11   Problem                                             1      1                 1     If n = 1000!, then the value of su...
12   Problem     If  and  2 are the two imaginary cube roots of unity, then the equation     whose roots are a 317 and ...
13   Problem                       14                                 (2k  1)           (2k  1)  is :     The value of...
14   Problem     1, a1, a2, ….., an –1 are roots of unity then the value of (1 – a1) (1 – a2) …..(1 – an-1)     is :     a...
15   Problem     If  ,  are the roots of ax2 + bx + c = 0;   h,   hare the roots of px2 + qx     + r = 0 ; and D1, D...
16   Problem     If a, b, c are three unequal numbers such that a, b, c are in A.P. and b – a, c – b, a     are in G.P., t...
17   Problem     The number of divisors of 3 x 73, 7 x 112 and 2 x 61 are in :     a. A.P.     b. G.P.     c. H.P.     d. ...
18   Problem     Suppose a, b, c are in A.P. and | a |, | b |, | c | < 1, If     x = 1+ a + a2 + ….. to      y = 1 + b + ...
19   Problem          4   7  10             is :     1      2  3  .....to           5 5    5           16     a.    3...
20   Problem     If the sum of first n natural numbers is   1    times the sum of their cubes, then                       ...
21   Problem     If p = cos 550, q = cos 650 and r = cos 1750, then the value of 1  1  r   is :                         ...
22   Problem     The value of sin 200(4 + sec200) (4 + sec 200) is :     a. 0     b. 1          2     c.     d.     3
23   Problem     If 4 sin x  cos x   , then x is equal to :             1      1     a. 0          1     b.          ...
24   Problem                                                 1    1   1     If the line x  y  1 moves such that       a2...
25   Problem     The straight line whose sum of the intercepts on the axes is equal to half of the     product of the inte...
26   Problem     If the circle x2 + y2 + 4x + 22y + c = 0 bisects the circumference of the circle     x2 + y2 –2x + 8y –d ...
27   Problem     the radius of the circle whose tangents are x + 3y – 5 = 0, 2x + 6y + 30 = 0 is :     a.    5     b.   10...
28   Problem                                    The latus rectum of the parabola y2 = 4ax whose focal chord is PSQ such t...
29   Problem     If M1 and M2 are the feet of the perpendiculars from the foci S1 and S2 of the                           ...
30   Problem     If the chords of contact of tangents from two points (x1,y1) and (x2,y2) to the                          ...
31   Problem     In a chess tournament where the participants were to play one game with one     another, two players fell...
32   Problem     the coefficient of x2 term in the Binomial expansion of  1 x1 / 2  x 1 / 4  is :                     ...
33   Problem     The solution set of the equation                                     log2 x                              ...
34   Problem                  x2   x3   x4     If y = x -               .....     and if | x | < 1, then :             ...
35   Problem                                                              x   y 1   z 2     The length of perpendicular ...
36   Problem     The plane 2x + 3y + 4z = 1 meets the co-ordinate axes in A, B, C. the centroide of     the triangle ABC i...
37   Problem     The vector equation of the sphere whose centre is the point (1, 0, 1) and radius is     4, is :     a.   ...
38   Problem     the plane 2x – (1 +  ) y + 3  z = 0 passes through the intersection of the planes :     a. 2x – y = 0 a...
39   Problem                                                                         a  b  c  0 and a    37, b ...
40   Problem                                                                                        i   j   ˆ        i...
41   Problem                                                              The total work done by two forces          i  ...
42   Problem                                                                                    Let   a, b and c    ...
43   Problem          xn   x n 2   x n 3     If                                                     1   1   1   then n...
44   Problem     If a1, a2, ……, an ….. are in G.P. and ai > 0 for each i, then the determinant             log an      log...
45   Problem     The value of a for which the system of equations x + y + z = 0, x + ay + az = 0, x –     ay + z = 0, pros...
46   Problem     If a square matrix A is such that AAT = I = ATA, then |A| is equal to :     a.   0     b.  1     c.    ...
47   Problem     The Boolean function of the input/output table as given below is :     a. f(x1, x2, x3) = x1.x2.x3+x1.x2....
48   Problem     A and B are two events. Odds against A are A  B 2 to 1. Odds in favour of are     3 to 1. If    x  P(B)...
49   Problem     In a series of three trials the probability of exactly two successes in nine time as     large as the pro...
50   Problem     An integer is chosen at random from first two hundred digits. Then the     probability that the integer c...
51   Problem                                                              x 2     Let A = R – {3}, B = R – {1}, Let be de...
52   Problem                      cos2 x  sin2 x  1     Let    f (x)                          (x  0)   = a (x = 0) the...
53   Problem     If f(2) = 4 and f’(2) = 1, then lim xf (2)  2f ( x ) is equal to :                                     x...
54   Problem     Let f(x) = sin x, g (x) = x2 and h (x) = log ex. If F(x) = (h o g o f) (x), then F’’(x) is     equal to :...
55   Problem     The length of subnormal to the parabola y2 = 4ax at any point is equal to :     a.   2         a     b. 2...
56   Problem     The function f(x) = a sin x + 1 sin 3x has maximum value of at x = . The value of                        ...
57   Problem     If  cos 4x  1 dx    A cos 4x  B,   then :         cot x  tan x                1     a. A = -        ...
58   Problem       |x|         b          a  x           dx , a < 0 < b, is equal to :     a. | b | - | a |     b. | b | ...
59   Problem                                           For any integer n, the integral 0                                ...
60   Problem                    sin x  sin2 x  sin3x   sin2 x   sin3x                           /2     If              ...
61   Problem     If   f : R  R, g : R  R          are continuous functions, then the value of the integral       /2    ...
62   Problem                     3           x         x2  1      The integral   1                                 ...
63   Problem                                2a          f (x)     The value of the integral 0    f (x)  f (2a  x)      ...
64   Problem     The area bounded by the curves y = x2, y = xe-x and the line x = 1, is :            2     a.            e...
65   Problem     The solution of x dy – y dx + x2 ex dx = 0 is :          y             ex  c     a.   x          x     ...
66   Problem     The degree and order of the differential equation of all parabolas whose axis is                        ...
67   Problem     Three forces P, Q, R act along the sides BC, CA, AB of a triangle ABC taken in     order. The condition t...
68   Problem     The resultant of two forces P and Q is R. If Q is doubled, R is doubled and if Q is     reversed, R is ag...
69   Problem     A particle is dropped under gravity from rest from a height h (g = 9.8 m/sec2) and     it travels a dista...
70   Problem     A man can throw a stone 90 metres. The maximum height to which it will rise in     metres, is :     a. 30...
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  • AMU - Mathematics - 2004

    1. 1. AMU –PAST PAPERSMATHEMATICS - UNSOLVED PAPER - 2004
    2. 2. SECTION – I CRITICAL REASONING SKILLS
    3. 3. 01 Problem The units place digit in the number 1325 + 1125 – 325 is : a. 0 b. 1 c. 2 d. 3
    4. 4. 02 Problem The angle of intersection of the curves y = x2, 6y = 7 – x3 at (1, 1) is : a.  /4 b.  /3 c.  /2 d. none of these
    5. 5. 03 Problem the value of x for which the equation 1 + r + r2 + …+ rx (1 + r2) (1 + r2) (1 + r4) (1 + r8) holds is : a. 12 b. 13 c. 14 d. 15
    6. 6. 04 Problem 2 If f (x)  x  1 for energy real number x, then minimum value of f(x) : 2 x 1 a. Does not exist b. Is equal to 1 c. Is equal to 0 d. Is equal to - 1
    7. 7. 05 Problem The value of a for which the sum of the squares of the roots of the equation x2 – (a - 2) x – a – 1 = 0 assumes the least value is : a. 0 b. 1 c. 2 d. 3
    8. 8. 06 Problem Suppose A1, A2….,A30 are thirty sets each having 5 elements and B1, B2, ….., Bn are 30 n n sets each with 3 elements, let i Ai  j 1 B j  Sand each element of S 1  belongs to exactly 10 of the Ai’s and exactly 9 of the Bj’S. Then n is equal to : a. 115 b. 83 c. 45 d. none of these
    9. 9. 07 Problem The number of onto mappings from the set A = {1, 2……., 100}to set B = {1, 2} is : a. 2100 – 2 b. 2100 c. 299 – 2 d. 299
    10. 10. 08 Problem Which of the following functions is inverse of itself ? 1 x a. f(x) = 1 x b. f(x) = 3logx c. f(x) = 3x(x +1) d. none of these
    11. 11. 09 Problem If f(x) = log (x + x 2  1 ), then f(x) is : a. Even function b. Odd function c. Periodic function d. None of these
    12. 12. 10 Problem The solution of log99(log2 (log3x)) = 0 is : a. 4 b. 9 c. 44 d. 99
    13. 13. 11 Problem 1 1 1 If n = 1000!, then the value of sum   ....  is : log2 n log3 n log1000 n a. 0 b. 1 c. 10 d. 103
    14. 14. 12 Problem If  and  2 are the two imaginary cube roots of unity, then the equation whose roots are a 317 and a 382 is : a. x2 + ax + a2 = 0 b. x2 + a2x + a = 0 c. x2 – ax + a2 = 0 d. x2 – a2x + a = 0
    15. 15. 13 Problem 14  (2k  1) (2k  1)  is : The value of1   cos  k 0 15   i sin 15   a. 0 b. - 1 c. 1 d. i
    16. 16. 14 Problem 1, a1, a2, ….., an –1 are roots of unity then the value of (1 – a1) (1 – a2) …..(1 – an-1) is : a. 0 b. 1 c. n d. n2
    17. 17. 15 Problem If  ,  are the roots of ax2 + bx + c = 0;   h,   hare the roots of px2 + qx + r = 0 ; and D1, D2 the respective discriminants of these equations, then D1 : D2 is equal to : a2 a. p2 b2 b. q2 2 c. c r2 d. none of these
    18. 18. 16 Problem If a, b, c are three unequal numbers such that a, b, c are in A.P. and b – a, c – b, a are in G.P., then a : b : c is : a. 1 : 2 : 3 b. 1 : 3 : 4 c. 2 : 3 : 4 d. 1 : 2 : 4
    19. 19. 17 Problem The number of divisors of 3 x 73, 7 x 112 and 2 x 61 are in : a. A.P. b. G.P. c. H.P. d. None of these
    20. 20. 18 Problem Suppose a, b, c are in A.P. and | a |, | b |, | c | < 1, If x = 1+ a + a2 + ….. to  y = 1 + b + b2 + …… to  z = 1 + c + c2 +……..to  then x, y, z are in : a. A.P. b. G.P. c. H.P. d. None of these
    21. 21. 19 Problem 4 7 10 is : 1  2  3  .....to  5 5 5 16 a. 35 11 b. 8 35 c. 16 d. 7 16
    22. 22. 20 Problem If the sum of first n natural numbers is 1 times the sum of their cubes, then 78 the value of n is : a. 11 b. 12 c. 13 d. 14
    23. 23. 21 Problem If p = cos 550, q = cos 650 and r = cos 1750, then the value of 1  1  r is : p q pq a. 0 b. -1 c. 1 d. none of these
    24. 24. 22 Problem The value of sin 200(4 + sec200) (4 + sec 200) is : a. 0 b. 1 2 c. d. 3
    25. 25. 23 Problem If 4 sin x  cos x   , then x is equal to : 1 1 a. 0 1 b. 2 1 c. - 2 d. 1
    26. 26. 24 Problem 1 1 1 If the line x  y  1 moves such that a2  2  2 b c where c is a constant, then a b the locus of the foot of the perpendicular from the origin to the line is : a. Straight line b. Circle c. Parabola d. Ellipse
    27. 27. 25 Problem The straight line whose sum of the intercepts on the axes is equal to half of the product of the intercepts, passes through the point : a. (1, 1) b. (2, 2) c. (3, 3) d. (4, 4)
    28. 28. 26 Problem If the circle x2 + y2 + 4x + 22y + c = 0 bisects the circumference of the circle x2 + y2 –2x + 8y –d = 0, then c + d is equal to : a. 60 b. 50 c. 40 d. 30
    29. 29. 27 Problem the radius of the circle whose tangents are x + 3y – 5 = 0, 2x + 6y + 30 = 0 is : a. 5 b. 10 c. 15 d. 20
    30. 30. 28 Problem  The latus rectum of the parabola y2 = 4ax whose focal chord is PSQ such that SP = 3 and SQ = 2 is given by : a. 24 5 b. 12 5 6 c. 5 d. 1 5
    31. 31. 29 Problem If M1 and M2 are the feet of the perpendiculars from the foci S1 and S2 of the x2 y2 ellipse on the tangent at any point P  1 on the ellipse, then (S1M1) 9 16 (S2M2) is equal to : a. 16 b. 9 c. 4 d. 3
    32. 32. 30 Problem If the chords of contact of tangents from two points (x1,y1) and (x2,y2) to the x1 x2 hyperbola 4x2 – 9y2 – 36 = 0 are at right angles, then is equal to : y1y2 a. 9 4 9 b.  4 81 c. 16 81 d. - 16
    33. 33. 31 Problem In a chess tournament where the participants were to play one game with one another, two players fell ill having played 6 games each, without playing among themselves. If the total number of games is 117, then the number of participants at the beginning was : a. 15 b. 16 c. 17 d. 18
    34. 34. 32 Problem the coefficient of x2 term in the Binomial expansion of  1 x1 / 2  x 1 / 4  is :   3  70 a. 243 60 b. 423 50 c. 13 d. none of these
    35. 35. 33 Problem The solution set of the equation log2 x logx 2   1 1 1    1 1 1  4 1     ....   54  1     ....    3 9 27    3 9 27  is :  1 a. 4,   4 b.  1 2,   2 c. {1, 2}  1 8,  d.  8
    36. 36. 34 Problem x2 x3 x4 If y = x -    ..... and if | x | < 1, then : 2 3 4 a. y2 y3 x 1 y    .... 2 3 y2 y3 b. x 1 y    .... 2 3 c. y2 y3 y4 x y     .... 2! 3! 4! y2 y3 y4 d. x  y     .... 2! 3! 4!
    37. 37. 35 Problem x y 1 z 2 The length of perpendicular from (1, 6, 3) to the line   is : 1 2 3 a. 3 b. 11 c. 13 d. 5
    38. 38. 36 Problem The plane 2x + 3y + 4z = 1 meets the co-ordinate axes in A, B, C. the centroide of the triangle ABC is : a. (2, 3, 4) 1 1 1 b. 2 ,3, 4   1 1 1  c.  6 , 9 12     3 3 3 d.  2 3 4   
    39. 39. 37 Problem The vector equation of the sphere whose centre is the point (1, 0, 1) and radius is 4, is : a.  ˆ | r  (ˆ  k ) |  4 i  b. i ˆ | r  (ˆ  k ) |  42  ˆ c. r  (ˆ  k )  4 i  ˆ d. r  (ˆ  k )  42 i
    40. 40. 38 Problem the plane 2x – (1 +  ) y + 3  z = 0 passes through the intersection of the planes : a. 2x – y = 0 and y + 3z = 0 b. 2x – y = 0 and y – 3z = 0 c. 2x + 3z = 0 and y = 0 d. none of these
    41. 41. 39 Problem        a  b  c  0 and a  37, b  3, c  4,  then the angle between b and c is : 300 450 600 900
    42. 42. 40 Problem     i j ˆ i ˆ  a  ˆ  ˆ  k , b  ˆ  k , c  2ˆ  ˆ i j the value of  such that a  c is perpendicular to is a. 1 b. - 1 c. 0 d. none of these
    43. 43. 41 Problem   The total work done by two forces i j i j ˆ F1  2ˆ  ˆ and F2  3ˆ  2 ˆ  k acting on a particle when it is displaced from the point j ˆ i j ˆ 3ˆ  2 ˆ  k to 5ˆ  5ˆ  3k i is : a. 8 units b. 9 units c. 10 units d. 11 units
    44. 44. 42 Problem       Let a, b and c be three non-coplanar vectors, and let p, q and r be        bxc  cxa  axb vectors defined by the relations P     , q     and r     Then the [abc ] [abc ] [abc ]          value of the expression (a  b)  p  (b  c )  q  (c  a)  r is equal to : a. 0 b. 1 c. 2 d. 3
    45. 45. 43 Problem xn x n 2 x n 3 If 1 1 1 then n is equal to : yn y n 2 y n 3  (y  z )( z  x )( x  y )     x y z zn z n 2 z n 3 a. 2 b. - 2 c. - 1 d. 1
    46. 46. 44 Problem If a1, a2, ……, an ….. are in G.P. and ai > 0 for each i, then the determinant log an log an 2 log an  4 is equal to :   log an  6 log an  8 log an 10 log an 12 log an 14 log an 16 a. 0 b. 1 c. 2 d. n
    47. 47. 45 Problem The value of a for which the system of equations x + y + z = 0, x + ay + az = 0, x – ay + z = 0, prossesses nonzero solutions, are given by : a. 1, 2 b. 1, -1 c. 1, 0 d. none of these
    48. 48. 46 Problem If a square matrix A is such that AAT = I = ATA, then |A| is equal to : a. 0 b.  1 c.  2 d. none of these
    49. 49. 47 Problem The Boolean function of the input/output table as given below is : a. f(x1, x2, x3) = x1.x2.x3+x1.x2.x3’+x1.x2’x3 + x1’.x2’.x3’ b. f(x1, x2, x3) = x1’.x2’.x3’+x1’.x2’.x3+x1’.x2.x3’ + x1.x2.x3 c. f(x1, x2, x3) = x1.x2’.x3’+x1’.x2.x3’ d. f(x1, x2, x3) = x1’.x2.x3+x1.x2’.x3
    50. 50. 48 Problem A and B are two events. Odds against A are A  B 2 to 1. Odds in favour of are 3 to 1. If x  P(B)  y, then ordered pair (x, y) is :  5 3 a.  ,   12 4  2 3 b. 3, 4   1 3 c. 3, 4   d. None of these
    51. 51. 49 Problem In a series of three trials the probability of exactly two successes in nine time as large as the probability of three successes. Then the probability of success in each trail is : 1 a. 2 1 b. 3 c. 1 4 3 d. 4
    52. 52. 50 Problem An integer is chosen at random from first two hundred digits. Then the probability that the integer chosen is divisible by 6 or 8 is : a. 1 4 2 b. 4 3 c. 4 d. none of these
    53. 53. 51 Problem x 2 Let A = R – {3}, B = R – {1}, Let be defined by f(x) = . Then : x 3 a. f is bijective b. f is one-one but not onto c. f is onto but not one-one d. none of these
    54. 54. 52 Problem cos2 x  sin2 x  1 Let f (x)  (x  0) = a (x = 0) then the value of a in order 2 x 4 2 that f(x) may be continuous at x = 0 is : a. - 8 b. 8 c. - 4 d. 4
    55. 55. 53 Problem If f(2) = 4 and f’(2) = 1, then lim xf (2)  2f ( x ) is equal to : x 2 x 2 a. 2 b. - 2 c. 1 d. 3
    56. 56. 54 Problem Let f(x) = sin x, g (x) = x2 and h (x) = log ex. If F(x) = (h o g o f) (x), then F’’(x) is equal to : a. a cosec3 x b. 2 cot x2 – 4x2 cosec2x2 c. 2x cot x2 d. - 2 cosec2 x
    57. 57. 55 Problem The length of subnormal to the parabola y2 = 4ax at any point is equal to : a. 2 a b. 2 2 a a c. 2 d. 2a
    58. 58. 56 Problem The function f(x) = a sin x + 1 sin 3x has maximum value of at x = . The value of 3 a is : a. 3 1 b. 3 c. 2 1 d. 2
    59. 59. 57 Problem If  cos 4x  1 dx  A cos 4x  B, then : cot x  tan x 1 a. A = - 2 b. A = - 1 8 c. A = - 1 4 d. none of these
    60. 60. 58 Problem |x| b  a x dx , a < 0 < b, is equal to : a. | b | - | a | b. | b | + | a | c. | a – b | d. none of these
    61. 61. 59 Problem  For any integer n, the integral 0 2 ecos x cos3(2n + 1)x dx has the value : a.  b. 1 c. 0 d. none of these
    62. 62. 60 Problem sin x  sin2 x  sin3x sin2 x sin3x  /2 If , then the value of 0 f (x) dx is : f (x)  3  4 sin x 3 4 sin x 1  sin x sin x 1 a. 3 2 b. 3 c. 1 3 d. 0
    63. 63. 61 Problem If f : R  R, g : R  R are continuous functions, then the value of the integral  /2 is :   /2 (f (x)  f ( x))(g(x)  g( x))dx a.  b. 1 c. - 1 d. 0
    64. 64. 62 Problem 3  x x2  1  The integral 1   tan1 2 x 1  tan1 x   dx is equal to : a.  /4  b. /2 c.  d. 2 
    65. 65. 63 Problem 2a f (x) The value of the integral 0 f (x)  f (2a  x) dx is : a. 0 b. a c. 2a d. none of these
    66. 66. 64 Problem The area bounded by the curves y = x2, y = xe-x and the line x = 1, is : 2 a. e 2 b. - e 1 c. e 1 d. e
    67. 67. 65 Problem The solution of x dy – y dx + x2 ex dx = 0 is : y  ex  c a. x x b.  ex  c y c. x + ey = c d. y + ex = c
    68. 68. 66 Problem The degree and order of the differential equation of all parabolas whose axis is  x- axis, are a. 2, 1 b. 1, 2 c. 3, 2 d. none of these
    69. 69. 67 Problem Three forces P, Q, R act along the sides BC, CA, AB of a triangle ABC taken in order. The condition that the resultant passes through the incentre, is : a. P + Q + R = 0 b. P cos A + Q cos B + R cos C = 0 c. P sec A + Q sec B + R sec C = 0 P Q R d.   0 sin A sin B sin C
    70. 70. 68 Problem The resultant of two forces P and Q is R. If Q is doubled, R is doubled and if Q is reversed, R is again doubled. If the ratio P2 : Q2 : R2 = 2 : 3 : x then x is equal to : a. 5 b. 4 c. 3 d. 2
    71. 71. 69 Problem A particle is dropped under gravity from rest from a height h (g = 9.8 m/sec2) and it travels a distance in the last second, the height h is : a. 100 mts b. 122.5 mts c. 145 mts d. 167.5 mts
    72. 72. 70 Problem A man can throw a stone 90 metres. The maximum height to which it will rise in metres, is : a. 30 b. 40 c. 45 d. 50
    73. 73. FOR SOLUTIONS VISIT WWW.VASISTA.NET

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