AMU –PAST PAPERSMATHEMATICS - UNSOLVED PAPER - 2005
SECTION – I   CRITICAL REASONING SKILLS
01   Problem       ~p      q    is logically equivalent to :     a. P      q     b. q      p     c. ~ (p       q)     d. ~...
02   Problem     The angle of elevation of the top of an incomplete vertical pillar at a horizontal     distance of 100 m ...
03   Problem                                          1   2   3   8     What must be the matrix X, if   2X                ...
04   Problem     The value of          1           1           1       is :                        bc             ca      ...
05   Problem                    441    441   443     The value of   445    446   447                                      ...
06    Problem                       ˆ ˆ     (a ˆ)ˆ        i i   (a ˆ)ˆ                 j j   (a k )k   is equal to :   ...
07   Problem                                cos 2    sin 2     Inverse of the matrix                       is :           ...
08   Problem                              a       3, b       4,                                     If                ...
09   Problem                                        The projection a   2i  j  ˆ                         ˆ 3ˆ 2k on b   i...
10   Problem     Let A = {x : x is a multiple of 3} and B = {x : x is a multiple of 5}. Then A B is given     by :     a. ...
11   Problem     The maximum of the function 3 cos x – 4 sin x is :     a. 2     b. 3     c. 4     d. 5
12   Problem     If the distance ‘s’ metres traversed by a particle in t seconds is given by s = t3 – 3     t2, then the v...
13   Problem     For the curve yn = an-1 x if the subnormal at any point is a constant, then n is     equal to :     a. 1 ...
14   Problem                                        d2 x     If x = A cos 4t + B sin 4t, then          is equal to :      ...
15   Problem     If tangent to the curve x = at2, y = 2at is perpendicular to x-axis, then its point of     contact is :  ...
16   Problem                                                         dy   1   cos 2y     The general solution of the diffe...
17   Problem                                                              3/4          1/3                                ...
18   Problem     The area enclosed between the curves y = x3 and y = is , (in sq unit) :     a.   5          3          5 ...
19   Problem              /8                   cos3 4 d   is equal to :          0              5     a.              3   ...
20   Problem          /2    cos x sin x    dx is equal to :     0         1 cos x sin x     a. 0     b.   2     c.    4   ...
21   Problem     If ax2 – y2 + 4x – y = 0 represents a pair of lines, then a is equal to :     a. -16     b. 16     c. 4  ...
22   Problem     What is the equation of the locus of a point which moves such that 4 times its     distance from the x-ax...
23   Problem     Equation of the straight line making equal intercepts on the axes and passing     through the point (2, 4...
24   Problem     If the area of the triangle with vertices (x, 0) (1, 1) and (0, 2) is 4 square unit, then     the value o...
25   Problem               is equal to :     lim 2         cot          2     a. 0     b. - 1     c. 1     d.
26   Problem     The co-axial system of circles given by x2 + y2 + 2gx + c = 0 for c < 0 represents :     a. Intersecting ...
27   Problem     The radius of the circle passing through the point (6, 2) and two of whose     diameters are x + y = 6 an...
28   Problem     If (0, 6) and (0, 3) are respectively the vertex and focus of a parabola, then its     equation is     a....
29   Problem     For the ellipse 24x2 + 9y2 – 120x –90y + 225 = 0, the eccentricity is equal to :     a.   2          5   ...
30   Problem                                  x2   y2                            x2   y2   1     If the foci of the ellips...
31   Problem     The differential coefficient of f(sin x) with respect to x where f(x) = log x is :     a. tan x     b. co...
32   Problem                  1   cos x     If f(x) =              ,   x   0   is continuous at x = 0, then the value of k...
33   Problem     If        1       3i                 2        is :                            then (3   3       )4       ...
34   Problem     If y = tan-1 (sec x – tan x), then is equal to :     a. 2     b. -2          1     c.   2              1 ...
35   Problem                 1                      1     If x            2 cos   then x n        is equal to :           ...
36   Problem       1           |1   x | dx is equal to :        1     a. - 2     b. 0     c. 2     d. 4
37   Problem             dx    is equal to :              7          x(x   1)     a.            x7          log           ...
38   Problem           xe    x                     dx is equal to :     a. 2 x       e   x                          4 xe  ...
39   Problem                dx       is equal to :            2        x       2x   2     a. sin-1 (x + 1) + c     b. sin-...
40   Problem     If a tangent to the curve y = 6x – x2 is parallel to the line 4x – 2y – 1 = 0, then     point of tangency...
41   Problem     0.5737373……… is equal to :     a.   284          497          284     b.          495          568     c....
42   Problem     The number of solutions for the equation x2 – 5 |x| + 6 = 0 is :     a. 4     b. 3     c. 2     d. 1
43   Problem     How many numbers of 6 digits can be formed from the digits of the number     112233 ?     a. 30     b. 60...
44   Problem     The last digit in 7300     a. 7     b. 9     c. 1     d. 3
45   Problem          log x   log y   log z     If                          then   xyz is equal to :          a b     b c ...
46   Problem     The smallest positive integer n for which (1 + i)2n = (1 - i)2n is :     a. 1     b. 2     c. 3     d. 4
47   Problem     If cos-1 p + cos-1 q + cos-1 r = then p2 + q2 + r2 + 2ppr is equal to :     a. 3     b. 1     c. 2     d....
48   Problem                    x             5     If   sin   1                        cosec-1           ,   then x is eq...
49   Problem     If 0      x                           2                   and 81sin   x       2                          ...
50   Problem                                                            x2   y2     The equation of the director circle of...
51   Problem     The normals at the extremities of the latus rectum of parabola intersects the axis     at an angle of :  ...
52   Problem     The circle x2 + y2 – 8x + 4y + 4 = 0 touches :     a. x-axis     b. y-axis     c. both axes     d. neithe...
53   Problem     If A = {1, 2, 3} and B = {3, 8}, then   (A   B) x ( A   B)   is :     a. {(3, 1), (3, 3), (3, 8)}     b. ...
54   Problem     The condition that one root of the equation ax2 + bx + c = 0 may be double of the     other, is     a. b2...
55   Problem     The value of k so that x2 + y2 + kx + 4y + 2 = 0 and 2 (x2 + y2) – 4x – 3y + k = 0 cut     orthogonally i...
56   Problem                           (3 x 1)                   4                 is equal to :     lim 1     x         X...
57   Problem     If A + B + C = 1800, then         A     B   is equal to :                                 tan     tan    ...
58   Problem     In a triangle ABC, If b = 2, B = 300 then the area of the circumcircle of triangle ABC     in square unit...
59   Problem     If sin x + sin2 x = 1, then cos12 x + 3 cos10 x + 3 cos8 x + cos6 x is equal to :     a. 1     b. 2     c...
60   Problem     If R denotes the set of all real number, then the function   f :R   R   defined     f(x) = |x| is :     a...
61   Problem     If f(x) = 2x3 + mx2 – 13x + n and 2,3 are roots of the equation f(x) = 0, then the     values of m and n ...
62   Problem     If p1, p2, p3 are respectively the perpendiculars from the vertices of a triangle to     the opposite sid...
63   Problem     If 5     cos         2 cos2          1   0,   , then   is equal to :                                     ...
64   Problem     The two forces acting at a point, the maximum effect is obtained when their     resultant is 4N If they a...
65   Problem     The resultant R of two forces P and Q act at right angles to P. Then the angle     between the forces is ...
66   Problem     A body starts from rest and moves with a uniform acceleration. The ratio of the     distance covered in n...
67   Problem     Two points move in the same straight line starting at the same moment from the     same point in the same...
68   Problem     The equation of the plane containing the line x    1   y       3   z       2   and                       ...
69   Problem     A plane passes through a fixed point (p, q) and cut the axes in A, B, C. Then the     locus of the centre...
70   Problem     The value of 12.C1 + 32 . C3 + 52 . C5 + … is :     a. n (n -1)n-2 + n. 2n-1     b. n(n - 1)2n-2     c. n...
FOR SOLUTIONS VISIT WWW.VASISTA.NET
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AMU - Mathematics - 2005

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AMU - Mathematics - 2005

  1. 1. AMU –PAST PAPERSMATHEMATICS - UNSOLVED PAPER - 2005
  2. 2. SECTION – I CRITICAL REASONING SKILLS
  3. 3. 01 Problem ~p q is logically equivalent to : a. P q b. q p c. ~ (p q) d. ~ (q p)
  4. 4. 02 Problem The angle of elevation of the top of an incomplete vertical pillar at a horizontal distance of 100 m from its base is 450. If the angle of elevation of the top of the complete pillar at the same point is to be 600, then the height of the incomplete pillar is to be increased by : a. 50 2 b. 100 m c. 100 ( 3 -1)m d. 100 ( 3 + 1)m
  5. 5. 03 Problem 1 2 3 8 What must be the matrix X, if 2X 3 4 7 2 ? 1 3 a. 2 1 1 3 b. 2 1 2 6 c. 4 2 2 6 d. 4 2
  6. 6. 04 Problem The value of 1 1 1 is : bc ca ab b c c a a b a. 1 b. 0 c. (a - b) (b - c) (c - a) d. (a + b) (b + c) (c + a)
  7. 7. 05 Problem 441 441 443 The value of 445 446 447 is : 449 450 451 a. 441 x 446 x 4510 b. 0 c. - 1 d. 1
  8. 8. 06 Problem    ˆ ˆ (a ˆ)ˆ i i (a ˆ)ˆ j j (a k )k is equal to :  a. a  b. 2 a  c. 3 a  d. 0
  9. 9. 07 Problem cos 2 sin 2 Inverse of the matrix is : sin 2 cos 2 a. cos 2 sin 2 sin 2 cos 2 b. cos 2 sin2 sin2 cos 2 cos 2 sin2 c. sin2 cos 2 cos 2 sin 2 sin 2 cos 2 d.
  10. 10. 08 Problem   a 3, b 4,   If then a value of for which a b is perpendicular to   a b is : 9 a. 16 3 b. 4 3 c. 2 4 d. 3
  11. 11. 09 Problem   The projection a 2i j ˆ ˆ 3ˆ 2k on b i j ˆ ˆ 2ˆ 3k is : 1 a. 14 2 b. 14 c. 14 2 d. 14
  12. 12. 10 Problem Let A = {x : x is a multiple of 3} and B = {x : x is a multiple of 5}. Then A B is given by : a. {3, 6, 9, ………… } b. {5, 10, 15, 20, ………..} c. {15, 30, 45, …………….} d. none of the above
  13. 13. 11 Problem The maximum of the function 3 cos x – 4 sin x is : a. 2 b. 3 c. 4 d. 5
  14. 14. 12 Problem If the distance ‘s’ metres traversed by a particle in t seconds is given by s = t3 – 3 t2, then the velocity of the particle when the acceleration is zero, (in m/s) is : a. 3 b. - 2 c. - 3 d. 2
  15. 15. 13 Problem For the curve yn = an-1 x if the subnormal at any point is a constant, then n is equal to : a. 1 b. 2 c. -2 d. -1
  16. 16. 14 Problem d2 x If x = A cos 4t + B sin 4t, then is equal to : dt 2 a. - 16 x b. 16 x c. x d. - x
  17. 17. 15 Problem If tangent to the curve x = at2, y = 2at is perpendicular to x-axis, then its point of contact is : a. (a, a) b. (0, a) c. (0, 0) d. (a, 0)
  18. 18. 16 Problem dy 1 cos 2y The general solution of the differential equation 0 is dx 1 cos 2x given by : a. tan y + cot x = c b. tan y – cot x = c c. tan x – cot y = c d. tan x + cot y = c
  19. 19. 17 Problem 3/4 1/3 2 The degree of the differential equation : dy d2y is : 1 dx dx 2 a. 1/3 b. 4 c. 9 d. 3/4
  20. 20. 18 Problem The area enclosed between the curves y = x3 and y = is , (in sq unit) : a. 5 3 5 b. 4 5 c. 12 12 d. 5
  21. 21. 19 Problem /8 cos3 4 d is equal to : 0 5 a. 3 5 b. 4 1 c. 3 1 d. 6
  22. 22. 20 Problem /2 cos x sin x dx is equal to : 0 1 cos x sin x a. 0 b. 2 c. 4 d. 6
  23. 23. 21 Problem If ax2 – y2 + 4x – y = 0 represents a pair of lines, then a is equal to : a. -16 b. 16 c. 4 d. -4
  24. 24. 22 Problem What is the equation of the locus of a point which moves such that 4 times its distance from the x-axis is the square of its distance from the origin ? a. x2 – y2 – 4y = 0 b. x2 + y2 – 4|y| = 0 c. x2 + y2 – 4x = 0 d. x2 + y2 – 4|x| = 0
  25. 25. 23 Problem Equation of the straight line making equal intercepts on the axes and passing through the point (2, 4) is : a. 4x – y – 4 = 0 b. 2x + y – 8 = 0 c. x + y – 6 = 0 d. x + 2y – 10 = 0
  26. 26. 24 Problem If the area of the triangle with vertices (x, 0) (1, 1) and (0, 2) is 4 square unit, then the value of x is : a. - 2 b. - 4 c. - 6 d. 8
  27. 27. 25 Problem is equal to : lim 2 cot 2 a. 0 b. - 1 c. 1 d.
  28. 28. 26 Problem The co-axial system of circles given by x2 + y2 + 2gx + c = 0 for c < 0 represents : a. Intersecting circles b. Non-intersecting circles c. Touching circles d. Touching or non-intersecting circles
  29. 29. 27 Problem The radius of the circle passing through the point (6, 2) and two of whose diameters are x + y = 6 and x + 2y = 4 is : a. 4 b. 6 c. 20 d. 20
  30. 30. 28 Problem If (0, 6) and (0, 3) are respectively the vertex and focus of a parabola, then its equation is a. x2 + 12y = 72 b. x2 - 12y = 72 c. x2 – 12x = 72 d. y2 + 12x = 72
  31. 31. 29 Problem For the ellipse 24x2 + 9y2 – 120x –90y + 225 = 0, the eccentricity is equal to : a. 2 5 b. 3 5 15 c. 24 1 d. 5
  32. 32. 30 Problem x2 y2 x2 y2 1 If the foci of the ellipse 1 and the hyperbola 16 b2 144 81 25 coincide, then the value of b2 is : a. 1 b. 7 c. 5 d. 9
  33. 33. 31 Problem The differential coefficient of f(sin x) with respect to x where f(x) = log x is : a. tan x b. cot x c. f(cos x) 1 d. x
  34. 34. 32 Problem 1 cos x If f(x) = , x 0 is continuous at x = 0, then the value of k is : x k, x 0 a. 0 1 b. 2 1 c. 4 1 d. - 2
  35. 35. 33 Problem If 1 3i 2 is : then (3 3 )4 2 a. 16 b. - 16 c. 16 d. 16 2
  36. 36. 34 Problem If y = tan-1 (sec x – tan x), then is equal to : a. 2 b. -2 1 c. 2 1 d. - 2
  37. 37. 35 Problem 1 1 If x 2 cos then x n is equal to : 2 xn a. 2n cos b. 2n cos n c. 2i sin n d. 2 cos n
  38. 38. 36 Problem 1 |1 x | dx is equal to : 1 a. - 2 b. 0 c. 2 d. 4
  39. 39. 37 Problem dx is equal to : 7 x(x 1) a. x7 log c x7 1 b. 1 x7 log 7 c 7 x 1 x7 1 c. log c x7 1 x7 1 log c d. 7 x7
  40. 40. 38 Problem xe x dx is equal to : a. 2 x e x 4 xe x c b. x (2 x 4 x 4)e c x c. (2x 4 x 4)e c x (1 4 x )e c d.
  41. 41. 39 Problem dx is equal to : 2 x 2x 2 a. sin-1 (x + 1) + c b. sin-1 (x + 1) + c c. tan-1 (x - 1) + c d. tan-1 (x + 1) + c
  42. 42. 40 Problem If a tangent to the curve y = 6x – x2 is parallel to the line 4x – 2y – 1 = 0, then point of tangency on the curve is : a. (2, 8) b. (8, 2) c. (6, 1) d. (4, 2)
  43. 43. 41 Problem 0.5737373……… is equal to : a. 284 497 284 b. 495 568 c. 999 567 d. 990
  44. 44. 42 Problem The number of solutions for the equation x2 – 5 |x| + 6 = 0 is : a. 4 b. 3 c. 2 d. 1
  45. 45. 43 Problem How many numbers of 6 digits can be formed from the digits of the number 112233 ? a. 30 b. 60 c. 90 d. 120
  46. 46. 44 Problem The last digit in 7300 a. 7 b. 9 c. 1 d. 3
  47. 47. 45 Problem log x log y log z If then xyz is equal to : a b b c c a a. 0 b. 1 c. - 1 d. 2
  48. 48. 46 Problem The smallest positive integer n for which (1 + i)2n = (1 - i)2n is : a. 1 b. 2 c. 3 d. 4
  49. 49. 47 Problem If cos-1 p + cos-1 q + cos-1 r = then p2 + q2 + r2 + 2ppr is equal to : a. 3 b. 1 c. 2 d. - 1
  50. 50. 48 Problem x 5 If sin 1 cosec-1 , then x is equal to : 5 4 2 a. 1 b. 4 c. 3 d. 5
  51. 51. 49 Problem If 0 x 2 and 81sin x 2 81cos x 30, then x is equal to : a. 6 b. 2 c. 4 3 d. 4
  52. 52. 50 Problem x2 y2 The equation of the director circle of the hyperbola 1 is given by : 16 4 a. x2 + y2 =16 b. x2 + y2 = 4 c. x2 + y2 = 20 d. x2 + y2 = 12
  53. 53. 51 Problem The normals at the extremities of the latus rectum of parabola intersects the axis at an angle of : a. Less than 900 b. Greater than 900 c. 900 d. none of the above
  54. 54. 52 Problem The circle x2 + y2 – 8x + 4y + 4 = 0 touches : a. x-axis b. y-axis c. both axes d. neither x-axis nor y-axis
  55. 55. 53 Problem If A = {1, 2, 3} and B = {3, 8}, then (A B) x ( A B) is : a. {(3, 1), (3, 3), (3, 8)} b. {(1, 3), (2, 3), (3, 3), (8, 3)} c. {(1, 2), (2, 2), (3, 3), (8, 8)} d. {(8, 3), (8, 2), (8, 1), (8, 8)}
  56. 56. 54 Problem The condition that one root of the equation ax2 + bx + c = 0 may be double of the other, is a. b2 = 9ac b. 2b2 = 9ac c. 2b2 = ac d. b2 = ac
  57. 57. 55 Problem The value of k so that x2 + y2 + kx + 4y + 2 = 0 and 2 (x2 + y2) – 4x – 3y + k = 0 cut orthogonally is : a. 10 3 8 b. 3 10 c. - 3 8 d. 3
  58. 58. 56 Problem (3 x 1) 4 is equal to : lim 1 x X 1 a. e12 b. e-12 c. e4 d. e3
  59. 59. 57 Problem If A + B + C = 1800, then A B is equal to : tan tan 2 2 a. 0 b. 1 c. 2 d. 3
  60. 60. 58 Problem In a triangle ABC, If b = 2, B = 300 then the area of the circumcircle of triangle ABC in square unit is : a. b. 2 c. 4 d. 6
  61. 61. 59 Problem If sin x + sin2 x = 1, then cos12 x + 3 cos10 x + 3 cos8 x + cos6 x is equal to : a. 1 b. 2 c. 3 d. 0
  62. 62. 60 Problem If R denotes the set of all real number, then the function f :R R defined f(x) = |x| is : a. One-one only b. Onto only c. Both one-one and onto d. Neither one-one nor onto
  63. 63. 61 Problem If f(x) = 2x3 + mx2 – 13x + n and 2,3 are roots of the equation f(x) = 0, then the values of m and n are a. - 5, - 30 b. - 5, 30 c. 5, 30 d. none of these
  64. 64. 62 Problem If p1, p2, p3 are respectively the perpendiculars from the vertices of a triangle to the opposite sides, then p1p2p3 is equal to : a. a2b2c2 b. 2 a2b2c2 4a2 b2c 3 c. R2 a2 b2 c 2 d. 8R2
  65. 65. 63 Problem If 5 cos 2 cos2 1 0, , then is equal to : 2 a. 3 1 3 , cos b. 3 5 1 3 c. cos 5 1 3 , cos d. 3 5
  66. 66. 64 Problem The two forces acting at a point, the maximum effect is obtained when their resultant is 4N If they act at right angles, then their resultant is 3N. Then the forces are : a. 1 1 (2 3)N and (2 3)N 2 2 b. (2 3)N and (2 3)N 1 1 c. (2 2)N and (2 2)N 2 2 (2 2)N and (2 2)N d.
  67. 67. 65 Problem The resultant R of two forces P and Q act at right angles to P. Then the angle between the forces is : 1 P a. cos Q 1 P cos b. Q 1 P sin c. Q 1 P d. sin Q
  68. 68. 66 Problem A body starts from rest and moves with a uniform acceleration. The ratio of the distance covered in nth sec to the distance covered in n second is : 2 1 a. n n2 1 1 b. n2 n 2 1 c. n2 n 2 1 d. n n2
  69. 69. 67 Problem Two points move in the same straight line starting at the same moment from the same point in the same direction. The first moves with constant velocity u and the second starts from rest with constant acceleration f, then distance between the two points will be maximum at time : 2u u a. t C. t f 2f u u2 t t b. f D. f
  70. 70. 68 Problem The equation of the plane containing the line x 1 y 3 z 2 and 3 2 1 the point (0, 7, -7) is : a. x + y + z = 1 b. x + y + z = 2 c. x + y + z = 0 d. none of these
  71. 71. 69 Problem A plane passes through a fixed point (p, q) and cut the axes in A, B, C. Then the locus of the centre of the sphere OABCi. : p q r a. 2 x y z b. p q r 1 x y z c. p q r 3 x y z d. None of these
  72. 72. 70 Problem The value of 12.C1 + 32 . C3 + 52 . C5 + … is : a. n (n -1)n-2 + n. 2n-1 b. n(n - 1)2n-2 c. n(n -1) . 2n-3 d. none of the above
  73. 73. FOR SOLUTIONS VISIT WWW.VASISTA.NET

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