1. AMU –PAST PAPERS
MATHEMATICS- UNSOLVED PAPER - 2002

2. SECTION – I
 Single Correct Answer Type
 There are five parts in this question. Four choices are given for each part and one of them is
correct. Indicate you choice of the correct answer for each part in your answer-book by
writing the letter (a), (b), (c) or (d) whichever is appropriate

3. 01 Problem
The area of a parallelogram whose adjacent sides are ˆ
i 2ˆ
j 3k and 2ˆ
i ˆ
j ˆ
4k
is :
a. 10 3
b. 5 3
c. 5 6
d. 10 6

4. 02 Problem
 
If a j ˆ
iˆ
2 ˆ 3k and b 3ˆ
i ˆ
j ˆ
2k , then the unit vector perpendicular
 
to a and b is :
i j ˆ
ˆ ˆ k
a.
3
ˆ
i ˆ
j ˆ
k
b.
3
ˆ
i ˆ
j ˆ
k
c.
3
ˆ
i ˆ
j ˆ
k
d.
3

5. 03 Problem
If the 10th term of a geometric progression is a and the 4th term it is 4, then its 7th
term is :
a. 9/4
b. 4/9
c. 36
d. 6

6. 04 Problem
a a
the harmonic mean of and is :
1 ab 1 ab
a. 1
1 a2b2
a
b.
1 a2b2
c. a
a
d.
(1 a2 b2 )

7. 05 Problem
The general value of obtained from the equation cos 2 sin is :
a. 2n
2
b. n ( 1)n
2
c. n
2 2
d. 2
2

8. 06 Problem
1 5
The principal value of sin is :
3
a. 5
3
b. 5
3
c.
3
d. 4
3

9. 07 Problem
The equation of the plane passes through (2,3,4) and parallel to the plane x + 2y
+ 4z = 5 is :
a. x + 2y + 4z = 10
b. x + 2y + 4z = 3
c. x + 2y + 4z = 24
d. x + y + 2z = 2

10. 08 Problem
The distance of the point (2, 3, 4) from the plane 3x – 5y + 2z + 11 = 0 is :
a. 2
b. 9
c. 10
d. 1

11. 09 Problem
For the function f(x) = x2 – 6x + 8 2 x 4, the value of x for which f’(x)
vanishes has :
a. 9/4
b. 5/2
c. 3
d. 7/2

12. 10 Problem
1
From Mean value theoren f(b) – f(a) = (b - af)’(x1) a < x1 < b is : f(x) = x
then x
equal to
2ab
a.
a b
b a
b. b a
c. ab
a b
d.
2

13. 11 Problem
If 1 2 1 0 then
A and B
3 0 2 3
a. AB = BA
b. B2 = B
c. AB BA
d. A2 = A

14. 12 Problem
1
If for real values of x, cos x ' then :
x
a. is an acute angle
b. is a right angle
c. is an obtuse angle
d. no value is possible

15. 13 Problem
One of the equations of the lines passing through the point (3, -2) and inclined
at 600 at the line 3x y 1:
a. x – y = 3
b. x + 2 = 0
c. x + y = 0
d. y + 2 = 0

16. 14 Problem
The lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 are perpendicular to each
other :
a. a1b1 – b1a2 = 0
b. a12b2 – b12a2 = 0
c. a1b1 – b2a2 = 0
d. a1a2 + b1b2 = 0

17. 15 Problem
/2 dx is equal to :
0 1 tan x
a.
b. /2
c.
3
d. /4

18. 16 Problem
cos3 x dx is equal to :
0
a.
b. 1
c. 0
d. - 1

19. 17 Problem
The angle between the vectors (2ˆ
i 6ˆ
j ˆ
3k ) and (12ˆ
i 4ˆ
j ˆ
3k ) is :
a. 1 1
cos
9
b. 1 9
cos
11
1 9
c. cos
91
1 1
d. cos
10

20. 18 Problem
If the vectors ˆ
ai 2ˆ
j ˆ
3k and ˆ
i 5ˆ
j ˆ
9k are perpendicular to each other
then a is equal to :
a. 5
b. -6
c. - 5
d. 6

21. 19 Problem
200
If i2 = -1 the value of i n is :
n 1
a. 0
b. 50
c. - 50
d. 100

22. 20 Problem
If c i
a ib , where a, b, c ae real, then a2 + b2 is equal to :
c i
a. 7
b. 1
c. c2
d. - c2

23. 21 Problem
The direction cosines to the normal plane x + 2y – 3z + 9 = 0 are :
1 2 3
a. , '
14 14 14
1 2 3
b. , ,
10 10 14
1 2 1
c. , ,
10 14 14
1 2 3
, ,
d. 14 14 14

24. 22 Problem
The equation of a plane which cuts equal intercepts of unit lengths on the axes, is
:
a. x + y + z = 0
b. x + y + z = 1
c. x + y – z = 1
x y z
d. 1
a b a

25. 23 Problem
Performing 3 iterations of bisection method of smallest positive approximate
roots of equation x3 – 5x + 1 = 0 is :
a. 0.25
b. 0.125
c. 0.05
d. 0.1875

26. 24 Problem
For the smallest positive root of transcendental equation x – e-x = 0 interval is :
a. (1, 2)
b. (0, 1)
c. (2, 3)
d. (-1, 0)

27. 25 Problem
The equation of the sides of a triangle are x + y – 5 = 0, x – y + 1 = 0 and y – 1 =
0 then the coordinate of the circumcentre are :
a. (2, 1)
b. (1, 2)
c. (2, -2)
d. (1, -2)

28. 26 Problem
A stick of length ‘l’ rests against the floor and a wall of a room. If the stick begins
to slide on the floor, then the locus of its middle point is :
a. A straight line
b. Circle
c. Parabola
d. Ellipse

29. 27 Problem
The probability, that a leap year has 53 Sundays is :
a. 2/7
b. 3/7
c. 4/7
d. 1/7

30. 28 Problem
In tossing 10 coins the probability of getting 5 heads is :
a. 1
2
63
b.
256
c. 193
250
9
d. 128

31. 29 Problem
After second iteration of Newton-Raphson method the positive root of equation
x2 = 3 is, (taking initial approximation 3 )
2
a. 7
4
97
b.
56
3
c.
2
d. 347
200

32. 30 Problem
By false positioning the second approximation of a root of equation f(x) = 0 is
(where x0, x1 are initial and first approximations respectively) :
a. x1 - f (x0 )
f (x1 ) f (x0 )
f (x0 )
b. x0 –
f (x1 ) f (x0 )
x0 f (x 1 ) x1f (x0 )
c.
f (x1 ) f (x0 )
d. x0 f (x 0 ) x1f (x1 )
f (x1 ) f (x0 )

33. 31 Problem
13 16 19
14 17 20
is equal to :
15 18 21
a. 57
b. - 39
c. 96
d. 0

34. 32 Problem
Which one of the following statements is true ?
a. If | A | 0, then
b. | A adj A | = | A | (n - 1) where A = | an | n x n
c. If A’ = A, then A is a square matrix
d. Determination of non-square matrix is zero
e. Non-singular square matrix does not have a unique inverse

35. 33 Problem
1
3
dx is equal to :
x x
1 x2
a. log c
2 (1 x 2 )
b. log x (1 – x2) + c
(1 x)
c. log c
x(1 x)
1 (1 x 2 )
d. log c
2 x2

36. 34 Problem
x 1
2
e x dxis equal to :
x
1
a. e x c
x
x
b. e c
x
x
c. e ce x
2
x
1
d. log x
x
c

37. 35 Problem
dy
Solution of differential equation dx
+ ay = emx is :
a. (a + m)y = emx + c
b. y = emx + ce-ax
c. (a + m) = emx + c
d. (a + m) y = emx + ce-ax

38. 36 Problem
Integrating factor of differential equation cos x + y sin x = 1 is :
a. sec x
b. sin x
c. cos x
d. tan x

39. 37 Problem
Solution of differential equation dy – sin x sin y dx = 0 is :
a. cos x tan y = c
b. cos x sin y = c
y
c. ecos x tan =c
2
d. ecos x tan y = c

40. 38 Problem
Solution of differential equation x dy – y dx = 0 represents :
a. Rectangular hyperbala
b. Parabola whose vertex is at origin
c. Circle whose centre is at origin
d. Straight line passing through origin

41. 39 Problem
With the help of Trapezoidal rule for numerical integration and the following
table :
x 0 0.25 0.50 0.75 1
f(x) 0 0.0625 0.2500 0.5625 1
1
The value of f (x) dx is :
0
a. 0.33334
b. 0.34375
c. 0.34457
d. 0.35342

42. 40 Problem
By the application of Simpson’s one-third rule for numerical integrations, with
1 dx
two sub-intervals, the value of 0 1 x is :
a. 17
36
17
b. 25
25
c. 36
17
d.
24

43. 41 Problem
1 1
If matrix A = 1 1
, then :
1 1
a. A’ = 1 1
1 1
b. A-1 = 1 1
1 1
c. Adj A = 1 1
d. A= 1 1
Where is a non zero scalar

44. 42 Problem
9 3 1/2 1/2
If for AX = B, B = 52
and A-1 = 4 3/4 5/4 then X is equal to :
0 2 3/4 3/4
3
a. 3/4
3/4
1/2
b. 1/2
2
c. 4
2
3
d. 1
3
5

45. 43 Problem
The sum of first n terms of the series 1 3 7 15 is :
...
2 4 8 16
a. n + 2 - n – 1
b. n2 – n – 1
c. 1 – 2- n
d. 2n – 1

46. 44 Problem
1
The coefficient of x in the expansion of 1 x2 x in ascending powers of x,
when | x | < 1, is :
a. 0
1
b. 2
1
c. 2
d. 1

47. 45 Problem
The angle between two lines is :
1 1
a. cos
9
1 4
b. cos
9
1 2
c. cos
9
1 3
cos
d. 9

48. 46 Problem
 
a 2ˆ
i ˆ
j ˆ
8k and b ˆ
i 3ˆ
j ˆ
4k ,
If then the magnitude of is equal to :
a. 13
13
b. 3
3
c.
13
4
d. 13

49. 47 Problem
The line y = 2x + c is tangent to the parabola y2 = 4x then c is equal to :
1
a. 2
1
b. 2
1
c. 3
d. 4

50. 48 Problem
The eccentricity of the ellipse, 4x2 + 9y2 + 8x + 36y + 4 = 0 is :
3
a.
5
b. 5
3
5
c. 6
2
d. 3

51. 49 Problem
sin2 x cos2 x
dx is equal to :
sin2 x cos2 x
a. tan x + cot x + c
b. cosec x + sec x + c
c. tan x + sec x + c
d. tan x + cosec x + c

52. 50 Problem
If sum of two numbers is 3, the maximum value of the product of first and the
square of second is :
a. 4
b. 3
c. 2
d. 1

53. 51 Problem
Three lines ex – y = 2, 5x + ay = 3 and 2x + y = 3 are concurrent, then a is equal to
:
a. 2
b. 3
c. - 2
d. 1

54. 52 Problem
j i j ˆ
The pair of straight line joining the origin to the points ˆ intersection2of the line y
iof 2 ˆ 3k and ˆ ˆ 4k
= 2 x + c and the circles x2 + y2 = 2 are at right angles, if :
a. c2 = 9 = 0
b. c2 = 10 = 0
c. c2 – 4 = 0
d. c2 – 8 = c

55. 53 Problem
The direction ratio of the diagonals of a cube which joins the origin to the
opposite corner are (when the three con current edges of the cube are co-
ordinate axes) :
a. 1, 2, 3
b. 2, -2, 1
c. 1, 1, 1
2 2 2
d. , ,
3 3 3

56. 54 Problem
The cosine of the angle between any two diagonals of a cube is :
1
a.
3
1
b. 3
1
c. 2
2
d. 3

57. 55 Problem
If x f (a) is equal to :
f (x) then
x 1 f (a 1)
a. f(a2)
1
b. f
a
c. f(-a)
a
f
d. a 1

58. 56 Problem
If the domain of the function f(x) = x2 – 6x + 7 is (- , ), then the range of
function is,
a. (-2, )
b. (- , )
c. (- 2, 1)
d. (- , -2)

59. 57 Problem
  
a (a x b) is equal to :
a. 0
b. a2
c. a2 + ab
 
d. a, b

60. 58 Problem
  
If a 3ˆ 7ˆ 5k, b
i j 3ˆ 3ˆ 3k and c
i j i j ˆ
7ˆ 5ˆ 3k are the three coterminous
edges of a parallelepiped, then its volume is :
a. 210
b. 108
c. 308
d. 272

61. 59 Problem
(2x 3)(3x 4) is equal to :
lim
x (4x 5)(5x 6)
a. 1
10
b. 0
1
c. 5
3
d. 10

62. 60 Problem
loge x
lim is equal to :
x 1 x 1
a. 1
b. 2
1
c. 2
d. 0

63. 61 Problem
Differential coefficient of sec x is :
1
a. sec x sin x
4 x
1
b. (sec x )3 / 2 sin x
4 x
1
c. x . sec x sin x
2
1
x (sec x )3 / 2 sin x
d. 2

64. 62 Problem
If y e1 log e x then the value dy
is equal to :
dx
a. 2
b. 1
c. 0
d. loge xe

65. 63 Problem
The co-ordinate of a point P are (3, 12, 4) with respect to origin O. Then the
direction cosines of OP are :
3 1 2
a. , ,
13 13 13
3 12 4
b. , ,
13 13 13
1 1 1
c. , ,
4 3 2
d. 2, 12, 4

66. 64 Problem
2
The equation of the normal to the hyperbola x y2 at the point (8, 3 3) is :
1
16 9
a. 3 x + 2y = 25
b. 2x + 3 y = 25
c. y + 2x = 25
d. x + y = 25

67. 65 Problem
Differential equation for y A cos x sin x, where a and B are arbitary
constants, is
2
a. d y2
y 0
dx
d2y 2
y 0
b. dx 2
d2y 2
c. y 0
dx 2
d2y
y 0
d. dx 2

68. 66 Problem
1/4
2 2
Order and degree of differential equation d y y
dy are :
dx 2 dx
a. 4 and 2
b. 2 and 4
c. 1 and 2
d. 1 and 4

69. 67 Problem
The function which is continuous for all real values of x and differentiable at x =
0 is :.
a. x1/2
b. | x |
c. log x
d. sin x

70. 68 Problem
x 1 x 2
Function f (x) is an continuous function :
2x 3 x 2
a. For x = 2 only
b. For all real value of x such that x 2
c. For all real value of x
d. For all integral value of x only

71. 69 Problem
The area of the curve x2 + y2 = 2ax is :
a. 4 a2
b. a2
1 2
c. a
2
d. 2 a2

72. 70 Problem
By graphical method, the solution of linear programming problem maxmize
z = 3x1 + 5x2 subjecdt to 3x1 2x2 18, x1 4, x2 6, x1 0, x2 0 is :
a. x1 = 4, x2 = 6, z = 4.2
b. x1 = 4, x2 = 3, z = 2.7
c. x1 = 2, x2 = 6, z = 36
d. x1 = 2, x2 = 0, z = 6

73. 71 Problem
Maximum value of f(x) = sin x + cos x is :
a. 2
b. 2
c. 1
d. 1
2

74. 72 Problem
For all real values of x, increasing function f(x) is :
a. x-1
b. x3
c. x2
d. x4

75. 73 Problem
2x2 + 7xy + 3y2 + 8x + 14y + = 0 will represent a pair of straight lines when is
equal to :
a. 8
b. 6
c. 4
d. 2

76. 74 Problem
The gradient of one of the lines x2 + hxy + 2y2 = 0 is twice that of the other, then
h is equal to :
a. 2
3
b. 2
c. 3
d. 1

77. 75 Problem
A coin is tossed three times is succession. If E is the event that three are at least
two heads and F is the event in which first throw is a head, then P(E/F) equal to :
a. 3
4
3
b. 8
1
c. 2
1
d. 8

78. 76 Problem
In a box there are 2 rad, 3 black and 4 white balls. Out of these three balls are
drawn together. The probability of these being of same colour is :
5
a.
84
1
b.
21
1
c.
84
d. none of these

79. 77 Problem
If 7th term of a Harmonic progression is 8 and the 8th term is 7, then its 5th term is
56
a.
15
b. 14
27
c.
14
d. 16

80. 78 Problem
If the sum of the first n terms of a series be 5n2 + 2n, then its second term is :
a. 42
b. 17
c. 24
d. 7

81. 79 Problem
If the conjugate (x + iy) (1 – 2i) be 1 + i, then :
3
a. y = 5
1 i
b. x iy
1 2i
1 i
c. x iy
1 2i
1
x
d. 5

82. 80 Problem
If a is an imaginary cube root of unity, then for the value of 3n 1 3n 3 3n 5
a. -1
b. 0
c. 3
d. 1

83. 81 Problem
The value of 15C 2 2
15C1 2
15C2 2
... 15C15 is :
0
a. 0
b. - 15
c. 15
d. 51

84. 82 Problem
(loge n)2 (loge n)4
1 ... is equal to :
2! 4!
1
a. 2 (n + n-1)
b. 1
n
c. n
1
d. (en + e-n)
2

85. 83 Problem
ax da is equal to :
ax 1
c
a. x 1
ax 1
b. e
loge a e
c. ax loge a + c
d. none of these

86. 84 Problem
Two numbers with in the bracket denote the ranks of 10 students of a class in
two subjects – (1, 10), (2, 9), (3, 8), (4, 7), (5, 6), (6, 5)(7, 4), (8, 3), (9, 2), (10, 1)
then rank correlation coefficient is :
a. - 1
b. 0
c. 0.5
d. 1

87. 85 Problem
There are 5 roads leading to a town from a village. The number of different
ways in which a villager can go to the town return backs is :
a. 20
b. 25
c. 5
d. 10

88. 86 Problem
If (1 + x)n = C0 + C1x + C2x2 + ….Cnxn then the value of Cn = 2C1 + 3C2 + ….+ (n + 1) Cn
will be :
a. (n + 2)2n
b. (n + 1)2n – 1
c. (n + 1)2n
d. (n + 2)2n - 1

89. 87 Problem
In a triangle ABC, 2 cos A = sin B cosec C then
a. 2a = bc
b. a = b
c. b = c
d. c = a

90. 88 Problem
If tan x b then the value of a cos 2x + b sin 2x is :
a'
a. a
b. a – b
c. a + b
d. b

91. 89 Problem
In angle of a triangle are in the ratio of 2 : 3 : 7, then the sides are in the ratio of :
a. 2 : ( 3+ 1) : 2
b. 2 : 2 : ( 3+ 1)
c. 2 : 2 : ( 3+ 1)
d. 2 : ( 3 + 1) : 2

92. 90 Problem
1 1 ac 1 bc
is equal to :
1 1 ad 1 bc
1 1 ac 1 bc
a. a + b + c
b. 1
c. 0
d. 3

93. 91 Problem
The angle of elevation of the sum if the length of the shadow of a tower is 3
times the height of the pole is :
a. 1500
b. 300
c. 600
d. 450

94. 92 Problem
If the sides of triangle are 3, 5, 7 then :
a. Triangle is right-angled
b. One angle is obtuse
c. All its angles are acute
d. None of these

95. 93 Problem
Area bounded by lines y = 2 + x, y = 2- x and x = 2 is :
a. 16
b. 8
c. 4
d. 3

96. 94 Problem
Area bounded by parabola y2 = x and straight line 2y = x is :
a. 4
3
b. 1
2
c. 3
1
d. 3

97. 95 Problem
The latusrectum of parabola y2 = 5x + 4y + 1 is :
a. 10
b. 5
5
c. 4
5
d. 2

98. 96 Problem
x = 7 touches the circles x2 + y2 – 4x – 6y – 12 = 0 then the co-ordinates of the
point of contact :
a. (7, 4)
b. (7, 3)
c. (7, 2)
d. (7, 8)

99. 97 Problem
If the roots of the equation, (a2 + b2) t2 – 2(ac + bd) t + (c2 + d2) = 0 are equal
then :
a. ad + bc = 0
a c
b.
b d
c. ab = dc
d. ac = bc

100. 98 Problem
2
If the roots a, of the equation x bx 1 are such that a 0 ,
ax c 1
then the value of is :
1
a. c
b. 0
a b
c. a b
a b
d. a b

101. 99 Problem
The equation of the tangents of the ellipse 9x2 + 16y2 = 144 from the point (2, 3)
are :
a. y = 3, x + y = 5
b. y = 3, x = 2
c. y = 2, x = 3
d. y = 3, x = 5

102. 100 Problem
The latus rectum of the hyperbola 9x2 – 16y2 – 18x – 32y – 151 = 0
9
a. 4
3
b. 2
c. 9
9
d. 2

103. FOR SOLUTIONS VISIT WWW.VASISTA.NET