1. AMU –PAST PAPERS
MATHEMATICS - UNSOLVED PAPER - 2003

2. SECTION – I
 CRITICAL REASONING SKILLS

3. 01 Problem
If var (x) = 8.25, var (y) = 33.96 and cov (x, y) = 10.2, then the correlation
coefficient is :
a. 0.89
b. - 0.98
c. 0.61
d. - 0.16

4. 02 Problem
If the standard deviation of a variable x is then the standard deviation of another
variable ax b is :
c
ax b
a.
c
a
b.
c
c.
d. none of these

5. 03 Problem
If x 15, y 36, xy 110, x 5 then cov(x,y) equals :
a. 1/5
b. -1/5
c. 2/5
d. -2/5

6. 04 Problem
The two lines of regression are given by 3x + 2y = 26 and 6x + y = 31. the
coefficient of correlation between x and y = is :
a. - 1/3
b. 1/3
c. - 1/2
d. 1/2

7. 05 Problem
The A.M of 4, 7, y and 9 is 7. Then the value of y is :
a. 8
b. 10
c. 28
d. 0

8. 06 Problem
The median from the table
Value 7 8 10 9 11 12 13
Frequency 2 1 4 5 6 1 3
Is :
a. 100
b. 10
c. 110
d. 1110

9. 07 Problem
The mode of the following series 3, 4, 2, 1, 7, 6, 7, 6, 8, 9, 5 is :
a. 5
b. 6
c. 7
d. 8

10. 08 Problem
The mean-deviation and coefficient of mean deviation from the data. Weight (in
kg) 54, 50, 40, 40, 42, 51, 45, 47, 55, 57 is :
a. 0.0900
b. 0.0956
c. 0.0056
d. 0.0946

11. 09 Problem
The S.D, coefficient of S.D, and coefficient of variation from the given data
Class interval 0-10 10-20 20-30 30-40 40-50
Frequency 2 10 8 4 6
is
a. 50
b. 51.9
c. 47.9
d. 51.8

12. 10 Problem
There are n letters and n addressed envelopes, the probability that all the letters
are not kept in the right envelope, is :
1
a. n!
1
b. 1 - n !
1
c. 1 - n
d. n!

13. 11 Problem
If A and B are two events such that A
P( A) 0 and P(B) 1, then P
B
a. A
1 P
B
A
1 P
b. B
1 P( A B)
c. P(B)
P( A)
d. P(B)

14. 12 Problem
A coin is tossed 3 times, the probability of getting exactly two heads is :
a. 3/8
b. 1/2
c. 1/4
d. none of these

15. 13 Problem
Six boys and six girls sit in a row. What is the probability that the boys and girls
sit alternatively :
a. 1/462
b. 1/924
c. 1/2
d. none of these

16. 14 Problem
3 8
If A and B are two independent events such that P( A B ') and P( A ' B)
25 25'
then P(A) is equal to :
1
a. 5
3
b. 8
2
c. 5
4
5
d.

17. 15 Problem
In a college of 300 students every student read 5 newspapers and every
newspaper is read by 60 students. The number of newspaper is :
a. At least 30
b. At most 20
c. Exactly 25
d. None of these

18. 16 Problem
1 x is equal to :
dx
1 x
a. 1 1
sin x 1 x2 c
2
b. sin 1 x 1 1 x 2 c
2
c. sin 1 x 1 x2 c
d. sin 1 x 1 x2 c

19. 17 Problem
x 1 is equal to :
3
ex d x
(x 1)
ex
a. c
(x 1)2
ex
b. c
(x 1)2
ex
c
c. (x 1)3
ex
c
d. (x 1)3

20. 18 Problem
1
dx is equal to :
x 2 (x 4 1)3 / 4
(x 4 1)1 / 4
a. c
x
(x 4 1)1 / 4
b. - c
x
3 (x 4 1)3 / 4
c
c. y x
4 (x 4 1)3 / 4
d. 3 c
x

21. 19 Problem
If y = cos (sin x2) then at x = dy is equal to :
,
2 dx
a. - 2
b. 2
c. - 2 2
d. 0

22. 20 Problem
dy
If y log x log x log x log x ... then dx
is equal to :
x
a.
2y 1
x
b. 2y 1
1
c. x(2y 1)
1
d. x(1 2y )

23. 21 Problem
If f(x + y) = f(x).f(y) for all x and y and f (5) = 2, f’(0) = 3 then f’(5) will be :
a. 2
b. 4
c. 6
d. 8

24. 22 Problem
On the interval [0, 1] the function x25 (1 - x)75 takes its maximum value at the
point :
a. 9
b. 1/2
c. 1/3
d. 1/4

25. 23 Problem
If y = x log x 3 d2y is equal to :
then x
a bx dx 2
dy
a. x dx –y
2
b. dy
x y
dx
dy
c. y -x
dx
2
dy
d. y x
dx

26. 24 Problem
The first derivative of the function cos 1 sin 1 x xx with respect to x at x =
2
1 is :
a. 3/4
b. 0
c. 1/2
d. - 1/2

27. 25 Problem
Function f(x) = 2x3 – 9x2 + 12x + 99 is monotonically decreasing when :
a. x < 2
b. x > 2
c. x > 1
d. 1 < x < 2

28. 26 Problem
x x
If f (x) and g(x) , where 0 < x 1, then in this interval :
sin x tan x
a. Both f(x) and g(x) and increasing function
b. Both f(x) and g(x) and decreasing function
c. f(x) is an increasing function
d. g(x) is an increasing function

29. 27 Problem
1
If x sin x 0 is equal to :
f (x) x then lim f (x)
x 0
0 x 0
a. 1
b. 0
c. - 1
d. none

30. 28 Problem
log(1 ax ) log(1 bx )
The function f (x) is not5 defined at x = 0, the value
x
of which should be assigned to f at x = 0, so that it is continuous at x = 0, is :
a. a – b
b. a + b
c. log a + log b
d. log a – log b

31. 29 Problem
Domain of the function f ( x) 2 2x x 2 is :
a. 3 x 3
b. 1 3 x 1 3
c. 2 x 2
d. 2 3 x 2 3

32. 30 Problem
The function f(x) = [x] cos 2x 1 where [.] denotes the greatest integer
2
function, is discontinuous at :
a. All x
b. No x
c. All integer points
d. x which is not an integer

33. 31 Problem
x a2 2 sin x 0 x /4
If the function , is continuous in the
f (x) x cot x b /4 x /2
b sin2x a cos 2x /2 x
interval [0, ] then the value of (a, b) are :
a. (-1, -1)
b. (0, 0)
c. (-1, 1)
d. (1)

34. 32 Problem
a.{(b + c) x (a + b+ c)} is equal to :
a. 0
b. [abc]
c. [abc] + [bca]
d. none of these

35. 33 Problem
The number of vectors of unit length perpendicular to vectors a = (1, 1, 0) and b =
(0, 1, 1) is :
a. Three
b. One
c. Two
d.

36. 34 Problem
The points with position vectors 60i + 3j, 40i – 8j, ai – 52j are collinear, if a is
equal to :
a. - 40
b. 40
c. 20
d. -20

37. 35 Problem
If a = 4i + 6j and b = 3j + 4k, then the component of along b is :
18
a. (3 j 4k )
10 3
b. 18
(3 j 4k )
35
c. 18
(3 j 4k )
3
d. 3j + 4k

38. 36 Problem
The unit vector perpendicular to the vectors 6i +2j + 3k and 3i – 6j – 2k is :
2i 3j 6k
a.
7
2i 3j 6k
b.
7
2i 3j 6k
c. 7
2i 3j 6k
d. 7

39. 37 Problem
A unit vector a makes an angle 4
with z-axis, if a + i + j as a unit vector, then a is
equal to :
i j k
a.
2 2 2
i j k
b. 2 2 2
c. i j k
2 2 2
i j k
d. 2 2 2

40. 38 Problem
   
If AO OB BO OC , then A, B, C from :
a. Equilateral triangle
b. Right angled triangle
c. Isosceles triangle
d. Line

41. 39 Problem
The area of a parallelogram whose adjacent sides are i – 2j + 3k and 2i + j – 4k, is :
a. 5 3
b. 105 3
c. 10 6
d. 5 6

42. 40 Problem
The sum of the first n terms of the series 1 3 7 15
... is :
2 4 8 16
a. 2n – n – 1
b. 1 – 2-n
c. n + 2-n – 1
d. 2n – 1

43. 41 Problem
If the first and (2n -1)th terms of an A.P, G.P. and H.P. are equal and their nth
terms are respectively a, b and c, then :
a. a b c
b. a c=b
c. ac – b2 = 0
d. a and c both

44. 42 Problem
If a1, a2, a3 … an are in H.P., then a1a2 + a2a3 + ….an-1an will be equal to :
a. a1an
b. na1an
c. (n -1)a1an
d. none of these

45. 43 Problem
The sum of n terms of the following series 1 + (1 + x) + (1 + x + x2) + .. will be :
1 xn
a. 1 x
(x(1 x n )
b. 1 x
n(1 x) x(1 xn)
c.
(1 x)2
d. none of these

46. 44 Problem
If Sn = nP + n (n -1)8, where Sn denotes the sum of the first n terms of an A.P. then
the common difference is :
a. P + Q
b. 2P + 3Q
c. 2Q
d. Q

47. 45 Problem
For any odd integer n 1 n3 – (n -1)3 + ….+ (-1)n -1, 13 is equal to :
1
a. (n - 1)2 (2n -1)
2
b. 1 (n - 1)2 (2n -1)
4
c. 1 (n + 1)2 (2n -1)
2
1
d. 4 (n + 1)2 (2n -1)

48. 46 Problem
If P = (1, 0), Q = (1, 0) and R = (2, 0) are three given points then the locus of a
point S satisfying the relation SQ2 + SR2 = 2SP2 is :
a. A straight line parallel to x-axis
b. A circle through origin
c. A circle with centre at the origin
d. A straight line parallel to y-axis

49. 47 Problem
If the vertices of a triangle be (2, 1), (5, 2) and (3, 4) then its circumcentre is :
a. 13 9
,
2 2
13 9
b. ,
4 4
9 13
c. ,
4 4
d. None of this

50. 48 Problem
The line x + y = 4 divides the line joining the points (-1, 1) and (5, 7) in the ratio :
a. 2 : 1
b. 1 : 2
c. 1 : 2 externally
d. none of these

51. 49 Problem
Given the points A (0, 4) and B (0, - 4) then the equation of the locus of the point
P (x, y) such that, | AP – BP | = 6, is :
x2 y2
a. 1
7 9
x2 y2
b. 1
9 7
x2 y2
1
c. 7 9
y2 x2
1
d. 9 7

52. 50 Problem
If P (1, 2), Q(4, 6), R(5, 7) and S(a, b) are the vertices of a parallelogram
PQRS, then :
a. a = 2, b = 4
b. a = b, b = 4
c. a = 2, b = 3
d. a = 3, b = 5

53. 51 Problem
The incentre of the triangle formed by (0, 0), (5, 12), (16, 12) is :
a. (7, 9)
b. (9, 7)
c. (-9, 7)
d. (-7, 9)

54. 52 Problem
1 4 20
The roots of the equation are :
1 2 5 0
1 2x 5x 2
a. - 1, - 2
b. - 1, 2
c. 1, -2
d. 1, 2

55. 53 Problem
x + ky – z = 0, 3x – ky – z = 0 and x – 3y + z = 0 has non-zero solution for k is equal
to :
a. - 1
b. 0
c. 1
d. 2

56. 54 Problem
x y z 9
If x y 5 then the value of (x, y, z) is :
y z 7
a. (4, 3, 2)
b. (3, 2, 4)
c. (2, 3, 4)
d. none

57. 55 Problem
If A is n x n matrix, then adj(adj A) =
a. | A |n – 1 A
b. | A |n – 2 A
c. | A | n
d. none

58. 56 Problem
a2 a 1
The value of determinant cos(nx ) cos(n 1)x cos(n 2)x is independent of
sin(nx ) sin(n 1)x sin(n 2)x
:
a. n
b. a
c. x
d. none of these

59. 57 Problem
3 1 5 1
If X then X is equal to :
4 1 2 3
a. 3 4
14 13
3 4
b.
14 13
3 4
c. 14 13
3 4
d. 14 13

60. 58 Problem
1 0 1
If matrix A = and its inverse is denoted by a11 a12 a13 then the
3 4 5
1
0 6 7 A a21 a22 a23
value of a23 is equal to : a31 a32 a33
a. 21/20
b. 1/5
c. -2/5
d. 2/5

61. 59 Problem
1 2 5
The rank of the matrix 2 4 a 4 is :
1 2 a 1
a. 1 if a = 6
b. 2 if a = 1
c. 3 if a = 2
d. 4 if a = - 6

62. 60 Problem
cos sin + sin cos is equal to :
cos sin
sin cos cos cos
0 0
a. 0 0
1 0
b.
0 0
0 1
c. 1 0
1 0
d. 0 1

63. 61 Problem
The solution of differential equation (sin x + cos x)dy + (cos x – sinx)dx = 0 is :
a. ex (sin x + cos x) + c = 0
b. ey (sin x + cos x) = c
c. ey (cos x + sin x) = c
d. ex (sin x - cos x) = c

64. 62 Problem
ab
z is a group of all positive rationals under the operation* defined by a * b = 2
the
identity of z is :
a. 2
b. 1
c. 0
1
d. 2

65. 63 Problem
The average of 5 quantities is 6, the average of the them is 4, then the average of
remaining two numbers is :
a. 9
b. 6
c. 10
d. 5

66. 64 Problem
A man can swim down stream at 8 km/h and up stream at 2 km/h, then the
man’s rate in still water and the speed of current is :
a. (5, 3)
b. (5, 4)
c. (3, 5)
d. (3, 3)

67. 65 Problem
Equation of curve through point (1, 0) which satisfies the differential equation
(1 + y2)dx – xydy = 0 is :
a. x2 + y2 = 4
b. x2 - y2 = 1
c. 2x2 + y2 = 4
d. none of these

68. 66 Problem
The order of the differential equation whose general solution is given by
y (c1 c2 ) cos(x c3 ) c 4e x c5
, where, c1, c2, c3, c4 and c5 are
arbitrary constants, is :
a. 5
b. 6
c. 3
d. 2

69. 67 Problem
The equation of the base of an equilateral triangle is x + y = 2 and the vertex is (2,
-1) the length of the side of the triangle is :
a. 3 /2 / 2 /3
b. 2
c. 2 /3
3 /2
d.

70. 68 Problem
The distance between the line 3x + 4y = 9 and 6x + 8y = 15 is :
a. 3/2
b. 3/10
c. 6
d. 0

71. 69 Problem
The area enclosed with in the curve | x | | + |y| = 1 is :
a. 2
b. 1
c. 3
d. 2

72. 70 Problem
Co-ordinates of the foot of the perpendicular drawn from (0, 0) to the line joining
(a cos , a sin ) and (a cos , a sin ) are :
a b
a.
2' 2
a a
b. (cos cos ), (sin sin )
2 2
cos , sin
c. 2 2
b
0,
d. 2

73. 71 Problem
The lines a1x + b1y + c1 = 0 and z2x + b2y + c2 = 0 are perpendicular to each other
if :
a. a1b2 – b1a2 = 0
b. a1b2 + b1a2 = 0
2 2
c. a1 b2 b1 a2 0
d. a1b1 – b2a2 = 0

74. 72 Problem
The angle between the pair of straight lines. y 2 sin2 xy sin2 x 2 (cos2 1) 1, is :
a. /3
b. /4
c. 2 /3
d. none of these

75. 73 Problem
If two circles (x - 1)2 + (y - 3)2 = r2 and x2 + y2 – 8x +2y + 8 = 0 intersect in two
distinct point, then :
a. 2 < r < 8
b. r = 2
c. r < 2
d. r > 2

76. 74 Problem
The equation of circle passing through the points (0, 0) (0, b) and (a, b) is :
a. x2 + y2 + ax + by = 0
b. x2 + y2 - ax + by = 0
c. x2 + y2 - ax - by = 0
d. x2 + y2 + ax - by = 0

77. 75 Problem
Two circles S1 = x2 + y2 + 2g1x + 2f1y + c1 = 0 and S1 = x2 + y2 + 2g2x + 2f2y + c2 = 0
each other orthogonally, then :
a. 2g1g2 + 2f1f2 = c1 + c2
b. 2g1g2 - 2f1f2 = c1 + c2
c. 2g1g2 + 2f1f2 = c1 - c2
d. 2g1g2 - 2f1f2 = c1 - c2

78. 76 Problem
The equation 13[(x -1)2 + (y - 2)2] = 3 (2x + 3y - 2)2 represents :
a. Parabola
b. Ellipse
c. Hyperbola
d. None of these

79. 77 Problem
Vertex of the parabola 9x2 – 6x + 36y + 9 = 0 is :
a. (1/3, - 2/9)
b. (-1/3, - 1/2)
c. (- 1/3, 1/2)
d. (1/3, 1/2)

80. 78 Problem
If the foci and vertices of an ellipse be ( 1,0) and ( 2,0) then the minor axis of
the ellipse is :
a. 2 5
b. 2
c. 4
d. 2 3

81. 79 Problem
the one which does not represent a hyperbola is :
a. xy = 1
b. x2 – y2 = 5
c. (x - 1) (y – 3) = 0
d. x2 – y2 = 0

82. 80 Problem
Eccentricity of the conic 16x2 + 7y2 = 112 is :
a. 3/ 7
b. 7/16
c. 3/4
d. 4/3

83. 81 Problem
If f and g are continuous function on [0, a] satisfying f(x) = f(a - x) and g(x) + g(a -
a
x) = 2, then f (x) g (x) =
0
a
a. f (x) dx
0
0
b. f (x) dx
a
a
c. 2 f (x) dx
0
d. none

84. 82 Problem
a
The value of which satisfying sin x dx sin2 , [ (0, 2 )]
/2
equal to :
a. /2
b. 3 /2
c. 7 /6
d. all the above

85. 83 Problem
n
The value of | sin x | dx is :
0
a. 2n + 1 + cos
b. 2n +1 – cos
c. 2n +1
d. 2n + cos

86. 84 Problem
2
The value of | 2 sin x | dx [.] represents greatest integer
function, is :
a. -
b. - 2
c. -5 /3
d. 5 /3

87. 85 Problem
By Simpson's rule, the value of is :
a. 1.358
b. 1.958
c. 1.625
d. 1.458

88. 86 Problem
The value of x0 (the initial value of x) to get the solution in interval (0.5, 0.75) of
the equation x3 – 5x + 3 = 0 by Newton Raphson method is :
a. 0.5
b. 0.75
c. 0.625
d. 0.60

89. 87 Problem
By bisection method, the real root of the equation x3 – 9x + 1 = 0 lying between x
= 2 and x = 4 is nearer to :
a. 2.2
b. 2.75
c. 5.5
d. 4.0

90. 88 Problem
By false-poisitioning, the second approximation of a root of equation f(x) = 0 is
(where, x0, x1 are initial and first approximations resp.)
f (x0 )
a. x0 =
f (x1 ) f (x0 )
b. x0 f (x1 ) x1f (x0 )
f (x1 ) f (x0 )
c. x0 f (x0 ) x1f (x1 )
f (x1 ) f (x0 )
x0 f (x1 ) x1f (x0 )
d.
f (x1 ) f (x0 )

91. 89 Problem
Let f(0) = 1, f(1) = 2.72, then the Trapezoidal rule gives approximation value of
1
f (x) dx is :
0
a. 3.72
b. 1.86
c. 1.72
d. 0.86

92. 90 Problem
If and are imaginary cube roots of unity, then 4 4 1
a. 3
b. 0
c. 1
d. 2

93. 91 Problem
z 5i
the complex number z = x + iy, which satisfy the equation 1 lies on :
z 5i
a. real axes
b. the line y = 5
c. a circle passing through the origin
d. none of these

94. 92 Problem
m
If 1 i
1 , then the least integeral value of m is :
1 i
a. 2
b. 4
c. 8
d. none of these

95. 93 Problem
If 1 , , 2 are the cube roots of unity, then (a + b +c 2)3 + (a + b 2 + c )3 is
equal to a + b + c = 0
a. 27 abc
b. 0
c. 3 abc
d. none of these

96. 94 Problem
If is an imaginary cube root of unity, then (1 + w – w2)7 equals :
a. 128
b. - 128
c. 128 2
d. - 128 2

97. 95 Problem
13
The value of the (i n i n 1 ) where i 1, equals :
n 1
a. i
b. i – 1
c. - i
d. 0

98. 96 Problem
The number of ways in which 6 rings can be worm on four fingers of one hand is
:
a. 46
b. 6c4
c. 64
d. 24

99. 97 Problem
Number greater than 1000 but not greater than 4000 which can be formed with
the digits 1, 2, 3, 4, are :
a. 350
b. 357
c. 450
d. 576

100. 98 Problem
How many can be formed from the letters of the word DOGMATIC if all the
vowels remain together:
a. 4140
b. 4320
c. 432
d. 43

101. 99 Problem
If A = sin2 + cos4 , then for all real values of :
a. 1 A 2
3
b. A 1
4
13
A 1
c. 16
3 13
A
d. 4 16

102. 100 Problem
In triangle ABC, if A = 450, B = 750, then a c 2 =
a. 0
b. 1
c. b
d. 2b

103. FOR SOLUTIONS VISIT WWW.VASISTA.NET