1. This document contains an unsolved mathematics paper from 1999 containing 46 multiple choice problems related to topics like matrices, calculus, probability, and vectors. 2. The problems cover a wide range of mathematical concepts including properties of matrices, limits, derivatives, integrals, probability, and vectors. 3. Multiple choice options are provided for each problem testing conceptual understanding of mathematical definitions, properties, and procedures.

1. AMU –PAST PAPERS
MATHEMATICS - UNSOLVED PAPER - 1999

2. SECTION – I
 CRITICAL REASONING SKILLS

3. 01 Problem
If A and B are non-zero square matrices of the same order such that AB = 0, then :
a. Adj A = 0 or adj B = 0
b. | A | = 0 or | B | = 0
c. adj A = 0 and adj B = 0
d. | A | = 0 and | B | = 0

4. 02 Problem
3 3 4
If A 2 3 4 ,
then A-1 equal to :
0 1 1
a. A
b. A2
c. A3
d. A4

5. 03 Problem
If A, B, C are square matrices of the same order, then which of the following is
true ?
a. AB = AC
b. (AB)2 = A2B2
c. AB = 0 A = 0 or B = 0
d. AB = I AB = BA

6. 04 Problem
The value of is
a. 0
b. abc
c. 4a2b2c2
d. none of these

7. 05 Problem
A speaks truth in 75% and B in 80% of the cases. In what percentage of cases are
they likely to contradict each other narrating the same incident ?
a. 35%
b. 45%
c. 15%
d. 5%

8. 06 Problem
The matrix (a1x1+a2x2+a3x3) is of order :
a. 1 x 3
b. 1 x 1
c. 2 x 1
d. 1 x 2

9. 07 Problem
Which of the following correct for A – B
a. A B
b. A’ B
c. A B’
d. A’ B’

10. 08 Problem
If S denotes the sum to infinity and Sn the sum of n tersm of the series
1 1 1 1
1 ......, such that S Sn , then the least value of n is :
2 4 4 1000
a. 8
b. 9
c. 10
d. 11

11. 09 Problem
The series ( 2 + 1), 1, ( 2 -1)…. is in :
a. A.P.
b. G.P.
c. H.P.
d. None of these

12. 10 Problem
2 sin2 3x is equal to :
lim
x 0 x2
a. 12
b. 18
c. 0
d. 6

13. 11 Problem
sin m2 is equal to :
lim
0
a. 0
b. 1
c. m
d. m2

14. 12 Problem
Let f(x + y) = f(x) + f(y) and f(x) = x2g(x) for all x, y R, where g(x) is continuous
function. Then f’(x) is equal to :
a. g'(x)
b. g(0)
c. g(0) + g’(x)
d. 0

15. 13 Problem
1 1 1 1
The value of is equal to :
r2 r12 r22 r33
a2 b2 c2
a.
a2 b2 c2
b. 2
a2 b2 c2
c. 3
a2 b2 c2
d.

16. 14 Problem
The function x2 1; x 1
f (x) x 1; x 1
2; x 1
a. Continuous for all x
b. Discontinuous at x = -1
c. Discontinuous for all x
d. Continuous x = -1

17. 15 Problem
In the expansion of (1+ x)n, then binomial coefficients of three consecutive terms
are respectively 220, 495 and 792. The value of n is :
a. 10
b. 11
c. 12
d. 13

18. 16 Problem
The number of roots of the quadratic equation 8 sec - sec + 1 = 0 is :
a. Infinite
b. 2
c. 1
d. 0

19. 17 Problem
If 12Pr = 11P6 + 6. 11P5 then r is equal to :
a. 6
b. 5
c. 7
d. none of these

20. 18 Problem
1
The value of the expression ( 3 sin 750 cos 750 ) is :
2
a. 1
b. 2
c. 2
d. 2 2

21. 19 Problem
the number of numbers consisting of four different digits that can be formed with
the digits 0, 1, 2, 3 is :
a. 16
b. 24
c. 30
d. 72

22. 20 Problem
For the curve y = xex, the point :
a. x = -1 is a point of local minimum
b. x = 0 is a point of maximum
c. x = -1 is a point of maximum
d. x = 0 is a point of maximum

23. 21 Problem
the function y = x – cot-1 x – log (x x2 1) is increasing on :
a. (- , 0)
b. ( , 0)
c. (0, )
d. (- , )

24. 22 Problem
If x denotes displacement in time t and x = a cos t, then acceleration is given by :
a. - a sin t
b. a sin t
c. a cos t
d. - a cos t

25. 23 Problem
Let f differentiable for all x. If f (1) = - 2 and f’(x) 2 for all x [1, 6],
2 for all x [1, 6], then :
a. f(6) < 8
b. f(6) 8
c. f(6) 5
d. f(6) 5

26. 24 Problem
0 1
The matrix 1 0
is the matrix of reflection in the line :
a. x = 1
b. y = 1
c. x = y
d. x + y = 1

27. 25 Problem
Let A and B be two matrices then (AB)’ equals :
a. A’B’
b. A’B
c. - AB
d. 1

28. 26 Problem
If at any point on a curve the subtangent and subnormal are equal, then the
tangent is equal to :
a. Ordinate
b. 2 ordinate
c. 2(ordinate)
d. none of these

29. 27 Problem
If f(x) = (x + 1) tan-1 (e-2x), then f’(0) is :
a. 1
2
b. 1
4
c. 5
6
d. none of these

30. 28 Problem
dy
If y = x log x, then dx
is :
a. 1 + log x
b. log x
c. 1 – log x
d. 1

31. 29 Problem
If xy + yz + zx = 1, then :
a. tan-1 x + tan-1 y + tan-1 z = 0
b. tan-1 x + tan-1 y + tan-1 z =
c. tan-1 x + tan-1 y + tan-1 z = 4
d. tan-1 x + tan-1 y + tan-1 z = 2

32. 30 Problem
The order of the differential equation whose solution is : y = a cos x + b sin x + ce-
x is :
a. 3
b. 2
c. 1
d. none of these

33. 31 Problem
If y = a cos px + b sin px, then :
d2y
a. dx 2 + p2y = 0
d2y
b. dx 2 - p2y = 0
d2y
c. dx 2 + py2 = 0
d2y
d. dx 2 - py = 0

34. 32 Problem
1/2 1 x
cos x log dx is equal to :
1/2 1 x
1
a.
2
1
b. - 2
c. 0
d. none of these

35. 33 Problem
dx
equals :
3
x 1 x
1
log( 1 x3 ) c
a. 3
1 1 x3 1
log c
b. 3 1 x 3
1
2 1
log c
c. 3 1 x 3
2 1 x3 1
d. log c
3 1 x3 1

36. 34 Problem
ex (sin h x + cos h x) dx equal to :
a. ex sec h x + c
b. ex cos h x + c
c. sin h 2x + c
d. cos h 2x + c

37. 35 Problem
A man can row 4.5 km/hr in still water and he finds that it takes him twice as long
to row up as to row down the river. The rate of the stream is :
a. 1.5 km/hr
b. 2 km/hr
c. 2.25 km/hr
d. 1.75 km/hr

38. 36 Problem
3 4 10
If m n , then :
4 3 11
a. m = - 2, n = 1
b. m = 22, n = 1
c. m = - 2, n = -23
d. m = 9, n = -10

39. 37 Problem
The area between the curve y = 2x4 – x2, the x-axis and the ordinates of two
minima of the curve is :
a. 7
120
9
b. 120
11
c. 120
15
d. 120

40. 38 Problem
If each of the variable in the matrix a b is doubled, then the value of the
c d
determinant of the matrix is :
a. Not changed
b. Doubled
c. Multiplied by 4
d. Multiplied by 8

41. 39 Problem
A fair coin is tossed repeatedly. If tail appears on first four tosses, then the
probability of head appearing on fifth toss equals :
a. 1
32
1
b. 2
3
c. 2
1
d. 5

42. 40 Problem
The reciprocal of the mean of the reciprocals of n observations is the
a. G.M
b. H.M
c. Median
d. Average

43. 41 Problem
If the area bounded by the parabola x2 = 4y, the x-axis and the line x = 4 is divided
into two equal area by the line x = , then the value of is :
a. 21/3
b. 22/3
c. 24/3
d. 25/3

44. 42 Problem
       
(a 2b c ) {(a b x (a b c )} is equal to :
a.  
[abc ]
 
b. 2 [abc ]
 
c. 3 [abc ]
d. 0

45. 43 Problem
The unit vector perpendicular to the plane determined by A (1, -1, 2), B (2, 0, -1)
and R (0, 2, 1) is :
1
i j ˆ
(2ˆ ˆ k )
a. 6
1 ˆ
b. (2ˆ
i ˆ
j k)
3
1 ˆ
c. (2ˆ
i ˆ
j k)
32
d. none of these

46. 44 Problem
The probability of occurance of an even A is 0.3 and that of occurance of an event
B is 0.4. If A and B are mutually exclusive, then the probability that neither occurs
nor B occurs is :
a. 0.2
b. 0.35
c. 0.3
d. none of these

47. 45 Problem
the probability that a man who is x years old will die in a year in P. Then amongst
n persons A1, A2,…., An each x years old now, the probability that A1 will die in one
year is
1
a. n2
b. 1 – (1 - P)n
1
c. n2
[1 – (1 - P)n]
1
d. n2 [1 – (1 - P)n]

48. 46 Problem
  
the vector a x (b x c )is :

a. parallel to a

b. perpendicular to a
c. parallel to
d. perpendicular to

49. 47 Problem
the next term of the series 3 + 7 + 13 + 21 + 31 + ….
a. 43
b. 45
c. 51
d. 64

50. 48 Problem
If log3 2, log3 (2x - 5) and log3 7 are in A.P., then x is equal to:
2x
2
a. 2
b. 3
c. 4
d. 2, 3

51. 49 Problem
If the radius of a spherical balloon increases by 0.2%. Find the percentage
increase in its volume :
a. 0.8%
b. 0.12%
c. 0.6%
d. 0.3%

52. 50 Problem
3 5 6 x 10 5
If 7 8 9 , then 5 3 6 equal to :
10 x 5 8 7 9
a.
b. -
c. x
d. 0

53. 51 Problem
1 1 5
The positive value of sin sin is :
2 3
a. 5
6
3
b.
5
2
c. 5
2
d. 5

54. 52 Problem
three numbers form an increasing G.P. If the middle number is doubled, then the
new numbers are in A.P. The common ratio of the G.P. is :
a. 2 - 3
b. 2 + 3
c. 3 -2
d. 3 + 3

55. 53 Problem
The nth term of the series 1 (1 2) (1 2 3) ….. is equal to :
2 3
a. n2 (n -1)
(n 1)(2n 1)
b. 2
n 1
c.
2
n(n 1)
d. 2

56. 54 Problem
Two finite sets have m and n element. The total number of subsets of the first
set is 56 more than the total number of subsets of the second set. The value of m
and n are :
a. m = 7, n = 6
b. m = 6, n = 3
c. m = 5, n = 1
d. m = 8, n = 7

57. 55 Problem
1
The domain of f ( x) 1 x2 is :
2x 1
1
a. ,1
2
b. [- 1, [
c. [1, [
d. none of these

58. 56 Problem
sec2 (log x)
The value of dx is :
x
a. tan (log x) + c
b. tan x + c
c. log (tan x) + c
d. none of these

59. 57 Problem
The period of f(x) = cos (x2) is :
a. 2
b. 4 2
2
c.
4
d. none of these

60. 58 Problem
The orthocenter of the triangle formed by the lines xy = 0 and x + y = 1 is :
1 1
a. ,
2 2
1 1
b. ,
3 3
c. (0,0)
1 1
,
d. 4 4

61. 59 Problem
The is acute angle and 4 x 2 sin2 1 = x, then tan is :
2
a. x2 1
b. x2 1
c. x 2
d. none of these

62. 60 Problem
The equation of the locus of a point whose abscissa and ordinate are always
equal is :
a. y + x = 0
b. y – x = 0
c. y + x – 1 = 0
d. y – x + 1 = 0

63. 61 Problem
The distance between the parallel lines y = 2x + 4 and 6x = 3y + 5 is :
a. 17/ 3
b. 1
c. 3/ 5
d. 17 5 /15

64. 62 Problem
The equation y2 – x2 + 2x – 1 = 0, represents :
a. A pair of straight lines
b. A circle
c. A parabola
d. An ellipse

65. 63 Problem
The intercepts made by the circle x2 + y2 –5x – 13y – 14 = 0 on x-axs and y- axis
are respectively:
a. 5,15
b. 6,15
c. 9,15
d. none of thes

66. 64 Problem
The intercepts made by the circle x2 + y2 –5x – 13y – 14 = 0 which are
perpendicular to 3x – 4y –1 = 0 are :
a. 3x + 4y = 3, 3x + 4y + 25 = 0
b. 4x + 3y = 5, 3x + 4y - 25 = 0
c. 3x - 4y = 5, 3x - 4y + 25 = 0
d. none of these

67. 65 Problem
three identical dice are rolled. The probability that the same number will appear
on each of them as :
a. 1
6
1
b. 18
1
c. 9
1
d. 36

68. 66 Problem
3
The principal value of sin is :
2
a. - 6
b. 6
2
c. - 3
2
d. 3

69. 67 Problem
1 3 cos x 4 sin x dy
If y cos then equals :
5 dx
1
a.
1 x3
b. 1
1
c. 1 x3
d. - 1

70. 68 Problem
The point on y2 = 4ax nearest to the focus has its abscissa equal to :
a. a
b. - a
a
c. 2
d. 0

71. 69 Problem
The vertex of the parabola x2 + 8x + 12y + 4 = 0 is :
a. (- 4, 1)
b. (4, - 1)
c. (- 4, -1)
d. (4, 1)

72. 70 Problem
The standard deviation for the data : 7, 9, 11, 13, 15 is :
a. 2.4
b. 2.5
c. 2.7
d. 2.8

73. 71 Problem
While dividing each entry in a data by a non-zero number a, the arithmetic mean
of the new data :
a. Is multiplied by a
b. Does not change
c. Is divided by a
d. Is diminished by a

74. 72 Problem
Two circles which passes through the points A (0, a) and B (0, -a) an touch the
line
y = mx + c will cut orthogonally if :
a. c = a 2 m2
b. a = a 2 m2
c. m2 = a2 (1+ c2)
d. m = - a 1 c2

75. 73 Problem
2 2
If , are the roots of ax2 + bx + c = 0, then equals :
a. c(a b)
a2
b. 0
bc
c. a2
d. abc

76. 74 Problem
The maximum value of 5 sin 3 sin 3 is :
3
a. 11
b. 10
c. 9
d. 12

77. 75 Problem
If x= y = 15, x2 = y2 = 49 xy = 44 and x = 5, then byx is equal to:
a. 1
3
2
b.
3
1
c. 4
1
d. 2

78. 76 Problem
The number of terms which are free from radical sings in the expansion of (x1/5 +
y1/10)55 is :
a. 5
b. 6
c. 11
d. 9

79. 77 Problem
The sum of the co-efficient in the expansion of (x + 2y + x)10 is :
a. 10C
x+y
b. x+yC
10
c. 26.4Cx
d. none of these

80. 78 Problem
There are 10 points in a plane, out of which 4 points are collinear. The number of
triangles formed with vertices as there points is :
a. 20
b. 120
c. 40
d. 116

81. 79 Problem
If the co-ordinate of the centroid of a triangle are (3, 2) and co-ordinates of two
vertices are (4, 1) and (2, 5), then co-ordinates to the third vertex are :
a. (6, 8)
b. (2, 8/3)
c. (0, - 4)
d. (6, 0)

82. 80 Problem
the argument of 1 i 3 is :
1 i 3
4
a. 3
2
b.
3
7
c. 6
d. 3

83. 81 Problem
In how many ways can a constant and a vowel be chosen out of the word
COURAGE ?
a. 7C
2
b. 7P
2
c. 4P x 3P1
1
d. 4P x 3P1
1

84. 82 Problem
The length of the latusrectum of the ellipse 5x2 + 9y2 = 45 is :
a. 5
3
b. 10
3
c. 2 5
5
5
d. 3

85. 83 Problem
The projections of a line segment on the coordinate axes are 12, 4, 3. The
direction cosine of the line are :
12 4 3
a. , ,
13 13 13
12 4 3
b. , ,
13 13 13
12 4 3
c. , ,
13 13 13
d. None of these

86. 84 Problem
n
The least positive value of n if i(1 3) is positive integer, is :
1 i2
a. 1
b. 2
c. 3
d. 4

87. 85 Problem
lim sec loge (2x ) is equal to :
x
1 4x
2
a. 0
b.
2
2
c.
4
d. 2

88. 86 Problem
The distance between the planes gives by ,
 ˆ  ˆ
r .(ˆ
i 2ˆ
j 2k ) 5 0 and r .(ˆ
i 2ˆ
j 2k ) 8 0 is :
a. 1 unit
13
b. 3
units
c. 13 units
d. none of these

89. 87 Problem
If the coefficient of correlation between X and Y is 0.28, covariance between X
and Y is 7.6 and the variance X is 9, then the standard deviation of Y series is :
a. 9.8
b. 10.1
c. 9.05
d. 10.05

90. 88 Problem
1 1 3
If sin tan , then equals :
4
3
a.
5
b. 1
2
c. 5
3
d. 4

91. 89 Problem
(x 1) 2
If 4 , then the value of x 1 is :
x x 2
a. 4
b. 10
c. 16
d. 18

92. 90 Problem
(x 1) x3 1
If 2 cos , then equals :
x x3
1
cos 3
a. 2
b. 2 cos
c. cos3
1
cos 3
d. 3

93. 91 Problem
The mode of the given distribution is :
Weight (in kg) 40 43 46 49 52 55
Number of children 5 8 16 9 7 3
a. 40
b. 55
c. 49
d. 46

94. 92 Problem
The factors of x a b are :
a x b
a b x
a. x – a, x – b and x + a + b
b. x + a, x + b and x + a + b
c. x + a, x + b and x - a - b
d. x – a, x – b and x - a - b

95. 93 Problem
7 1
The equation of a curve passing through 2, and having gradient 1 at (x, y)
2 x2
is :
a. y = x2 + x + 1
b. xy = x2 + x + 1
c. xy = x + 1
d. none of these

96. 94 Problem
The general value of x satisfying is given by cos x = 3 (1 – sin x ) :
a. x n
2
b. x n x
x m ( 1)n
c. 3 6
x n
d. 3

97. 95 Problem
The angle of elevation of the tops of two vertical tower as seen from the middle
point of the line joining the foot of the towers are 600 and 300 respectively. The
ratio of the height of the tower is :
a. 1 : 2
b. 2 : 4
c. 4 : 2
d. 2 : 1

98. 96 Problem
If an angle is divided into two parts A and B such that A – B = x and
tan A : tan B = k : 1, then the value of sin x :
a. k 1
sin
k 1
k
sin
b. k 1
k 1
sin
c. k 1
d. None of this

99. 97 Problem
In triangle ABC and DEF, AB = DE, AC = EF and A 2 E . Two triangles will
have the same area if angle A is equal to :
a.
3
b. 2
2
c. 3
5
d. 6

100. 98 Problem
the even function is :
a. f(x) = x2 (x2 + 1)
b. f(x) = x (x + 1)
c. f(x) = tan x + c
d. f(x) = sin2 x + 2

101. 99 Problem
The middle term in the expansion of (1 + x)2n will be :
a. (n + 1)th
b. (n - 1)th
c. nth
d. (n + 2)th

102. 100 Problem
For the equation | x |2 | | x | - 6 = 0
a. There is only one root
b. There are only two distinct roots
c. There are only three distinct roots
d. There are four distinct roots

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