1. AMU –PAST PAPERS
MATHEMATICS - UNSOLVED PAPER - 1998

2. SECTION – I
 CRITICAL REASONING SKILLS

3. 01 Problem
If a, b,c are in A.P., then the value of x 1 x 2 x a is :
x 2 x 3 x b
x 3 x 4 x c
a. 3
b. - 3
c. 0
d. none of these

4. 02 Problem
The system of simultaneous equations kx + 2y – z = 1, (k - 1) y – 2z = 2 and (k + 2)
z = 3 have a unique solution if k equals :
a. -1
b. -2
c. 0
d. 1

5. 03 Problem
If A and B are Hermition matrices of the same order, then (AB - BA) is :
a. A null matrix
b. A Hermitian matrix
c. A Skew-Hermitian matrix
d. None of these

6. 04 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. {15, 30, 45, …….}
b. {3, 6, 9, ………..}
c. {15, 10, 15, 20, ……}
d. {5, 10, 20, ………..}

7. 05 Problem
A tree is broken by wind, its upper part touches the ground at a point 10 metres
from the foot of the tree and makes an angle of 450 with the ground. The entire
length of the tree is
a. 15 metres
b. 20 metres
c. 10 (1 + 2 ) metres
3
1
d. 10 2 metres

8. 06 Problem
The length of the shadow of a pole is times of the length of the pole. The length of
elevation of the sun is :
a. 450
b. 300
c. 900
d. 600

9. 07 Problem
If sin A = sin B, cos A = cos B, then the value of A in terms of B is :
a. n +B
b. n + (-1)n B
c. 2n + B
d. 2n -B

10. 08 Problem
Cos . cos (90 - ) – sin sin (90 - ) equals :
a. 1
b. 2
c. - 1
d. 0

11. 09 Problem
If in a triangle r r , then the triangle is :
1 1
1 1
2
r2 r3
a. Right angled
b. Isosceles
c. Equilateral
d. None of these

12. 10 Problem
The maximum value of 3 cos x + 4 sin x + 5 is :
a. 5
b. 9
c. 7
d. none of these

13. 11 Problem
A box contains 10 mangoes out of which 4 are rotten. 2 mangoes are taken out
together. If one of them is found to be good, the probability that the other is also
good is :
2
a.
3
5
b. 13
8
c. 13
7
d.
13

14. 12 Problem
Ten different letters of an alphabet are given words with five letters are formed
with three letters. The number of words which atleast one letter repeated is :
a. 69760
b. 30240
c. 99748
d. 37120

15. 13 Problem
arg z arg z; z 0 is equal to :
a.
4
b.
c. 0
d. 2

16. 14 Problem
If a,b,c,d,e,f are in A.P., then e-c is equal to :
a. 2(c - a)
b. 2 (d -c)
c. 2 (f - d)
d. (d - c)

17. 15 Problem
a
51 x
51 x
, , 52 x 5 2x
are in A.P, then the value of a is :
2
a. a < 12
b. a 12
c. a 12
d. none of these

18. 16 Problem
The harmonic mean and geometric mean of two positive number be in the ratio
4 : 5, then two numbers are in the ratio is :
a. 1 : 4
b. 4 : 1
c. 3 : 2
d. 2 ; 3

19. 17 Problem
1
The probability of safe arrival of one ship out of five is 5
. The probability of
safe arrival of atleast 3 ship is :
a. 3
52
1
b. 31
184
c.
3125
181
d. 3125

20. 18 Problem
10
The coefficient of x4 in the expansion of x 3 is :
2 x2
405
a.
256
504
b.
259
450
c. 263
540
d. 276

21. 19 Problem
The product of n positive number is unity, then their sum is :
a. Divisible by n
b. A positive integer
1
c. Equal to n
n
d. Never less than n

22. 20 Problem
A sum of money lent on simple interest becomes double in 8 years the same sum
will triple in :
a. 24 years
b. 16 years
c. 32 years
d. 12 years

23. 21 Problem
The period of the function f (x) = sin4 x + cos4 x is :
a.
b.
2
c. 2
d. none of these

24. 22 Problem
2x
Let f ( x) sin 1
, where 0 < x < 1 < f (x) < , then f’(x) is equal to :
1 x2 2
2
a. 1 x2
x
b. 1 x2
2x
c. 1 x2
x
d. 1 x2

25. 23 Problem
tan 2x x is equal to :
lim
x 0 3x sin x
a. 1
2
1
b. - 2
3
c. 2
3
d. - 2

26. 24 Problem
x
If a function f(x) is defined as , x 0
f (x) x 2 then :
0, x 0
a. f(x) is continuous at x = 0 but not differentiable at x = 0
b. f(x) is continuous as well as differentiable at x = 0
c. f(x) is discontinuous at x = 0
d. none of these

27. 25 Problem
Let [x] denotes the greatest inter function and f(x) = [tan2 x,] then :
a. lim f(a) does not exist
x 0
b. f(x) is continuous at x = 0
c. f(x) is discontinuous at x = 0
d. f(0) = 1

28. 26 Problem
If f(x) = (x + 1) tan-1 (e-2x), then f’(0) is :
a. 2 +1
b. 4 -1
c. 6 +5
d. none of these

29. 27 Problem
The angle of intersection to the curve y = x2 , 6y = 7 – x3 at (1, 1) is :
a. 2
b. 4
c. 3
d.

30. 28 Problem
y = [x (x - 3)]2 increases for all values of x lying in the interval :
3
a. 0 < x < 2
b. 0 < x <
c. <x<0
d. 1 < x < 3

31. 29 Problem
The value of function for which the function f(x) = 1 + 2 sin 2x + 3 cos 2x has
maximum value :
a. 3
13
b. 3
c. 13
d. 0

32. 30 Problem
If the line ax + by + c = 0 is a normal to the curve xy = 1, then :
a. a < 0, b > 1
b. a < 0, b < 0
c. a > 0, b > 0
d. a > 0, b < 0

33. 31 Problem
The greatest value of f (x) cos(xe( x ) 7x 2 3x), x [ 1, ) is :
a. - 1
b. 1
c. 0
d. none of these

34. 32 Problem
dx equal to :
x x
e e
a. log (ex + 1) + c
b. log (ex + e-x) + c
c. tan-1 ex + c
d. sin-1 ex + c

35. 33 Problem
/3 x sin x
2
dx is equal to :
/3 cos x
1
a. (4 1)
3
4 5
2 log tan
b. 3 12
4 5
c. log tan
3 12
d. none of these

36. 34 Problem
for any integer n the integral 1 2
ecos x
[cos3 (2x 1)]x dx has the value :
1
a. 0
b.
c. 1
d. 2

37. 35 Problem
The differential equation of y = Ae2x + Be-2x is :
dy
a. dx - 4y = 0
d2y
b. - 4y = 0
dx 2
c. d2y = y2
dx 2
d. d2y -y=0
dx 2

38. 36 Problem
The compound interest on Rs. 800 at 8% per annum compounded annually for 2
years is :
a. Rs. 133.12
b. Rs. 137.38
c. Rs. 130.15
d. Rs. 125. 25

39. 37 Problem
The area of the figure bounded by y = sin x, y = cos x in the first quadrant is :
a. 2( 2 - 1)
b. 3+ 1
c. 2 ( 3 + 1)
d. none of these

40. 38 Problem
The ratio dose the x- axis divide the area of the region bounded by the parabola
y = 4x – x2 and y=x2-x is
a. 12
5
125
b.
4
52
c.
4
15
d. 4

41. 39 Problem
m 3 2n m 0 7
If , then the value of m, n, p, q are
p 1 4p 6 3 22
a. 3, - 4, 2, - 3
b. 4, 2, 3, - 3
c. - 3, - 2, 4, 5
d. - 4, 2, 3, - 3

42. 40 Problem
   
If a, b, c, d be the position vectors of four points A, B, C, D such that :
       
(a d ).(d c) (b d ).(c a) 0, then D is the :
a. centroid of ABC
b. incentre ABC
c. circumcentre of ABC
d. orthocentre ABC

43. 41 Problem
The vectors A 3ˆ
j ˆ
k, B ˆ
i 2 ˆ are adjacent sides of a parallelogram
j
then its area is :
a. 17
b. 41
c. 14
d. 7

44. 42 Problem

A force F 2ˆ
i ˆ
j ˆ
5k is applied at the point A (1, 2, 5). If moment about
the point
ˆ 6ˆ ˆ
B (-1, - 2, 3) is (16i j 2 k ) , then is equal to :
a. 2
b. - 1
c. 0
d. - 2

45. 43 Problem
The value of loga 1 y2 is :
2
(1 y y )
a. loga (1 - y)
b. loga (1 + y)
c. loga (1+ y2)
d. loga (1 – y2)

46. 44 Problem
In ABC the angle B is greater than angle A. If the value of the angles A and
B satisfy, the equation 3 sin x – 4 sin3 x – x = 0. Then the value of angle C is :
2
a. 3
b. 3
c. 5
5
d. 6

47. 45 Problem
1
A and B are two independent events the probability that both A and B occurs is 6
1
and the probability that neither of them occur 3 is , then probability of the
occurane of A is :
1
a.
5
1
b. 3
1
c. 4
1
d. 6

48. 46 Problem
Pair of dice is rolled together till a sum of either 5 or 7 is obtained, then the
probability that 5 comes before 7 is :
4
a.
7
3
b. 7
2
c. 5
5
7
d.

49. 47 Problem
A father has 3 children with atheist one he The probability that he has 2 boys
and one girl is :
1
a.
3
2
b.
3
1
c. 4
2
d. 5

50. 48 Problem
If a, b, c are any real number, then :
a. max (a, b) < max (a, b, c)
b. min (a, b) = (a + b + |a - b|)
c. max (a, b) < min (a, b)
d. max (a, b) < max (a, b, c)

51. 49 Problem
The A.M. of the series 1, 2, 4, 8, 16, …. , 2n is :
a. 2n 1
n
2n 1 1
b. n 1
2n 1
c. n
2n 1
d. n 1

52. 50 Problem
The variance of first n natural numbers is :
n2 1
a.
12
b. (n 1)(2n 1)
n 1
n2 n
c.
n
2n 1
d. n 1

53. 51 Problem
The observation which occur most frequently is known as :
a. Mode
b. Median
c. Weighted mean
d. Mean

54. 52 Problem
i 0
If A = 0 i
then A4n when n is a natural number equals : number equals :
a. I
b. - A
c. - I
d. A

55. 53 Problem
The standard deviation of 35, 40, 42, 36, 27 :
a. 25.8
b. 26.9
c. 26.8
d. 27.8

56. 54 Problem
Which one of the following is a true statement :
1
a. 2 (bxy + byx) < r
1
b. 2 (bxy + byx) = r
1
c. 2
(bxy + byx) > r
d. none of these

57. 55 Problem
If A and B are finite sets then (A - B) (B - A) equals :
a. (A B) – A
b. (A - B) B
c. (A B) – (A B)
d. (A - B) A

58. 56 Problem
Two lines of regression between x and y are given by
y y yyx (x x) andx x bxy (y y), then bxy x byx is :
a. x * y
x
b.
y
c.
x y
x y
d.

59. 57 Problem
The equation of a circle two of whose diameters are 2x – 3y + 12 = 0 and x + 4y –
5 = 0 and whose are a is 154 sq. units, is :
a. x2 + y2 –6x + 46 – 36 = 0
b. x2 + y2 + 6x - 46 – 36 = 0
c. x2 + y2 – 6x + 46 + 25 = 0
d. none of these

60. 58 Problem
Which of the lines are coplanar ?
(x 1) (y 2) (z 3)
(i) 2 3 4
(x-2) (y 3) (z 4)
(ii) 3 4 3
(x-3) (y 4) (z 5)
(iii)
4 5 6
a. (i) only
b. (i) only
c. (i) only
d. all the lines are coplanar

61. 59 Problem
If a line joining two points A (2, 0) and B (3, 1) is rotated about A I anti-clockwise
direction through an angle 150, then the equation of the line in the new position
is :
a. 3x y 2 3
b. 3x y 2 3
c. x 3y 2 3
d. None of these

62. 60 Problem
Area of the quadrilateral formed by the lines | x | + | y | = 1 is :
a. 4
b. 2
c. 8
d. none of these

63. 61 Problem
If a plane meets the coordinate axes at A, B and C, in such a way that the centroid
of ABC is at the point (1, 2, 3), the equation of the plane is :
x y z
1
a. 1 2 3
x y z
b. 1
3 6 9
x y z 1
c. 1 2 3 3
d. none of these

64. 62 Problem
The planes a1x + b1y + c1z = 0 and a2x + b2y + c2z + d2 = 0 and parallel of :
a1 b1 c1
a. a b2 c2
2
a1 b1 c1
b. a2 b2 c2
a1 b1 c1
c.
a2 b2 c2
a1 b1 c1
d. a2 b2 c2

65. 63 Problem
The value of loga 1 y3 is :
1 y y2
a. loga (1 - y)
b. loga (1 + y)
c. loga (1 + y2)
d. loga (1 - y2)

66. 64 Problem
2
If the circles x y2 2ax 2b ' y c 0 and 2x2 + 2y2 + 2ax + 2by + c =
0 intersect orthogonally, then :
a. aa' + bb’ = c + c’
c'
b. aa’ + bb’ = c + 2
c'
c. aa’ + bb ‘ = 2 + c’
d. none of these

67. 65 Problem
The lines 3x – 4y + 4 = 0 and 6x – 8y – 7 = 0 are tangents to the same circle. Then
its radius is :
a. 1
4
1
b.
2
3
c. 4
5
d. 6

68. 66 Problem
If tan2 A = 2 tan2 B + 1, then cos 2A + sin 2B equals :
a. - 1
b. 1
c. 0
d. 2

69. 67 Problem
A function f : R [ 1 ]
,1 defined by f ( x) sin x, R, where R is
the subset of real numbers in one-one and onto if R is the interval :
a. [0,2 ]
2
b. ,
2 2
c. [ , ]
d. [0, ]

70. 68 Problem
On the ellipse 4x2 + 9y2 = 1, the points at which the tangent are parallel to the
line 8x = 9y are :
2 1
a. ,
5 5
3 1
b. ,
5 5
3 1
,
c. 5 5
2 1
,
d. 5 5

71. 69 Problem
The eccentricity of the conic x2 – 4x + 4y2 = 12 is :
3
a.
2
2
b. 3
c. 3
d. none of these

72. 70 Problem
The number of solutions of the equations | x | - 3 | x | + 2 = 0 is :
a. 4
b. 1
c. 3
d. 2

73. 71 Problem
If x is real the function x2 bc has no real values between :
2x b c
a. b and c
b
b. bc and c
c. b2 and c
d. b and c2

74. 72 Problem
an equilateral triangle is inscribed in the parabola y2 = 4ax whose vertex is at the
vertex of the parabola the length of side the triangle is :
a. 12a 3
b. 8a 3
c. 6a 3
d. 10a 3

75. 73 Problem
Focus of the middle points of all chords of the parabola y2 = 4x which are drawn
through the vertex is :
a. y2 = 8x
b. y2 = 2x
c. x2 + 4y2 = 16
d. x2 = 2y

76. 74 Problem
The equation of the conic with focus at (1, -1), directrix along x – y + 1 = 0 and
with eccentricity is :
a. xy = 1
b. x2 – y2 = 1
c. 2xy – 4x + 4y + 1 = 0
d. 2xy + 4x - 4y - 1 = 0

77. 75 Problem
The slope of the tangent at the point (h, k) of the circle x2 + y2 = a2 is :
a. 0
b. 1
c. -1
d. depends on h

78. 76 Problem
Let b the range being all real numbers except a, and b =
(x) ax
x a'
a2 . Then its inverse is :
a. (ax - b)/(x - a)
b. (ax - a)/(ax - b)
c. (bx - a)/(x - a)
d. (a - bx)/(1 - ax)

79. 77 Problem
1 tan x is equal to ;
lim
x
4
1 2 sin x
a. -1
b. 1
c. 2
d. - 2

80. 78 Problem
The value of b for which the function f(x) = sin x – bx + c is decreasing in the
interval is ( , ) given by :
a. b > 1
b. b < 1
c. b 1
d. b 1

81. 79 Problem
d
{log2 (x2 1)} is :
dx
a. x/(x2 + 1) log2
b. x log 2/x2 + 1
c. log2 e/x2 + 1
d. 1
(x 2 1)

82. 80 Problem
The function f(x) = x4 – 62x2 + ax + 9 attains its maximum value on the interval
[0, 2] at x = 1. Then the value of a is :
a. 120
b. - 120
c. 52
d. none of these

83. 81 Problem
If a1, a2, a3, …. an-1 are positive numbers in A.P. and d is their common difference
then
an-1 – a1 equals :
a. nd
b. (n - 2)d
c. (n + 1)d
d. (n - 1)d

84. 82 Problem
1
The probability of India winning a test match against Australia is 2 . Assuming
independence from match to match to match the probability that in a 5 match
series India’s second win occurs at the third test is :
a. 1
2
b. 1
3
c. 1
4
1
d. 5

85. 83 Problem
The degree of the differential equation y32/3 + 2 + 3y2 + y1 = 0 is :
a. 1
b. 2
c. 3
d. none of these

86. 84 Problem
The solution of ydx xdy
3
3x 2 y 2e x dx 0 is :
x 3
a. ex c
y
b. x ex
3
0
y
x 3
c. - ex c
y
d. none of these

87. 85 Problem
For a moderately skewed distribution mean = 34, median = 36, then the mode is
:
a. 35
b. 45
c. 30
d. 40

88. 86 Problem
 
The component of a 4ˆ
i 6 ˆ along
j b 3ˆ
j ˆ
4k is :
1 ˆ ˆ
a. (3 j 4k )
5
18 ˆ ˆ
(3 j 4k )
b. 25
18 ˆ
(3ˆ
j 4k )
c. 13
18 ˆ
(3ˆ
j 4k )
d. 10 13

89. 87 Problem
The equation of the sphere passing through the origin and the points A (a, 0, 0), B
(0, b, 0) and C (0, 0, c) is :
a. x2 + y2 + z2 + ax + by + cz = 0
b. x2 + y2 + z2 - ax - by - cz = 0
c. x2 + y2 + z2 - 2ax - 2by - 2cz = 0
d. none of these

90. 88 Problem
A quadratic equation with rational coefficient can have :
a. Both roots equal and irrational
b. One root real and other imaginary
c. Both roots real and irrational
d. None of these

91. 89 Problem
How many words beginning with T and ending with E can be made (with no letter
repeated) out of the letters of the word ‘TRIANGLE’ ?
a. 1440
b. 8P
6
c. 720
d. 722

92. 90 Problem
If f(x) = ex (a cos x + b sin x) where a, b are constant then f’(x) + 2f(x) is equal to :
a. f'(x)
b. 2f’(x)
c. 3f’(x)
d. 0

93. 91 Problem
dy
If x sin cos , y cos cos 2 then the value of at is :
dx 4
a. 2
b. 1
c. 3
d. 0

94. 92 Problem
1 1 1
If be a complex cube root of unity, then the value of 1 2 1 2 1 2
is
:
a. 0
b. 1
c. -1
d. 2

95. 93 Problem
If A , then Adj. A is equal to :
a.
b.
c.
d.

96. 94 Problem
The A.M. between two quantities a and b is twice as large as the G.M. then a, b is
:
a. 3 /2
b. 2 + 3/2 - 2
c. 2 + 3 /2 - 3
d. 2/ 3

97. 95 Problem
The perpendicular distance of a corner of a unit cube from a diagonal not passing
through it is equal to :
a. 2
b. 3
c. 1/ 3
2 /3
d.

98. 96 Problem
The quartile deviation of daily wages of 7 persons which are RS. 12, 7, 15, 10, 17,
17, 26 is :
a. 7
b. 14.5
c. 9
d. 3.5

99. 97 Problem
x 2y 2, x 2y 8, x, y 0
The maximum value of z = 3x + 2y subjected to is
:
a. 32
b. 24
c. 40
d. none of these

100. 98 Problem
If sin + cosec = 2, sin2 + cosec2 is equal to :
a. 1
b. 4
c. 2
d. none of these

101. 99 Problem
For , 0 and x cos2n y sin2n .z cos2n .sin2n
2 n 0 n 0 n 0
a. xyz = xz + y
b. xyz = xy – z
c. x + y + z = xyz
d. xyz = yz + x

102. 100 Problem
In the group G = {0, 1, 2, 3, 4, 5} under addition modulo 6, (2 + 3-1 + 4)-1 is equal
to :
a. 0
b. 2
c. 3
d. 5

103. FOR SOLUTIONS VISIT WWW.VASISTA.NET