The document contains 48 math problems from an unsolved past paper from 2000. The problems cover topics like relations, equations of curves, derivatives, integrals, vectors, probability, and series. They range from single-step problems to multi-part conceptual questions.

1. AMU –PAST PAPERS
MATHEMATICS - UNSOLVED PAPER - 2000

2. SECTION – I
 CRITICAL REASONING SKILLS

3. 01 Problem
Let A = {1, 2, 3, 4} and let R = {(2, 2), (3, 3,), (4, 4), (1, 2)} be a relation on A. Then
R is
a. Reflexive
b. Symmetric
c. Transitive
d. None of these

4. 02 Problem
The equation of family of curve for which the length of the normal is equal to the
radius vector is :
a. y2 x2 = k2
b. y x=k
c. y2 = kx
d. none of these

5. 03 Problem
dy
y = eax cos bx, dx
equals :
a. eax (a cos bx + b sin ax)
b. eax (a cos bx - b sin ax)
c. eax (a sin bx + b sin ax)
d. eax (a sin ax - a cos ax)

6. 04 Problem
dy ax h
The solution of represents a parabola when :
dx by k
a. a = 1, b = 2
b. a = 0, b = 0
c. a = 0, b 0
d. a = 2, b = 1

7. 05 Problem
2
The equation x 2a a(1 ) where a is constant in the parametric
2
,y 2
1 1
equation of the curves :
a. x2 + y2 = - a2
b. x2 - y2 = a2
c. x2 + y2 = a2
d. x2 + y2 - 2a2 = 0

8. 06 Problem
Let z be the set of integers and 0 be binary operation of z defined as a 0 b =a + b -
ab for all a, b z. The inverse of an element a( 1) z is :
a
a.
a 1
b. 1
1 a
c. a 1
a
d. none of these

9. 07 Problem
Which term of the G.P. 2,2 2 , 4, ….. is 64 :
a. 9th term
b. 7th term
c. 4th term
d. 11th term

10. 08 Problem
If two events a and b such that a – b = 6, then the solution of m x a = b for m is :
a. Unique
b. Does not exist
c. Exist when a b
d. None of these

11. 09 Problem
x3
The value of sin x x is :
lim 6
x 0 x5
a. 0
b. 1
60
c. 1
120
d. 1

12. 10 Problem
cos x equals :
lim
x
2 x
2
a. - 6
b. - 1
c.
d. -

13. 11 Problem
kx 2 , if x 2 If (x) is continuous at x = 2, then the value of k :
f (x)
3, if x 2
a. 2
b. 3
2
c. 3
3
d. 4

14. 12 Problem
f (m) f (n)
For which of the following function m n
is constant for all
numbers m and n m n:
a. f(x) = log x
b. f (x) = cos x
c. f(x) = 4x + 7
d. f(x) = x2 + 1

15. 13 Problem
If x and y are two unit vectors and is the angle between them, then |x y|
2
is equal to
a. | sin |
sin
b. 2
c. | 2 sin |
cot
2
d.

16. 14 Problem
     
   a.(b x c ) b.(a x c )
If a, b, c are non-coplanar vector, then       is equal to :
(c x a).b c.(a x b)
a. 0
b. 1
c. 2
d. 13

17. 15 Problem
the general solution of the equation, 3(sin cos ) (sin cos ) 2 is :
a. 2n
4 3
b. 2n
4 12
c. 2n
4 3
d. 2n
6 12

18. 16 Problem
1/2 x sin 1 x
The value of the integral dx, is
0 2
1 x
a. 1 3
2 2
1
b. 2 12 3
c. 1 3
2 12
1 3
d.
2 2

19. 17 Problem
Suppose that the velocity of a moving particle is = 30 – 2t m/sec. The total
distance in metres it travels between the times t = 0 and t = 20 seconds is :
a. 200
b. 225
c. 250
d. 275

20. 18 Problem
9 2
The least value of the function f (x) 4x sin x is :
x
a. 10 -1
b. 11 -1
c. 12 -1
d. 14 -1

21. 19 Problem
The distance between the line 3x + 4y = 9 and 6x + 8y = 15, is :
3
a.
2
3
b.
10
c. 6
d. none of these

22. 20 Problem
The angel between the tangent from the point (4, 3) to the circle x2 + y2 –2x – 4y
= 0 is :
a. 300
b. 450
c. 600
d. 900

23. 21 Problem
The value of dx is :
2 4 3/4
x (x 1)
1/4
a. 1
1 c
x4
b. (x4 + 1)1/4 + c
1/4
1
c. - 1 c
x4
1/4
1
d. 1 c
x4

24. 22 Problem
If standard deviation of a variate x is , then standard deviation of ax b
c
where a,b,c are constant is :
a. a
c
c
b. a
2
c. c
a
b
d. c

25. 23 Problem
The value of the determinant x 1 x 2 x 4 is :
x 3 x 5 x 8
x 7 x 10 x 14
a. - 2
b. x2 + 2
c. 2
d. 3

26. 24 Problem
Given 12 points in a plane, no three of which are collinear. Then number of line
segments can be determined, are :
a. 76
b. 66
c. 60
d. 80

27. 25 Problem
There are 10 true-false questions in a examination. Then these questions can be
answered in :
a. 100 ways
b. 20 ways
c. 512 ways
d. 1024 ways

28. 26 Problem
1 2
The value of ex dx lies in the interval :
0
a. [0, 1]
b. [1, 2]
c. [1, e]
d. [1, 3]

29. 27 Problem
If 30Cn + 2 = 30Cn - 2, then n equals :
a. 8
b. 15
c. 30
d. 32

30. 28 Problem
|x a | equals :
lim
x a x a
a. 2
b. - 1
c. 1
d. 0

31. 29 Problem
If x, y, z are positive integers then (x + y) (y + z) (z + x) is :
a. < 8xyz
b. = 8xyz
c. > 8xyz
d. none of these

32. 30 Problem
The nth term of the series, 1 + 3 + 6 + 10 …….. is :
n(n 1)
a.
2
n 1
b.
2
c. n(n 1)
2
d. n 1
2

33. 31 Problem
If cos , cos , cos are direction cosines of line, then value of
sin2 sin2 sin2 is :
a. 1
b. 2
c. - 1
d. 3

34. 32 Problem
The line lx + my + n = 0 touches the circle x2 + y2 = 1 if :
1
a. l2 + m2 =
n2
b. l2 + m2 = 2n2
n2
c. l2 + m2 = 2
d. l2 + m2 = n2

35. 33 Problem
The value of /2
( tan x cot x )dx, is
0
a.
2
b. 2
c.
2
d. 2

36. 34 Problem
 
If a 3i ˆ
k, b ˆ
i 2ˆ
j are and joint sides of a parallelogram, then its area
is :
a. 1 17
2
1
b. 7
2
c. 41
1
d. 41
2

37. 35 Problem
Forces acting on a particle are represented in magnitude and direction by the



 

sides AB,BC ,CD, and DE , of regular pentagon ABCDE. The resultant of
there forces is :
a. EA
b. AE
c. AE 5
d. EA 5

38. 36 Problem
The value of a third order determined is 5, then this value of the square of the
determinant formed by its co-factors will be :
a. 125
b. 250
c. 25
d. 5

39. 37 Problem
Out of 40 consecutive integers, two are chosen at random, the probability that
their sum is odd is :
a. 14
29
21
b. 29
22
c. 39
20
d. 39

40. 38 Problem
an anti-aircraft gun takes a maximum of four shots at an enemy plane moving
away from it. The probability of hitting the plane at the first, second, third and
fourth shot are 0.4, 0.3, 0.2 and 0.1 respectively. The probability that the gun hits
the plane is :
a. 0.2412
b. 0.21
c. 0.16
d. 0.6976

41. 39 Problem
The area enclosed by the curve y2 = x2 (1 – x2) is :
1
a. 3 sq. units
b. 2 sq. units
3
c. 1 sq. units
4
d. 3 sq. units

42. 40 Problem
The value of cos 200 - 2 cot 200 is :
a. 0
b. -1
c. 2
d. 3

43. 41 Problem
The function f(x) = x4 – 62x2 + ax + 9 attains its maximum value in the interval
[0, 2] at x = 1. Then the value of a is :
a. 120
b. - 120
c. 52
d. 102

44. 42 Problem
If , are the roots of the quadratic equation ax2 + bx + c = 0, then
2 2
2 equals :
a. 0
bc
b.
a2
c. Abc
c(a b)
d. a2

45. 43 Problem
Two equals circle of radius r intersect such that each passes through the centre of
the other. The length of the common chord is :
a. 2
b. 2r
c. 3 r
d. 3

46. 44 Problem
The angle of intersection of the curves y = x2, 6y = 7- x3 at (1, 1) is :
a.
4
b.
3
c. 2
d.

47. 45 Problem
The maximum value of sin x cos x in the interval 0, is attained
6 6 2
at :
a. 12
b.
6
c.
3
d.
2

48. 46 Problem
Origin is a limiting point of a coaxial system of which x2 + y2 – 6x – 8y + 1 = 0 is a
member. The other limiting point is :
a. (- 2, - 4)
3 4
,
b. 25 25
3 4
,
c. 25 25
4 3
d. ,
25 25

49. 47 Problem
A vector has constant magnitude but its direction varies with time. The derivative
of such a vector is always :
a. 0
b. perpendicular to itself
c. parallel to itself
d. a unit of vector

50. 48 Problem
2 3
a b 1 a b 1 a b
Sum of the series : ..... is :
a 2 a 3 a
a. log a – log b
b. log (a - b)
c. e(a - b)/a –1
d. e1 – b/a

51. 49 Problem
If one vertex of an equilateral triangle is at (2, - 1) and the base is x + y – 2 =
0, then the length of each side is :
a. 3
2
b. 2
3
c. 2
3
d. 3
2

52. 50 Problem
The eccentricity of an ellipse whose pair of a conjugate diameter are y = x and 3y
= -2x is
a. 2
3
b. 1
3
1
c.
3
2
d.
3

53. 51 Problem
(8C1 + 8C2 + 8C3 - 8C4 + 8C5 - 8C6 + 8C7 ) equals :
a. 0
b. 1
c. 70
d. 256

54. 52 Problem
The equation of the circle which has its centre at (a, b) and which touches the y-
axis is :
a. x2 + y2 = b2
b. (x - a)2 + (y - b)2 = b2
c. x2 + y2 = a2
d. (x - a)2 + (y - b)2 = a2

55. 53 Problem
The focus of the parabola y2 – 4y – 8x + 4 = 0 is :
a. (1, 1)
b. (1, 2)
c. (2, 1)
d. (0, 2)

56. 54 Problem
Two dice are tossed 6 times. Then the probability that 7 will show an exactly four
of the tosses is :
225
a.
18442
116
b. 20003
125
c. 15552
117
d. 17442

57. 55 Problem
The standard deviation of 7, 9, 11, 13, 15 is :
a. 2.82
b. 2.4
c. 2.7
d. 2.5

58. 56 Problem
x
The period of the function f(x) = 2 sin 2
is :
a.
b. 2
c. 3
d. 6

59. 57 Problem
If A is a square matrix of order n x n and is scalar. Then Adj (A ) is equal to :
a. (Adj.A) n
b. (Adj.A) -n
c. (Adj.A) n-1
d. none of these

60. 58 Problem
cos 1 x
The domain of f (x) is :
[x]
a. [-1, 1]
b. [-1, 0]
c. [-1, 0] {1}
d. [- 1, - 1]

61. 59 Problem
If z is a complex number, then arg z + arg ( z)is equal to :
a. 0
b. 2
c.
2
d. 4

62. 60 Problem
If p, q are the roots of the equation. x2 + mx + m2 + a = 0, then p2 + pq + p2 + a will
be equal to :
a. 0
b. 1
c. - m
d. m2 + a

63. 61 Problem
The co-ordinate of the centre of the sphere, 2x2 + 2y2 + 2z2 – 4x + 6y – 8z – 10 = 0
are :
a. 3
,1, 2
2
3
b. 1, ,2
2
3
c. 1,2,
2
3
d. ,2,1
2

64. 62 Problem
A point moves so that its distance from the x-axis half of its distance from the
origin. The equation of its locus is ;
a. x2 = 2y2
b. x2 = 3y2
c. x = 2y
d. 2x = y

65. 63 Problem
x x2 x3
If x y z and y y2 y3 0 , then xyz is equal to :
z z2 z3
a. 1
b. -1
c. 0
d. x + y + z

66. 64 Problem
On the set I, binary operation * is defined as follows : a*b = a + b + 1 Then
identity element of the group (I, *) is :
a. 1
b. -1
c. 0
d. 2

67. 65 Problem
If n is a positive integer, then (n + 1) (n + 2) (n + 3) ………..(2n) is a multiple of :
a. 2n
b. 2(n + 1)
c. 2(n + 1)
d. 2n

68. 66 Problem
P.I. of the differential equation (D2 – 4D + 3) y = ex, is :
a.
b. ex
1
c. 2
ex
e 4e 3
d. 1
xe x
2

69. 67 Problem
x 1 y z 1 x 4 y z z 5
The value of for which the lines and ,
2 3 4 3 3
are perpendicular is :
a. 6
b. 1
6
c. - 6
1
d. -
6

70. 68 Problem
The area bounded by the parabola y = 2 - x2 and the line x + y = 0 is :
9
a. 2
7
b. 2
17
c. 6
34
d. 7

71. 69 Problem
If f (x) x (1 t) then f(x) is :
log dt,
0 (1 t)
a. An odd function
b. A period function
c. A symmetric function
d. None of these

72. 70 Problem
The pedal equation of the curve r 2 a2 cos 2 is :
a. p = ar3
b. a2p = r3
c. p2 = ar3
d. p = a2r3

73. 71 Problem
n
px 1 5
If the 4th term in the binomial expansion of is , then :
x 2
a. n = 8, p = 6
1
b. n = 8, p = 2
1
c. n = 6, p = 2
d. n = 6, p = 6

74. 72 Problem
If is the angle between the plane 4x – y – 12 = 1 and the line whose direction
ratio’s are (1, -1, 1) then sin given by :
a. 3 6
6
b.
3
3
c.
2
3
d.
6

75. 73 Problem
    
A straight line r a b meets the plane r n 0 in P. The position
vector of P is :
 
 a n
a. a  b
b n
 
b.  a n
a  b
b n
 
c.  a n
a  a
b n
 
 a n 
d. a  a
b n

76. 74 Problem
The arithmetic mean of a set of observations is . If each observation is divided by
then is increased by 10, then the man of the new series is :
a. x
b. x 10
c. x 10
d. x 10

77. 75 Problem
The maximum area of rectangle inscribed in a circle of diameter R is :
a. R2
R2
b. 2
R2
c.
4
R2
d. 8

78. 76 Problem
     
Let holds a (ˆ
i ˆ
j ˆ) and b
pk (ˆ
i j ˆ
ˆ k ) then| a b | | a | | b | for :
a. p = - 1
b. p = 1
c. all real p
d. no real p

79. 77 Problem
The equation whose roots are twice the roots of the equation, x2 – 3x + 3 = 0 is :
a. 4x2 + 6x + 3 = 0
b. 2x2 - 3x + 3 = 0
c. x2 - 3x + 6 = 0
d. x2 - 6x + 12 = 0

80. 78 Problem
x 3 7
If (x + 9) = 0 is a factor of 2 x 2 = 0, then the other factor is :
7 6 x
a. (x - 2) (x - 7)
b. (x - 2) (x - a)
c. (x + 9) (x - a)
d. (x + 2) (x + a)

81. 79 Problem
If cos sin 2 cos , then cos sin is equal to :
a. 2 sin
b. 2 cos
c. 2 tan
d. 2 sec

82. 80 Problem
The total number of ways of selecting six coins out of 20 one rupee coins, 10
fifty paise coins an 7 twenty five paise coins is :
a. 37C
6
b. 56
c. 28
d. 29

83. 81 Problem
The sum of the coefficients of the polynomial (1+x3x2)2143is :
a. 1
b. -1
c. 0
d. 2

84. 82 Problem
The radius of the incircle triangle whose sides are 18, 24 and 30 cm is:
a. 2cms
b. 4cms
c. 6cms
d. 9cms

85. 83 Problem
The equations of tangent to the hyperbola 4x2-3y2=24 which make an angle of
600 with x-axis are:
a. y 3x 10
b. y 10x 3
c. y 10x 3
y 3x 3
d.

86. 84 Problem
Suppose n people enter a chess tournament in which each person is to play one
game against each of the others. The total number of games that will be played in
the tournament is :
n n 1
a. 2
n n 1
b. 2
c. n(n+1)
d. n(n-1)

87. 85 Problem
If the sides of a triangle are 7cm, 4 3 cm and 13 cm, then the smallest angel of
the triangle is :
a. 150
b. 450
c. 300
d. 600

88. 86 Problem
A curve has the parametric equation x- t2 1 and y= b t 2 1 , then
2t 2t
its equation in rectangular Cartesian co-ordinate is :
x2 y2
a. a2 14
b2
b. x2+y2=a2b2
c. b2x2-a2y2=a2b2
d. none of these

89. 87 Problem
1 4 20
The solution set of the equation 1 2 5 0 is :
1 2x 5x 2
a. {0,1}
b. {1,2}
c. {1,5}
d. {2,-1}

90. 88 Problem
If a square matrix satisfies the relation A2+A-I=0 then A-1
a. Exists and equals I+A
b. Exists and equals I-A
c. Exists and equals A2
d. None of these

91. 89 Problem
  2   2
equals :
axb b a
a. 0
 
a.b
b.
2 2
c. 2 a .b
2 2
d. a .b

92. 90 Problem
If x then the (r+1)th term the expansion of (1-x)2 is :
a. (r+1)xr
b. rxr-1
c. rx-r+1
d. (r+1)xr-1

93. 91 Problem
When m varies, the locus of the point of intersection of the straight lines
x y x y 1 is :
m and
a b a b m
a. A parabola
b. A hyperbola
c. An ellipse
d. A circle

94. 92 Problem
1 sin x cos x
The differential coefficient of tan w.r.t x is :
cos x sin x
a. 0
1
b. 2
c. 1
d. 2

95. 93 Problem
The coefficient of correlation between x and y ;
x : 65 66 67 67 69 70 72
y : 67 68 65 68 72 69 71
is given by :
a. 0.5
b. 0.53
c. 0.6
d. 0.7

96. 94 Problem
the length of the subnormal at the point (1, 3) of the curve, y = x2 + x + 1 is :
a. 1
b. 3
c. 9
d. 3 10

97. 95 Problem
The differential equation y dy x a (a is any constant) represents :
dx
a. A set of circles having centre on the y-axis
b. A set of circles centre on the x-axis
c. A set of ellipse
d. None of these

98. 96 Problem
1 2 3 4
The value of the infinte series ……. Is :
2.3 2.5 2.7 2.9
2
a.
3
b. 2e
e
c. 2
1
d. 2e

99. 97 Problem
Distance between the parallel planes 2x – y + 3z + 4 = 0 and 6x – 3y + 9z –3 = 0 is
:
5
a. 3
4
b. 6
5
c. 14
3
d. 2 3

100. 98 Problem
Three numbers m + 2, 4m –6, 3m – 2 are in A.P. in m equals to :
a. 3
b. 2
c. 1
d. 0

101. 99 Problem
The first derivative of the expression (xx + ax) is :
a. xx log x + ax log e
b. xx log x + ax log a
c. xx log x - ax log a
d. xx log x - ax log e

102. 100 Problem
Counters marked 1, 2, 3 are placed in a bag and one is drawn and replaced. The
operation is repeated three times. The chance of obtaining a total of 6 is :
7
a. 27
20
b. 27
13
c. 27
14
d. 27

103. FOR SOLUTIONS VISIT WWW.VASISTA.NET