The document contains 42 math problems related to probability, statistics, trigonometry, and calculus. The problems cover topics such as probability, mean, median, mode, standard deviation, functions, limits, and trigonometric functions.

1. AMU –PAST PAPERS
MATHEMATICS - SOLVED PAPER - 1997

2. SECTION – I
 CRITICAL REASONING SKILLS

3. 01 Problem
The probability that a card drawn from a pack of 52 cards will be a diamond or
king is :
a. 1
52
2
b.
13
c. 4
13
d. 1
13

4. 02 Problem
N cadets have to stand in a row if all possible permutations are equally likely, the
probability of two particular cadets standing side by side is :
a. 4
N
b. 3
N2
c. 1
2N
d. 2
N

5. 03 Problem
In a simultaneous throw of 2 coins, the probability of having 2 heads is :
a. 1
4
1
b. 2
1
c.
8
1
d. 6

6. 04 Problem
The probability of getting more than 7 when a pair of dice are thrown is :
7
a. 36
5
b. 12
c. 7
12
d. none of these

7. 05 Problem
If sets A and B are defined as A = {(x, y) | y = ex , x R} B = {B = {(x, y)| y = x, x R},
then :
a. B A
b. A B
c. A B
d. A B A

8. 06 Problem
The co-efficient of variation is computed by :
mean
a. standard deviation
standard deviation
b. mean
mean
c. standard deviation x 100
standard deviation
d. mean x 100

9. 07 Problem
If r is the correlation coefficient, then
a. r 1
b. r 1
c. |r| 1
d. |r| 1

10. 08 Problem
The reciprocal of the mean of the reciprocals of n observation is the :
a. Geometric mean
b. Median
c. Harmonic mean
d. Average

11. 09 Problem
Find the mode from the data given below :
Marks obtained 0-5 5-10 10-15 15-20 20-25 20-30
No. of students 18 20 25 30 16 14
a. 16.3
b. 15.3
c. 16.5
d. none of these

12. 10 Problem
find the median of 18, 35, 10, 42, 21 :
a. 20
b. 19
c. 21
d. 22

13. 11 Problem
The quartile deviation from the following data
x 2 3 4 5 6
f 3 4 8 4 1 is
1
a. 2
1
b. 4
3
c.
4
d. 1

14. 12 Problem
If z k k , then z1z2z3z4 is equal to :
k cos i sin
10 10
a. 1
b. -1
c. 2
d. -2

15. 13 Problem
n
The value of n
pr
r 1 r!
a. 2n
b. 2n – 1
c. 2n –1
d. 2n + 1

16. 14 Problem
the number of parallelograms that can be formed from a set of four parallel lines
intersecting another set of three parallel lines :
a. 6
b. 9
c. 18
d. 12

17. 15 Problem
The probability that in a random arrangement of the letter of the word
‘UNIVERSITY’, the two I’s do not come together is :
a. 4
5
1
b. 10
9
c. 10
1
d. 5

18. 16 Problem
The coefficient of x4 in expansion of (a + x + x2 + x3)n is ;
a. nC
n
b. nC + n C2
n
c. nC + n C1 + n Cn + n C2
n
d. none of these

19. 17 Problem
n 2n
If 1, , 2 are the cube roots of unity, the 1 has the
2n n
1
value : n 2n
1
a. 1
b.
c. 2
d. 0

20. 18 Problem
If x2 x2 y2 z2 z2 for all positive value of x, y and z then :
y x y z x
a. x < y < z
b. x < y > z
c. x < y > z
d. x > y < z

21. 19 Problem
If A and B are independent events such that P (A) > 0, P (B) > 0, then :
a. A and B re mutually exclusive
b. A and B are independent
c. A and B are dependent
d. P(A/B) + P ( A /B) = 1

22. 20 Problem
The least value of the expression 2 log10 0.001- logx0.01 for x > 1 :
a. 2
b. 1
c. 4
d. 3

23. 21 Problem
If p and q are respectively the sum and the sum of the square of n successive
integers beginning with a, nq – p2 is :
a. Independent of a
b. Independent of n
c. Dependent of a
d. None of these

24. 22 Problem
If a + b + c = 0, then the quadratic equation 3ax2 + 2bx + c has :
a. atleast on root in (0, 1)
b. one root is (1, 2) other in (-1, 0)
c. both imaginary
d. none of these

25. 23 Problem
If the root of the equations (x - c) (x - b) – k = 0 are c and b, then roots of the
equation (x - a) (x - d) + k are :
a. a and c
b. b and c
c. a and d
d. a and b

26. 24 Problem
If f :R R be a mapping defined by f(x) = x3 + 5, then f-1 (x) is equal
to :
a. (x + 3)1/3
b. (x - 5)1/3
c. (5 – x)1/3
d. (5 – x)

27. 25 Problem
If ax = b, by = c and cz = a, then xyz is equal to :
a. 1
b. 2
c. -3
d. -1

28. 26 Problem
If the natural numbers are divided into groups as (1, 2, 3), (4, 5, 6).., then 1st
terms of the 10th group will be :
a. 40
b. 45
c. 46
d. 48

29. 27 Problem
The value of 1 3 is :
cos sin
5
3
a.
5
4
b. 5
4
c. 6
5
d. 3

30. 28 Problem
The modulus of 2i 2i is :
a. 2
b. 2
c. 0
d. 2 2

31. 29 Problem
The value of [ 2 {cos (560 15’) + i sin (560 15’)}]8 is :
a. 4i
b. 8i
c. 16i
d. -16i

32. 30 Problem
The sum of two irrational number is always :
a. An irrational number
b. A rational number
c. Both rational number and irrational
d. None of these

33. 31 Problem
1 is equal to :
sin cos
6
a.
2
b. 6
c. 3
3
d. 2

34. 32 Problem
The family of curves represented by dy x2 x 1 and the family
dx y2 y 1
represented by dy y 2 y 1
2
0 :
dx x x 1
a. Touch each other
b. Are orthogonally
c. Are one and the same
d. None of these

35. 33 Problem
dy
The family of curves represented by the differential equation x dx
= cot y is :
a. x cos y = log x
b. x cos y = constant
c. log (x cos y) = x
d. cos y = log x

36. 34 Problem
The differential equation of all parabolas having their axis of symmetry coinciding
with the axis of x is :
2
d2y dy
y 0
a. dx 2 dx
2
d2 x dx
x 0
b. dy 2 dy
d2y dy
c. y 0
dx 2 dx
d. none of these

37. 35 Problem
d3y d2y
For which of the following functions does the property holds :
dx 3 dx 2
a. y = ex
b. y = e-x
c. y = cos x
d. y = sin x

38. 36 Problem
The domain of definition of the function 1 is :
f (x)
|x| x
a. R
b. (0, )
c. (- , 0)
d. none of these

39. 37 Problem
1
The range of the function for real x of y is :
2 sin3 x
1
y 1
a. 3
1
y 1
b. - 3
1
c. - y 1
3
1
d. - y 1
3

40. 38 Problem
The period of the function f(x) = sin 2x 3 is :
6
a. 2
b. 6
c. 6 2
d. 3

41. 39 Problem
A, B, C are three consecutive milestone on a straight road from each of which a
distant spine is visible, the spine is observed to bear with north at A, east at B and
600 east of south at C. Then the shortest distance of the spine from the road is :
7 9 3
a. miles
7 5 3
b. 13 miles
7 5 3
c. 15 miles
7 5 3
d. 17
miles

42. 40 Problem
The smallest positive value of x in tan (x + 1000) = tan (x + 500) tan x .tan (x - 500) :
a. 150 or 300
b. 300 or 800
c. 300 or 450
d. 300 or 550

43. 41 Problem
1 2 n is equal to :
lim ...
n 1 n2 1 n2 1 n2
a. 2
1
b. - 2
c. e-1
d. e2

44. 42 Problem
x
1 equals to :
lim 1
x x
a. e
b. e-2
c. e-1
d. e2

45. 43 Problem
If f(x) = x ( x x 1) then :
a. f(x) is continuous but not differentiable at x = 0
b. f (x) is differentiable at x = 0
c. f(x) is differentiable but not continuous at x = 0
d. f(x) is not differentiable at x = 0

46. 44 Problem
the value of cos2 x dx equals :
1 1
x sin2x c
a. 2 2
1 1
b. x sin 2x +c
2 2
1 1
c. x sin2x
2 2
d. (x + sin 2x) + c

47. 45 Problem
is equal to :
a. x – log | 1 - ex| + c
b. x – log |1 - ex| + c
c. log |1 - ex| + ex + c
d. none of these

48. 46 Problem
Two vectors are said to be equal if :
a. They originate from the same point
b. They meet at the same point
c. They have same magnified and direction
d. None of these

49. 47 Problem
The solution of the differential equation dy 1 ex y is :
dx
a. (x + c)ex + y = 0
b. (x + c)ex - y = 0
c. (x - c)ex + y = 0
d. (x + c)e- x + y = 0

50. 48 Problem
     
If axb c and b x c a, then
a. a = 1, b = c
b. a = 1, c = 1
c. b = 1, c = a
d. b = 2, c = 2a

51. 49 Problem
If the vectors ˆ
(ai ˆ
j ˆ i
k ),(ˆ ˆ
bj ˆ
k ) and ˆ
i ˆ
j ˆ
ck(a b, c 1) are coplanar,
then the value of 1 1 1 is :
1 a 1 b 1 c
a. 1
b. 2
c. 0
d. none of these

52. 50 Problem
1 1 1
The following consecutive terms of a series are in :
1 x'1 x'1 x
a. H.P.
b. G.P.
c. A.P.
d. A.P., G.P.

53. 51 Problem
n
The sum of series S
(n n)!
is :
n 0
a. - e2
1
b. e
c. e2
d. e

54. 52 Problem
If 1, a1, a2, …. an-1 are n roots of unity, then the value of (1 – a1) (1 – a2) …..(1 – an -
1) is :
a. 0
b. 1
c. n
d. n2

55. 53 Problem
Let P (x) = a0 + a1x2 + a2x4 + ….. anx2n be a polynomial in a real variable x with 0 <
a0 < a1 …. < an. The function P(x) has :
a. Neither maximum nor minimum
b. Only one maximum
c. Only one minimum
d. Both maximum and minimum

56. 54 Problem
If A = [aij] is a skew-symmetric matrix of order x, then aij equal to :
a. 0 for some i
b. 0 for all i = 1, 2, …..
c. 1 for some i
d. 1 for all i = 1, 2, …, n

57. 55 Problem
If x and y are matrices satisfying x +y = I and 2x – 2y = I where I is the unit matrix
of order 3, then x equals :
3/4 0 0
0 3/4 0
a. 0 0 0
3 0 4
0 3 0
b. 0 0 0
1 0 1
0 0 0
c. 1 1 1
1 0 0
0 1 0
d. 0 0 1

58. 56 Problem
a b
If A and A2 , then
b a
a. a2 b2 , = ab
b. a2 b2 , = 2ab
c. a2 b2 , = a 2 – b2
d. 2ab, a2 b2

59. 57 Problem
If A is an invertible matrix and B is a matrix, then :
a. rank (AB) = rank (A)
b. rank (AB) = rank (B)
c. rank (AB) > rank (B)
d. rank (AB) > rank (A)

60. 58 Problem
Three lines ax + by + c = 0, cx + ay + b = 0 and bx + cy + a = 0 are concurrent only
when
a. a + b + c = 1
b. a2 + b2 + c2 = ab + bc + ca
c. a3 + b3 + c3 = abc
d. a2 + b2 + c2 = abc

61. 59 Problem
When number x is rounded to P, decimal digits, then magnitude of the relations
error cannot exceed :
a. 0.5 x 10-P+1
b. 0.05 x 10P+2
c. 0.5 x 10P+1
d. 0.05 x 10-P+1

62. 60 Problem
sin2 x cos2 x 1 equals to :
cos2 x sin2 x 1
10 12 2
a. 0
b. 12 cos2 x – 10 sin2 x
c. 12 cos2 x – 10 sin2 x -2
d. 10 sin x

63. 61 Problem
The equation of the sphere passing through the point (1, 3, - 2) and the circle y2 +
x2 = 25 and x = 0 is :
a. x2 + y2 + z2 – 11x + 25 = 0
b. x2 + y2 + z2 + 11x - 25 = 0
c. x2 + y2 + z2 + 11x + 25 = 0
d. x2 + y2 + z2 – 11x - 25 = 0

64. 62 Problem
For which of the following function does the property hold y d2y :
dx2
a. e-3x
b. y = ex
c. e-2x
d. y = e2x

65. 63 Problem
the length of common chord of the circle x2 + y2 + 2x + 3y + 1 = 0 and
x2 + y2 + 4x + 3y + 2 = 0 is :
a. 2 2
b. 4
c. 2
d. 3 2

66. 64 Problem
The radical centre of the circles x2 + y2 = 1, x2 + y2 – 2y = 1 and x2 + y2 – 2x = 1 is :
a. (1, 1)
b. (0, 0)
c. (1, 0)
d. (0, 1)

67. 65 Problem
The natural numbers are grouped as follows 1, (2, 3), (4, 5, 6), (7, 8, 9, 10) ….. the
1st term of the 20th group is :
a. 191
b. 302
c. 201
d. 56

68. 66 Problem
If the two pairs of lines x2 – 2mxy – y2 = 0 and x2 – 2nxy – y2 = 0 are such that one
of them represents the bisector of the angles between the other, then :
a. mn + 1 = 0
b. mn – 1 = 0
1 1
0
c. m n
1 1
0
d. m n

69. 67 Problem
Solution of the equation tan x + tan 2x + tan x . tan 2x = 1 will be :
n
a. x
3 12
b. x n
4
c. x n
4
x n
d. 4

70. 68 Problem
At what point on the parabola y2 = 4x the normal makes equal angles with the
axes ?
a. (4, 3)
b. (9, 6)
c. (4, -4)
d. (1, -2)

71. 69 Problem
The equation x3 + y3 – xy (x + y) + a2 (y - x) represents :
a. Three straight lines
b. A straight line and a rectangular hyperbola
c. A circle and an ellipse
d. A straight line and a ellipse

72. 70 Problem
2
The eccentricity of an ellipse x y2
1 whose latusrectum is half of its
a2 b2
major axis is :
1
a.
2
2
b.
3
3
c.
2
5
d. 2

73. 71 Problem
If cos , cos , cos are direction cosine of a line then value of
sin2 sin2 sin2 is :
a. 1
b. 2
c. 3
d. 4

74. 72 Problem
The curve y – exy + x = 0 has vertical tangent at the point :
a. (1, 1)
b. at no point
c. (0, 1)
d. (1, 0)

75. 73 Problem
x 2 y 3 z 1
The length of perpendicular from the point (3, 4, 5) on the line 2 5 3
is :
a. 17
3
b. 17
17
c. 2
17
d. 5

76. 74 Problem
The area bounded by f (x) x2 , 0 x 1, g(x) x 2,1 x 2 and x-axis is
:
3
a. 2
4
b. 3
8
c. 3
d. none of these

77. 75 Problem
The foot of the perpendicular from P ( , , ) on z-axis is :
a. ( , 0, 0)
b. (0, , 0)
c. (0, 0, )
d. (0, 0, 0)

78. 76 Problem
In a parabola semi-latusrectum is the harmonic mean of the :
a. Segment of a chord
b. Segment of focal chord
c. Segment of the directrix
d. None of these

79. 77 Problem
The plane 2x – 2y + z + 12 = 0 touches the sphere x2 + y2 + z2 – 2x – 4y + 2z – 3 = 0
at the point :
a. (1, 4, 2)
b. (-1, 4, 2)
c. (-1, 4, -2)
d. (1, -4, - 2)

80. 78 Problem
If sin2 x. sin 3x is an identity in x where C0, C1, C2, …. Cn are constant and then the
value of n is :
a. 6
b. 17
c. 27
d. 16

81. 79 Problem
xf (a) af (x)
If f’(a) = 2 and f(a) = 4, then lim equals :
x x a
a. 2a – 4
b. 4 – 2a
c. 2a + 4
d. 4a – 2

82. 80 Problem
If y = cex/(x - a), then dy equals :
dx
a. a (x - a)2
ay
b. -
(x a)2
c. a2 (x - a)2
d. none of these

83. 81 Problem
If cos( ).sin( ) cos( ).cos( ) , then the value of cos .cos .cos
is :
a. cot
b. cot
cot( )
c.
d. cot

84. 82 Problem
If f(x) = loga loga x the f’(x) is :
loga e
a. x loga x
log ea
b. x loga x
loga a
c.
x
x
d. loge a

85. 83 Problem
The equation of the tangent to the curve y = 1 – ex/2 at the point of intersection
with the y-axis is :
a. x + 2y = 0
b. 2x + y = 0
c. x – y = 2
d. none of these

86. 84 Problem
The vectors 2ˆ
i 3ˆ, 4ˆ
j i ˆ and 5ˆ
j i ˆ
yj have their initial points at the origin.
The value of y so that the vectors terminate on one straight line is :
a. -1
1
b. 2
c. 0
d. 1

87. 85 Problem
Let f(x) = ex in [0, 1]. Then, the value of c of the mean value theorem is :
a. 0.5
b. (e- 1)
c. log (e - 1)
d. none

88. 86 Problem
1 1 1
If r, r1, r2, r3 have their usual meanings, the value of is :
r1 r2 r3
a. 1
b. 0
1
c. r
d. r

89. 87 Problem
If then x is equal to :
a. 6
4
b. 3
5
c. 6
2
d. 3

90. 88 Problem
The distance between the foci of a hyperbola is 16 and its eccentricity is 2,
then equation of hyperbola is :
a. x2 + y2 = 32
b. x2 - y2 = 16
c. x2 + y2 = 16
d. x2 - y2 = 32

91. 89 Problem
4R sin A . sin B . sin C is equal to :
a. a + b + c
b. (a + b + c)r
c. (a + b + c)R
r
d. (a + b+ c)
R

92. 90 Problem
The measure of dispersion is :
a. Mean deviation
b. Standard deviation
c. Quartile deviation
d. All a, b and c

93. 91 Problem
The circles x2 + y2 – 4x – 6y – 12 = 0 and x2 + y2 + 4x + 6y + 4 = 0 :
a. Touch externally
b. Touch internally
c. Intersect at two points
d. Do not intersect

94. 92 Problem
If x = my + c is a normal to the prabola x2 = 4ay, then value of c is :
a. - 2am – am3
b. 2am + am3
2a a
c.
m m3
d. 2a a
m m3

95. 93 Problem
A dice is tossed twice. The probability of having a number greater than 3 on each
toss is
1
a. 4
1
b.
3
1
c.
2
d. 1

96. 94 Problem
If f(x) = 3x -1 + 3 - (x - 1) for real x, then the value of f(x) is :
2
a.
3
b. 2
c. 6
7
d. 9

97. 95 Problem
If a function f .[2, ] B defined by f (x) = x2 – 4x + 5 is a bijection, then
B is equal to :
a. R
b. [1, )
c. [2, )
d. [5, )

98. 96 Problem
The minimum value of px + qy when xy = r2 is
a. 2r pq
b. 2pq r
pq
c. -2r
d. none of these

99. 97 Problem
The area cut off from parabola y2 = px by the line y = px is :
a. p3/3
1
b. 2 P2
1
c.
6p
p
d. 6

100. 98 Problem
The graph of y = loga x is reflection of the graph of y = ax in the line :
a. y + x = 0
b. y - x = 0
c. ayx + 1
d. y – ax – 1 = 0

101. 99 Problem
Let Q+ be the set of all positive rational numbers. Let* be an operation on Q+
defined by
ab
a*b= a, b Q . Then, the identity element in Q+ for the operation *
2
is :
a. 0
b. 1
c. 2
1
d. 2

102. 100 Problem
the complex number 1 2i lies in :
1 i
a. I quadrant
b. II quadrant
c. III quadrant
d. IV quadrant

103. FOR SOLUTIONS VISIT WWW.VASISTA.NET