1. AMU –PAST PAPERS
MATHEMATICS- UNSOLVED PAPER - 2001

2. SECTION – I
 Single Correct Answer Type
 There are five parts in this question. Four choices are given for each part and one of them is
correct. Indicate you choice of the correct answer for each part in your answer-book by
writing the letter (a), (b), (c) or (d) whichever is appropriate

3. 01 Problem
The area of the triangle formed by the complex number z, iz, z + iz in the argand
diagram is :
1
a. | z |2
2
b. | z |2
c. 2 | z |2
d. none of these

4. 02 Problem
The points with position vectors 60ˆ 3ˆ, 40ˆ 8ˆ, ai 52ˆare collinear if
i j i j ˆ j
:
a. a = 40
b. a = 20
c. a = - 40
d. a = 30

5. 03 Problem
Which of the following is correct ?
a. A B A A'
(A B)' A' A'
b.
c. (A ' B ') A' A
d. (A B)' A' B '

6. 04 Problem
The locus of the point of intersection of lines x cos y sin a and x sin y cos b
( is a variable) :
a. 2(x2 + y2) = a2 + b2
b. x2 – y2 = a2 – b2
c. x2 + y2 = a2 + b2
d. none of these

7. 05 Problem
If l m m l 3 m3 n3 is :
, then 3
p p r p q3 r3
a. 1
b. lmn
c. - 1
d. 0

8. 06 Problem
The solution of 3x 2 2 = 2x – 1 are :
a. (2, 4)
b. (1, 4)
c. (3, 4)
d. (1, 3)

9. 07 Problem
The two circles x2 + y2 – 2x – 2y – 7 = 0 and (x2 + y2) – 8x + 29y = 0 :
a. Touch externally
b. Touch internally
c. Cut each orthogonally
d. Do not cut each other

10. 08 Problem
In a circle with centre O, AB and CD are two diameter perpendicular to each
other. The length of the Chord AC is :
a. 2AB
1
b. 2AB
c. AB
d. AB

11. 09 Problem
The eccentricity of the conic x2 – 4y = 1 is :
3
a. 2
2
b.
3
c. 2 5
5
d. 2

12. 10 Problem
x2 y2
the value of m for which y = mx + 6 is a tangent to the hyperbola 1 is
100 49
:
a. 20
17
17
b.
20
3
c. 20
20
d. 3

13. 11 Problem
the equation of an ellipse with one vertex at the point (3, 1) the nearer focus
2
at the point (1, 1) and e is :
3
a. (x 3)2 (y 1)2
1
36 20
(x 3)2 (y 1)2
b. 1
20 36
(x 3)2 (y 1)2
c. 1
36 20
(x 3)2 (y 1)2
d. 1
36 20

14. 12 Problem
The line x –y = 1 touches the hyperbola 3x2 – 4y2 = 12 at the point :
a. (- 4, 1)
b. (2, -3)
c. (4, 3)
d. (- 4, - 3)

15. 13 Problem
A rational number lying between 3 and 5 is :
a. 2.132
b. 1.413
c. 3.126
d. 2.561

16. 14 Problem
One root of the equation 5x2 + 13x + m = 0 is reciprocal of the other if m equals
:
a. 0
b. 5
1
c. 6
d. 6

17. 15 Problem
If the equation x2 + px + q = 0 has roots u and v where p,q are non-zero constants.
Then :
1 1
a. qx2 + px + 1 = has roots and
u
b. (x - p) (x + q) = 0 has roots u2 and v2
c. x2 + p2x + q2 = 0 has roots u2 and v2
u u
d. x2 + qx + p = 0 has roots and
u

18. 16 Problem
The number of proper subset of {a, b, c} is :
a. 3
b. 8
c. 6
d. 7

19. 17 Problem
Fifteen coupons are numbered 1, 2, 3, …., 15 respectively. Seven coupons are
selected at random one at a time with replacement. The probability that the
largest number appearing on a selected coupon is 9 is :
6
a. 9
16
6
3
b.
5
4
8
c.
15
5
3
d.
5

20. 18 Problem
If A and B are square matrix of the same order such that AB = A and BA = B, then
A and B are both :
a. Singular
b. Non-singular
c. Idempotent
d. Involutory

21. 19 Problem
a c b a2 c2 b2
If then :
b b a and ' b2 b2 a2
c a c c2 a2 c2
a. = a2b2c2 ’
b. ’ = a2b2c2
c. = abc ’
d. none of these

22. 20 Problem
If p + q + r = 0 = a + b + c, then the value of the determinant pa qb rc is :
qc ra pb
rb pc qa
a. 0
b. pa + qb + rc
c. 1
d. none of these

23. 21 Problem
If two angles of on triangle are 720 and 800 respectively and that of another
triangle are 280 and 720 respectively, then the triangle are :
a. Similar
b. Congruent
c. Obtuse angle
d. Equal in area

24. 22 Problem
x
The range of f ( x) is :
(1 x 2 )
1
a. ,1
2
1
,
b. 2
1 1
,
c. 2 2
,
d.

25. 23 Problem
the period of the function cos 3x is :
a.
b. 2
c. 3
d. 4

26. 24 Problem
A drawer contains 5 brown socks and 4 blue socks well mixed. A man reaches
the drawer and pulls out 2 socks at random, What is the probability that they
match ?
a. 2
9
4
b. 9
5
c. 9
5
d. 8

27. 25 Problem
11
the coefficient of x-3 in expansion of 1 is :
x
x
a. - 33
b. - 330
c. 330
d. 33

28. 26 Problem
If are the roots of ax2 + bx + c = 0 and if , then
a. a(b2 – 4ac) = 4c
b. b2 – 4ac = a
c. a(b2 + 4ac)
d. b2 + 4ac = a

29. 27 Problem
In 9th term of the series 1 + 5 + 13 + 29 ……….. is given by :
a. 505
b. 515
c. 525
d. 171

30. 28 Problem
x sin x
If then f (x) , lim f (x) :
x cos2 x x
a. 0
b.
c. 1
d. none of these

31. 29 Problem
x
Let f(x) be defined for all x > 0 and by continuous. Let f(x) satisfy f = f(x) –f(y)
y
for all x, y and f(o) = 1, Then :
1
f as x 0
a. x
b. f(x) is bounded
c. xf (x) 0 as x 0
d. f(x) = in x

32. 30 Problem
If m and n are any two odd numbers with n < m. The largest integer which divide
all possible numbers of the type m2 = n2, is :
a. 9
b. 8
c. 4
d. 6

33. 31 Problem
If a < 0, then f(x) = eax + e-ax is decreasing for :
a. x > 0
b. x < 0
c. x > 1
d. x < 1

34. 32 Problem
the function f(x) = x2 log x is the interval [1, e] has :
a. a point of maximum and minimum
b. a point of maximum only
c. no point of maximum and minimum [1, e]
d. no point of maximum and minimum

35. 33 Problem
The area of the circle exceeds the area of regular polygon of n sides of equal
perimeter in the ratio of :
a. cot :
n n
b. tan n
:
n
c. cos n
:
n
:
d. sin n n

36. 34 Problem
        1 
If a, b, c an three unit vectors such that b is not parallel to c and a x(b x c ) b,
2
 
then the angle between a and c :
a.
3
b. 2
c.
6
d. 4

37. 35 Problem
   
If a and b are two non-collinear vectors and is a vector coplanar with a and b
then :
a.   
r a b
  
b. r a b
  
c. r ma nb
  
d. r a b

38. 36 Problem
If x [0, 5], then what is the probability that x 2 – 3x + 27 = 0 ?
a. 4
5
1
b.
5
2
c. 5
3
d. 5

39. 37 Problem
b
If f(a + b - x) = f(x), then xf (x) dx is equal to :
a
a b b
a. xf (b x )dx
2 a
a b b
b. f (x)dx
2 a
a b b
c. f ( x )dx
2 a
d. none of these

40. 38 Problem
If 1 x then :
dx tan a b,
1 sin x 2
a. a ,b R
4
b. 5
a ,b R
4
c. a ,b R
4
d. none of these

41. 39 Problem
2 4
If A and B have change of solving problem in physics correctly are 5
and 11
respectively. If the probability of their matching a common is . If the probability of
1
their answer is 105 correct is :
4
a.
5
14
b. 25
41
c. 23
40
d. 41

42. 40 Problem
A letter is known to have come either from LONDON or LEBANON on the
postmark only the two consecutive letter ‘ON’ are legible. The probability that it
come from LONDON is :
a. 12
17
15
b. 17
13
c. 17
14
d. 17

43. 41 Problem
b c c a a b
In a ABC , is equal to :
r1 r2 r3
a. 1
b. r1r2r3
c. 0
d. abc

44. 42 Problem
4 5
The value of cos , cos .cos .cos :
3 7 7 7
a. 16
1
b. 16
11
c. 16
16
d. 11

45. 43 Problem
The circles x2 + y2 – 10x + 16 = 0 and x2 + y2 = r2 intersect each other in two
distinct if :
a. r < 2
b. r > 8
c. 2 < r < 8
d. 2 r 8

46. 44 Problem
If two even A and B one such that P (A /B) = P(A), then
a. B is impossible
b. Vector only
c. B is implied by A
d. None of these

47. 45 Problem
The line y = mx + 1 is tangent to the parabola y2 = 4x if m equal to :
a. 1
b. 2
c. 3
d. 4

48. 46 Problem
The angle between the pair of tagents drawn to the ellipse 3x2 + 2y2 = 5 from the
point (1, 2) is :
6
a. tan 1
5
1 12
b. tan
5
1 12
c. tan
5
1 12
d. tan
5

49. 47 Problem
 ˆ
the radius of the circular section of the sphere by the plane r .(ˆ
i ˆ
j k) 3 3
is
a. 3
b. 4
c. 5
d. none of these

50. 48 Problem
2c
The integral |x c | dx equal :
0
a. C
c
b. 2
c. 4c
d. 2c

51. 49 Problem
tan2 A = 2 tan2 B + 1, then cos 2A equals :
a. sin2 B
b. - sin2 B
c. tan2 B
d. tan B

52. 50 Problem
The value of 3 cosec 200 – sec200 is equal to :
1. 2
2. 1
3. 4
4. -4

53. 51 Problem
There are 6 positive and 8 negative numbers. Four numbers are chosen at
random without replacement and multiplied the probability that the product is
a positive numbers :
a. 0.504
b. 0.503
c. 0.502
d. 0.501

54. 52 Problem
If the line of regression of Y and x and x on y are respectively y = kx + 9 and x = 4y
+ 5, then :
a. 0 k 4
1
b. 0 k
4
1
c. k
4
d. none of these

55. 53 Problem
The correlation coefficient between the variable xi and yi positive, then the
curve passing through (xi yi) :
a. Collinear
b. Straight line
c. Slopes downwards through (xi, yi)
d. Rising upwards to the right

56. 54 Problem
The mean of n items is X . If the first item is increased by 1, second by 2 and so
on, the new mean is :
a. x
X
2
b. X +x
n 1
c. X
2
d. none of these

57. 55 Problem
AB is vertical pole. The end A is on the level ground. C is the middle point of AB. P
is a point on the level ground. The portion BC subtends an angle B at P. If AP = n
AB, then tan B is equal to :
n
a. 2n 2
1
n
b.
n2 1
n
c. 2
n 1
d. none of these

58. 56 Problem
If | k | = 5 and 00 3600 , then the number of different solutions of
3cos 4 sin k is :
a. Zero
b. Two
c. One
d. Infinite

59. 57 Problem
1
If y 2x 1
for real x, then the least value of y is :
2x 1
a. 1
b. 2
c. - 1
d. - 2

60. 58 Problem
If f(x) = (x - 1) (x - 3) (x - 4) (x - 6) + 19 for all real value of x is :
a. Positive
b. Negative
c. Zero
d. None of these

61. 59 Problem
3
The equation of the line with gradient which 2
is concurrent with the lines
4x + 3y – 7 = 0 and 8x + 5y – 1 = 0 is :
a. 3x + 2y – 2 = 0
b. 3x + 2y – 63 = 0
c. 2y – 3x – 2 = 0
d. none of these

62. 60 Problem
         
If X. A 0, X. B 0 and X.C 0 for some non-zero vector X , then [A B C] is
equal to :
a. 2
b. 1
c. 0
d. - 2

63. 61 Problem
The system of simultaneous equations kx + 2y – z = 1, (k - 1)y – 2z = 2 and (k +
2)z = 3 have a solution if k equals :
a. 0
b. - 2
c. 1
d. - 1

64. 62 Problem
A single letter is selected from the word ‘ALIGARH MUSLIM UNIVERISTY’ the
Probability that it is A vowel is :
a. 11.16!
25!
b. 160!
161.10
c. 25!
16.10!
d. 13!

65. 63 Problem
1 2
The principal value of sin sin is :
3
2
a. 3
2
b. - 3
c. 3
4
d. 3

66. 64 Problem
The imaginary part of log 1 ei is :
a.
b. -
c. 2
d. - 2

67. 65 Problem
If the second term of G.P. is 1 and the sun of its infinte number is 4 then its first
term is :
a. 3
b. 2
c. 1
d. 4

68. 66 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. ay = x + 1
d. y – ax – 1 = 0

69. 67 Problem
In a binomial distribution, the mean is 4 and variances is 3. Then its mode is :
a. 4
b. 5
c. 6
d. 7

70. 68 Problem
If be a complex number and arg ( z ) then :
4
a. Re(z) = Im(z) only
b. Re(z) = Im(z) > 0
c. Re(z2) = Im(22)
d. None of these

71. 69 Problem
The number of number that can be formed with the digits 1, 2, 3, 4, 2, 1. so that
odd digits always occupy the odd planes is :
a. 36
b. 72
c. 144
d. 18

72. 70 Problem
1 1 1
If the product of the roots of the equation is 0, then sum
x a x b x c
of its root is :
2bc
a.
b c
2bc
b. b c
b c
c. 2bc
b c
d. 2bc

73. 71 Problem
The domain of f(x) = cos-1 (3x - 1) is :
a. 2
0,
3
b. 2 2
,
3 3
c. 2
0,
3
d. none of these

74. 72 Problem
If 2x2 + 4xy – y2 – 12x – 6y + 15 = 0, where x and y are real, then :
a. x lie between 1 and 2
b. y lie between 1 and 2
c. x and y real numbers
d. none of these

75. 73 Problem
1 3 3
If one end of a diameter of the sphere x2 + y2 + z2 – 2x – 2z – 2 = 0 is then 1 , ,
2 2 2
other end is :
1 1 1
a. 1 ,,
2 2 2
1 3 3
b. 1
2
,
2
,
2
c. 1 1 1
1 , ,
2 2 2
d. none of these

76. 74 Problem
The function, f(x) = log (1 + ax) – log (1 - bx) not defined at x = 0. The value
which should be assigned to at x = 0 so that it is continuous at x = 0 is :
a. a + b
b. a – b
c. a
d. b

77. 75 Problem
The rectangular garden is of 20 x5 metre and a circular garden is of radius 5
metre, the ratio of areas of rectangular garden top circular garden is :
a. 15 :
b. 10 :
c. 4 :
d. 20 :

78. 76 Problem
nC + nCr – 1 + nCr – 2 is equal to :
r
a. n + 1C
r
b. nC
r+1
c. n - 1C
r+1
d. none of these

79. 77 Problem
The greatest possible number of points of intersection of 8 straight lines and 4
circles is :
a. 32
b. 64
c. 76
d. 104

80. 78 Problem
The value of a and b such that x(1 a cos x) b sin x are :
lim 1
x 0 x3
a. (- 1, 1)
b. (- 2, 2)
c. (-3, 3)
d. none of these

81. 79 Problem
The length of the sub tangent to the curve x y 3 at the point (4, 1) is :
a. 2
1
b.
2
c. 3
d. 4

82. 80 Problem
The angel between the planes 2x – y + z = 6 and x + y + 2z = 7 is :
a.
6
b.
4
c. 3
d. 2

83. 81 Problem
A boat goes 70 km in 10 hours along the stream and returns back the same
distance in 14 hours. Find the speed of boat :
a. 10 km/hr
b. 6 km/hr
c. 7 km/hr
d. 5 km/hr

84. 82 Problem
20 men complete one-third of a pieces of work in 20 days. How many more men
should be employed to finish the rest of the work in 25 more days :
a. 10
b. 12
c. 15
d. 20

85. 83 Problem
A point on the curve y = x2 which is closets to the line 2x – y – 4 = 0 is :
a. (0, 0)
b. (1, 1)
c. (2, 4)
d. (3, 9)

86. 84 Problem

The angles of pentagon an in A.P. one of the angles in degree must be :
b
a. 54
b. 72
c. 90
d. 108

87. 85 Problem
If the function f(x) = x3 – 6x2 + ax + b defined on [1, 3], satisfies the Rolles
theorem for 2 3 1 , them :
c
3
a. a = 1, b = 6
b. a = -1, b = 6
c. a = 11, b R
d. none of these

88. 86 Problem
If y 1 / m [x 1 x 2 ] , then (1 + x2)y2 + xy1 is equal to :
a. m2y
b. my2
c. m2y2
d. none of these

89. 87 Problem
If f : R R is given as f(x) = | x | and A = {x R/X > 0}then f-1 (A) equal to
a. R
b. R- {0}
c.
d. A

90. 88 Problem
Four faces of a regular tetrahedron are to be painted with different colors the
number of way in which this can be down is:
a. 1
b. 2!
c. 3!
d. 4!

91. 89 Problem
21
a a
If the (r + 1)th term in the expansion of 3 contains a and b to
b 3 b
one and the same power, then the value of r is :
a. 9
b. 10
c. 8
d. 6

92. 90 Problem
the equation xy = 1 represents :
a. straight line
b. pair of straight line
c. an ellipse
d. a hyperbola

93. 91 Problem
The vertex of the parabola (x - 4)2 + 2y = 9 is :
a. (2, 8)
b. (7, 2)
9
4,
c. 2
9
4,
d. 2

94. 92 Problem
The value of 
2.357 is :
2355
a.
1001
2355
b. 999
2355
c. 1111
d. none of these

95. 93 Problem
After interesting x Ams between 2 and 38, the sum of the resulting progression
is 200. The value of x is :
a. 10
b. 8
c. 9
d. none of these

96. 94 Problem
(2 x) (y 2)
The equation 3, if x = -2, then y equal to :
2 2
a. - 2
b. - 7
c. 7
d. 5

97. 95 Problem
The graph of a linear equation our the set of real is :
a. A point
b. Circle
c. A curve
d. Straight line

98. 96 Problem
If x, y, z are real and distinct, then x2 + 4y2 + 9z2 – 6yz – 3zx – 2xy is always :
a. Non-negative
b. Non-positive
c. Zero
d. None of these

99. 97 Problem
If each of third order determinant of value is multiplied by 4, then value of the
new determinant is :
a.
b. 21
c. 64
d. 128

100. 98 Problem
The number of generators of an infinite cyclic group is :
a. 1
b. 2
c. infinite
d. none of these

101. 99 Problem
The radius of a sphere initially at zero increases at the rate of 5 cm/sec. Then its
volume after 1 sec. Is increasing at the rate of :
a. 50
b. 5
c. 500
d. none of these

102. 100 Problem
1 (x x 3 )1 / 3
The value of dx is :
1/3 x4
a. 6
b. 3
c. 2
d. 2/3

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