1. AMU –PAST PAPERS
MATHEMATICS - UNSOLVED PAPER - 2005

2. SECTION – I
 CRITICAL REASONING SKILLS

3. 01 Problem
~p q is logically equivalent to :
a. P q
b. q p
c. ~ (p q)
d. ~ (q p)

4. 02 Problem
The angle of elevation of the top of an incomplete vertical pillar at a horizontal
distance of 100 m from its base is 450. If the angle of elevation of the top of the
complete pillar at the same point is to be 600, then the height of the incomplete
pillar is to be increased by :
a. 50 2
b. 100 m
c. 100 ( 3 -1)m
d. 100 ( 3 + 1)m

5. 03 Problem
1 2 3 8
What must be the matrix X, if 2X
3 4 7 2
?
1 3
a. 2 1
1 3
b. 2 1
2 6
c. 4 2
2 6
d. 4 2

6. 04 Problem
The value of 1 1 1 is :
bc ca ab
b c c a a b
a. 1
b. 0
c. (a - b) (b - c) (c - a)
d. (a + b) (b + c) (c + a)

7. 05 Problem
441 441 443
The value of 445 446 447
is :
449 450 451
a. 441 x 446 x 4510
b. 0
c. - 1
d. 1

8. 06 Problem
   ˆ ˆ
(a ˆ)ˆ
i i (a ˆ)ˆ
j j (a k )k is equal to :

a. a

b. 2 a

c. 3 a

d. 0

9. 07 Problem
cos 2 sin 2
Inverse of the matrix is :
sin 2 cos 2
a. cos 2 sin 2
sin 2 cos 2
b. cos 2 sin2
sin2 cos 2
cos 2 sin2
c. sin2 cos 2
cos 2 sin 2
sin 2 cos 2
d.

10. 08 Problem
 
a 3, b 4,  
If then a value of for which a b is perpendicular to
 
a b is :
9
a.
16
3
b. 4
3
c. 2
4
d. 3

11. 09 Problem
 
The projection a 2i j ˆ
ˆ 3ˆ 2k on b i j ˆ
ˆ 2ˆ 3k is :
1
a.
14
2
b.
14
c. 14
2
d. 14

12. 10 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. {3, 6, 9, ………… }
b. {5, 10, 15, 20, ………..}
c. {15, 30, 45, …………….}
d. none of the above

13. 11 Problem
The maximum of the function 3 cos x – 4 sin x is :
a. 2
b. 3
c. 4
d. 5

14. 12 Problem
If the distance ‘s’ metres traversed by a particle in t seconds is given by s = t3 – 3
t2, then the velocity of the particle when the acceleration is zero, (in m/s) is :
a. 3
b. - 2
c. - 3
d. 2

15. 13 Problem
For the curve yn = an-1 x if the subnormal at any point is a constant, then n is
equal to :
a. 1
b. 2
c. -2
d. -1

16. 14 Problem
d2 x
If x = A cos 4t + B sin 4t, then is equal to :
dt 2
a. - 16 x
b. 16 x
c. x
d. - x

17. 15 Problem
If tangent to the curve x = at2, y = 2at is perpendicular to x-axis, then its point of
contact is :
a. (a, a)
b. (0, a)
c. (0, 0)
d. (a, 0)

18. 16 Problem
dy 1 cos 2y
The general solution of the differential equation 0 is
dx 1 cos 2x
given by :
a. tan y + cot x = c
b. tan y – cot x = c
c. tan x – cot y = c
d. tan x + cot y = c

19. 17 Problem
3/4 1/3
2
The degree of the differential equation : dy d2y is :
1
dx dx 2
a. 1/3
b. 4
c. 9
d. 3/4

20. 18 Problem
The area enclosed between the curves y = x3 and y = is , (in sq unit) :
a. 5
3
5
b. 4
5
c. 12
12
d. 5

21. 19 Problem
/8
cos3 4 d is equal to :
0
5
a.
3
5
b. 4
1
c. 3
1
d. 6

22. 20 Problem
/2 cos x sin x dx is equal to :
0 1 cos x sin x
a. 0
b. 2
c. 4
d. 6

23. 21 Problem
If ax2 – y2 + 4x – y = 0 represents a pair of lines, then a is equal to :
a. -16
b. 16
c. 4
d. -4

24. 22 Problem
What is the equation of the locus of a point which moves such that 4 times its
distance from the x-axis is the square of its distance from the origin ?
a. x2 – y2 – 4y = 0
b. x2 + y2 – 4|y| = 0
c. x2 + y2 – 4x = 0
d. x2 + y2 – 4|x| = 0

25. 23 Problem
Equation of the straight line making equal intercepts on the axes and passing
through the point (2, 4) is :
a. 4x – y – 4 = 0
b. 2x + y – 8 = 0
c. x + y – 6 = 0
d. x + 2y – 10 = 0

26. 24 Problem
If the area of the triangle with vertices (x, 0) (1, 1) and (0, 2) is 4 square unit, then
the value of x is :
a. - 2
b. - 4
c. - 6
d. 8

27. 25 Problem
is equal to :
lim 2
cot
2
a. 0
b. - 1
c. 1
d.

28. 26 Problem
The co-axial system of circles given by x2 + y2 + 2gx + c = 0 for c < 0 represents :
a. Intersecting circles
b. Non-intersecting circles
c. Touching circles
d. Touching or non-intersecting circles

29. 27 Problem
The radius of the circle passing through the point (6, 2) and two of whose
diameters are x + y = 6 and x + 2y = 4 is :
a. 4
b. 6
c. 20
d. 20

30. 28 Problem
If (0, 6) and (0, 3) are respectively the vertex and focus of a parabola, then its
equation is
a. x2 + 12y = 72
b. x2 - 12y = 72
c. x2 – 12x = 72
d. y2 + 12x = 72

31. 29 Problem
For the ellipse 24x2 + 9y2 – 120x –90y + 225 = 0, the eccentricity is equal to :
a. 2
5
b. 3
5
15
c. 24
1
d. 5

32. 30 Problem
x2 y2 x2 y2 1
If the foci of the ellipse 1 and the hyperbola
16 b2 144 81 25
coincide, then the value of b2 is :
a. 1
b. 7
c. 5
d. 9

33. 31 Problem
The differential coefficient of f(sin x) with respect to x where f(x) = log x is :
a. tan x
b. cot x
c. f(cos x)
1
d. x

34. 32 Problem
1 cos x
If f(x) = , x 0 is continuous at x = 0, then the value of k is :
x
k, x 0
a. 0
1
b. 2
1
c. 4
1
d. - 2

35. 33 Problem
If 1 3i 2 is :
then (3 3 )4
2
a. 16
b. - 16
c. 16
d. 16 2

36. 34 Problem
If y = tan-1 (sec x – tan x), then is equal to :
a. 2
b. -2
1
c. 2
1
d. - 2

37. 35 Problem
1 1
If x 2 cos then x n is equal to :
2 xn
a. 2n cos
b. 2n cos n
c. 2i sin n
d. 2 cos n

38. 36 Problem
1
|1 x | dx is equal to :
1
a. - 2
b. 0
c. 2
d. 4

39. 37 Problem
dx is equal to :
7
x(x 1)
a. x7
log c
x7 1
b. 1 x7
log 7 c
7 x 1
x7 1
c. log c
x7
1 x7 1
log c
d. 7 x7

40. 38 Problem
xe x
dx is equal to :
a. 2 x e x
4 xe x
c
b. x
(2 x 4 x 4)e c
x
c. (2x 4 x 4)e c
x
(1 4 x )e c
d.

41. 39 Problem
dx is equal to :
2
x 2x 2
a. sin-1 (x + 1) + c
b. sin-1 (x + 1) + c
c. tan-1 (x - 1) + c
d. tan-1 (x + 1) + c

42. 40 Problem
If a tangent to the curve y = 6x – x2 is parallel to the line 4x – 2y – 1 = 0, then
point of tangency on the curve is :
a. (2, 8)
b. (8, 2)
c. (6, 1)
d. (4, 2)

43. 41 Problem
0.5737373……… is equal to :
a. 284
497
284
b.
495
568
c. 999
567
d. 990

44. 42 Problem
The number of solutions for the equation x2 – 5 |x| + 6 = 0 is :
a. 4
b. 3
c. 2
d. 1

45. 43 Problem
How many numbers of 6 digits can be formed from the digits of the number
112233 ?
a. 30
b. 60
c. 90
d. 120

46. 44 Problem
The last digit in 7300
a. 7
b. 9
c. 1
d. 3

47. 45 Problem
log x log y log z
If ' then xyz is equal to :
a b b c c a'
a. 0
b. 1
c. - 1
d. 2

48. 46 Problem
The smallest positive integer n for which (1 + i)2n = (1 - i)2n is :
a. 1
b. 2
c. 3
d. 4

49. 47 Problem
If cos-1 p + cos-1 q + cos-1 r = then p2 + q2 + r2 + 2ppr is equal to :
a. 3
b. 1
c. 2
d. - 1

50. 48 Problem
x 5
If sin 1
cosec-1 , then x is equal to :
5 4 2
a. 1
b. 4
c. 3
d. 5

51. 49 Problem
If 0 x
2
and 81sin x 2
81cos x
30, then x is equal to :
a.
6
b. 2
c. 4
3
d. 4

52. 50 Problem
x2 y2
The equation of the director circle of the hyperbola 1 is given by :
16 4
a. x2 + y2 =16
b. x2 + y2 = 4
c. x2 + y2 = 20
d. x2 + y2 = 12

53. 51 Problem
The normals at the extremities of the latus rectum of parabola intersects the axis
at an angle of :
a. Less than 900
b. Greater than 900
c. 900
d. none of the above

54. 52 Problem
The circle x2 + y2 – 8x + 4y + 4 = 0 touches :
a. x-axis
b. y-axis
c. both axes
d. neither x-axis nor y-axis

55. 53 Problem
If A = {1, 2, 3} and B = {3, 8}, then (A B) x ( A B) is :
a. {(3, 1), (3, 3), (3, 8)}
b. {(1, 3), (2, 3), (3, 3), (8, 3)}
c. {(1, 2), (2, 2), (3, 3), (8, 8)}
d. {(8, 3), (8, 2), (8, 1), (8, 8)}

56. 54 Problem
The condition that one root of the equation ax2 + bx + c = 0 may be double of the
other, is
a. b2 = 9ac
b. 2b2 = 9ac
c. 2b2 = ac
d. b2 = ac

57. 55 Problem
The value of k so that x2 + y2 + kx + 4y + 2 = 0 and 2 (x2 + y2) – 4x – 3y + k = 0 cut
orthogonally is :
a. 10
3
8
b. 3
10
c. - 3
8
d. 3

58. 56 Problem
(3 x 1)
4 is equal to :
lim 1
x X 1
a. e12
b. e-12
c. e4
d. e3

59. 57 Problem
If A + B + C = 1800, then A B is equal to :
tan tan
2 2
a. 0
b. 1
c. 2
d. 3

60. 58 Problem
In a triangle ABC, If b = 2, B = 300 then the area of the circumcircle of triangle ABC
in square unit is :
a.
b. 2
c. 4
d. 6

61. 59 Problem
If sin x + sin2 x = 1, then cos12 x + 3 cos10 x + 3 cos8 x + cos6 x is equal to :
a. 1
b. 2
c. 3
d. 0

62. 60 Problem
If R denotes the set of all real number, then the function f :R R defined
f(x) = |x| is :
a. One-one only
b. Onto only
c. Both one-one and onto
d. Neither one-one nor onto

63. 61 Problem
If f(x) = 2x3 + mx2 – 13x + n and 2,3 are roots of the equation f(x) = 0, then the
values of m and n are
a. - 5, - 30
b. - 5, 30
c. 5, 30
d. none of these

64. 62 Problem
If p1, p2, p3 are respectively the perpendiculars from the vertices of a triangle to
the opposite sides, then p1p2p3 is equal to :
a. a2b2c2
b. 2 a2b2c2
4a2 b2c 3
c.
R2
a2 b2 c 2
d. 8R2

65. 63 Problem
If 5 cos 2 cos2 1 0, , then is equal to :
2
a.
3
1 3
, cos
b. 3 5
1 3
c. cos
5
1 3
, cos
d. 3 5

66. 64 Problem
The two forces acting at a point, the maximum effect is obtained when their
resultant is 4N If they act at right angles, then their resultant is 3N. Then the
forces are :
a. 1 1
(2 3)N and (2 3)N
2 2
b. (2 3)N and (2 3)N
1 1
c. (2 2)N and (2 2)N
2 2
(2 2)N and (2 2)N
d.

67. 65 Problem
The resultant R of two forces P and Q act at right angles to P. Then the angle
between the forces is :
1 P
a. cos
Q
1 P
cos
b. Q
1 P
sin
c. Q
1 P
d. sin
Q

68. 66 Problem
A body starts from rest and moves with a uniform acceleration. The ratio of the
distance covered in nth sec to the distance covered in n second is :
2 1
a. n n2
1 1
b.
n2 n
2 1
c. n2 n
2 1
d. n n2

69. 67 Problem
Two points move in the same straight line starting at the same moment from the
same point in the same direction. The first moves with constant velocity u and
the second starts from rest with constant acceleration f, then distance between
the two points will be maximum at time :
2u u
a. t C. t
f 2f
u u2
t t
b. f D. f

70. 68 Problem
The equation of the plane containing the line x 1 y 3 z 2 and
3 2 1
the point (0, 7, -7) is :
a. x + y + z = 1
b. x + y + z = 2
c. x + y + z = 0
d. none of these

71. 69 Problem
A plane passes through a fixed point (p, q) and cut the axes in A, B, C. Then the
locus of the centre of the sphere OABCi. :
p q r
a. 2
x y z
b. p q r
1
x y z
c. p q r
3
x y z
d. None of these

72. 70 Problem
The value of 12.C1 + 32 . C3 + 52 . C5 + … is :
a. n (n -1)n-2 + n. 2n-1
b. n(n - 1)2n-2
c. n(n -1) . 2n-3
d. none of the above

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