1. AMU –PAST PAPERS
MATHEMATICS - UNSOLVED PAPER - 2006

2. SECTION – I
 CRITICAL REASONING SKILLS

3. 01 Problem
If 4a2 + 9b2 + 16c2 = 2 (3ab + 6bc + 4ca), where a, b, c are non-zero numbers, then
a, b, c are in :
a. AP
b. GP
c. HP
d. None of these

4. 02 Problem
1 1 1 4 1 1 1
It is given that  4  4  ...  to   Then 4  4  4  ...  to  is equal to :
14 2 3 90 1 3 5
4
a.
96
4
b.
45
c. 89  4
90
d. none of these

5. 03 Problem
The polynomial (ax2 + bx + c) (ax2 – dx - c), ac  0 , has :
a. Four real roots
b. At least two real roots
c. At most two real roots
d. No real roots

6. 04 Problem
If (3 + i)z = (3 - i) z , then the complex number z is :
a. a(3  i), a  R
a
b. (3  i) , a  R
c. a(3  i), a  R
d. a(3  i), a  R

7. 05 Problem
If e  cos   i sin  , then for the ∆ ABC eiA eiB eiC is :
i
a. i
b. 1
c. - 1
d. none of these

8. 06 Problem
Numbers lying between 999 and 10000 that can be formed from the digits 0, 2, 3,
6, 7, 8 (repetition of digits not allowed) are :
a. 100
b. 200
c. 300
d. 400

9. 07 Problem
In a club election the number of contestants is one more than the number of
maximum candidates for which a voter can vote. If the total number of ways in
which a voter can vote be 126, then the number of contestants is :
a. 4
b. 5
c. 6
d. 7

10. 08 Problem
1 1 1 ……. is equal to :
  
1!(n  1) 3!(n  3)! 5!(n  5)!
2n 4
a. for even value of n only
n!
2n4  1
b.  1 for odd values of n only
n!
c. 2n1 for all values of n
n!
d. none of these

11. 09 Problem
+ ………+ to  is equal to :
2 3
a  b 1a  b 1a  b
    3 a 
a 2 a   
a. log ab
a
b. log b
b
c. log
a
d. none of these

12. 10 Problem
1 1 1
 
The sum of infinite terms of the series (1  a)(2  a) (2  a)(3  a) (3  a)(4  a)+ …..+
to  , where a is a constant, is :
1
a.
1 a
b. 2
1 a
c. 
d. none of these

13. 11 Problem
21
..
98.
The value of log2 log3…log 10099 is equal to :
log100
a. 0
b. 1
c. 2
d. 100!

14. 12 Problem
The value of tan 200 + 2 tan 500 – tan 700 is :
a. 1
b. 0
c. tan 50
d. none of these

15. 13 Problem
A circular ring of radius 3 cm is suspended horizontally from a point 4 cm
vertically above the centre by 4 string attached at equal intervals to its
circumference. If the angles between two consecutive strings be θ , then cos θ is
:
4
a.
5
4
b.
25
c. 16
25
d. none of these

16. 14 Problem
The number of positive integral solutions of the equation
y 3
tan1 x  cos1  sin1 is :
1  y2 10
a. One
b. Two
c. Zero
d. None of these

17. 15 Problem
sin(ax 2  bx  c)
If α is a repeated root of ax2 + bx + c = 0, then lim is :
x  (x   )2
a. 0
b. a
c. b
d. c

18. 16 Problem
log(1  x3 ) is equal to :
lim
x 0 sin3 x
a. 0
b. 1
c. 3
d. none of these

19. 17 Problem
the domain of the function f(x) = loge(x – [x]) is :
a. R
b. R – Z
c. (0 +  )
d. Z

20. 18 Problem
If f(x + y, x - y) = xy, then the arithmetic mean of f(x, y) and f(y, x) is :
a. x
b. y
c. 0
d. none of these

21. 19 Problem
The equations of the three sides of a triangle are x = 2, y + 1 = 0 and x + 2y = 4.
The coordinates of the circumecentre of the triangle are :
a. (4, 0)
b. (2, -1)
c. (0, 4)
d. (- 1, 2)

22. 20 Problem
If the point (a, a) falls between the lines | x + y| = 4, then :
a. | a | = 2
b. | a | = 3
c. | a | < 2
d. | a | < 3

23. 21 Problem
The equation of the image of the pair of rays y = | x | by the line y = 1 is :
a. y = | x | + 2
b. y = | x | - 2
c. y = | x | + 1
d. y = | x | - 1

24. 22 Problem
Let P = (1, 1) and Q = (3, 2). The point R on the x-axis such that PR + RQ is
minimum, is :
5 
a.  3 , 0 
 
b.  1 , 0 
 
3 
c. (3, 0)
d. (5, 0)

25. 23 Problem
L is a variable line such that the algebraic sum of the distances of the points (1,
1), (2, 0) and (0, 2) from the line is equal to zero. The line L will always pass
through :
a. (1, 1)
b. (2, 1)
c. (1, 2)
d. (2, 2)

26. 24 Problem
C1 is a circle of radius 2 touching the x-axis and the y-axis. C2 is another circle of
radius > 2 and touching the axes as well as the circle C1. Then the radius of C2 is :
a. 6 – 4 √2
b. 6 + 4 √2
c. 6 – 4 √3
d. 6 + 4 √3

27. 25 Problem
the locus of a point represented by x  a  t  1  y  a  t  1  is :
2 t 
  2 t 
 
a. an ellipse
b. a circle
c. a pair of straight lines
d. none of these

28. 26 Problem
the locus of the centre of the circle for which one end of a diameter is (1, 1) while
the other end is on the line x + y = 3, is :
a. x + y =1
b. 2 (x - y) = 5
c. 2x + 2y = 5
d. none of these

29. 27 Problem
The locus of the middle points of chords of a parabola which subtend a right
angle at the vertex of the parabola, is :
a. A circle
b. An ellipse
c. A parabola
d. A hyperbola

30. 28 Problem
  x 1 y  2 z  3 x  4 y 1
The point of intersection of the lines   and  z is :
2 3 4 5 2
a. (0, 0, 0)
b. (1, 1, 1)
c. (-1, -1, -1)
d. (1, 2, 3)

31. 29 Problem
The equation of the plane which meets the axes in A, B, C such that the centroid
of the triangle ABC is is given by :
a. x + y + z = 1
b. x + y + z = 2
c. x + y + z =

32. 30 Problem
The image of the point (5, 4, 6) in the plane x + y +2z – 15 = 0 is :
a. (3, 2, 2)
b. (2, 3, 2)
c. (2, 2, 3)
d. (-5, - 4,- 6 )

33. 31 Problem
The radius of the circle x + 2y + 2z = 15, x2 + y2 + z2 – 2y – 4z = 11 is :
a. 2
b. √7
c. 3
d. √5

34. 32 Problem
A straight line which makes angle of 600 with each of y and z – axes, is inclined
with x – axis at angle of :
a. 300
b. 450
c. 600
d. 750

35. 33 Problem
The value of 1 1 1 is :
x x 2 x x 2 x x 2
(2  2 ) (3  3 ) (5  5 )
(2x  2 x )2 (3x  3 x )2 (5x  5 x )2
a. 0
b. 30x
c. 30-x
d. 1

36. 34 Problem
4 1 0 0
The rank of the matrix   is :
 3 0 1 0
6 0 2 0
 
a. 4
b. 3
c. 2
d. 1

37. 35 Problem
The values of a for which the system of equations ax + y + z = 0, x – ay + z = 0, x +
y + z = 0 possesses non-zero solutions, are given by :
a. 1, 2
b. 1, -1
c. 0
d. none of these

38. 36 Problem
If A is skew symmetric matrix of order n and C is a column matrix of order n x 1,
then CTAC is :
a. An identity matrix of order n
b. An identity matrix of order 1
c. A zero matrix of order 1
d. None of these

39. 37 Problem
y z xz xy
If yz zx yx
= kxyz, then the value of k is :
xy zx xy
a. 2
b. 4
c. 6
d. 8

40. 38 Problem
Let f(x) be a polynomial function of the second degree. If f(1) = f(1) and a1, a2, a3
are in A.P., then f’(a1), f’(a2),f’(a3) are in :
a. A.P.
b. G.P.
c. H.P.
d. None of these

41. 39 Problem
The curve given by x + y = exy has a tangent parallel to the y-axis at the point :
a. (0, 1)
b. (1, 0)
c. (1, 1)
d. (- 1, -1)

42. 40 Problem
Let f(x) = 1 + 2x2 + 22 x4 + …+ 210x20. Then f(x) has :
a. More than one minimum
b. Exactly one minimum
c. At least one maximum
d. None of these

43. 41 Problem
4x 2  1
The interval in which the function f (x)  is decreasing, is :
x
a. (-1, -1)
b. (1, 1)
c. (-1, 1)
d. [- 1, 1]

44. 42 Problem
A right circular cylinder which is open at the top and has a given surface area, will
have the greatest volume if its height h and radius r are related by :
a. 2 h = r
b. h = 4 r
c. h = 2r
d. h = r

45. 43 Problem
If f(x) = x2 – 2x + 4 on [1, 5], then the value of a constant c such that is :
a. 0
b. 1
c. 2
d. 3

46. 44 Problem
1  cos 4 x
The value of k for which the function  , x 0 is continuous
f (x)   8x 2
 k x 0
at x = 0, is : 
a. k = 0
b. k = 1
c. k = - 1
d. none of these

47. 45 Problem
If f(x) = cos x cos 2x cos 4x cos 8x cos 16x, then   is :
f ' 
4
a. 2
b. 0
1
c.
2
d. 3
2

48. 46 Problem
Let f (x)  x 2dx and f(0) = 0. Then f(1) is :
 2
(1  x )(1  1  x )2
a. log (1 + 2 )
b. 
log(1  2) 
4

c. log(1  2) 
4
d. none of these

49. 47 Problem
2
The value of 1 [f {g(x)}]1 {g(x)}g’(x)dx, where g(1) = g (2), is equal to :
a. 1
b. 2
c. 0
d. none of these

50. 48 Problem
x 1
If 
0
f (t )dt  x  
x
t f (t ) dt , then the value of f(1) is :
1
a. 2
b. 0
c. 1
d. - 1
2

51. 49 Problem
a na
If f(x) = f(a + x) and  f (x)dx  k, then  f (x) dx is equal to :
0 0
a. n k
b. (n - 1) k
c. (n + 1)k
d. 0

52. 50 Problem
If  b x3dx  0 and if  b x2dx 2 then the values of a and b are respectively

a a 3'
:
a. 1, 1
b. -1, -1
c. 1, -1
d. - 1, 1

53. 51 Problem
A vector has components 2a and 1 with respect to a rectangular Cartesian
system. The axes are rotated through an angle about the origin in the
anticlockwise direction. If the vector has components a + 1 and 1 with respect to
the new system, then the values of a are :
a. 1, - 1/3
b. 0
c. - 1 , 1/3
d. 1, -1

54. 52 Problem
 
Let i j ˆ  j ˆ
a  ˆ  ˆ  k , c  ˆ  k , If b is a vector satisfying
    
a x b  c and a  b  3, then b is :
1 ˆ ˆ
a. (5i  2ˆ  2k)
j
3
1 ˆ ˆ
b. (5i  2ˆ  2k )
j
3
ˆ j ˆ
3i  ˆ  k
c.
d. none of these

55. 53 Problem
    
Let OA  a, OB  10a  2b and OC  b , where O, A and C are non-collinear points.
Let p denote the area of the quadrilateral OABC and q denote the area of the
p
parallelogram with OA and OC as adjacent sides. Then is equal to :
q
a. 4
b. 6
 
|a  b|
c. 
2| a|
 
|a  b|
d. 
2| a|

56. 54 Problem
The value of x so that the four points A = {0, 2, 0}, B = (1, x, 0), C = (1, 2, 0) and
D = (1, 2, 1) are coplanar, is :
a. 0
b. 1
c. 2
d. 3

57. 55 Problem
Constant forces i j ˆ i j ˆ j ˆ
P1  ˆ  ˆ  k , P2  ˆ  ˆ  k and P3  ˆ  k act on a particle at a point a.
The work done when the particle is displaced from the point A to B where
i j ˆ i j ˆ
A  4ˆ  ˆ  k and B  6ˆ  ˆ  3k is :
a. 3
b. 9
c. 20
d. none of these

58. 56 Problem
dy
The solution of the differential equation x  y  x cos x  sin x,
dx
 , is :
given that y = 1, when x = x 
2
a. y = sin x – cos x
b. y = cos x
c. y = sin x
d. y = sin x + cos x

59. 57 Problem
From a point on the ground at a distance 70 feet from the foot of a vertical wall, a
ball is thrown at an angle of 450 which just clears the top of the wall and
afterwards strikes the ground at a distance 30 feet on the other side of the wall.
The height of the wall is :
a. 20 feet
b. 21 feet
c. 10 feet
d. 105 feet

60. 58 Problem
Three coplanar forces acting on a particle are in equilibrium. The angle between
the first and the second is 600 and that between the second and the third is 1500.
The ratio of the magnitude of the forces are :
a. 1 : 1 : 3
b. 1 : 3: 1
c. 3: 1:1
d. 3: 3: 1

61. 59 Problem
A particle having simultaneous velocities 3 m/s, 5 m/s and 7 m/s, is at rest. The
angle between the first two velocities is :
a. 300
b. 450
c. 600
d. 900

62. 60 Problem
A cyclist is beginning to move with an acceleration of 1 m/s2 and a boy, who is 40
m behind the cyclist, starts running at 9 m/s to meet him. The boy will be able to
meet the cyclist after :
a. 6 sec
b. 8 sec
c. 9 sec
d. 10 sec

63. 61 Problem
Two bodies slide from rest down two smooth inclined planes commencing at the
same point and terminating in the same horizontal plane. The ratio of the
velocities attained if inclinations to the horizontal of the planes are 300 and 600
respectively, is :
a. 3 :1
b. 2 : 3
c. 1 : 1
d. 1 : 2

64. 62 Problem
A die is thrown 2n + 1 times. The probability that faces with even numbers show
odd number of times, is :
2n  1
a.
4n  3
n
b.
2n
c. n 1
2n  1
d. none of these

65. 63 Problem
The probability that exactly one of the independent events A and B occurs, is
equal to :
a. P (A) + P (B) + 2P (A B)
b. P (A) + P (B) – P (A B)
c. P(A’) + P (B’) = 2P (A’ B’)
d. None of these

66. 64 Problem
A bag contains 30 tickets, numbered from 1 to 30. Five tickets are drawn at
random and arranged in the ascending order. The probability that the third
number is 20, is :
20
a. C2 x 10C2
30
C5
19
C2 x 10C2
b. 30
C5
19
c. C2 x 11C2
30
C5
d. none of these

67. 65 Problem
The probability that at least one of the events A and B occur is 0.6 If A and B
occur simultaneously with probability 0.2, then P( A)  P(B) is :
a. 0.4
b. 0.8
c. 1.2
d. 1.4

68. 66 Problem
The relation of “congruence modulo” is : 
a. Reflexive only
b. Symmetric only
c. Transitive only
d. An equivalence relation

69. 67 Problem
If flow values of switches x1, x2 and x3- are respectively 0, 0 and 1, then the flow
value of the circuit s = (x’1.x’2.x3) + (x1.x’2.x’3) + (x’1.x2.x’3) is :
a. 0
b. 1
c. 2
d. none of these

70. 68 Problem
In a Boolean Algebra a v (a’ b) is equal to :
a. a v b
b. a  b
c. a’
d. b’

71. 69 Problem
The range of the function f (x)  ( x  1)(3  x ) is :
a. [- 1, 1]
b. (- 1, 1)
c. (- 3, 3)
d. (- 3, 1)

72. 70 Problem
The coefficients of x in the quadratic equation x2 + bx + c = 0 was taken as 17 in
place of 13, its roots were found to be – 2 and –15. The correct roots of the
original equation are :
a. - 10, - 3
b. - 9, - 4
c. - 8, - 5
d. - 7, - 6

73. FOR SOLUTIONS VISIT WWW.VASISTA.NET