The document contains 38 multiple choice questions from an unsolved mathematics past paper from 2007. The questions cover topics such as functions, relations, complex numbers, logarithms, trigonometry, matrices, integrals, conic sections, and coordinate geometry.

1. AMU –PAST PAPERS
MATHEMATICS - UNSOLVED PAPER - 2007

2. SECTION – I
 CRITICAL REASONING SKILLS

3. 01 Problem
The function f : R f :R R defined by f(x) = (x - 1) (x - 2) (x - 3) is
a. One-one but not onto
b. Onto but not one-one
c. Both one-one and onto
d. Neither one-one nor onto

4. 02 Problem
If R is an equivalence relation on a set A, then R-1 is
a. Reflexive only
b. Symmetric but not transitive
c. Equivalence
d. None of the above

5. 03 Problem
If the complex numbers z1,z2,z3 are in AP, then they lie on a
a. A circle
b. A parabola
c. Line
d. Ellipse

6. 04 Problem
Let a, b, c be in AP and |a| < 1, |b| < 1, |c| < 1. If
x = 1 + a + a2 + ….. To ,
y = 1 + b + b2 + …...to ,
z = 1 + c + c2 + …… to , then x, y, z are in
a. AP
b. GP
c. HP
d. None of these

7. 05 Problem
a b 1
If loge 2 2
(loge a + loge b), then
a. a = b
b
b. a = 2
c. 2a = b
d. a = b/3

8. 06 Problem
9
The number of real solutions the equation 10
= -3 + x – x2 is
a. 0
b. 1
c. 2
d. none of these

9. 07 Problem
If f(x) = ax + b and g (x) = cx + d, then f{g(x)} = g{(x)} is equivalent to
a. f(a) = g(c)
b. f(b) = g(b)
c. f(d) = g(b)
d. f(c) = g(a)

10. 08 Problem
(1+ i)8 + (1 - i)8 equal to
a. 28
b. 25
c. 24 cos 4
d. 28 cos 8

11. 09 Problem
The value of 3 cosec 200 – sec 200 is
a. 2
b. 4
c. - 4
d. none of these

12. 10 Problem
If x, y, z are in HP, then log (x + z) + log (x – 2y + z) is equal to
a. log (x - z)
b. 2 log (x - z)
c. 3 log (x - z)
d. 4 log (x - z)

13. 11 Problem
The lines 2x – 3y – 5 = 0 and 3x – 4y = 7 are diameters of circle of area 154 sq
unit, then the equation of the circle is
a. x2 + y2 + 2x – 2y – 62 = 0
b. x2 + y2 + 2x –2y – 47 = 0
c. x2 + y2 - 2x + 2y – 47 = 0
d. x2 + y2 - 2x + 2y – 62 = 0

14. 12 Problem
Which of the following is a point on the common chord of the circle
x2 + y2 + 2x – 3y + 6 = 0 ?
x2 + y2 + x – 8y – 13 = 0 ?
a. (1, -2)
b. (1, 4)
c. (1, 2)
d. (1, -4)

15. 13 Problem
The angle of depressions of the top and the foot of a chimney as seen from the
top of a second chimney, which is 150 m high and standing on the same level as
the first are and respectively, then the distance between their tops when
4 5 is
tan and tan
3 2
150
a. 3
M
b. 100 3m
c. 150 m
d. 100 m

16. 14 Problem
If one root is square of the other root of the equation x2 + px + q = 0, then the
relation between p and q is
a. p3 – (3p - 1)q + q2 = 0
b. p3 – (3p + 1)q + q2 = 0
c. p3 + (3p - 1)q + q2 = 0
d. p3 + (3p + 1)q + q2 = 0

17. 15 Problem
100
100
Cm (x - 3)100 – m. 2m is
m 0
a. 100C
47
b. 100C
53
c. -100C53
d. -100C100

18. 16 Problem
If (-3,2) lies on the circle x2 + y2 + 2gx + 2fy + c = 0 which is concentric with the
circle x2 + y2 + 6x + 8y – 5 = 0, then c is equal to
a. 11
b. - 11
c. 24
d. 100

19. 17 Problem
  
If a ˆ
i ˆ
j ˆ, b
k ˆ
i 3ˆ
j ˆ
5k and c 7ˆ
i j ˆ
9ˆ 11k , then the area of
parallelogram having diagonals is
a. 4 6 sq unit
1
b. 21 sq unit
2
c. 6 sq unit
2
d. 6 sq unit

20. 18 Problem
 ˆ  ˆ
The centre of the circle given by r .(ˆ 2ˆ 2k )
i j 15 and | r ( ˆ 2k ) |
j 4 is
a. (0, 1, 2)
b. (1, 3, 4)
c. (-1, 3, 4)
d. none of these

21. 19 Problem
1 5 7
If A = 0 7 9
, then trace of matrix A is
11 8 9
a. 17
b. 25
c. 3
d. 12

22. 20 Problem
The value of the determinant cos sin 1 is
sin cos 1
cos( ) sin( ) 1
a. Independent of
b. Independent of
c. Independent of and
d. None of the above

23. 21 Problem
A committee of five is to be chosen from a group of 9 people. The probability that
a certain married couple will either serve together or not at all, is
a. 1
2
b. 5
9
4
c.
9
d. 2
3

24. 22 Problem
The maximum value of 4 sin2 x – 12 sin x + 7 is
a. 25
b. 4
c. does not exit
d. none of these

25. 23 Problem
If a point P(4, 3) is shifted by a distance unit parallel to the line y = x, then
coordinates of P in new position are
a. (5, 4)
b. (5 + 2 ,4+ 2 )
c. (5 - 2 ,4- 2)
d. none of the above

26. 24 Problem
A straight line through the point A (3, 4) is such that its intercept between the
axis is bisected at A. Its equation is
a. 3x – 4y + 7 = 0
b. 4x + 3y = 24
c. 3x + 4y = 25
d. x + y = 7

27. 25 Problem
If (- 4, 5)is one vertex and 7 x – y + 8 = 0 is one diagonal of a square, then the
equation of second diagonal is
a. x + 3y = 21
b. 2x – 3y = 7
c. x + 7y = 31
d. 2x + 3y = 21

28. 26 Problem
The equation 2x2 – 24xy + 11y2 = 0 represents
a. Two parallel lines
b. Two perpendicular lines
c. Two lines passing through the origin
d. A circle

29. 27 Problem
The tangent at (1, 7) to the curve x2 = y – 6 touches the circle x2 + y2 + 16x +
12y + c = 0 at
a. (6, 7)
b. (-6, 7)
c. (6, -7)
d. (-6, - 7)

30. 28 Problem
The equation of straight line through the intersection of the lines x – 2y = 1 and x
+ 3y = 2 and parallel to 3x + 4y = 0 is
a. 3x + 4y + 5 = 0
b. 3x + 4y – 10 = 0
c. 3x + 4y – 5 = 0
d. 3x + 4y + 6 = 0

31. 29 Problem
dx
equals
sin x cos x 2
1 x
tan c
a. 2 2 8
1 x
tan c
b. 2 2 8
1 x
c. cot c
2 2 8
1 x
cot c
d. 2 2 8

32. 30 Problem
2x 2 3 x 1 1 x
If dx a log b tan c , then value of a and b are
(x 2 1)(x 2 4) x 1 2
a. (1, -1)
b. (-1, 1)
1 1
,
c. 2 2
1 1
,
d. 2 2

33. 31 Problem
cosec4 x dx is equal to
cot3 x
a. cot x + 3
+c
tan3 x
b. tan x + c
3
cot3 x
c. - cot x - 3
+c
tan3 x
d. - tan x - c
3

34. 32 Problem
The value of integral 1 1 x is
dx
0 1 x
a. 2 +1
b. -1
2
c. - 1
d. 1

35. 33 Problem
1 1
The value of I x x dx is
0 2
1
a.
3
1
b. 4
1
c.
8
d. none of these

36. 34 Problem
The slope of tangents drawn from a point (4, 10) to the parabola y2 = 9x are
1 3
a. ,
4 4
1 9
b. ,
4 4
c. 1 1
,
4 3
d. none of these

37. 35 Problem
x2 y2
The line x = at2 meets the ellipse 1 in the real points, iff
a2 b2
a. | t | < 2
b. | t | 1
c. | t | > 1
d. none of these

38. 36 Problem
x y
The eccentricity of the ellipse which meets the straight line 1on the
7 2
x y
axes of x and the straight line 1 on the axis of y and whose axes lie
3 5
along the axes of coordinates, is
3 2
a. 7
2 6
b.
7
c. 3
7
d. none of these

39. 37 Problem
2
If x y2 (a > b) and x2 – y2 = c2 cut at right angles, then
2
1
a b2
a. a2 + b2 = 2c2
b. b2 - a2 = 2c2
c. a2 - b2 = 2c2
d. a2b2 = 2c2

40. 38 Problem
The equation of the conic with focus at (1, -1) directrix along x – y +1 = 0 and with
eccentricity is
a. x2 – y2 = 1
b. xy = 1
c. 2xy – 4x + 4y + 1 = 0
d. 2xy + 4x – 4y – 1 = 0

41. 39 Problem
The sum of all five digit numbers that can be formed using the digits 1, 2, 3, 4, 5
when repetition of digits is not allowed, is
a. 366000
b. 660000
c. 360000
d. 3999960

42. 40 Problem
There are 5 letters and 5 different envelopes. The number of ways in which all the
letters can be put in wrong envelope, is
a. 119
b. 44
c. 59
d. 40

43. 41 Problem
12 22 12 22 32 12 22 32 42
The sum of the series 1 .... is
2! 3! 4!
a. 3e
17
b. 6 e
13
c. e
6
19
d. e
6

44. 42 Problem
The coefficient of xn in the expansion of loga(1 + x) is
( 1)n 1
a. n
b. ( 1)n 1 log e
a
n
n 1
c. ( 1) loge a
n
d. ( 1)a log e
a
n

45. 43 Problem
46 n
If the mean of n observation 12, 22, 32, …, n2 is , then n is equal to
11
a. 11
b. 12
c. 23
d. 22

46. 44 Problem
If a plane meets the coordinate axes at A, B and C in such a way that the centroid
of ABC is at the point (1, 2, 3) the equation of the plane is
x y z
1
a. 1 2 3
x y z
b. 1
3 6 9
x y z 1
c. 1 2 3 3
d. none of these

47. 45 Problem
The projections of a directed line segment on the coordinate axes are
12, 4, 3, The DC’s of the line are
a. 12 4 3
, ,
13 13 13
12 4 3
b. , ,
13 13 13
12 4 3
c. , ,
13 13 13
d. None of these

48. 46 Problem
     
The value of a (b c ) x (a b c) is
 
a. 2 [abc ]
 
b. [abc ]
c. 0
d. none of these

49. 47 Problem
   
Let a 2ˆ
i ˆ
j ˆ
k, b ˆ
i 2ˆ
j ˆ
k and a unit vector c be coplanar. If c is
 
perpendicular to a , then c is equal to
1 ˆ
a. ( ˆ
j k)
2
1
b. i j ˆ
( ˆ ˆ k)
3
1 ˆ
c. (i 2ˆ)
j
5
1 ˆ
d. (i j ˆ
ˆ k)
3

50. 48 Problem
  
If a, b, c are the position vectors of the vertices of an equilateral triangle
whose orthocenter is at the origin, then
  
a. a b c 
0
  
b. a2 b2 c2
  
c. a b c
d. none of these

51. 49 Problem
The points with position vectors 60ˆ
i 3ˆ 40ˆ
j, i 8 ˆ ai
j, ˆ 52 ˆ
j are
collinear, if
a. a = - 40
b. a = 40
c. a = 20
d. none of these

52. 50 Problem
Area lying in the first quadrant 3y and bounded by the circle x2 + y2 = 4, the
line x = and x-axis is
a. sq unit
b. 2 sq unit
c. 3
sq unit
d. none of these

53. 51 Problem
1/ x
The value of lim tan 1
x is
x 2
a. 0
b. 1
c. - 1
d. e

54. 52 Problem
If f(x) = mx 1, x is continuous at x = , then
2 2
sin x n, x
2
a. m = l, n = 0
n
b. m = 1
2
c. n = m
2
d. m = n =
2

55. 53 Problem
The domain of the function f ( x ) 4 x2 is
sin 1 (2 x)
a. [0, 2]
b. [0, 2)
c. [1, 2)
d. [1, 2]

56. 54 Problem
The general solution of the differential equation (1 + y2)dx + (1 + x2)dy = 0 is
a. x – y = c (1 - xy)
b. x – y = c (1 + xy)
c. x + y = c (1 - xy)
d. x + y = c (1 + xy)

57. 55 Problem
3/2
2
The order and degree of the differential equation dy are
1
dx
respectively
d2y
dx 2
a. 2, 2
b. 2, 3
c. 2, 1
d. none of these

58. 56 Problem
1 3 1 1
The matrix A satisfying the equation 0 1
A
0 1
is
1 4
a. 1 0
1 4
b. 1 0
1 4
c. 0 1
d. none of these

59. 57 Problem
The relation R defined on the set of natural numbers as {(a, b) : a differs from b
by 3} is given
a. {(1, 4), (2, 5), (3, 6), ….}
b. {(4, 1), (5, 2), (6, 3), … }
c. {(1, 3), (2, 6), (3, 9), ….}
d. none of the above

60. 58 Problem
dy dx h
The solution of dx by k
represents a parabola when
a. a = 0, b = 0
b. a = 1, b = 2
c. a = 0, b 0
d. a = 2, b = 1

61. 59 Problem
dy 2yx 1
The solution of the differential equation is
dx 1 x2 (1 x 2 )2
a. y (1 + x2) = c + tan-1 x
y
b. c + tan-1 x
1 x2
c. y log (1+ x2) = c + tan-1 x
d. y (1+ x2) = c + sin-1 x

62. 60 Problem
If x, y, z are all distinct and x x2 1 x3 = 0, then the value of xyz is
2 3
y y 1 y
2
z z 1 z3
a. - 2
b. - 1
c. - 3
d. none of these

63. 61 Problem
The probability that at least one of the events A and B occurs 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

64. 62 Problem
If A and B are two events such that P(A) > 0 and P(B) 1, then P( A / B) is equal to
a. 1- P (A/ B )
b. 1- P( A /B)
1 P( A B)
c. P(B)
P( A)
d. P(B)

65. 63 Problem
A letter is taken out at random from ‘ASSISTANT’ and another is taken out from
‘STATISTICS’. The probability that they are the same letters, is
1
a. 45
13
b. 90
19
c.
90
d. none of these

66. 64 Problem
If 3p and 4p are resultant of a force 5p, then angle between 3p and 5p is
1 3
a. sin
5
b. 1 4
sin
5
c. 900
d. none of these

67. 65 Problem
Resultant velocity of two velocities 30 km/h and 60 km/h making an angle 600
with each other is
a. 90 km/h
b. 30 km/h
c. 30 7 km/h
d. none of these

68. 66 Problem
A ball falls of from rest from top of a tower. If the ball reaches the foot of the
tower is 3s, then height of tower is (take g = 10 m/s2)
a. 45 m
b. 50 m
c. 40 m
d. none of these

69. 67 Problem
Two trains A and B 100 km apart are traveling towards each other with starting
speeds of 50 km/h. The train A is accelerating at 18 km/h2 and B deaccelerating
at 18 km/h2. The distance where the engines cross each other from the initial
position of A is
a. 50 km
b. 68 km
c. 32 km
d. 59 km

70. 68 Problem
If 2 tan-1 (cos x) = tan-1 (2 cosec x), then the value of x is
3
a. 4
b.
4
c.
3
d. none of these

71. 69 Problem
Let a be any element in a Boolean algebra B. If a + x = 1 and ax = 0, then
a. x = 1
b. x = 0
c. x = a
d. x = a’

72. 70 Problem
Dual of (x + y) . (x + 1) = x + x . y + y is
a. (x .y) + (x . 0) = x . (x + y) .y
b. (x .y) + (x .1) = x . (x + y) .y
c. (x .y) (x .0) = x . (x + y) .y
d. none of these

73. FOR SOLUTIONS VISIT WWW.VASISTA.NET