1. AMU –PAST PAPERS
MATHEMATICS - UNSOLVED PAPER - 2004

2. SECTION – I
 CRITICAL REASONING SKILLS

3. 01 Problem
The unit's place digit in the number 1325 + 1125 – 325 is :
a. 0
b. 1
c. 2
d. 3

4. 02 Problem
The angle of intersection of the curves y = x2, 6y = 7 – x3 at (1, 1) is :
a.  /4
b.  /3
c.  /2
d. none of these

5. 03 Problem
the value of x for which the equation 1 + r + r2 + …+ rx (1 + r2) (1 + r2) (1 + r4) (1 +
r8) holds is :
a. 12
b. 13
c. 14
d. 15

6. 04 Problem
2
If f (x)  x  1 for energy real number x, then minimum value of f(x) :
2
x 1
a. Does not exist
b. Is equal to 1
c. Is equal to 0
d. Is equal to - 1

7. 05 Problem
The value of a for which the sum of the squares of the roots of the equation
x2 – (a - 2) x – a – 1 = 0 assumes the least value is :
a. 0
b. 1
c. 2
d. 3

8. 06 Problem
Suppose A1, A2….,A30 are thirty sets each having 5 elements and B1, B2, ….., Bn are
30 n
n sets each with 3 elements, let i Ai  j 1 B j  Sand each element of S
1

belongs to exactly 10 of the Ai’s and exactly 9 of the Bj’S. Then n is equal to :
a. 115
b. 83
c. 45
d. none of these

9. 07 Problem
The number of onto mappings from the set A = {1, 2……., 100}to set B = {1, 2} is :
a. 2100 – 2
b. 2100
c. 299 – 2
d. 299

10. 08 Problem
Which of the following functions is inverse of itself ?
1 x
a. f(x) = 1 x
b. f(x) = 3logx
c. f(x) = 3x(x +1)
d. none of these

11. 09 Problem
If f(x) = log (x + x 2  1 ), then f(x) is :
a. Even function
b. Odd function
c. Periodic function
d. None of these

12. 10 Problem
The solution of log99(log2 (log3x)) = 0 is :
a. 4
b. 9
c. 44
d. 99

13. 11 Problem
1 1 1
If n = 1000!, then the value of sum   ....  is :
log2 n log3 n log1000 n
a. 0
b. 1
c. 10
d. 103

14. 12 Problem
If  and  2 are the two imaginary cube roots of unity, then the equation
whose roots are a 317 and a 382 is :
a. x2 + ax + a2 = 0
b. x2 + a2x + a = 0
c. x2 – ax + a2 = 0
d. x2 – a2x + a = 0

15. 13 Problem
14
 (2k  1) (2k  1)  is :
The value of1 
 cos

k 0 15
  i sin
15


a. 0
b. - 1
c. 1
d. i

16. 14 Problem
1, a1, a2, ….., an –1 are roots of unity then the value of (1 – a1) (1 – a2) …..(1 – an-1)
is :
a. 0
b. 1
c. n
d. n2

17. 15 Problem
If  ,  are the roots of ax2 + bx + c = 0;   h,   hare the roots of px2 + qx
+ r = 0 ; and D1, D2 the respective discriminants of these equations, then D1 : D2 is
equal to :
a2
a. p2
b2
b. q2
2
c. c
r2
d. none of these

18. 16 Problem
If a, b, c are three unequal numbers such that a, b, c are in A.P. and b – a, c – b, a
are in G.P., then a : b : c is :
a. 1 : 2 : 3
b. 1 : 3 : 4
c. 2 : 3 : 4
d. 1 : 2 : 4

19. 17 Problem
The number of divisors of 3 x 73, 7 x 112 and 2 x 61 are in :
a. A.P.
b. G.P.
c. H.P.
d. None of these

20. 18 Problem
Suppose a, b, c are in A.P. and | a |, | b |, | 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. A.P.
b. G.P.
c. H.P.
d. None of these

21. 19 Problem
4 7 10 is :
1  2  3  .....to 
5 5 5
16
a. 35
11
b.
8
35
c.
16
d. 7
16

22. 20 Problem
If the sum of first n natural numbers is 1 times the sum of their cubes, then
78
the value of n is :
a. 11
b. 12
c. 13
d. 14

23. 21 Problem
If p = cos 550, q = cos 650 and r = cos 1750, then the value of 1  1  r is :
p q pq
a. 0
b. -1
c. 1
d. none of these

24. 22 Problem
The value of sin 200(4 + sec200) (4 + sec 200) is :
a. 0
b. 1
2
c.
d. 3

25. 23 Problem
If 4 sin x  cos x   , then x is equal to :
1 1
a. 0
1
b.
2
1
c. -
2
d. 1

26. 24 Problem
1 1 1
If the line x  y  1 moves such that a2
 2  2
b c
where c is a constant, then
a b
the locus of the foot of the perpendicular from the origin to the line is :
a. Straight line
b. Circle
c. Parabola
d. Ellipse

27. 25 Problem
The straight line whose sum of the intercepts on the axes is equal to half of the
product of the intercepts, passes through the point :
a. (1, 1)
b. (2, 2)
c. (3, 3)
d. (4, 4)

28. 26 Problem
If the circle x2 + y2 + 4x + 22y + c = 0 bisects the circumference of the circle
x2 + y2 –2x + 8y –d = 0, then c + d is equal to :
a. 60
b. 50
c. 40
d. 30

29. 27 Problem
the radius of the circle whose tangents are x + 3y – 5 = 0, 2x + 6y + 30 = 0 is :
a. 5
b. 10
c. 15
d. 20

30. 28 Problem

The latus rectum of the parabola y2 = 4ax whose focal chord is PSQ such that SP =
3 and SQ = 2 is given by :
a. 24
5
b. 12
5
6
c.
5
d. 1
5

31. 29 Problem
If M1 and M2 are the feet of the perpendiculars from the foci S1 and S2 of the
x2 y2
ellipse on the tangent at any point P  1 on the ellipse, then (S1M1)
9 16
(S2M2) is equal to :
a. 16
b. 9
c. 4
d. 3

32. 30 Problem
If the chords of contact of tangents from two points (x1,y1) and (x2,y2) to the
x1 x2
hyperbola 4x2 – 9y2 – 36 = 0 are at right angles, then is equal to :
y1y2
a. 9
4
9
b. 
4
81
c. 16
81
d. - 16

33. 31 Problem
In a chess tournament where the participants were to play one game with one
another, two players fell ill having played 6 games each, without playing among
themselves. If the total number of games is 117, then the number of participants
at the beginning was :
a. 15
b. 16
c. 17
d. 18

34. 32 Problem
the coefficient of x2 term in the Binomial expansion of  1 x1 / 2  x 1 / 4  is :
 
3 
70
a. 243
60
b. 423
50
c. 13
d. none of these

35. 33 Problem
The solution set of the equation
log2 x logx 2
  1 1 1    1 1 1 
4 1     ....   54  1     .... 
  3 9 27    3 9 27 
is :
 1
a. 4, 
 4
b.  1
2, 
 2
c. {1, 2}
 1
8, 
d.  8

36. 34 Problem
x2 x3 x4
If y = x -    ..... and if | x | < 1, then :
2 3 4
a. y2 y3
x 1 y    ....
2 3
y2 y3
b. x 1 y    ....
2 3
c. y2 y3 y4
x y     ....
2! 3! 4!
y2 y3 y4
d. x  y     ....
2! 3! 4!

37. 35 Problem
x y 1 z 2
The length of perpendicular from (1, 6, 3) to the line   is :
1 2 3
a. 3
b. 11
c. 13
d. 5

38. 36 Problem
The plane 2x + 3y + 4z = 1 meets the co-ordinate axes in A, B, C. the centroide of
the triangle ABC is :
a. (2, 3, 4)
1 1 1
b. 2 ,3, 4
 
1 1 1 
c.  6 , 9 ' 12 
 
 3 3 3
d.  2' 3' 4 
 

39. 37 Problem
The vector equation of the sphere whose centre is the point (1, 0, 1) and radius is
4, is :
a.  ˆ
| r  (ˆ  k ) |  4
i

b. i ˆ
| r  (ˆ  k ) |  42
 ˆ
c. r  (ˆ  k )  4
i
 ˆ
d. r  (ˆ  k )  42
i

40. 38 Problem
the plane 2x – (1 +  ) y + 3  z = 0 passes through the intersection of the planes :
a. 2x – y = 0 and y + 3z = 0
b. 2x – y = 0 and y – 3z = 0
c. 2x + 3z = 0 and y = 0
d. none of these

41. 39 Problem
      
a  b  c  0 and a  37, b  3, c  4, 
then the angle between b and c
is :
300
450
600
900

42. 40 Problem
  

i j ˆ i ˆ 
a  ˆ  ˆ  k , b  ˆ  k , c  2ˆ  ˆ
i j the value of  such that a  c is
perpendicular to is
a. 1
b. - 1
c. 0
d. none of these

43. 41 Problem
 
The total work done by two forces i j i j ˆ
F1  2ˆ  ˆ and F2  3ˆ  2 ˆ  k acting on a
particle when it is displaced from the point j ˆ i j ˆ
3ˆ  2 ˆ  k to 5ˆ  5ˆ  3k
i is :
a. 8 units
b. 9 units
c. 10 units
d. 11 units

44. 42 Problem
     
Let a, b and c be three non-coplanar vectors, and let p, q and r be
     
 bxc  cxa  axb
vectors defined by the relations P     , q     and r     Then the
[abc ] [abc ] [abc ]
        
value of the expression (a  b)  p  (b  c )  q  (c  a)  r is equal to :
a. 0
b. 1
c. 2
d. 3

45. 43 Problem
xn x n 2 x n 3
If 1 1 1 then n is equal to :
yn y n 2 y n 3  (y  z )( z  x )( x  y )    
x y z
zn z n 2 z n 3
a. 2
b. - 2
c. - 1
d. 1

46. 44 Problem
If a1, a2, ……, an ….. are in G.P. and ai > 0 for each i, then the determinant
log an log an 2 log an  4 is equal to :
  log an  6 log an  8 log an 10
log an 12 log an 14 log an 16
a. 0
b. 1
c. 2
d. n

47. 45 Problem
The value of a for which the system of equations x + y + z = 0, x + ay + az = 0, x –
ay + z = 0, prossesses nonzero solutions, are given by :
a. 1, 2
b. 1, -1
c. 1, 0
d. none of these

48. 46 Problem
If a square matrix A is such that AAT = I = ATA, then |A| is equal to :
a. 0
b.  1
c.  2
d. none of these

49. 47 Problem
The Boolean function of the input/output table as given below is :
a. f(x1, x2, x3) = x1.x2.x3+x1.x2.x3’+x1.x2’x3 + x1’.x2’.x3’
b. f(x1, x2, x3) = x1’.x2’.x3’+x1’.x2’.x3+x1’.x2.x3’ + x1.x2.x3
c. f(x1, x2, x3) = x1.x2’.x3’+x1’.x2.x3’
d. f(x1, x2, x3) = x1’.x2.x3+x1.x2’.x3

50. 48 Problem
A and B are two events. Odds against A are A  B 2 to 1. Odds in favour of are
3 to 1. If x  P(B)  y, then ordered pair (x, y) is :
 5 3
a.  , 
 12 4 
2 3
b. 3, 4
 
1 3
c. 3, 4
 
d. None of these

51. 49 Problem
In a series of three trials the probability of exactly two successes in nine time as
large as the probability of three successes. Then the probability of success in each
trail is :
1
a. 2
1
b. 3
c. 1
4
3
d.
4

52. 50 Problem
An integer is chosen at random from first two hundred digits. Then the
probability that the integer chosen is divisible by 6 or 8 is :
a. 1
4
2
b.
4
3
c.
4
d. none of these

53. 51 Problem
x 2
Let A = R – {3}, B = R – {1}, Let be defined by f(x) = . Then :
x 3
a. f is bijective
b. f is one-one but not onto
c. f is onto but not one-one
d. none of these

54. 52 Problem
cos2 x  sin2 x  1
Let f (x)  (x  0) = a (x = 0) then the value of a in order
2
x 4 2
that f(x) may be continuous at x = 0 is :
a. - 8
b. 8
c. - 4
d. 4

55. 53 Problem
If f(2) = 4 and f’(2) = 1, then lim xf (2)  2f '( x ) is equal to :
x 2 x 2
a. 2
b. - 2
c. 1
d. 3

56. 54 Problem
Let f(x) = sin x, g (x) = x2 and h (x) = log ex. If F(x) = (h o g o f) (x), then F’’(x) is
equal to :
a. a cosec3 x
b. 2 cot x2 – 4x2 cosec2x2
c. 2x cot x2
d. - 2 cosec2 x

57. 55 Problem
The length of subnormal to the parabola y2 = 4ax at any point is equal to :
a. 2 a
b. 2 2 a
a
c. 2
d. 2a

58. 56 Problem
The function f(x) = a sin x + 1 sin 3x has maximum value of at x = . The value of
3
a is :
a. 3
1
b. 3
c. 2
1
d. 2

59. 57 Problem
If  cos 4x  1 dx  A cos 4x  B, then :
cot x  tan x
1
a. A = -
2
b. A = - 1
8
c. A = - 1
4
d. none of these

60. 58 Problem
|x|
b

a x
dx , a < 0 < b, is equal to :
a. | b | - | a |
b. | b | + | a |
c. | a – b |
d. none of these

61. 59 Problem

For any integer n, the integral 0
2
ecos x
cos3(2n + 1)x dx has the value :
a. 
b. 1
c. 0
d. none of these

62. 60 Problem
sin x  sin2 x  sin3x sin2 x sin3x  /2
If , then the value of 0 f (x) dx is :
f (x)  3  4 sin x 3 4 sin x
1  sin x sin x 1
a. 3
2
b. 3
c. 1
3
d. 0

63. 61 Problem
If f : R  R, g : R  R are continuous functions, then the value of the integral
 /2
is :

 /2
(f (x)  f ( x))(g(x)  g( x))dx
a. 
b. 1
c. - 1
d. 0

64. 62 Problem
3  x x2  1 
The integral 1 

tan1 2
x 1
 tan1
x 
 dx is equal to :
a.  /4

b. /2
c. 
d. 2 

65. 63 Problem
2a f (x)
The value of the integral 0 f (x)  f (2a  x)
dx is :
a. 0
b. a
c. 2a
d. none of these

66. 64 Problem
The area bounded by the curves y = x2, y = xe-x and the line x = 1, is :
2
a.
e
2
b. - e
1
c. e
1
d. e

67. 65 Problem
The solution of x dy – y dx + x2 ex dx = 0 is :
y
 ex  c
a. x
x
b.  ex  c
y
c. x + ey = c
d. y + ex = c

68. 66 Problem
The degree and order of the differential equation of all parabolas whose axis is 
x-
axis, are
a. 2, 1
b. 1, 2
c. 3, 2
d. none of these

69. 67 Problem
Three forces P, Q, R act along the sides BC, CA, AB of a triangle ABC taken in
order. The condition that the resultant passes through the incentre, is :
a. P + Q + R = 0
b. P cos A + Q cos B + R cos C = 0
c. P sec A + Q sec B + R sec C = 0
P Q R
d.   0
sin A sin B sin C

70. 68 Problem
The resultant of two forces P and Q is R. If Q is doubled, R is doubled and if Q is
reversed, R is again doubled. If the ratio P2 : Q2 : R2 = 2 : 3 : x then x is equal to :
a. 5
b. 4
c. 3
d. 2

71. 69 Problem
A particle is dropped under gravity from rest from a height h (g = 9.8 m/sec2) and
it travels a distance in the last second, the height h is :
a. 100 mts
b. 122.5 mts
c. 145 mts
d. 167.5 mts

72. 70 Problem
A man can throw a stone 90 metres. The maximum height to which it will rise in
metres, is :
a. 30
b. 40
c. 45
d. 50

73. FOR SOLUTIONS VISIT WWW.VASISTA.NET