AMU - Mathematics - 2004

AMU –PAST PAPERS
MATHEMATICS - UNSOLVED PAPER - 2004

SECTION – I

 CRITICAL REASONING SKILLS

01 Problem

The unit's place digit in the number 1325 + 1125 – 325 is :

a. 0
b. 1
c. 2
d. 3

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

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

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

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

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

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

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

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

10 Problem

The solution of log99(log2 (log3x)) = 0 is :

a. 4
b. 9
c. 44
d. 99

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

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

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

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

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

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

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

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

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

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

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

22 Problem

The value of sin 200(4 + sec200) (4 + sec 200) is :

a. 0
b. 1
2
c.
d. 3

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

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

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)

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

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

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

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

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

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

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

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

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!

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

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 
 

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 

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

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

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

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

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

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

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

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

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AMU - Mathematics - 2004

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