AMU - Mathematics - 2007

AMU –PAST PAPERS
MATHEMATICS - UNSOLVED PAPER - 2007

SECTION – I

 CRITICAL REASONING SKILLS

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

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

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

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

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

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

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)

08 Problem

(1+ i)8 + (1 - i)8 equal to

a. 28
b. 25
c. 24 cos 4

d. 28 cos 8

09 Problem

The value of 3 cosec 200 – sec 200 is

a. 2
b. 4
c. - 4
d. none of these

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)

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

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)

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

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

15 Problem
100
100
Cm (x - 3)100 – m. 2m is
m 0

a. 100C
47

b. 100C
53

c. -100C53
d. -100C100

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

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

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

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

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

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

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

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

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

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

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

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)

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

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

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

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

32 Problem

The value of integral 1 1 x is
dx
0 1 x

a. 2 +1

b. -1
2
c. - 1

d. 1

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

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

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

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

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

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

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

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

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

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

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

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

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

46 Problem
     
The value of a (b c ) x (a b c) is
 
a. 2 [abc ]
 
b. [abc ]

c. 0
d. none of these

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

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

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

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

51 Problem
1/ x
The value of lim tan 1
x is
x 2

a. 0
b. 1
c. - 1
d. e

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

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]

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)

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

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

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

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

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

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

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

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)

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

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

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

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

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

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

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’

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

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AMU - Mathematics - 2007

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