AMU - Mathematics - 2005

AMU –PAST PAPERS
MATHEMATICS - UNSOLVED PAPER - 2005

SECTION – I

 CRITICAL REASONING SKILLS

01 Problem
~p q is logically equivalent to :

a. P q
b. q p
c. ~ (p q)
d. ~ (q p)

02 Problem

The angle of elevation of the top of an incomplete vertical pillar at a horizontal
distance of 100 m from its base is 450. If the angle of elevation of the top of the
complete pillar at the same point is to be 600, then the height of the incomplete
pillar is to be increased by :

a. 50 2

b. 100 m
c. 100 ( 3 -1)m
d. 100 ( 3 + 1)m

03 Problem
1 2 3 8
What must be the matrix X, if 2X
3 4 7 2
?

1 3
a. 2 1

1 3
b. 2 1

2 6
c. 4 2

2 6
d. 4 2

04 Problem

The value of 1 1 1 is :
bc ca ab
b c c a a b

a. 1
b. 0
c. (a - b) (b - c) (c - a)
d. (a + b) (b + c) (c + a)

05 Problem
441 441 443
The value of 445 446 447
is :
449 450 451

a. 441 x 446 x 4510
b. 0
c. - 1
d. 1

06 Problem
   ˆ ˆ
(a ˆ)ˆ
i i (a ˆ)ˆ
j j (a k )k is equal to :


a. a

b. 2 a

c. 3 a

d. 0

07 Problem
cos 2 sin 2
Inverse of the matrix is :
sin 2 cos 2

a. cos 2 sin 2
sin 2 cos 2

b. cos 2 sin2
sin2 cos 2

cos 2 sin2
c. sin2 cos 2

cos 2 sin 2
sin 2 cos 2
d.

08 Problem
 
a 3, b 4,  
If then a value of for which a b is perpendicular to
 
a b is :

9
a.
16

3
b. 4

3
c. 2

4
d. 3

09 Problem
 
The projection a 2i j ˆ
ˆ 3ˆ 2k on b i j ˆ
ˆ 2ˆ 3k is :

1
a.
14

2
b.
14

c. 14

2
d. 14

10 Problem

Let A = {x : x is a multiple of 3} and B = {x : x is a multiple of 5}. Then A B is given
by :

a. {3, 6, 9, ………… }
b. {5, 10, 15, 20, ………..}
c. {15, 30, 45, …………….}
d. none of the above

11 Problem

The maximum of the function 3 cos x – 4 sin x is :

a. 2
b. 3
c. 4
d. 5

12 Problem

If the distance ‘s’ metres traversed by a particle in t seconds is given by s = t3 – 3
t2, then the velocity of the particle when the acceleration is zero, (in m/s) is :

a. 3
b. - 2
c. - 3
d. 2

13 Problem

For the curve yn = an-1 x if the subnormal at any point is a constant, then n is
equal to :

a. 1
b. 2
c. -2
d. -1

14 Problem
d2 x
If x = A cos 4t + B sin 4t, then is equal to :
dt 2

a. - 16 x
b. 16 x
c. x
d. - x

15 Problem

If tangent to the curve x = at2, y = 2at is perpendicular to x-axis, then its point of
contact is :

a. (a, a)
b. (0, a)
c. (0, 0)
d. (a, 0)

16 Problem
dy 1 cos 2y
The general solution of the differential equation 0 is
dx 1 cos 2x
given by :

a. tan y + cot x = c
b. tan y – cot x = c
c. tan x – cot y = c
d. tan x + cot y = c

17 Problem
3/4 1/3
2
The degree of the differential equation : dy d2y is :
1
dx dx 2

a. 1/3
b. 4
c. 9
d. 3/4

18 Problem

The area enclosed between the curves y = x3 and y = is , (in sq unit) :

a. 5
3

5
b. 4

5
c. 12

12
d. 5

19 Problem
/8
cos3 4 d is equal to :
0

5
a.
3

5
b. 4

1
c. 3

1
d. 6

20 Problem
/2 cos x sin x dx is equal to :
0 1 cos x sin x

a. 0

b. 2

c. 4

d. 6

21 Problem

If ax2 – y2 + 4x – y = 0 represents a pair of lines, then a is equal to :

a. -16
b. 16
c. 4
d. -4

22 Problem

What is the equation of the locus of a point which moves such that 4 times its
distance from the x-axis is the square of its distance from the origin ?

a. x2 – y2 – 4y = 0
b. x2 + y2 – 4|y| = 0
c. x2 + y2 – 4x = 0
d. x2 + y2 – 4|x| = 0

23 Problem

Equation of the straight line making equal intercepts on the axes and passing
through the point (2, 4) is :

a. 4x – y – 4 = 0
b. 2x + y – 8 = 0
c. x + y – 6 = 0
d. x + 2y – 10 = 0

24 Problem

If the area of the triangle with vertices (x, 0) (1, 1) and (0, 2) is 4 square unit, then
the value of x is :

a. - 2
b. - 4
c. - 6
d. 8

25 Problem

is equal to :
lim 2
cot
2

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

26 Problem

The co-axial system of circles given by x2 + y2 + 2gx + c = 0 for c < 0 represents :

a. Intersecting circles
b. Non-intersecting circles
c. Touching circles
d. Touching or non-intersecting circles

27 Problem

The radius of the circle passing through the point (6, 2) and two of whose
diameters are x + y = 6 and x + 2y = 4 is :

a. 4
b. 6
c. 20
d. 20

28 Problem

If (0, 6) and (0, 3) are respectively the vertex and focus of a parabola, then its
equation is

a. x2 + 12y = 72
b. x2 - 12y = 72
c. x2 – 12x = 72
d. y2 + 12x = 72

29 Problem

For the ellipse 24x2 + 9y2 – 120x –90y + 225 = 0, the eccentricity is equal to :

a. 2
5

b. 3
5

15
c. 24

1
d. 5

30 Problem
x2 y2 x2 y2 1
If the foci of the ellipse 1 and the hyperbola
16 b2 144 81 25
coincide, then the value of b2 is :

a. 1
b. 7
c. 5
d. 9

31 Problem

The differential coefficient of f(sin x) with respect to x where f(x) = log x is :

a. tan x
b. cot x
c. f(cos x)
1
d. x

32 Problem
1 cos x
If f(x) = , x 0 is continuous at x = 0, then the value of k is :
x
k, x 0

a. 0
1
b. 2

1
c. 4

1
d. - 2

33 Problem

If 1 3i 2 is :
then (3 3 )4
2

a. 16
b. - 16
c. 16
d. 16 2

34 Problem

If y = tan-1 (sec x – tan x), then is equal to :

a. 2
b. -2
1
c. 2
1
d. - 2

35 Problem
1 1
If x 2 cos then x n is equal to :
2 xn

a. 2n cos
b. 2n cos n
c. 2i sin n
d. 2 cos n

36 Problem
1
|1 x | dx is equal to :
1

a. - 2
b. 0
c. 2
d. 4

37 Problem
dx is equal to :
7
x(x 1)

a. x7
log c
x7 1

b. 1 x7
log 7 c
7 x 1

x7 1
c. log c
x7

1 x7 1
log c
d. 7 x7

38 Problem

xe x
dx is equal to :

a. 2 x e x
4 xe x
c

b. x
(2 x 4 x 4)e c

x
c. (2x 4 x 4)e c

x
(1 4 x )e c
d.

39 Problem
dx is equal to :
2
x 2x 2

a. sin-1 (x + 1) + c
b. sin-1 (x + 1) + c
c. tan-1 (x - 1) + c
d. tan-1 (x + 1) + c

40 Problem

If a tangent to the curve y = 6x – x2 is parallel to the line 4x – 2y – 1 = 0, then
point of tangency on the curve is :

a. (2, 8)
b. (8, 2)
c. (6, 1)
d. (4, 2)

41 Problem

0.5737373……… is equal to :

a. 284
497

284
b.
495

568
c. 999

567
d. 990

42 Problem

The number of solutions for the equation x2 – 5 |x| + 6 = 0 is :

a. 4
b. 3
c. 2
d. 1

43 Problem

How many numbers of 6 digits can be formed from the digits of the number
112233 ?

a. 30
b. 60
c. 90
d. 120

44 Problem

The last digit in 7300

a. 7
b. 9
c. 1
d. 3

45 Problem
log x log y log z
If ' then xyz is equal to :
a b b c c a'

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

46 Problem

The smallest positive integer n for which (1 + i)2n = (1 - i)2n is :

a. 1
b. 2
c. 3
d. 4

47 Problem

If cos-1 p + cos-1 q + cos-1 r = then p2 + q2 + r2 + 2ppr is equal to :

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

48 Problem
x 5
If sin 1
cosec-1 , then x is equal to :
5 4 2

a. 1
b. 4
c. 3
d. 5

49 Problem

If 0 x
2
and 81sin x 2
81cos x
30, then x is equal to :

a.
6

b. 2

c. 4

3
d. 4

50 Problem
x2 y2
The equation of the director circle of the hyperbola 1 is given by :
16 4

a. x2 + y2 =16
b. x2 + y2 = 4
c. x2 + y2 = 20
d. x2 + y2 = 12

51 Problem

The normals at the extremities of the latus rectum of parabola intersects the axis
at an angle of :

a. Less than 900
b. Greater than 900
c. 900

52 Problem

The circle x2 + y2 – 8x + 4y + 4 = 0 touches :

a. x-axis
b. y-axis
c. both axes
d. neither x-axis nor y-axis

53 Problem

If A = {1, 2, 3} and B = {3, 8}, then (A B) x ( A B) is :

a. {(3, 1), (3, 3), (3, 8)}
b. {(1, 3), (2, 3), (3, 3), (8, 3)}
c. {(1, 2), (2, 2), (3, 3), (8, 8)}
d. {(8, 3), (8, 2), (8, 1), (8, 8)}

54 Problem

The condition that one root of the equation ax2 + bx + c = 0 may be double of the
other, is

a. b2 = 9ac
b. 2b2 = 9ac
c. 2b2 = ac
d. b2 = ac

55 Problem

The value of k so that x2 + y2 + kx + 4y + 2 = 0 and 2 (x2 + y2) – 4x – 3y + k = 0 cut
orthogonally is :

a. 10
3

8
b. 3

10
c. - 3

8
d. 3

56 Problem
(3 x 1)
4 is equal to :
lim 1
x X 1

a. e12
b. e-12
c. e4
d. e3

57 Problem

If A + B + C = 1800, then A B is equal to :
tan tan
2 2

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

58 Problem

In a triangle ABC, If b = 2, B = 300 then the area of the circumcircle of triangle ABC
in square unit is :

a.
b. 2
c. 4
d. 6

59 Problem

If sin x + sin2 x = 1, then cos12 x + 3 cos10 x + 3 cos8 x + cos6 x is equal to :

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

60 Problem

If R denotes the set of all real number, then the function f :R R defined
f(x) = |x| is :

a. One-one only
b. Onto only
c. Both one-one and onto
d. Neither one-one nor onto

61 Problem

If f(x) = 2x3 + mx2 – 13x + n and 2,3 are roots of the equation f(x) = 0, then the
values of m and n are

a. - 5, - 30
b. - 5, 30
c. 5, 30
d. none of these

62 Problem

If p1, p2, p3 are respectively the perpendiculars from the vertices of a triangle to
the opposite sides, then p1p2p3 is equal to :

a. a2b2c2

b. 2 a2b2c2
4a2 b2c 3
c.
R2
a2 b2 c 2
d. 8R2

63 Problem

If 5 cos 2 cos2 1 0, , then is equal to :
2

a.
3

1 3
, cos
b. 3 5

1 3
c. cos
5

1 3
, cos
d. 3 5

64 Problem

The two forces acting at a point, the maximum effect is obtained when their
resultant is 4N If they act at right angles, then their resultant is 3N. Then the
forces are :

a. 1 1
(2 3)N and (2 3)N
2 2

b. (2 3)N and (2 3)N

1 1
c. (2 2)N and (2 2)N
2 2

(2 2)N and (2 2)N
d.

65 Problem

The resultant R of two forces P and Q act at right angles to P. Then the angle
between the forces is :

1 P
a. cos
Q

1 P
cos
b. Q

1 P
sin
c. Q

1 P
d. sin
Q

66 Problem

A body starts from rest and moves with a uniform acceleration. The ratio of the
distance covered in nth sec to the distance covered in n second is :

2 1
a. n n2

1 1
b.
n2 n

2 1
c. n2 n

2 1
d. n n2

67 Problem

Two points move in the same straight line starting at the same moment from the

same point in the same direction. The first moves with constant velocity u and

the second starts from rest with constant acceleration f, then distance between

the two points will be maximum at time :

2u u
a. t C. t
f 2f

u u2
t t
b. f D. f

68 Problem

The equation of the plane containing the line x 1 y 3 z 2 and
3 2 1
the point (0, 7, -7) is :

a. x + y + z = 1
b. x + y + z = 2
c. x + y + z = 0
d. none of these

69 Problem

A plane passes through a fixed point (p, q) and cut the axes in A, B, C. Then the
locus of the centre of the sphere OABCi. :

p q r
a. 2
x y z

b. p q r
1
x y z

c. p q r
3
x y z

d. None of these

70 Problem

The value of 12.C1 + 32 . C3 + 52 . C5 + … is :

a. n (n -1)n-2 + n. 2n-1
b. n(n - 1)2n-2
c. n(n -1) . 2n-3

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AMU - Mathematics - 2005

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