AMU - Mathematics - 2003

AMU –PAST PAPERS
MATHEMATICS - UNSOLVED PAPER - 2003

SECTION – I

 CRITICAL REASONING SKILLS

01 Problem

If var (x) = 8.25, var (y) = 33.96 and cov (x, y) = 10.2, then the correlation
coefficient is :

a. 0.89
b. - 0.98
c. 0.61
d. - 0.16

02 Problem

If the standard deviation of a variable x is then the standard deviation of another
variable ax b is :
c

ax b
a.
c
a
b.
c

c.

d. none of these

03 Problem

If x 15, y 36, xy 110, x 5 then cov(x,y) equals :

a. 1/5
b. -1/5
c. 2/5
d. -2/5

04 Problem

The two lines of regression are given by 3x + 2y = 26 and 6x + y = 31. the
coefficient of correlation between x and y = is :

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

05 Problem

The A.M of 4, 7, y and 9 is 7. Then the value of y is :

a. 8
b. 10
c. 28
d. 0

06 Problem

The median from the table
Value 7 8 10 9 11 12 13
Frequency 2 1 4 5 6 1 3
Is :

a. 100
b. 10
c. 110
d. 1110

07 Problem

The mode of the following series 3, 4, 2, 1, 7, 6, 7, 6, 8, 9, 5 is :

a. 5
b. 6
c. 7
d. 8

08 Problem

The mean-deviation and coefficient of mean deviation from the data. Weight (in
kg) 54, 50, 40, 40, 42, 51, 45, 47, 55, 57 is :

a. 0.0900
b. 0.0956
c. 0.0056
d. 0.0946

09 Problem

The S.D, coefficient of S.D, and coefficient of variation from the given data
Class interval 0-10 10-20 20-30 30-40 40-50
Frequency 2 10 8 4 6
is

a. 50
b. 51.9
c. 47.9
d. 51.8

10 Problem

There are n letters and n addressed envelopes, the probability that all the letters
are not kept in the right envelope, is :

1
a. n!

1
b. 1 - n !
1
c. 1 - n

d. n!

11 Problem

If A and B are two events such that A
P( A) 0 and P(B) 1, then P
B

a. A
1 P
B

A
1 P
b. B

1 P( A B)
c. P(B)

P( A)
d. P(B)

12 Problem

A coin is tossed 3 times, the probability of getting exactly two heads is :

a. 3/8
b. 1/2
c. 1/4
d. none of these

13 Problem

Six boys and six girls sit in a row. What is the probability that the boys and girls
sit alternatively :

a. 1/462
b. 1/924
c. 1/2
d. none of these

14 Problem
3 8
If A and B are two independent events such that P( A B ') and P( A ' B)
25 25'
then P(A) is equal to :

1
a. 5

3
b. 8

2
c. 5

4
5
d.

15 Problem

In a college of 300 students every student read 5 newspapers and every
newspaper is read by 60 students. The number of newspaper is :

a. At least 30
b. At most 20
c. Exactly 25
d. None of these

16 Problem
1 x is equal to :
dx
1 x

a. 1 1
sin x 1 x2 c
2

b. sin 1 x 1 1 x 2 c
2

c. sin 1 x 1 x2 c

d. sin 1 x 1 x2 c

17 Problem
x 1 is equal to :
3
ex d x
(x 1)
ex
a. c
(x 1)2

ex
b. c
(x 1)2

ex
c
c. (x 1)3

ex
c
d. (x 1)3

18 Problem
1
dx is equal to :
x 2 (x 4 1)3 / 4

(x 4 1)1 / 4
a. c
x
(x 4 1)1 / 4
b. - c
x
3 (x 4 1)3 / 4
c
c. y x
4 (x 4 1)3 / 4
d. 3 c
x

19 Problem

If y = cos (sin x2) then at x = dy is equal to :
,
2 dx

a. - 2
b. 2
c. - 2 2

d. 0

20 Problem
dy
If y log x log x log x log x ... then dx
is equal to :

x
a.
2y 1

x
b. 2y 1

1
c. x(2y 1)

1
d. x(1 2y )

21 Problem

If f(x + y) = f(x).f(y) for all x and y and f (5) = 2, f’(0) = 3 then f’(5) will be :

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

22 Problem

On the interval [0, 1] the function x25 (1 - x)75 takes its maximum value at the
point :

a. 9
b. 1/2
c. 1/3
d. 1/4

23 Problem

If y = x log x 3 d2y is equal to :
then x
a bx dx 2

dy
a. x dx –y
2
b. dy
x y
dx
dy
c. y -x
dx
2
dy
d. y x
dx

24 Problem

The first derivative of the function cos 1 sin 1 x xx with respect to x at x =
2
1 is :

a. 3/4
b. 0
c. 1/2
d. - 1/2

25 Problem

Function f(x) = 2x3 – 9x2 + 12x + 99 is monotonically decreasing when :

a. x < 2
b. x > 2
c. x > 1
d. 1 < x < 2

26 Problem
x x
If f (x) and g(x) , where 0 < x 1, then in this interval :
sin x tan x

a. Both f(x) and g(x) and increasing function
b. Both f(x) and g(x) and decreasing function
c. f(x) is an increasing function
d. g(x) is an increasing function

27 Problem
1
If x sin x 0 is equal to :
f (x) x then lim f (x)
x 0
0 x 0

a. 1
b. 0
c. - 1
d. none

28 Problem
log(1 ax ) log(1 bx )
The function f (x) is not5 defined at x = 0, the value
x

of which should be assigned to f at x = 0, so that it is continuous at x = 0, is :

a. a – b
b. a + b
c. log a + log b
d. log a – log b

29 Problem

Domain of the function f ( x) 2 2x x 2 is :

a. 3 x 3

b. 1 3 x 1 3

c. 2 x 2

d. 2 3 x 2 3

30 Problem

The function f(x) = [x] cos 2x 1 where [.] denotes the greatest integer
2
function, is discontinuous at :

a. All x
b. No x
c. All integer points
d. x which is not an integer

31 Problem
x a2 2 sin x 0 x /4
If the function , is continuous in the
f (x) x cot x b /4 x /2
b sin2x a cos 2x /2 x

interval [0, ] then the value of (a, b) are :

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

32 Problem

a.{(b + c) x (a + b+ c)} is equal to :

a. 0
b. [abc]
c. [abc] + [bca]
d. none of these

33 Problem

The number of vectors of unit length perpendicular to vectors a = (1, 1, 0) and b =
(0, 1, 1) is :

a. Three
b. One
c. Two
d.

34 Problem

The points with position vectors 60i + 3j, 40i – 8j, ai – 52j are collinear, if a is
equal to :

a. - 40
b. 40
c. 20
d. -20

35 Problem

If a = 4i + 6j and b = 3j + 4k, then the component of along b is :

18
a. (3 j 4k )
10 3

b. 18
(3 j 4k )
35
c. 18
(3 j 4k )
3
d. 3j + 4k

36 Problem

The unit vector perpendicular to the vectors 6i +2j + 3k and 3i – 6j – 2k is :

2i 3j 6k
a.
7

2i 3j 6k
b.
7

2i 3j 6k
c. 7

2i 3j 6k
d. 7

37 Problem

A unit vector a makes an angle 4
with z-axis, if a + i + j as a unit vector, then a is
equal to :

i j k
a.
2 2 2

i j k
b. 2 2 2

c. i j k
2 2 2

i j k
d. 2 2 2

38 Problem
   
If AO OB BO OC , then A, B, C from :

a. Equilateral triangle
b. Right angled triangle
c. Isosceles triangle
d. Line

39 Problem

The area of a parallelogram whose adjacent sides are i – 2j + 3k and 2i + j – 4k, is :

a. 5 3

b. 105 3

c. 10 6

d. 5 6

40 Problem

The sum of the first n terms of the series 1 3 7 15
... is :
2 4 8 16

a. 2n – n – 1
b. 1 – 2-n
c. n + 2-n – 1
d. 2n – 1

41 Problem

If the first and (2n -1)th terms of an A.P, G.P. and H.P. are equal and their nth
terms are respectively a, b and c, then :

a. a b c

b. a c=b
c. ac – b2 = 0
d. a and c both

42 Problem

If a1, a2, a3 … an are in H.P., then a1a2 + a2a3 + ….an-1an will be equal to :

a. a1an
b. na1an
c. (n -1)a1an
d. none of these

43 Problem

The sum of n terms of the following series 1 + (1 + x) + (1 + x + x2) + .. will be :

1 xn
a. 1 x

(x(1 x n )
b. 1 x

n(1 x) x(1 xn)
c.
(1 x)2

d. none of these

44 Problem

If Sn = nP + n (n -1)8, where Sn denotes the sum of the first n terms of an A.P. then
the common difference is :

a. P + Q
b. 2P + 3Q
c. 2Q
d. Q

45 Problem

For any odd integer n 1 n3 – (n -1)3 + ….+ (-1)n -1, 13 is equal to :

1
a. (n - 1)2 (2n -1)
2

b. 1 (n - 1)2 (2n -1)
4

c. 1 (n + 1)2 (2n -1)
2
1
d. 4 (n + 1)2 (2n -1)

46 Problem

If P = (1, 0), Q = (1, 0) and R = (2, 0) are three given points then the locus of a
point S satisfying the relation SQ2 + SR2 = 2SP2 is :

a. A straight line parallel to x-axis
b. A circle through origin
c. A circle with centre at the origin
d. A straight line parallel to y-axis

47 Problem

If the vertices of a triangle be (2, 1), (5, 2) and (3, 4) then its circumcentre is :

a. 13 9
,
2 2

13 9
b. ,
4 4

9 13
c. ,
4 4

d. None of this

48 Problem

The line x + y = 4 divides the line joining the points (-1, 1) and (5, 7) in the ratio :

a. 2 : 1
b. 1 : 2
c. 1 : 2 externally
d. none of these

49 Problem

Given the points A (0, 4) and B (0, - 4) then the equation of the locus of the point
P (x, y) such that, | AP – BP | = 6, is :

x2 y2
a. 1
7 9

x2 y2
b. 1
9 7

x2 y2
1
c. 7 9

y2 x2
1
d. 9 7

50 Problem

If P (1, 2), Q(4, 6), R(5, 7) and S(a, b) are the vertices of a parallelogram
PQRS, then :

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

51 Problem

The incentre of the triangle formed by (0, 0), (5, 12), (16, 12) is :

a. (7, 9)
b. (9, 7)
c. (-9, 7)
d. (-7, 9)

52 Problem
1 4 20
The roots of the equation are :
1 2 5 0
1 2x 5x 2

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

53 Problem

x + ky – z = 0, 3x – ky – z = 0 and x – 3y + z = 0 has non-zero solution for k is equal
to :

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

54 Problem
x y z 9
If x y 5 then the value of (x, y, z) is :
y z 7

a. (4, 3, 2)
b. (3, 2, 4)
c. (2, 3, 4)
d. none

55 Problem

If A is n x n matrix, then adj(adj A) =

a. | A |n – 1 A
b. | A |n – 2 A
c. | A | n
d. none

56 Problem
a2 a 1
The value of determinant cos(nx ) cos(n 1)x cos(n 2)x is independent of
sin(nx ) sin(n 1)x sin(n 2)x
:

a. n
b. a
c. x
d. none of these

57 Problem
3 1 5 1
If X then X is equal to :
4 1 2 3

a. 3 4
14 13

3 4
b.
14 13

3 4
c. 14 13

3 4
d. 14 13

58 Problem
1 0 1
If matrix A = and its inverse is denoted by a11 a12 a13 then the
3 4 5
1
0 6 7 A a21 a22 a23
value of a23 is equal to : a31 a32 a33

a. 21/20
b. 1/5
c. -2/5
d. 2/5

59 Problem
1 2 5
The rank of the matrix 2 4 a 4 is :
1 2 a 1

a. 1 if a = 6
b. 2 if a = 1
c. 3 if a = 2
d. 4 if a = - 6

60 Problem
cos sin + sin cos is equal to :
cos sin
sin cos cos cos

0 0
a. 0 0

1 0
b.
0 0
0 1
c. 1 0
1 0
d. 0 1

61 Problem

The solution of differential equation (sin x + cos x)dy + (cos x – sinx)dx = 0 is :

a. ex (sin x + cos x) + c = 0
b. ey (sin x + cos x) = c
c. ey (cos x + sin x) = c
d. ex (sin x - cos x) = c

62 Problem
ab
z is a group of all positive rationals under the operation* defined by a * b = 2
the
identity of z is :

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

63 Problem

The average of 5 quantities is 6, the average of the them is 4, then the average of
remaining two numbers is :

a. 9
b. 6
c. 10
d. 5

64 Problem

A man can swim down stream at 8 km/h and up stream at 2 km/h, then the
man’s rate in still water and the speed of current is :

a. (5, 3)
b. (5, 4)
c. (3, 5)
d. (3, 3)

65 Problem

Equation of curve through point (1, 0) which satisfies the differential equation
(1 + y2)dx – xydy = 0 is :

a. x2 + y2 = 4
b. x2 - y2 = 1
c. 2x2 + y2 = 4
d. none of these

66 Problem

The order of the differential equation whose general solution is given by
y (c1 c2 ) cos(x c3 ) c 4e x c5
, where, c1, c2, c3, c4 and c5 are
arbitrary constants, is :

a. 5
b. 6
c. 3
d. 2

67 Problem

The equation of the base of an equilateral triangle is x + y = 2 and the vertex is (2,
-1) the length of the side of the triangle is :

a. 3 /2 / 2 /3

b. 2

c. 2 /3

3 /2
d.

68 Problem

The distance between the line 3x + 4y = 9 and 6x + 8y = 15 is :

a. 3/2
b. 3/10
c. 6
d. 0

69 Problem

The area enclosed with in the curve | x | | + |y| = 1 is :

a. 2

b. 1

c. 3

d. 2

70 Problem

Co-ordinates of the foot of the perpendicular drawn from (0, 0) to the line joining
(a cos , a sin ) and (a cos , a sin ) are :

a b
a.
2' 2

a a
b. (cos cos ), (sin sin )
2 2

cos , sin
c. 2 2

b
0,
d. 2

71 Problem

The lines a1x + b1y + c1 = 0 and z2x + b2y + c2 = 0 are perpendicular to each other
if :

a. a1b2 – b1a2 = 0
b. a1b2 + b1a2 = 0
2 2
c. a1 b2 b1 a2 0
d. a1b1 – b2a2 = 0

72 Problem

The angle between the pair of straight lines. y 2 sin2 xy sin2 x 2 (cos2 1) 1, is :

a. /3
b. /4
c. 2 /3
d. none of these

73 Problem

If two circles (x - 1)2 + (y - 3)2 = r2 and x2 + y2 – 8x +2y + 8 = 0 intersect in two
distinct point, then :

a. 2 < r < 8
b. r = 2
c. r < 2
d. r > 2

74 Problem

The equation of circle passing through the points (0, 0) (0, b) and (a, b) is :

a. x2 + y2 + ax + by = 0
b. x2 + y2 - ax + by = 0
c. x2 + y2 - ax - by = 0
d. x2 + y2 + ax - by = 0

75 Problem

Two circles S1 = x2 + y2 + 2g1x + 2f1y + c1 = 0 and S1 = x2 + y2 + 2g2x + 2f2y + c2 = 0
each other orthogonally, then :

a. 2g1g2 + 2f1f2 = c1 + c2
b. 2g1g2 - 2f1f2 = c1 + c2
c. 2g1g2 + 2f1f2 = c1 - c2
d. 2g1g2 - 2f1f2 = c1 - c2

76 Problem

The equation 13[(x -1)2 + (y - 2)2] = 3 (2x + 3y - 2)2 represents :

a. Parabola

b. Ellipse

c. Hyperbola

d. None of these

77 Problem

Vertex of the parabola 9x2 – 6x + 36y + 9 = 0 is :

a. (1/3, - 2/9)
b. (-1/3, - 1/2)
c. (- 1/3, 1/2)
d. (1/3, 1/2)

78 Problem

If the foci and vertices of an ellipse be ( 1,0) and ( 2,0) then the minor axis of
the ellipse is :

a. 2 5

b. 2
c. 4
d. 2 3

79 Problem

the one which does not represent a hyperbola is :

a. xy = 1
b. x2 – y2 = 5
c. (x - 1) (y – 3) = 0
d. x2 – y2 = 0

80 Problem

Eccentricity of the conic 16x2 + 7y2 = 112 is :

a. 3/ 7

b. 7/16
c. 3/4
d. 4/3

81 Problem

If f and g are continuous function on [0, a] satisfying f(x) = f(a - x) and g(x) + g(a -
a
x) = 2, then f (x) g (x) =
0

a

a. f (x) dx
0

0
b. f (x) dx
a

a
c. 2 f (x) dx
0

d. none

82 Problem
a
The value of which satisfying sin x dx sin2 , [ (0, 2 )]
/2
equal to :

a. /2
b. 3 /2
c. 7 /6
d. all the above

83 Problem
n
The value of | sin x | dx is :
0

a. 2n + 1 + cos
b. 2n +1 – cos
c. 2n +1
d. 2n + cos

84 Problem
2
The value of | 2 sin x | dx [.] represents greatest integer
function, is :

a. -
b. - 2
c. -5 /3
d. 5 /3

85 Problem

By Simpson's rule, the value of is :

a. 1.358
b. 1.958
c. 1.625
d. 1.458

86 Problem

The value of x0 (the initial value of x) to get the solution in interval (0.5, 0.75) of
the equation x3 – 5x + 3 = 0 by Newton Raphson method is :

a. 0.5
b. 0.75
c. 0.625
d. 0.60

87 Problem

By bisection method, the real root of the equation x3 – 9x + 1 = 0 lying between x
= 2 and x = 4 is nearer to :

a. 2.2
b. 2.75
c. 5.5
d. 4.0

88 Problem

By false-poisitioning, the second approximation of a root of equation f(x) = 0 is
(where, x0, x1 are initial and first approximations resp.)

f (x0 )
a. x0 =
f (x1 ) f (x0 )

b. x0 f (x1 ) x1f (x0 )
f (x1 ) f (x0 )

c. x0 f (x0 ) x1f (x1 )
f (x1 ) f (x0 )
x0 f (x1 ) x1f (x0 )
d.
f (x1 ) f (x0 )

89 Problem

Let f(0) = 1, f(1) = 2.72, then the Trapezoidal rule gives approximation value of
1
f (x) dx is :
0

a. 3.72
b. 1.86
c. 1.72
d. 0.86

90 Problem

If and are imaginary cube roots of unity, then 4 4 1

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

91 Problem
z 5i
the complex number z = x + iy, which satisfy the equation 1 lies on :
z 5i

a. real axes
b. the line y = 5
c. a circle passing through the origin
d. none of these

92 Problem
m
If 1 i
1 , then the least integeral value of m is :
1 i

a. 2
b. 4
c. 8
d. none of these

93 Problem

If 1 , , 2 are the cube roots of unity, then (a + b +c 2)3 + (a + b 2 + c )3 is
equal to a + b + c = 0

a. 27 abc
b. 0
c. 3 abc
d. none of these

94 Problem

If is an imaginary cube root of unity, then (1 + w – w2)7 equals :

a. 128
b. - 128
c. 128 2

d. - 128 2

95 Problem
13
The value of the (i n i n 1 ) where i 1, equals :
n 1

a. i
b. i – 1
c. - i
d. 0

96 Problem

The number of ways in which 6 rings can be worm on four fingers of one hand is
:

a. 46
b. 6c4
c. 64
d. 24

97 Problem

Number greater than 1000 but not greater than 4000 which can be formed with
the digits 1, 2, 3, 4, are :

a. 350
b. 357
c. 450
d. 576

98 Problem

How many can be formed from the letters of the word DOGMATIC if all the
vowels remain together:

a. 4140
b. 4320
c. 432
d. 43

99 Problem

If A = sin2 + cos4 , then for all real values of :

a. 1 A 2

3
b. A 1
4

13
A 1
c. 16

3 13
A
d. 4 16

100 Problem

In triangle ABC, if A = 450, B = 750, then a c 2 =

a. 0

b. 1

c. b

d. 2b

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