AMU - Mathematics - 1999

AMU –PAST PAPERS
MATHEMATICS - UNSOLVED PAPER - 1999

SECTION – I

 CRITICAL REASONING SKILLS

01 Problem

If A and B are non-zero square matrices of the same order such that AB = 0, then :

a. Adj A = 0 or adj B = 0
b. | A | = 0 or | B | = 0
c. adj A = 0 and adj B = 0
d. | A | = 0 and | B | = 0

02 Problem
3 3 4
If A 2 3 4 ,
then A-1 equal to :
0 1 1

a. A
b. A2
c. A3
d. A4

03 Problem

If A, B, C are square matrices of the same order, then which of the following is
true ?

a. AB = AC
b. (AB)2 = A2B2
c. AB = 0 A = 0 or B = 0
d. AB = I AB = BA

04 Problem

The value of is

a. 0
b. abc
c. 4a2b2c2
d. none of these

05 Problem

A speaks truth in 75% and B in 80% of the cases. In what percentage of cases are
they likely to contradict each other narrating the same incident ?

a. 35%
b. 45%
c. 15%
d. 5%

06 Problem

The matrix (a1x1+a2x2+a3x3) is of order :

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

07 Problem

Which of the following correct for A – B

a. A B
b. A’ B
c. A B’
d. A’ B’

08 Problem

If S denotes the sum to infinity and Sn the sum of n tersm of the series
1 1 1 1
1 ......, such that S Sn , then the least value of n is :
2 4 4 1000

a. 8
b. 9
c. 10
d. 11

09 Problem

The series ( 2 + 1), 1, ( 2 -1)…. is in :

a. A.P.
b. G.P.
c. H.P.
d. None of these

10 Problem
2 sin2 3x is equal to :
lim
x 0 x2

a. 12
b. 18
c. 0
d. 6

11 Problem
sin m2 is equal to :
lim
0

a. 0
b. 1
c. m
d. m2

12 Problem

Let f(x + y) = f(x) + f(y) and f(x) = x2g(x) for all x, y R, where g(x) is continuous
function. Then f’(x) is equal to :

a. g'(x)
b. g(0)
c. g(0) + g’(x)
d. 0

13 Problem
1 1 1 1
The value of is equal to :
r2 r12 r22 r33

a2 b2 c2
a.

a2 b2 c2
b. 2

a2 b2 c2
c. 3

a2 b2 c2
d.

14 Problem

The function x2 1; x 1
f (x) x 1; x 1
2; x 1

a. Continuous for all x
b. Discontinuous at x = -1
c. Discontinuous for all x
d. Continuous x = -1

15 Problem

In the expansion of (1+ x)n, then binomial coefficients of three consecutive terms
are respectively 220, 495 and 792. The value of n is :

a. 10
b. 11
c. 12
d. 13

16 Problem

The number of roots of the quadratic equation 8 sec - sec + 1 = 0 is :

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

17 Problem

If 12Pr = 11P6 + 6. 11P5 then r is equal to :

a. 6
b. 5
c. 7
d. none of these

18 Problem
1
The value of the expression ( 3 sin 750 cos 750 ) is :
2

a. 1
b. 2
c. 2

d. 2 2

19 Problem

the number of numbers consisting of four different digits that can be formed with
the digits 0, 1, 2, 3 is :

a. 16
b. 24
c. 30
d. 72

20 Problem

For the curve y = xex, the point :

a. x = -1 is a point of local minimum
b. x = 0 is a point of maximum
c. x = -1 is a point of maximum
d. x = 0 is a point of maximum

21 Problem

the function y = x – cot-1 x – log (x x2 1) is increasing on :

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

22 Problem

If x denotes displacement in time t and x = a cos t, then acceleration is given by :

a. - a sin t
b. a sin t
c. a cos t
d. - a cos t

23 Problem

Let f differentiable for all x. If f (1) = - 2 and f’(x) 2 for all x [1, 6],
2 for all x [1, 6], then :

a. f(6) < 8
b. f(6) 8
c. f(6) 5
d. f(6) 5

24 Problem
0 1
The matrix 1 0
is the matrix of reflection in the line :

a. x = 1
b. y = 1
c. x = y
d. x + y = 1

25 Problem

Let A and B be two matrices then (AB)’ equals :

a. A’B’
b. A’B
c. - AB
d. 1

26 Problem

If at any point on a curve the subtangent and subnormal are equal, then the
tangent is equal to :

a. Ordinate
b. 2 ordinate
c. 2(ordinate)

d. none of these

27 Problem

If f(x) = (x + 1) tan-1 (e-2x), then f’(0) is :

a. 1
2

b. 1
4

c. 5
6

d. none of these

28 Problem
dy
If y = x log x, then dx
is :

a. 1 + log x
b. log x
c. 1 – log x
d. 1

29 Problem

If xy + yz + zx = 1, then :

a. tan-1 x + tan-1 y + tan-1 z = 0
b. tan-1 x + tan-1 y + tan-1 z =
c. tan-1 x + tan-1 y + tan-1 z = 4

d. tan-1 x + tan-1 y + tan-1 z = 2

30 Problem

The order of the differential equation whose solution is : y = a cos x + b sin x + ce-
x is :

a. 3
b. 2
c. 1
d. none of these

31 Problem

If y = a cos px + b sin px, then :

d2y
a. dx 2 + p2y = 0
d2y
b. dx 2 - p2y = 0
d2y
c. dx 2 + py2 = 0
d2y
d. dx 2 - py = 0

32 Problem
1/2 1 x
cos x log dx is equal to :
1/2 1 x

1
a.
2
1
b. - 2

c. 0
d. none of these

33 Problem
dx
equals :
3
x 1 x

1
log( 1 x3 ) c
a. 3

1 1 x3 1
log c
b. 3 1 x 3
1

2 1
log c
c. 3 1 x 3

2 1 x3 1
d. log c
3 1 x3 1

34 Problem

ex (sin h x + cos h x) dx equal to :

a. ex sec h x + c
b. ex cos h x + c
c. sin h 2x + c
d. cos h 2x + c

35 Problem

A man can row 4.5 km/hr in still water and he finds that it takes him twice as long
to row up as to row down the river. The rate of the stream is :

a. 1.5 km/hr
b. 2 km/hr
c. 2.25 km/hr
d. 1.75 km/hr

36 Problem
3 4 10
If m n , then :
4 3 11

a. m = - 2, n = 1
b. m = 22, n = 1
c. m = - 2, n = -23
d. m = 9, n = -10

37 Problem

The area between the curve y = 2x4 – x2, the x-axis and the ordinates of two
minima of the curve is :

a. 7
120

9
b. 120

11
c. 120

15
d. 120

38 Problem

If each of the variable in the matrix a b is doubled, then the value of the
c d
determinant of the matrix is :

a. Not changed
b. Doubled
c. Multiplied by 4
d. Multiplied by 8

39 Problem

A fair coin is tossed repeatedly. If tail appears on first four tosses, then the
probability of head appearing on fifth toss equals :

a. 1
32

1
b. 2

3
c. 2

1
d. 5

40 Problem

The reciprocal of the mean of the reciprocals of n observations is the

a. G.M
b. H.M
c. Median
d. Average

41 Problem

If the area bounded by the parabola x2 = 4y, the x-axis and the line x = 4 is divided
into two equal area by the line x = , then the value of is :

a. 21/3
b. 22/3
c. 24/3
d. 25/3

42 Problem
       
(a 2b c ) {(a b x (a b c )} is equal to :

a.  
[abc ]
 
b. 2 [abc ]
 
c. 3 [abc ]
d. 0

43 Problem

The unit vector perpendicular to the plane determined by A (1, -1, 2), B (2, 0, -1)
and R (0, 2, 1) is :

1
i j ˆ
(2ˆ ˆ k )
a. 6
1 ˆ
b. (2ˆ
i ˆ
j k)
3
1 ˆ
c. (2ˆ
i ˆ
j k)
32
d. none of these

44 Problem

The probability of occurance of an even A is 0.3 and that of occurance of an event
B is 0.4. If A and B are mutually exclusive, then the probability that neither occurs
nor B occurs is :

a. 0.2
b. 0.35
c. 0.3
d. none of these

45 Problem

the probability that a man who is x years old will die in a year in P. Then amongst
n persons A1, A2,…., An each x years old now, the probability that A1 will die in one
year is

1
a. n2
b. 1 – (1 - P)n
1
c. n2
[1 – (1 - P)n]
1
d. n2 [1 – (1 - P)n]

46 Problem
  
the vector a x (b x c )is :


a. parallel to a

b. perpendicular to a
c. parallel to
d. perpendicular to

47 Problem

the next term of the series 3 + 7 + 13 + 21 + 31 + ….

a. 43
b. 45
c. 51
d. 64

48 Problem

If log3 2, log3 (2x - 5) and log3 7 are in A.P., then x is equal to:
2x
2

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

49 Problem

If the radius of a spherical balloon increases by 0.2%. Find the percentage
increase in its volume :

a. 0.8%
b. 0.12%
c. 0.6%
d. 0.3%

50 Problem
3 5 6 x 10 5
If 7 8 9 , then 5 3 6 equal to :
10 x 5 8 7 9

a.
b. -
c. x
d. 0

51 Problem
1 1 5
The positive value of sin sin is :
2 3

a. 5
6

3
b.
5

2
c. 5

2
d. 5

52 Problem

three numbers form an increasing G.P. If the middle number is doubled, then the
new numbers are in A.P. The common ratio of the G.P. is :

a. 2 - 3

b. 2 + 3
c. 3 -2
d. 3 + 3

53 Problem

The nth term of the series 1 (1 2) (1 2 3) ….. is equal to :
2 3

a. n2 (n -1)
(n 1)(2n 1)
b. 2

n 1
c.
2

n(n 1)
d. 2

54 Problem

Two finite sets have m and n element. The total number of subsets of the first
set is 56 more than the total number of subsets of the second set. The value of m
and n are :

a. m = 7, n = 6
b. m = 6, n = 3
c. m = 5, n = 1
d. m = 8, n = 7

55 Problem
1
The domain of f ( x) 1 x2 is :
2x 1

1
a. ,1
2

b. [- 1, [
c. [1, [
d. none of these

56 Problem
sec2 (log x)
The value of dx is :
x

a. tan (log x) + c
b. tan x + c
c. log (tan x) + c
d. none of these

57 Problem

The period of f(x) = cos (x2) is :

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

58 Problem

The orthocenter of the triangle formed by the lines xy = 0 and x + y = 1 is :

1 1
a. ,
2 2

1 1
b. ,
3 3

c. (0,0)

1 1
,
d. 4 4

59 Problem

The is acute angle and 4 x 2 sin2 1 = x, then tan is :
2

a. x2 1

b. x2 1

c. x 2

d. none of these

60 Problem

The equation of the locus of a point whose abscissa and ordinate are always
equal is :

a. y + x = 0
b. y – x = 0
c. y + x – 1 = 0
d. y – x + 1 = 0

61 Problem

The distance between the parallel lines y = 2x + 4 and 6x = 3y + 5 is :

a. 17/ 3
b. 1
c. 3/ 5

d. 17 5 /15

62 Problem

The equation y2 – x2 + 2x – 1 = 0, represents :

a. A pair of straight lines
b. A circle
c. A parabola
d. An ellipse

63 Problem

The intercepts made by the circle x2 + y2 –5x – 13y – 14 = 0 on x-axs and y- axis
are respectively:

a. 5,15
b. 6,15
c. 9,15
d. none of thes

64 Problem

The intercepts made by the circle x2 + y2 –5x – 13y – 14 = 0 which are
perpendicular to 3x – 4y –1 = 0 are :

a. 3x + 4y = 3, 3x + 4y + 25 = 0
b. 4x + 3y = 5, 3x + 4y - 25 = 0
c. 3x - 4y = 5, 3x - 4y + 25 = 0
d. none of these

65 Problem

three identical dice are rolled. The probability that the same number will appear
on each of them as :

a. 1
6

1
b. 18

1
c. 9

1
d. 36

66 Problem
3
The principal value of sin is :
2

a. - 6

b. 6

2
c. - 3

2
d. 3

67 Problem
1 3 cos x 4 sin x dy
If y cos then equals :
5 dx

1
a.
1 x3

b. 1
1
c. 1 x3

d. - 1

68 Problem

The point on y2 = 4ax nearest to the focus has its abscissa equal to :

a. a
b. - a
a
c. 2

d. 0

69 Problem

The vertex of the parabola x2 + 8x + 12y + 4 = 0 is :

a. (- 4, 1)
b. (4, - 1)
c. (- 4, -1)
d. (4, 1)

70 Problem

The standard deviation for the data : 7, 9, 11, 13, 15 is :

a. 2.4
b. 2.5
c. 2.7
d. 2.8

71 Problem

While dividing each entry in a data by a non-zero number a, the arithmetic mean
of the new data :

a. Is multiplied by a
b. Does not change
c. Is divided by a
d. Is diminished by a

72 Problem

Two circles which passes through the points A (0, a) and B (0, -a) an touch the
line
y = mx + c will cut orthogonally if :

a. c = a 2 m2

b. a = a 2 m2

c. m2 = a2 (1+ c2)
d. m = - a 1 c2

73 Problem
2 2
If , are the roots of ax2 + bx + c = 0, then equals :

a. c(a b)
a2

b. 0

bc
c. a2

d. abc

74 Problem

The maximum value of 5 sin 3 sin 3 is :
3

a. 11
b. 10
c. 9
d. 12

75 Problem

If x= y = 15, x2 = y2 = 49 xy = 44 and x = 5, then byx is equal to:

a. 1
3

2
b.
3

1
c. 4

1
d. 2

76 Problem

The number of terms which are free from radical sings in the expansion of (x1/5 +
y1/10)55 is :

a. 5
b. 6
c. 11
d. 9

77 Problem

The sum of the co-efficient in the expansion of (x + 2y + x)10 is :

a. 10C
x+y

b. x+yC
10

c. 26.4Cx
d. none of these

78 Problem

There are 10 points in a plane, out of which 4 points are collinear. The number of
triangles formed with vertices as there points is :

a. 20
b. 120
c. 40
d. 116

79 Problem

If the co-ordinate of the centroid of a triangle are (3, 2) and co-ordinates of two
vertices are (4, 1) and (2, 5), then co-ordinates to the third vertex are :

a. (6, 8)
b. (2, 8/3)
c. (0, - 4)
d. (6, 0)

80 Problem

the argument of 1 i 3 is :
1 i 3

4
a. 3

2
b.
3

7
c. 6

d. 3

81 Problem

In how many ways can a constant and a vowel be chosen out of the word
COURAGE ?

a. 7C
2

b. 7P
2

c. 4P x 3P1
1

d. 4P x 3P1
1

82 Problem

The length of the latusrectum of the ellipse 5x2 + 9y2 = 45 is :

a. 5
3

b. 10
3

c. 2 5
5

5
d. 3

83 Problem

The projections of a line segment on the coordinate axes are 12, 4, 3. The
direction cosine of the line are :

12 4 3
a. , ,
13 13 13

12 4 3
b. , ,
13 13 13

12 4 3
c. , ,
13 13 13

d. None of these

84 Problem
n
The least positive value of n if i(1 3) is positive integer, is :
1 i2

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

85 Problem

lim sec loge (2x ) is equal to :
x
1 4x
2

a. 0

b.
2
2
c.
4
d. 2

86 Problem

The distance between the planes gives by ,
 ˆ  ˆ
r .(ˆ
i 2ˆ
j 2k ) 5 0 and r .(ˆ
i 2ˆ
j 2k ) 8 0 is :

a. 1 unit
13
b. 3
units
c. 13 units
d. none of these

87 Problem

If the coefficient of correlation between X and Y is 0.28, covariance between X
and Y is 7.6 and the variance X is 9, then the standard deviation of Y series is :

a. 9.8
b. 10.1
c. 9.05
d. 10.05

88 Problem
1 1 3
If sin tan , then equals :
4

3
a.
5

b. 1

2
c. 5

3
d. 4

89 Problem
(x 1) 2
If 4 , then the value of x 1 is :
x x 2

a. 4
b. 10
c. 16
d. 18

90 Problem
(x 1) x3 1
If 2 cos , then equals :
x x3

1
cos 3
a. 2

b. 2 cos

c. cos3

1
cos 3
d. 3

91 Problem

The mode of the given distribution is :
Weight (in kg) 40 43 46 49 52 55
Number of children 5 8 16 9 7 3

a. 40
b. 55
c. 49
d. 46

92 Problem

The factors of x a b are :
a x b
a b x

a. x – a, x – b and x + a + b
b. x + a, x + b and x + a + b
c. x + a, x + b and x - a - b
d. x – a, x – b and x - a - b

93 Problem
7 1
The equation of a curve passing through 2, and having gradient 1 at (x, y)
2 x2
is :

a. y = x2 + x + 1
b. xy = x2 + x + 1
c. xy = x + 1
d. none of these

94 Problem

The general value of x satisfying is given by cos x = 3 (1 – sin x ) :

a. x n
2

b. x n x

x m ( 1)n
c. 3 6

x n
d. 3

95 Problem

The angle of elevation of the tops of two vertical tower as seen from the middle
point of the line joining the foot of the towers are 600 and 300 respectively. The
ratio of the height of the tower is :

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

96 Problem

If an angle is divided into two parts A and B such that A – B = x and
tan A : tan B = k : 1, then the value of sin x :

a. k 1
sin
k 1

k
sin
b. k 1

k 1
sin
c. k 1

d. None of this

97 Problem

In triangle ABC and DEF, AB = DE, AC = EF and A 2 E . Two triangles will
have the same area if angle A is equal to :

a.
3

b. 2

2
c. 3

5
d. 6

98 Problem

the even function is :

a. f(x) = x2 (x2 + 1)
b. f(x) = x (x + 1)
c. f(x) = tan x + c
d. f(x) = sin2 x + 2

99 Problem

The middle term in the expansion of (1 + x)2n will be :

a. (n + 1)th
b. (n - 1)th
c. nth
d. (n + 2)th

100 Problem

For the equation | x |2 | | x | - 6 = 0

a. There is only one root
b. There are only two distinct roots
c. There are only three distinct roots
d. There are four distinct roots

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AMU - Mathematics - 1999

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