AMU - Mathematics - 1996

AMU –PAST PAPERS
MATHEMATICS - UNSOLVED PAPER - 1996

SECTION – I

 CRITICAL REASONING SKILLS

01 Problem
2
The differential equation of all ‘Simple Harmonic Motions’ of given period n is :
x = a cos (nt + b)

d2 x
a. dt 2
+ nx = 0
d2 x
b. + n 2x = 0
dt 2
d2 x
c. - n2 x = 0
dt 2

d2 x 1
d. + x=0
dt 2 n2

02 Problem

The elimination of constants A, B and C from y = A + Bx – Ce-x leads the
differential equation :

a. y'’ + y’’ = 0
b. y’’ – y’’’ = 0
c. y’ + ex = 0
d. y’’ + ex = 0

03 Problem

The solution of the differential equation xdy + ydx = 0 is :

a. xy = c
b. x2 + y2 = 0
c. xy logx = 1
d. log xy = c

04 Problem
dy
The solution of the equation +2y tan x = sin x, is :
dx

a. y sec2 x = sec x + c
b. y sec x = tan x + c
c. y = sec x + c sec2 x
d. none of these

05 Problem
x
a is equal to:
dx
x

a x
a. c
log a

b. 2a x
c
log a

x
c. 2a .log a  c

d. None of these

06 Problem
1/2 dx
 1/4
xx 2
is equal to :


a. 6


b. 3


c. 4

d. 0

07 Problem
 /2 1
The value of

0 3
1  tan x
dx is :

a. 0

b. 1

c.  /4

d. 2 

08 Problem
Q dx
The value of

P x
, (P  Q) is :

a. log (Q - P)
b. log P
c. log Q
d. log (P Q)

09 Problem

For a biased dice the probabilities of different faces to turn up are given below :
Face: 1 2 3 4 5 6
Probabili 0.1 0.32 0.21 0.15 0.05 0.17
ty
The dice is tossed and you are told that either the face 1 or 2 has turned up. The
probability that it is face is :

a. 1
7

4
b. 7

5
c. 21

4
d. 21

10 Problem
1 x /2
If A 
1
tan x dx and B  
x / 2
cot x dx then A + B is equals :

a. 0
b. 1
c. 2

d. 4

11 Problem

If f’(x) = (x - a)2n (x - b)2p+1 where x and p are positive integers, then :

a. x = a is a point of minimum
b. x = a is a point of maximum
c. x is not a point of maximum or minimum
d. none of these

12 Problem

the inverse of a symmetric matrix is :

a. diagnol matrix
b. skew symmetric
c. square matrix
d. a symmetric matrix

13 Problem
 a  p
The matrix product     equals :
 b  x[xyz ]x q 
c 
  r 
 

pqr  abc
a.
xyz

b. xyz  pqr
abc
c. pqr  abc
xyz
d. none of these

14 Problem

In a group of 52 persons, 16 drink tea but not coffee, while 33 drink tea. How
many persons drink coffee but not tea :

a. 17
b. 36
c. 23
d. 19

15 Problem

If the length of major axis of an ellipse is three times the length of minor
axis, then the eccentricity is :

1
a. 3

1
b.
2

1
c.
3

2 2
d.
3

16 Problem

A conic section with centricity e is a parabola if :

a. e = 0
b. e < 1
c. e > 1
d. e = 1

17 Problem

For the ellipse 3x2 + 4y2 = 12 length of the latusrectum is :

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

18 Problem

If (2, 0) is the vertex and y-axis is the direction of a parabola, then its focus is

a. (2, 0)
b. (-2, 0)
c. (4, 0)
d. (-4, 0)

19 Problem

The length of tangent from the point (6,-7) to the circle : 3x2 + 3y2 – 7x – 6y = 12
is

a. 6
b. 9
c. 7
d. 13

20 Problem

Which of the following lines is fathest from the origin ?

a. x – y + 1 = 0
b. 2x – y + 3 = 0
c. x + 2y – 2 = 0
d. x + y – 2 = 0

21 Problem

The normal to a given curve is parallel to x-axis if :

dy
a. dx =0
dy
b. dx
=1

c. dx =0
dy
dx
d. =1
dy

22 Problem

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

3
a. 2

3
b.
10

c. 6

d. 9
4

23 Problem

a line passes through (2, 2) and is perpendicular to the line 3x + y = 3. Its y-
intercept is :

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

24 Problem

If the volume in cm3 and surface and surface area is cm2 of a sphere are
numerically equal Then the radius of the sphere in cm is :

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

25 Problem

the locus of a point represented by x  a  t  1  y  a  t  1  is :
2 t 
  2 t 
 

a. an ellipse
b. a circle
c. a pair of straight lines
d. none of these

26 Problem

The equation of a tangent parallel to y = x drawn x2 y 2 is :
 1
3 2

a. x – y + 1 = 0
b. x – y + 2 = 0
c. x + y – 1 = 0
d. x – y + 2 = 0

27 Problem

Which of the following is not correct :

a. A function f such that f(x) = x – 1 for every integer x is binary operation
b. The binary operation positive on IR is commutative
c. The set{(a, b)/a I and b = - a} is a binary operation on I
d. The division on IR – [0] is not as associative operation

28 Problem

If cos   - 0.6 and 1800 <  < 2700, then tan 4 is equal to :

a. 5
2

(1  5)
b. 2

c. (1  5)
2

( 5  1)
d.
2

29 Problem
z1 z
Let z1 and z2 be two complex number such that  2  1. Then :
z2 z1

a. z1, z2 are collinear
b. z1, z2 and the origin form a right angled triangle
c. z1, z2 and the origin form an equilateral triangle
d. none of these

30 Problem
1i 3
The amplitude of is :
3 1


a.
6


b. 4


c. 3


d. 2

31 Problem

In a G.P. if (m + n)th terms is p and (m - 1)th term is q then mth term is :

p
a. q
q
b. p

c. Pq

d. pq

32 Problem

In a G.P. of positive terms, any term is equal to the sum of the next two terms.
Then the common ratio of the G.P. is

a. 2 cos 180
b. sin 18
c. cos 18
d. 2 sin 18

33 Problem

If n+2C8 n-2P4 = 57 : 16, then the value of x is :

a. 20
b. 19
c. 18
d. 17

34 Problem

A library has a copies of one book, b copies of each of two books, c copies of each
of three books, and single copies of d books. The total number of ways in which
these books can be distributed is :

(a  b  c  d)!
a. a! b ! c !

(a  2b  3c  d)!
b. a !(b !)2 (c !)3

(a  2b  3c  d)!
c.
a! b ! c !

d. None of these

35 Problem

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

1
a. 6

1
b. 36

1
c. 18

3
d. 28

36 Problem

If a, b, c are real numbers a  0. If , is a root of a2x2 + bx + c = 0, is a
root of a2x2 – bx + c = 0 and 0 < a < b, then the equation a2x2 + 2bx + 2c = 0 has a
root that always satisfied :

a.     
2

b.     
2

c.  

d.     

37 Problem

The solution of log7log5 ( x 5  x ) = 0 is :

a. 4
b. 5
c. 6
d. 9

38 Problem
1
The most general value of  , satisfying the two equations, cos  = , tan   1
2
is

5
a. 2n 
4


b. 2n 
4

5
c. n 
4


d. (2n  1) 
4

39 Problem

Three faces of a fair dice are yellow, two faces red and only one is blue. The dice
is tosses three times. The probability that the colour yellow, red and blue appears
in the first, second and the third tosses respectively is :

a. 1
30

b. 1
25

1
c. 36

1
d. 32

40 Problem

If a2 + b2 + c2 = 1, then ab + bc + ca lies in the interval :

a. 1 
 2 ,2
 

b. [-1, 2]
 1 
c.   2 ,1
 
 1
d.  1, 2 
 

41 Problem

The common tangent to the parabola y2 = 4ax and x2 = 4ay is :

a. x + y + a = 0
b. x + y - a = 0
c. x - y + a = 0
d. x - y - a = 0

42 Problem

The solution of the equation cos2  + sin  + 1 = 0, lies in the interval :

  
a.  4 , 4 
 

 3 
b.  4 , 4 
 

 3 5 
c.  4 , 4 
 

 5 7 
d.  4 , 4 
 

43 Problem

If cosec  - cot = q, then the value of cot  is :

1  q2
a.
q

1  q2
b. q

q
c. 1  q2

q
d. 1  q2

44 Problem

The length of the intercept on the normal at the point (at2, 2at) of the parabola y2
= 4ax made by the circle on the line joining the focus and p as diameter is :

a. a 1  t2

b. a 1  t2

c. a 1t

d. a 1t

45 Problem

From the top of a light house 60 metres high with its base at the sea level, the
angle of depression of a boat is 150. The distance of the boat from the foot of the
light house is :

 3  1

 3  1

a.   60 metres
 3  1

 3  1

b.   60 metres
 3  1
c. 
 3  1
 metres
 

d. none of these

46 Problem

The expression 1  sin  1  sin   equal to :
 for  
1  sin  1  sin  2

2 cos 
a.
cos2 

2 sin 
b.
sin2 
2 cos 
c.
sin2 

d. none of these

47 Problem
x2 y2
If m1 and m2 are the slopes of the tangent to the parabola  1 , which
15 16
passes through the point (6, 2) the value of (m1 + m2) is :

a. 14
11

4
b.
11

11
c. 4

24
d. 11

48 Problem

If f’(x) = sin (log x) and y = f  2 x  3  , then dy equals :
 3  2x 
  dx

1
a. sin (log x) . x log x

12   2x  3  
b. 2
sin log  
(3  2x)   3  2x  
  2x  3  
c. sin log  
  3  2x  

d. none of these

49 Problem

The derivatives of tan-1  2x  w.r.t. sin-1 2x  is :
 2 
1  x 2 
1  x   

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

50 Problem
x
 m equals to :
lim 1 
x 
 x 


a. em
b. e-m
c. m-e
d. me

51 Problem
x 2
If y  f (x )  then :
x 1

a. f(1) = 3
b. 2 = f(4)
c. f is rational function of x
d. y increases with x for x < 1

52 Problem
x
The co-ordinates of the points on the curve, f (x)  1  x where the tangent to
2

the curve has greatest slope is :

a. (0, 2)
b. (0, 0)
c. (0, 1)
d. (1, 1)

53 Problem

the maximum value of xy subjected to x + y = 8 is :

a. 8
b. 16
c. 20
d. 24

54 Problem
df
If f = xy then is equal to :
dx

a. x log f
y
b. xy . x

c. xy
log x
d. yx

55 Problem

The curve y - exy + x = 0 has a vertical tangent at the point :

a. (1, 1)
b. at no point
c. (0, 1)
d. (1, 0)

56 Problem

The sum of the series 1 1 :
1   ...
(1  2) (1  2  3)

a. 1
2

b. 1

c. 2
3
d. 2

57 Problem

Let S be a finite set containing n elements. Then the total number of binary
operations on S is :

2
a. nn
b. nn
c. n2
2
d. 2n

58 Problem

The ratio of the volume of a cube to that of the sphere which will fit inside the
cube :

a. 5 : 
b. 7 : 
c. 6 : 
d. 3 : 

59 Problem

The function f(x) = cos (log (x + x2  1 )) is :

a. Even
b. Constant
c. Odd
d. None of these

60 Problem

The perimeter of a given rectangle is x, its area will maximum of its sides are :

a. x x
,
2 2

x x
b. ,
3 3

x x
c. ,
6 3

x x
d. ,
4 4

61 Problem
d 3y dy
Which of the following functions hold the property of 3
 =0?
dx dx

a. y = ex
b. y = cos x
c. y = tan x
d. y = sin x

62 Problem
0.5
the domain of definition of the function : x.1  2( x  4) + (x + 4)0.5 +
2  ( x  4)0.5
4(x + 4)0.5 is :

a. R
b. (- 4, 4)
c. R+
d. (-4, 0)  (0, )

63 Problem

If a, b, c, d are in H.P. then :

a. ab > cd
b. ad > bc
c. ac > bd
d. none of these

64 Problem

If  1 x  then the value of cos   x is :
tan  tan ,
2 1 x 2 1  x cos 

a. sin x
b. cos x
c. cos 
d. sin 

65 Problem

A round balloon of radius r subtends an angle at the eye of the observer while
the angle of elevation of its centre is  , then the height of the centre of balloon
is :

a. r tan  sin 
 
b. r cosec  2 
 
c. r tan  and 
 
d. r sin 2
  cosec , 

66 Problem
1 a 1 1
If a-1 + b-1 + c-1 = 0 such that 1 1 b 1 
then the value of 
1 1 1 c
is :

a. 0
b. - abc
c. abc
d. none of these

67 Problem
4 5 2 
If A =   then adj. (A) equals :
5 4 2 
2
 2 8 

36 36 18 
 
a. 36 36 18
18 18
 9 

 36 36 18 
b.  
 36 36 18
 18
 18 9 


0 0 0
c.  
0 0 0
0 0 0
 

d. none of these

68 Problem

The statement of k = 1 10 by 2, do remit in :

a. 3 cycles
b. 2 cycles
c. 8 cycles
d. 9 cycles

69 Problem

The Newton’s method converges first of f’(x) is :

a. Small
b. mid
c. large
d. zero

70 Problem

are unit vectors and  is the angle between them then
  
If a and b a  b
2
is :


a. sin 2

b. sin 
c. 2 sin 
d. sin 2 

71 Problem

The area of ABC with vertices A (1, -1, 2), B (2, 1, -1) and C (3, -1, 2) is :

a. 13 2 sq. units
b. 3 6 sq. units
c. 13 sq. units
d. 15 3 sq. units

72 Problem

If p, q, r be three non-zero vectors, then equation p . q = p . r implies :

a. q = r
b. p is orthogonal to both q and r
c. p is orthogonal to q – r
d. either q = r or p is orthogonal to q – r.

73 Problem
 
j ˆ i j ˆ  i j ˆ
ˆ  a2 ˆ  a3 k, b  b1ˆ  b2 ˆ  b3 k, c  c1ˆ  c2 ˆ  c3 k, be three non-
Let a  a1i
 
zero vectors such that is a unit vector perpendicular to both a and b . If the
   a1 a2 a3
a and b is
angle between 6' then b1 b2 b3 is equal to :
c1 c2 c3

a. 0

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

74 Problem
     
The scalar a.{(b  c ) x (a  b  c )} equals :

a. 0
  
b. 2 [a b c ]
  
c. [a b c ]

d. none of these

75 Problem

If f(x) = x1/x-1 for all positive x = 1 and if f is continuous at 1, then equals :

a. 0
1
b. e

c. e
d. e2

76 Problem

the point equidistant from the four points (0, 0, 0), (3/2, 0, 0), (0,5/2, 0) and
(0, 0, 7/2) is :

 2 1 2
a.  3' 3' 5 
 

 3
b.  3,2, 5 
 

3 5 7
c. 4, 4, 4
 

1 
 2 , 0, 1 
d.  

77 Problem

The area bounded by the curve y = x sin x and the x-axis between x = 0 and x = 2
 is :

a. 2 
b. 3 
c. 4 
d. 

78 Problem

The value of (i)I where i2 = -1 is :

a. Whole imaginary number
b. Whole real number
c. Zero
d. E  /2

79 Problem

The equations of the sphere with A (2, 3, 5) and B (4, 9, -3) as the ends of a
diameter is :

a. x2 + y2 + z2 – 6x – 12y – 2z + 20 = 0
b. 2x2 + 2y2 + 2z2 – x - y – z + 1 = 0
c. 3x2 + 3y2 + 3z2 – 2x – 2y –2z –1 = 0
d. none of these

80 Problem

If (2, 0) is the vertex and y-axis the direction of a parabola, then its focus is :

a. (2, 0)
b. (-2, 0)
c. (4, 0)
d. (-4, 0)

81 Problem

If f(x) = log (x + a), for x  a , then the value of f’(x) is equal to :

1
a. x a

1
b. | x  a|

1
c. | x  a|

1
d. | x  a|

82 Problem

In a collection of 6 mathematics books and 4 physics books, the probability that 3
particular mathematics books will be together is :

1
a. 8

1
b. 10

1
c. 15

1
d. 20

83 Problem

Find the mode from the data given below :
Marks 0-5 10-15 15-20 15-20 20-25 25-30
Obtained 18 20 25 30 16 14

a. 16.1
b. 16.2
c. 16.7
d. 16.3

84 Problem

If the mean of four observations is 20 and when a constant c is added to each
observation, the mean becomes 22. The value of c is :

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

85 Problem

The area of surface generated by rotating the circle x = b =  cos y = a + b sin  ,
0    2 abut x-axis is :

a.  ab sq. units
b. ab sq. units
x
c. 2 loge x c

d. ( x )a loge x c

86 Problem

The eigin values of the matrix 5 4 are :
A   
1 2

a. 2, 4
b. 1, 6
c. 3, -3
d. 1, -2

87 Problem
x y 2 z 3
the length of perpendicular from the point P(3, -1, 11) to the line  
2 3 4
is :

a. 2 13

b. 53

c. 2 14
d. 8

88 Problem

the angle A, B, C of a triangle are in the ratio of 3 : 5 : 4, then a + i 2 is equal to :

a. 3b
b. 2b

c. 2b
d. 4b

89 Problem

If the angle between the lines joining the foci of an ellipse to an extremity of the
minor axis is 900, the eccentricity of the ellipse is :

1
a.
8

11
b.
4

c. 3
2

1
d. 2

90 Problem

If a man running at the rate of 15 km per hour crosses a bridge in 5 minutes, the
length of the bridge in metres is :

a. 7500
b. 1250
c. 1000
1
d. 1333 3

91 Problem

If  x   y  15,  x 2   y 2  49,  xy  44 and n = 5, then byx is equal to :

1
a. - 3

2
b. - 3

c. - 1
4

1
d. -
2

92 Problem

If (x + 1) is a factor of x4 + (p -3)x3 – (3p - 5)x2 + (2p -10) x + 5 then the value of p is
:

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

93 Problem

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

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

94 Problem

The circle x2 = y2 – 2ax – 2ay + a2 = 0 touches :

a. x-axis only
b. y-axis only
c. both axis
d. none of these

95 Problem

If |A| = 2 and A is a 2 x 2 matrix, what is the value of det, {adj. {adj. (adj. A2)}} is
equal to

a. 4
b. 16
c. 64
d. 128

96 Problem

If y = (4x - 5) is a tangent to the curve y2 = px3 + q at (2, 3), then :

a. p = -2, q = -7
b. p = -2, q = 7
c. p = 2, q = -7
d. p = 2, q = 7

97 Problem

given that x > 0, y > 0, x > y and z 0. the inequality which is not always correct is :

a. x + z > y + z
b. x – z > x – z
c. xz > y > z
x y
 2
d. z 2
z

98 Problem

Given that log10343 = 2.5353. The least n such that 7n > 105 is :

a. 4
b. 3
c. n
d. 6

99 Problem

the smallest angle of the triangle whose sides are 6 12, 48, 24 is :


a. 4


b. 6

c. 
3
d. none of these

100 Problem
1 p  
If f’(x) = (1  cos x) for 0  x  2 and that f(0) = 3, then f 2
 
lies in the interval :

a. 1 
 2 ,1
 

  
b. 4 , 2
 

  
c. 3, ,3  
 4
 2

  
d. 3  ,3  
 4 2

FOR SOLUTIONS VISIT WWW.VASISTA.NET

AMU - Mathematics - 1996

More Related Content

What's hot (20)

Viewers also liked (9)

Similar to AMU - Mathematics - 1996 (18)

More from Vasista Vinuthan (20)

Recently uploaded (20)

AMU - Mathematics - 1996

Editor's Notes