MATHEMATICS 1. If A, B and C are any three sets, then A – (B C) is equal to

(a) (A – B) (A – C) (b) (A – B) (A – C)

(c) (A – B) C (d) (A – B) C

2. Let f = {(1, 5), (2, 6), (3, 4)}, g = {(4, 7), (5, 8), (6, 9)}. Then gof is

(a) {(4, 7), (5, 8), (6, 9), (1, 5), (2, 6), (3, 4)}

(b) null set

(c) {(1, 8), (2, 9), (3, 7)}

(d) none of these

3. If x = ,721 then x is equal to

(a) 21

26 (b)

14

21 (c)

99

126 (d)

99

127

4. The range of the function f (x) = 2

21

x

x is equal to

(a) [0, 1] (b) (0, 1) (c) (1, ) (d) [1, )

5. If z = ,2

3 i then

69z is equal to

(a) – i (b) i (c) 1 (d) – 1

6. The domain of the function f (x) = )1(log 23 xxx is

(a) (– 3, 1) (1, ) (b) (– 3, – 1) (– 1, )

(c) (– 2, – 1) (1, ) (d) none of these

7. The three vertices of a triangle are represented by the complex number 0, 1z and 2z . If the

triangle is equilateral, then

(a) 2122

21 zzzz (b) 21

21

21 zzzz

(c) 2122

21 zzzz (d) 021

22

21 zzzz

8.

12

2

1lim

x

x x

x is

(a) e (b) 2e (c) 1e (d) 1

9. A straight line moves so that the sum of the reciprocals of its intercepts on the co-ordinate

axes is unity. Then

(a) the straight line always passes through fixed point (1, 1)

(b) it does not pass through any fixed point

(c) it passes through the origin

(d) none of these

10. The angle between two equal forces acting on a particle when the square of their resultant is

three times their product is

(a) 90º (b) 60º (c) 120º (d) none of these

11. If the quadrilateral formed by the lines

ax + by + c = 0, a'x + b'y + c = 0, ax + by + c' = 0, a'x + b'y + c' = 0

have perpendicular diagonals, then

(a) 2222 '' cbcb (b) 2222 '' acac

(c) 2'222 ' baba (d) none of these

12. The distance between the pair of parallel lines 046344 22 yxyxyx is

(a) 5 (b) 5

2 (c)

5

1 (d)

2

5

13. The tangent to the circle 522 yx at the point (1, –2) also touches the circle

0206822 yxyx . Then its point of contact is

(a) (– 2, 1) (b) (– 1, – 1)

(c) (– 3, 0) (d) (3, – 1)

14. The locus of the mid-points of chords of the circle 422 yx which subtends a right angle

at the origin is

(a) x + y = 2 (b) 122 yx

(c) 222 yx (d) x + y = 1

15. The equation of the circle whose radius is 5 and which touches the circle

0204222 yxyx at the point (5, 5) is

(a) 0120161822 yxyx (b) 0120161822 yxyx

(c) 0120161822 yxyx (d) 0120161822 yxyx

16. Solution of (x + y – 1) dx + (2x + 2y – 3) dy = 0 is

(a) y + x + log (x + y – 2) = c (b) y + 2x + log (x + y – 2) = c

(c) 2y + x + log (x + y – 2) = c (d) 2y + 2x + log (x + y – 2) = c

17. The equation of the directrix of the parabola 02442 xyy is

(a) x = – 1 (b) x = 1

(c) x = 2

3 (d) x =

2

3

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

(a) 72122 yx (b) 72122 yx

(c) 72122 xy (d) 72122 xy

19. 0242 axy is a

(a) normal chord of axy 42 (b) normal chord of a circle 222 ayx

(c) tangent to axy 42 (d) none of these

20. The eccentricity of the curve represented by the equation 02322 22 yxyx is

(a) 0 (b) 2

1 (c)

2

1 (d) 2

21. The equations Rtt

tby

t

tax ;

1

2;

1

122

2

represent

(a) a circle (b) an ellipse (c) a parabola (d) a hyperbola

22. ,}{

][

0

0

n

n

dxx

dxx (where [x] and {x} denote the integral and fractional parts of x and n N) is

equal to

(a) 1

1

n (b)

n

1 (c) n (d) n – 1

23. A common tangent to 144169 22 yx and 922 yx is

(a) 7

5

7

3xy (b)

7

15

7

23 xy

(c) 7157

32 xy (d) none of these

24. If the chords of contact of tangents from two points ),( 11 yx and ),( 22 yx to the hyperbola

12

2

2

2

b

y

a

x are at right angles, then

21

21

yy

xx is equal to

(a) 2

2

b

a (b)

2

2

a

b (c)

4

4

a

b (d)

4

4

b

a

25. If a circle cuts the rectangular hyperbola xy = 1 in the points ),( rr yx where r = 1, 2, 3, 4,

then

(a) 24321 xxxx (b) 14321 xxxx

(c) 04321 xxxx (d) 04321 yyyy

26. dxxxx )log1(

(a) Cxxx log (b) Cexx

(c) Cxx (d) none of these

27. 8

5cos1

8

3cos1

8cos1

8

7cos1 is equal to

(a) 2

1 (b)

8 (c)

8

1 (d)

22

21

28. Let sin x + sin y = a and cos x + cos y = b. Then sin (x + y) is equal to

(a) ba

ab (b)

22 ba

ab (c)

22

2

ba

ab (d)

)(2 22 ba

ab

29. The maximum value of 33

cos2cos5 , is

(a) 5 (b) 10 (c) 11 (d) – 1

30. If tan + tan 4 + tan 7 = tan tan 4 tan 7 , then =

(a) 4

n, n Z (b)

7

n, n Z

(c) 12

n, n Z (d) n , n Z

31. Solution of the equation sin x – cos x = 2 is

(a) 4

32n , n Z (b) 2n , n Z

(c) n , n Z (d) (2n + 1) , n Z

32. The value of 3

2tan

5

4costan 11 is

(a) 17

6 (b)

6

17

(c) 7

16 (d) none of these

33. The expression 224

)()()()(

cb

cbabacacbcba is equal to

(a) cos 2 A (b) 1 – cos A (c) A2sin (d) 1 + cos A

34. 2

)1(

2

2

x

xd

(a) Cx 22 2 (b) Cx 22

(c) Cx 2/32 )2(

1 (d) none of these

35. If in a triangle ABC, cos A cos B + sin A sin B sin C = 1, then the triangle is

(a) isosceles (b) right angled

(c) isosceles right angled (d) equilateral

36. A flagstaff on the top of a house subtends the same angle at two points at distance a and b

from the house and on the same side of it, then the length of flagstaff is

(a) (a + b) sin (b) (a + b) cos

(c) (a + b) tan (d) (a + b) cot

37. If x satisfies |x – 1| + |x – 2| + |x – 3| 6, then

(a) 0 x 4 (b) x – 2 or x 4

(c) x 0 or x 4 (d) none of these

38. For all x R, if 01592 mmxmx then m lies in the interval

(a) 0,61

4 (b)

61

4,0 (c)

4

61,

61

4 (d) 0,

4

61

39. If roots of an equation 01nx are 1, ,.....,, 121 naaa then the value of

)1().....1()1()1( 1321 naaaa will be

(a) n (b) 2n (c) nn (d) 0

40. If ..........,,, 321 aaa are in A.P. such that 2252420151051 aaaaaa , then

2423321 ..... aaaaa is equal to

(a) 909 (b) 75 (c) 750 (d) 900

41. .....4

1

3

1

2

11 to equals

(a) e (b) log 2 (c) 1e (d) none of these

42. If .....5.4

1

4.3

1

3.2

1

2.1

1S , then Se equals

(a) e

e

4log (b)

e

4 (c)

4log

ee (d)

4

e

43. If n

r

rr

n xCx0

)1( , then 11

2

0

1 1.....11n

n

C

C

C

C

C

C

(a) )!1(

1

n

nn

(b) !1

)1( 1

n

n n

(c) !

)1(

n

n n

(d) !

)1( 1

n

n n

44. If A = )( ija is a 4 × 4 matrix and ijc is the co-factor of the element ija in Det (A), then the

expression 1414131312121111 CaCaCaCa equals

(a) 0 (b) – 1 (c) 1 (d) Det (A)

45. The angle of intersection of the curves 2xy and 376 xy at (1, 1) is

(a) 4

(b) 3

(c) 2

(d) none of these

46. If cba

,, are three non-coplanar vectors, then ],,[ bacacba

is equal to

(a) ][3 cba

(b) ][2 cba

(c) ][4 cba

(d) ][2 cba

47. If ABCDEF is a regular hexagon inscribed in a circle with centre O, then

)( AFAEADACAB equals

(a) AO4 (b) AO5 (c) AO6 (d) AO8

48. The length of the perpendicular from the origin to the line

)543()424( kjikjir

is

(a) 52 (b) 2 (c) 25 (d) 6

49. The radius of the circular section of the sphere 1142222 zyzyx by the plane

x + 2y + 2z = 15 is

(a) 4 (b) 7 (c) 5 (d) 7

50. The image of the point (1, 2, – 1) in the plane 5)ˆ4ˆ5ˆ3( kjir

is

(a) 25

39,

5

6,

25

73 (b)

25

39,

5

6,

25

73

(c) (– 1, – 2, 1) (d) none of these

51. The function f (x) =

1,0

11,

1,1

x

xx

x

is

(a) differentiable for all x (b) f is continuous everywhere

(c) f is differentiable at x = – 1 (d) f is continuous at x = – 1

52. Which of the following is not a measure of dispersion?

(a) Variance (b) Mean Deviation

(c) Standard Deviation (d) Mode

53. The standard deviation of the first n natural numbers is

(a) 2

1n (b)

2

)1(nn

(c) 12

12n (d) none of these

54. If at each point of the curve y = ,123 xaxx the tangent is inclined at an acute angle

with the positive direction of x-axis, then

(a) a > 0 (b) a < 3

(c) 33 a (d) none of these

55. f (x) = sin x – a sin 2x – 3

1 sin 3x + 2ax increases for all x R if

(a) a < 0 (b) 0 < a < 1

(c) a = 1 (d) a > 1

56. If 122 )()()(' mn bxaxxf where m, n N, then

(a) x = b is a point of minima (b) x = b is a point of maxima

(c) x = b is a point of inflexion (d) none of these

57. If the sub-normal at any point on nn xay 1 is of constant length, then the value of n is

(a) 2

1 (b) 1 (c) 2 (d) – 2

58. dxxx2/

2/

3coscos

(a) 4

3 (b)

4

3 (c)

3

4 (d) 0

59. The area bounded by the curves 2yx and 223 yx , then

(a) 2 sq. units (b) 3 sq. units (c) 4 sq. units (d) 5 sq. units

60. A thief when detected jumps out of a running train at right angles to its direction with a

velocity of 5 m/min. If the velocity of the train is 60 km/hr, the direction in which the thief

falls inclined with direction of train at angle of

(a) 200

1tan 1 (b)

12

1tan 1

(c) 200tan 1 (d) none of these

61. The solution of the equation )0(,2sinsin4cos2cos2 xxxxx is

(a) 2

1cot 1

(b) 2tan 1

(c) 2

1tan 1

(d) None of these

62. Let xxf cos)( , then which of the following is true

(a) )(xf is periodic with period 2

(b) )(xf is periodic with period

(c) )(xf is periodic with period 24

(d) )(xf is not a periodic function

63. In a triangle ABC, 3

B and 4

C and D divides BC internally in the ratio 1 : 3. Then

CAD

BAD

sin

sin is equal to

(a) 3

1 (b)

3

1

(c) 6

1 (d)

3

2

64. If 0,tancossin 111 xxxx , then the smallest interval in which lies is given by

(a) 4

3

4 (b) 0

(c) 04

(d) 24

65. If 2

3sinsinsin 111 zyx , then the value of

101101101

100100100 9

zyxzyx is

equal to

(a) 0 (b) 3 (c) – 3 (d) 9

66. Three vertices of parallelogram taken in order, are (1, 3), (2, 0) and (5, 1). Then its fourth

vertex is

(a) (3, 3) (b) (4, 4) (c) (4, 0) (d) (0, –4)

67. The line joining two points )0,2(A and )1,3(B is rotated about A in anti-clockwise

direction through an angle of 15°. The equation of the line in the new position, is

(a) 0323 yx (b) 023 yx

(c) 0323 yx (d) 023 yx

68. In the equation )( 11 xxmyy if m and x1 are fixed and different lines are drawn for

different values of y1, then

(a) the lines will pass through a single point

(b) there will be a set of parallel lines

(c) there will be one line only

(d) none of these

69. The intercept of a line between the coordinate axes is divided by point (– 5, 4) in the ratio

1 : 2. The equation of the line will be

(a) 06085 yx (b) 06058 yx

(c) 03052 yx (d) None of these

70. The equation 4)2()2( 2222 yxyx represents a

(a) circle

(b) pair of straight lines

(c) parabola

(d) ellipse

71. The straight line 0)3()2( yx cuts the circle 11)3()2( 22 yx at

(a) no points (b) one point

(c) two points (d) none of these

72. A circle lies in the second quadrant and touches both the axes. If the radius of the circle be 4,

then its equation is

(a) 0168822 yxyx (b) 0168822 yxyx

(c) 0168822 yxyx (d) 0168822 yxyx

73. The locus of the intersection point of ayx sincos and byx cossin is

(a) ellipse (b) hyperbola

(c) parabola (d) none of these

74. 05222 yxy represents

(a) a circle whose centre is (1, 1) (b) a parabola whose focus is (1, 2)

(c) a parabola whose directrix is 2

3x (d) a parabola whose directrix is

2

1x

75. The curve described parametrically by .12 ttx 12 tty represents

(a) a pair of straight lines (b) an ellipse

(c) a parabola (d) a hyperbola

76. If the module of the vectors cba

,, are 3, 4, 5 respectively and a

and bcb

, and cac

,

and ba

are mutually perpendicular, then the modulus of cba

is

(a) 12 (b) 12

(c) 25 (d) 50

77. What will be the length of longer diagonal of the parallelogram constructed on ba

25 and

ba

3 , if it is given that 3,22 ba

and angle between a

and b

is 4

(a) 15 (b) 113

(c) 593 (d) 369

78 Image point of (5, 4, 6) in the plane 0152zyx is

(a) (3, 2, 2) (b) (2, 3, 2)

(c) (2, 2, 3) (d) (–5, –4, –6)

79. If a line makes angles , , , with the four diagonals of a cube, then the value of 2222 coscoscoscos is equal to

(a) 1 (b) 4/3

(c) Variable (d) none of these

80. Suppose 2)1()( xxf for 1x . If )(xg is the function whose graph is the reflection of

the graph of )(xf with respect to the line xy , then )(xg equals

(a) 0,1 xx (b) 1,)1(

12

xx

(c) 1,1 xx (d) 0,1 xx

81. Let xxf sin)( and ||ln)( xxg . If the ranges of the composite functions fog and gof are

1R and 2R respectively, then

(a) }0:{},11:{ 21 vvRuuR

(b) }11:{},0:{ 21 vvRuuR

(c) }0:{},11:{ 21 vvRuuR

(d) }0:{},11:{ 21 vvRuuR

82. If 225)( xxG , then1

)1()(lim

1 x

GxG

x=

(a) 24

1 (b)

5

1

(c) 24 (d) none of these

83. Let kgf )1()1( and their nth derivatives )1(nf and )1(ng exist and are not equal for some

n. If )()(

)()()()()()(lim

xfxg

agxfagafxgaf

ax = 4, then the value of k is

(a) 4 (b) 2

(c) 1 (d) 0

84. xedx

d x tan.21

(a) 2

21

1

tansec

2

x

xxxe x

(b) 2

21

1

tansec

2

x

xxxe x

(c) 2

21

1

tansec

2

x

xxe x

(d) none of these

85. If qpqp yxyx )( , then

2

2

dx

yd =

(a) 0 (b) 1

(c) 2 (d) none of these

86. The distance travelled by a particle moving in a straight line in time t is cbtats 2 .

Acceleration of the particle is

(a) proportional to t (b) proportional to s

(c) proportional to s–3 (d) none of these

87. The length of subtangent to the curve 422 ayx at the point ),( aa is

(a) 3a (b) 2a

(c) a (d) 4a

88. 22 1)1( xx

dx

(a) cx

x

2

1tan

2

1 21 (b) c

x

x

2

1

1

2tan

2

1

(c) cx

x

2

1tan2

21 (d) c

x

x

2

1tan2

21

89. dxxa

xa

(a) cxaaxa 221 )/(cos (b) cxaaxa 221 )/(cos

(c) cxaaxa 221 )/(cos (d) cxaaxa 221 )/(cos

90. Let

xx

xx

xxxxx

xf22

222

2

coscos1

coseccoscos

coseccotseccoscos

)( , then 2/

0

)( dxxf

(a) 15

8

4 (b)

15

8

4

(c) 15

8

4 (d)

15

8

4

91. Let ][)( xxxf , for every real number x, where [x] is the integral part of x. Then 1

1

)( dxxf

(a) 1 (b) 2

(c) 0 (d) 2

1

92. The value of

3

1

33 dxx lies in the interval

(a) (1, 3) (b) (2, 30) (c) (4, 302 ) (d) none of these

93. The differential equation 3

2/12

2

22

ydx

dy

dx

yd has the degree

(a) 1/2 (b) 2 (c) 3 (d) 4

94. The solution of the differential equation 0)()1(1tan2

dx

dyexy y , is

(a) ykex1tan)2( (b) kexe yy 11 tan2tan2

(c) kyxe y 1tan tan1

(d) kyexe y 1tan2 tan1

95. A kite of weight W is flying with its string along a straight line. If the ratios of the resultant

air pressure R to the tension T in the string and to the weight of the kite are 2 and )13(

respectively, then

(a) WT )26( (b) 2

)13( WT

(c) WT )26(2

1 (d) WT )13(

96. To a man running at a speed of 20 km/hr, the rain drops appear to be falling at an angle of 30°

from the vertical. If the rain drops are actually falling vertically downwards, their velocity in

km/hr, is

(a) 310 (b) 10 (c) 320 (d) 40

97. Two squares are chosen at random on a chess-board. The probability that they a side in

common, is

(a) 1/9 (b) 2/7 (c) 1/18 (d) none of these

98. A car completes the first half of its journey with a velocity v1 and the rest half with a velocity

v2. Then the average velocity of the car for the whole journey is

(a) 2

21 vv (b) 21vv (c)

21

212

vv

vv (d) none of these

99. Two lines of regression are 0743 yx and 054 yx . Then correlation coefficient

between x and y is

(a) 4

3 (b)

4

3 (c)

16

3 (d)

16

3

100 For the following feasible region, the linear constraints except 0x and 0y , are

2x + y = 600 x = 250

y = 350

O X

Y

(a) 6002,350,250 yxyx (b) 6002,350,250 yxyx

(c) 6002,350,250 yxyx (d) 6002,350,250 yxyx