Question Bank In Math 2x6x2f

CONTENTS CHAPTER NO.

CHAPTER NAME

PAGE NO.

1

Trigonometric Ratios & Identities

1-5

2

Trigonometric Equations & Inverse Circular Functions

6 - 10

3

Properties and Solution of Triangles

11 - 14

4

Heights and Distances

15 - 16

5

Sequence and Progressions

17 - 20

6

Quadratic Equations and Inequations

21 - 24

7

Complex Numbers

25 - 29

8

Permutation and Combination

30 - 32

9

Binomial Theorem

33 - 36

10

Infinite Series

37 - 40

11

Straight lines and Pair of Straight lines

41 - 44

12

Circles

45 - 48

13

Conic Sections - Parabola

49 - 52

14

Ellipse and Hyperbola

53 - 56

15

Functions, Limit and Continuity

57 - 62

16

Differentiability and Differentiation

63 - 67

17

Application of Derivatives

68 - 71

18

Indefinite Integration

72 - 77

19

Definite Integration

78 -83

20

Area Under the Curve

84 - 85

21

Differential Equations

86 - 88

22

Determinants

89 - 93

23

Matrices

94 - 97

24

Vectors

98 - 100

25

Three Dimensional Geometry

101 - 103

26

Probability

104 - 106

27

Statistics

107 - 110

28

Correlation and Regression analysis

111 - 112

29

Sets, Relations and Mappings/Functions

113 - 116

30

Linear Programming

117 - 119

31

Numerical Methods

120 - 121

TRIGONOMETRIC RATIOS & INDENTITIES

1 − tan 2 15° 1. The value of is 1 + tan 2 15° a) 1

3

b)

c)

d) 2

3/2

2. The equation a sin x + b cos x = c, where | c | > a 2 + b 2 has a) a unique solution b) infinite no. of solutions

c) no solution

d) none of these

3. If sin x + cosec x = 2, then sin x + cosec x is equal to a) 2 b) 2n

c) 2n - 1

d) 2n - 2

c) 11

d) none of these

n

4. If cos A =

FG IJ FG IJ H K H K

A 5A 3 sin = , then 32 sin 2 2 4

a) 7

b) 8

5. The value of sin

a)

n

3π 5π 7π 9π 11 π 13 π π is sin sin sin sin sin sin 14 14 14 14 14 14 14

1 16

b)

1 64

c)

1 128

d) none of these.

6. If sin (α + β ) = 1, sin (α − β ) = 1 / 2; α , β ∈ [0, π / 2] then tan (α + 2β) tan ( 2α + β) is equal to a) 1 7. If a)

b) -1

c) 0

d) none of these

1 + sin x + sin 2 x + sin 3 x +....+.... ∞ is equal to 4 + 2 3 , 0 < x < π , then x = π 6

b)

π 4

c)

π π or 3 6

d)

π 2π . or 3 3

8. The value of tan 5 θ is

5 tan θ − 10 tan 3 θ + tan5 θ a) 1 − 10 tan 2 θ + 5 tan 4 θ

5 tan θ + 10 tan 3 θ − tan5 θ b) 1 + 10 tan 2 θ − 5 tan 4 θ

5 tan5 θ − 10 tan 3 θ + tan θ c) 1 − 10 tan 2 θ + 5 tan 4 θ

d) none of these.

3

3

9. For what and only what values of α lying between 0 and π is the inequality sin α cos α > sin α cos α valid ?

FG π IJ U  π , 3π  H 4K  2 4 

a) α ∈ 0,

10. If sin A + cos A = m and

FG π IJ H 2K

b) α ∈ 0,

c) α ∈

d) none of these

sin 3 A + cos3 A = n, then

m3 − 3m + n = 0 3 c) m − 3m + 2n = 0

n 3 − 3n + 2m = 0 3 d) m + 3m + 2n = 0

a)

11. For x ∈ R , tan x +

FG π , π IJ H 4 2K

b)

FG IJ H K

1 x 1 x 1 x tan + 2 tan 2 +...+ n −1 tan n −1 is equal to 2 2 2 2 2 2 Quest Tutorials Head Office : 44C,Kalusarai New Delhi-16; Ph.(011) 46080363

1

1

a) 2 cot 2x -

c)

cot

12. If y =

a)

2

n −1

cot

FG x IJ H2 K

b)

n −1

FG x IJ − cot 2 x H2 K

1 2

n −1

cot

FG x IJ − 2 cot 2 x H2 K n −1

d) none of these

n −1

sec 2 θ − tan θ then sec 2 θ + tan θ '

1 ≤ y≤3 3

b) y ∉

LM 1 , 3OP N3 Q

c) −3 < y < −

1 3

d) none of these.

13. If α, β, γ , δ are the smallest positive angles in ascending order of magnitude which have their sines equal to the positive quantity k, then the value of 4 sin

α β γ δ + 3 sin + 2 sin + sin is equal to 2 2 2 2

2 1+ k

b)

a) 2 1− k

c)

1+ k 2

d) none of these

14. The expression cosec A cot A − sec A tan A − (cot A − tan A) (sec A cosec A − 1) is equal to 2

a) 1

2

2

2

2

b) -1

2

2

2

c) 0

d) none of these.

c) 210°, 240°

d) 210°, 330°

15. The values of θ ( 0 < θ < 360° ) satisfying cosec θ + 2 = 0 are a) 210°, 300°

16. If

b) 240°, 300°

12 10 8 6 4 2 sin x + sin 2 x = 1, then the value of cos x + 3 cos x + 3 cos x + cos x + 2 cos x + cos x − 2, is equal to

a) 0

b) 1

d) sin2 x.

c) 2

17. If A, B, C are acute positive angles such that A + B + C = π and cot A cot B cot C = K, then a) K ≤

1 3 3

b) K ≥

1 3 3

c) K <

1 9

d) K >

1 3

18. If sin β is the GM between sin α and cos α , then cos 2 β = a) 2 sin 2

FG π − αIJ H4 K

b) 2 cos2

FG π − αIJ H4 K

19. sin2 A + sin2 ( A - B ) + 2 sin A cos B sin ( B - A ) is equal to b) sin2 B a) sin2 A

c) 2 sin 2

FG π + αIJ H4 K

c) cos2 A

d) none of these

d) cos2 B

sin 2 13° sin 2 77° tan 135° 2 2 20. The value of the determinant sin 77° tan 135° sin 13° is equal to tan 135° sin 2 13° sin 2 77° a) -1

b) 0

c) 1

Quest Tutorials Head Office : 44C,Kalusarai New Delhi-16; Ph.(011) 46080363

d) 2 2

21. If tan A + sec A = 2 and A is less than 90°, then sin A + cos A is equal to a) 2 / 5

b) 3 / 5

c) 4 / 5

22. Let n be a fixed positive integer such that sin ( π / 2 n) + cos( π / 2 n) = a) n = 4

b) n = 5

d) 7 / 5

n / 2, then

c) n = 6

d) none of these

23. Number of solutions of the equations tan x + sec x = 2 cos x lying in the interval [ 0, 2 π ] is a) 0

b) 1

c) 2

1 , then 4 a) sin 2 A = 8 / 17 c) sin A + cos A = 23 / 17

d) 3

24. If sec A − tan A =

25. If 2 sin ( x + 2

b) cos A = 15 / 17 d) cos A − sin A = 7 / 17

π ) + 3 cos 2 x > 0, then 4

a) cos( 2 x −

π 1 )>− 6 2

b) sin( 2 x −

1 π )<− 6 2

c) sin( 2 x −

π 1 )>− 6 2

d) cos( 2 x −

1 π )<− 6 2

26. sec θ = 2

a)

4 xy , where x ∈ R, y ∈ R , is true if and only if ( x + y) 2

x+ y≠0

27. If tan

b)

x = y, x ≠ 0

x>y

b)

28. The minimum value of

cos 2θ + cos θ for real values of θ is

a) −

29. If

d)

x ≠ 0, y ≠ 0

π π 5π 7π are in AP and tan , y and tan are also in AP then , x and tan 9 9 18 18

2x = y

a)

c) x = y

9 8

b) 0

0° < θ < 180° then

a) 2 cos

θ 2n

c) x = y

d) none of these

c) - 2

(d)none of these

2 + 2 + 2 +...+ 2 (1 + cos θ) , there being n number of 2's, is equal to b) 2 cos

θ 2

n−1

c) 2 cos

θ 2 n+1


d) none of these

3

30. If tan

a) −

31. If

a)

α β and tan are the roots of the equation 8 x 2 − 26 x + 15 = 0 then cos(α + β) is equal to 2 2 627 725

627 725

b)

d) none of these

2 sin α 1 + sin α − cos α is equal to = λ then 1 + sin α + cos α 1 + sin α 1 λ

b) λ

32. If |tan A | < 1, and | A | is acute then a) tan A

c)

b) - tan A

LM0, π IJ N 2K

b)

FG H

34. If tan θ + tan θ +

IJ K

FG H

1− λ

d)

1+ λ

1 + sin 2 A + 1 − sin 2 A is equal to 1 + sin 2 A − 1 − sin 2 A

33. The set of all possible values of α in [ −π, π] such that

a)

c) - 1

c) cot A

d) - cot A

1 − sin α is equal to sec α - tan α is 1 + sin α

LM0, π IJ ∪ FG π , πIJ N 2K H 2 K

c)

[ −π, 0]

d)

FG − π , π IJ H 2 2K

IJ K

π π + tan θ − = k tan 3θ then k is equal to 3 3

a) 1

b) 3

35. Let n be an odd integer. If sin nθ =

n

∑

c) 1/3

d) none of these

br sin r θ for all real θ then

r =0

a) b0 = 1, b1 = 3

b) b0 = 0, b1 = n

c) b0 = −1, b1 = n

d) b0 = 0, b1 = n 2 − 3n − 3

36. The value of tan 63° - cot 63° is equal to a)

2 . 10 − 2 5 5 +1

b)

2 . 10 + 2 5 5 +1

c)

5 −1 . 10 − 2 5 4

37. The sum of the real roots of cos6 x + sin 4 x = 1 in the interval − π ≤ x ≤ π is equal to a) 0 b) π c) - π

d) none of these

d)

none of these

38. If ABCD is a convex quadrilateral such that 4 sec A + 5 = 0 then the quadratic equation whose roots are tan A and cosec A is a) 12 x 2 − 29 x + 15 = 0

b) 12 x 2 − 11x − 15 = 0

c) 12 x 2 + 11x − 15 = 0


d) none of these

4

39. log tan 1° + log tan 2° + .... + log tan 89° = a) 1 40.

b) 0

c) π / 4

d) none of these

b) 2 sin 20° / sin 40°

c) 4

d) 4 sin 20° / sin 40°

3 cos ec 20° − sec 20° is equal to a) 2

41. If tan A = a) 0

a 1 and tan B = , then the value of A + B is a +1 2a + 1 c) π / 3 b) π / 2

d) π / 4

42. tan 5 x − tan 3x − tan 2 x = a) tan 5x tan 3x tan 2x

b) cos 5x cos 3x cos 2 x

c) sin 5x sin 3x sin 2x

d) tan 8x tan 2x tan x

43. The value of tan 81° − tan 63° − tan 27° + tan 9° a) 1 b) 2 8

c) 3

d) 4

c) 1 < 2A ≤ 3

d) none of these

14

44. If A = sin θ + cos θ , then for all values of θ , b) 0 < A ≤ 1 a) A ≥ 1

45. If sin(θ + α) = a and sin(θ + β) = b , then cos 2(α − β) − 4ab cos(α − β) is equal to a) 1 − a 2 − b 2

b) 1 − 2a 2 − 2b 2

c) 2 + a 2 + b 2

d) 2 − a 2 − b 2

46. If log 3 sin x − log 3 cos x − log 3 (1 − tan x ) − log 3 (1 + tan x ) = −1 , then tan 2 x = a) - 2

b) 3/2

c) 2 / 3

d) 6

c) (1 / 8) cot 10°

d) (1 / 8) sec 10°

47. If x = cos10° cos 20° cos 40° , then the value of x is a) (1 / 8) tan 10°

3

48. If sin x sin 3x =

b) (1 / 8) cos ec10° n

∑C

m

cos m x is an identity in x , where C 0 , C1 , ......., C n are constants and C n ≠ 0 , then the value of n is

m =0

a) 17

b) 27

c) 16

d) 6

ANSWERS 1. 8. 15. 22. 29. 36. 43.

c a d c a a d

2. 9. 16. 23. 30. 37. 44.

c a d d a a b

3. 10. 17. 24. 31. 38. 45.

a c a c b b b

4. 11. 18. 25. 32. 39. 46.

c b a a c b c

5. 12. 19. 26. 33. 40. 47.

b a c b d c c


6. 13. 20. 27. 34. 41. 48.

a b b a b d d

7. 14. 21. 28. 35. 42.

5

d c d a b a

Trigonometric Equations & Inverse Circular Functions

1. The general solution of sin x − 3 sin 2 x + sin 3 x = cos x − 3 cos 2 x + cos 3 x is a) nπ +

π 8

nπ π + 2 8

b)

c)

( −1) n

FG nπ + π IJ H 2 8K

d)

2 n π + cos−1

d)

n π + ( −1) n

FG 3IJ H 2K

2. The general value of θ satisfying the equation 2 sin 2 θ − 3 sin θ − 2 = 0 is a) n π + ( −1) n

π 6

b) n π + ( −1)

3. The set of values of x for which

n

π 2

c)

b)

RSn π + π , n = 1, 2, 3,....UV W T 4

d)

b) a - b 2

24 25

b) −

FG α + β IJ = H 2 K

c) b / a

5. If α is a root of 25 cos θ + 5 cos θ − 12 = 0, a)

7π 6

RS π UV T4W RS2 n π + π , n = 1, 2, 3,....UV W T 4

4. If α, β are different values of x satisfying a cos x + b sin x = c, then tan a) a + b

5π 6

tan 3x − tan 2 x = 1 is 1 + tan 3x tan 2 x

a) φ

c)

n π + ( −1) n

d) a / b.

π < α < π , then sin 2α is equal to 2

24 25

c)

13 18

d)

6. The number of pairs ( x, y ) satisfying the equations sin x + sin y = sin ( x + y ) and | x | + | y | = 1 is a) 2 b) 4 c) 6

−

13 18

d) infinite.

7. The expression ( 1 + tan x + tan2 x ) ( 1 - cot x + cot2 x ) has the positive values for x, given by a) 0 ≤ x ≤

π 2

b) 0 ≤ x ≤ π

c) for all x ∈ R

d)

x ≥ 0.

8. If the complex numbers (sin x + i cos 2x) and (cos x - i sin 2x) are conjugate to each other, then x is equal to a) n π

b)

FG n + 1 IJ π H 2K

c) 0

9. The values of x & y satisfying the system of equations a) x = n π + ( −1)

n

c) x = n π + ( −1)

n

2 sin x + cos y = 1, 16sin

d) none of these 2

x + cos2 y

π π and y = 2n π ± 6 3

b) x = nπ + (−1)

π 2π and y = 2n π ± 3 6

d)

n +1

= 4 are given by

π π and y = 2nπ ± 6 6

x = n π + ( −1) n+1

2π π and y = 2n π ± 6 3


6

10. The number of all possible ordered pairs ( x, y ),

x+y=

x , y ∈ R satisfying the system of equations

2π 3 , cos x + cos y = is 3 2

a) 0

b) 1

c) infinite

8 cos2 x = 1 are in AP with common difference

11. The value of x between 0 and 2 π which satisfy the equation sin x a)

12. If

π 4

b)

π 8

d) none of these.

c)

3π 8

c)

n π + ( −1) n

d)

5π 8

1 sin x ,cos x , tan x are GP, then x is equal to 6

a) n π ±

π , n ∈Z 3

13. From the identity

b) 2n π ±

π , n ∈Z 3

π , n ∈Z 3

d) none of these

sin 3x = 3 sin x − 4 sin 3 x it follows that if x is real and | x | < 1, then

a) | 3 x − 4 x 3 | > 1

b) | 3 x − 4 x 3 | ≤ 1

c) | 3 x − 4 x 3 | < 1

d) Nothing can be said about

3x − 4 x 3 .

14. 4 tan −1

1 1 is equal to − tan −1 5 239

a) π

b) π / 2

c)

15. The value of sin ( 2 sin-1 (0.8)) is equal to a) sin 1.2° b) sin 1.6°

16. If sin

−1

d)

π/4

c) 0.48

d) 0.96

c) 1

d) 3

FG IJ HK

x 5 π + cosec −1 = , then x = 5 4 2

a) 4 17. The value of sin (cot a)

π/3

x2 + 2 x2 + 1

b) 5 −1

(cot (tan −1 x ))) is b)

x2 + 1 x2 + 2

18. The greatest and least values of (sin

π π a) − , 2 2

−1

x c)

x2 + 1

d)

1 x2 + 2

x ) 3 + (cos−1 x ) 3 are

π3 π3 , b) − 8 8

π3 7 π3 , c) 32 8


d) none of these

7

19. If sin

−1

x + sin −1 y + sin −1 z =

a) 0

3π 9 is , the value of x 100 + y 100 + z100 − 101 2 x + y101 + z101

b) 1

c) 2

20. The number of real solutions of tan −1 a) zero

d) 3

x ( x + 1) + sin −1 x 2 + x + 1 = π / 2 is

b) one

c) two

21. If sin α sin ( 60° - α ) sin ( 60° + α ) = 1 / 8, and

d) infinite

n ∈I, then

a) α = nπ + ( −1) π / 6

b) α = ( nπ / 3) + ( −1) π / 18

c) α = nπ = ( −1) π / 3

d) α = ( nπ / 3) + ( −1) n π / 9

n

n

n

22. The value of sin ( tan-1 x ) is equal to a)

23.

x

b)

1− x2

1

c)

1+ x2

x 1+ x

d)

2

1 1− x2

2 cos2 x + 4 cos x = 3 sin 2 x. If a) cos x =

−2 − 19 5

b) cos x =

−2 + 19 5

c) sin x =

−2 − 19 5

d)

24. The number of all possible triplets ( a1 , a2 , a3 ) such that a1 + a2 cos 2x + a3 sin2 x = 0 for all x is a) 0 b) 1 c) 3

sin x =

−2 + 19 5

d) infinite

25. In a triangle ABC, the angle A is greater than the angle B. If the values of the angles A and B satisfy the equation 3 sin x - 4 sin3 x - k = 0, 0 < k < 1, then the measure of angle C is a) π / 3 b) π / 2 c) 2 π / 3 d) 5 π / 6. 26. The principal value of sin a) 5 π / 6

−1

F − 3 I + cos FG cos 7π IJ is GH 2 JK H 6 K −1

b) π / 2

c) 3 π / 2

d) none of these

27. If sin θ + 7 cos θ = 5, then tan( θ / 2 ) is a root of the equation a)

28.

x 2 − 6x + 1 = 0

b) 6 x − x − 1 = 0 2

c)

6x 2 + x + 1 = 0

d)

x2 − x + 6 = 0

sin 3θ 1 = if 2 cos 2θ + 1 2 a) θ = n π +

π 6

b) θ = 2n π −

π 6

c) θ = n π + ( −1)

n

π 6

d) θ = n π −

π 6

29. A solution of the equation log 2 sin x − log 2 cos x − log 2 (1 − tan x ) − log 2 (1 + tan x ) + 1 = 0 is given by a) tan x = -1

b) tan x = 1

c) tan 2 x = -1 Quest Tutorials

Head Office : 44C,Kalusarai New Delhi-16; Ph.(011) 46080363

d) tan 2 x = 1 8

30.

6 tan 2 x − 2 cos2 x = cos 2 x if a) cos 2 x = -1

b) cos 2 x = 1

c) cos 2 x = -1/2

d)

cos 2 x = 1/2

d)

a is any positive integer

31. The equation cos4 x − ( a + 2) cos2 x − ( a + 3) = 0 possess a solution if a) a > - 3

b) a < - 2 −1

c)

−1

32. If A = tan 1 / 7, B = tan 1 / 3, then a) cos 2 A = sin 2 A b) cos 2 A = sin 2 B 33. If u = cot −1

tan α − tan −1 tan α , then tan

tan α

a)

−1

−1

c)

a) 1

π 8

d)

c) 24 / 7

cot α .

d) 5 / 3

3 − tan cot −1 3 ) is equal to b) π / 6

c)

36. If sin A + cos A = 1 , then sin 2 A is equal to a) 1 b) 2

sin 2

tan α

d) cos 2 A = sin 4 B

7 + cos−1 (3 / 5) then the value of x is b) 125 / 117

35. 2 tan −1 ( cosec tan −1 a) π / 16

c) cos 2 A = cos 2 B

FG π − u IJ is equal to H 4 2K

cot α

b)

34. If cosec x = 2 cot a) 44 / 117

37. The value of

− 3 ≤ a ≤ −2

+ sin 2

π/3

d)

c) 0

π/2

d) 1/2

3π 5π 7π + sin 2 + sin 2 is 8 8 8 c) 1

b) 2

1 8

d)

2

1 8

38. The value of cos1° cos 2° cos 3° ..... cos 179° is equal to a)

1 2

b) 0

c) 1

d) 2

39. The number of solutions of the equation 3(sin x + cos x) − 2(sin3 x + cos3 x) = 8 are a) 0 b) 1 c) 2

d) 3

40. If tan pθ − tan qθ = 0 , then the values of θ form a series in a) A.P b) G.P c) H.P

d) none of these

41. If α and β are the solutions of a tan θ + b sec θ = c , then tan (α + β) = a)

2ac 2

a −c

2

2ac

b)

2

2

c)

2ac 2

2

c −a a +c 42. The number of roots of the equation x + 2 tan x = π / 2 in the interval [0, 2π] is a) 1

b) 2

c) 3


d) none of these

d) infinite

9

43. tan [tan −1 (1 / 7) + tan −1 (1 / 13)] is equal to a) 2/9 b) 9/2

c) 7/9

d) 9/7

44. If sin (sin −1 1 / 5 + cos −1 x ) = 1 , then x = a) 1 b) 0

c) 4/5

d) 1/5

45. 2 sin −1 x = cos −1 (1 − 2 x 2 ) is true for b) − 1/ 2 ≤ x ≤ 1/ 2

a) − 1 ≤ x ≤ 1

c)

0 ≤ x ≤1

d) none of these

c)

( π / 3) (4k + 1)

d)

46. If cos 3x + sin(2 x + 5π / 6) = −2 , then x = a) (π / 3) (6k − 1)

b) (π / 3) (6k + 1)

(π / 3) ( 2k + 1)

ANSWERS 1. 8. 15. 22. 29. 36. 43.

b d d c d c a

2. 9. 16. 23. 30. 37. 44.

d c d b d b d

3. 10. 17. 24. 31. 38. 45.

c a c d c b c

4. 11. 18. 25. 32. 39. 46.

c a c c d a b

5. 12. 19. 26. 33. 40.

b b a b a a


6. 13. 20. 27. 34. 41.

c b c b b a

7. 14. 21. 28. 35. 42.

10

c d b c c c

PROPERTIES AND SOLUTION OF TRIANGEL 1. The perimeter of a ∆ ABC is 6 times the arithmetic mean of the sines of its angles. If the side a is 1, then the angle A is a)

π 6

b)

π 3

c)

π 2

d)

π

2. The area of the circle and the area of a regular polygon of n sides and of perimeter equal to that of the circle are in the ratio of a) tan

FG π IJ : π H nK n

b) cos


c) sin

π π : n n

d) cot


3. In a triangle ABC, the line ing the circumcentre to the incentre is parallel to BC, then cos B + cos C = a) 3 / 2 b)1 c) 3 / 4 d)1 / 2. 4. The ex - radii of a triangle r1, r2, r3 are in harmonic progression, then the sides a, b, c are a) in H.P. b) in A.P. c) in G.P. 5. If a cos A = b cos B, then the triangle is a) equilateral c) isosceles

d) none of these

b) right angled d) isosceles or right angled

6. Two straight roads intersect at an angle of 60°. A bus on one road is 2 km. away from the intersection and a car on the other road is 3 km. away from the intersection. Then the direct distance between the two vehicle is a) 1 km 7. In any triangle ABC, ∑

b)

2 km

c) 4 km

d)

7 km

sin 2 A + sin A + 1 is always greater than sin A

a) 9

b) 10

8. In any ∆ ABC if 2 cos B = a / c then the triangle is a) right angled b) equilateral

c) 27

d) none of these

c) isosceles

d) none of these.

9. In an equilateral triangle, the in - radius, circum - radius and one of the ex - radii are in the ratio a) 2 : 3 : 5

b) 1 : 2 : 3

c) 1 : 3 : 7

d) 3 : 7 : 9.

c) 8 / 17

d) 1 / 2.

10. If in a ∆ ABC, ∆ = a2 - ( b - c )2 , then tan A = a) 15 / 16

b) 8 / 15

11. In a triangle the length of the two larger sides are 24 and 22, respectively. If the angles are in AP, then the third side is a) 12 + 2 13

b) 12 - 2

2

3

c) 2

2

3 +2

d) 2

3-2

2

12. If in a triangle ABC, sin A = sin B and 2 cos A = 3 cos B , then the ∆ ABC is a) right angled

b) obtuse angled

c) isosceles


d) equilateral

11

13. Points D, E are taken on the side BC of a triangle ABC, such that BD = DE = EC. If ∠BAD = x , ∠DAE = y , ∠EAC = z , then the value of

sin ( x + y ) sin ( y + z ) is equal to sin x sin z

a) 1

b) 2

c) 4

d) none of these

14. If A + B + C = π , n ∈ Z , then the tan nA + tan nB + tan nC is equal to a) 0

b) 1

c)

tan nA tan nB tan nC

d) none of these

15. If p1 , p2 , p3 are altitudes of a triangle ABC from the vertices A, B, C and ∆ , the area of the triangle, then p1−2 + p2−2 + p3−2 is equal to a)

a +b+c ∆

16. If in a triangle ABC,

b)

a 2 + b2 + c2 4 ∆2

c)

a 2 + b2 + c2 ∆2

d) none of these

sin A sin ( A − B ) = , then sin C sin ( B − C )

a) a, b, c are in AP

b) a

2

, b 2 , c 2 are in AP

c) a, b, c are in HP

d)

a 2 , b 2 , c 2 are in HP

17. If R is the radius of circumscribing circle of a regular polygon of n-sides having length of side as a , then R = a)

FG IJ H K

π a sin 2 n

b)

FG IJ H K

π a cos 2 n

c)

FG IJ H K

π a cosec 2 n

d)

18. If p1 , p2 , p3 are respectively the perpendiculars from the vertices of a triangle to the opposite sides, then

FG IJ H K

π a cosec 2 2n

cos A cos B cos C is + + p1 p2 p3

equal to a)

19. If

1 r

b)

1 R

c)

r r2 = , then r1 r3

a) A = 90°

1 ∆

d) none of these

c) C = 90o

b) B = 90°

d) none of these

20. If the angles of a triangle are 30° and 45° and the included side is ( 3 + 1) , then the area of the triangle is a)

1 3 −1

21. In a triangle ABC, ∠ B =

a)

1 6

3 +1

b)

c)

1 3 +1

d) none of these

π π sin ∠BAD and ∠ C = . Let D divide BC internally in the ratio 1 : 3. Then equals 3 4 sin ∠CAD b)

1 3

c)

1 3

d)

2 3

22. The two adjacent sides of a cyclic quadrilateral are 2 and 5 and the angle between them is 60°. If the third side is 3, the remaining fourth side is Quest Tutorials Head Office : 44C,Kalusarai New Delhi-16; Ph.(011) 46080363

12

a) 2

b) 3

c) 4

d) 5

23. If the data given to construct a triangle ABC are a = 5, b = 7, sin A = 3 / 4, then it is possible to construct a) only one triangle b) two triangles c) infinitely many triangles d) no trianlges 24. In triangle ABC, A = 30°, b = 8, a = 6, then B = sin-1 x, where x = a) 1 / 2 b) 1 / 3

c) 2 / 3

d) 1

25. If A = 30°, a = 7, b = 8 in ∆ ABC, then B has a) one solution b) two solutions

c) no solution

d) none of these.

26. In a triangle the angles are in A.P. and the lengths of the two larger sides are 10 and 9 respectively, then the length of the third side can be a) 5 ± 6

b) 0.7

c)

27. The smallest angle of the triangle whose sides are 6 + 12 , a)

π 3

b)

π 4

5− 6

d) none of these

48 , 24 is c)

π 6

d) none of these

28. In a ∆ ABC, a, b, A are given and c1 , c2 are two values of the third side c. The sum of the areas of two triangles with sides a, b, c1 and a, b, c2 is b) (1/2) a2 sin 2A c) b2 sin 2 A d) none of these a) (1/2) b2 sin 2A 29. If in a triangle ABC, ( a + b + c ) ( b + c - a ) = bc, then the triangle is a) equilateral b) right anlged c) obtuse angled

d) none of these

30. If in a triangle ABC, a2 , b2 , c2 , the squares of the lengths of the sides of the triangles are in arithmetical progression, then b3 is equal to a) 2abc cos A b) 2abc cos B c) 2abc cos C d) none of these 31. If the angles of a triangle are in the ratio 1 : 3 : 5 and θ denotes the smallest angle, then the ratio of the largest side to the smallest side of the triangle is a)

3 sin θ + cos θ 2 sin θ

b)

3 cos θ − sin θ 2 sin θ

c)

cos θ + 3 sin θ 2 sin θ

d)

3 cos θ + sin θ 2 sin θ

32. The vertical angle of a triangle is divided into two parts, such that the tangent of one part is 3 times the tangent of the other and the difference of these parts is 30°, then the triangle is a) isosceles b) right angled c) obtuse angled d) none of these 33. In a cyclic quadrilateral ABCD; a, b, c, d denote the length of the sides AB, BC, CD and DA respectively, then cos A is equal to

a 2 + b2 − c2 − d 2 b) a) 2 (ab + cd )

b2 + c2 − d 2 − a 2 2 (bc + da )

c2 + d 2 − a 2 − b2 c) 2 (cd + ab)

34. In a triangle ABC, sin A + sin B + sin C is maximum when the triangle is a) right angled b) isosceles c) equilateral 35. If the angles A and B of the triangle ABC satisfy the equation sin A + sin B = a) π / 6

b)

π/3

d 2 + a 2 − b2 − c2 d) 2 (da + bc) d) obtuse angle

3 (cos B-cos A ) then they differ by

c) π / 4 Quest Tutorials


d)

π/2 13

36. The sides of a triangle are 17, 25, 28. The length of the largest altitude is a) 15 b) 84 / 5 c) 420 / 17

37. In a triangle ABC if

d) 210 / 17

cos A cos B cos C a + + = , then A is a b c bc

a) an acute angle

b) an obtuse angle

c) a right angle

d) equal to B-C

38. If in a triangle ABC, the line ing the circumcentre 0 and the incentre I is parallel to BC, then a) r = R cos A b) r = R sin A c) R = r cos A 39. Suppose that sin 3 x. sin 3 x =

d) R = r sin A

n

∑ cm cos mx, is an identity in x, where c0 , c1, c 2 are constants and c n ≠ 0 , then the value of n is

m =0

a) 4

b) 5

40. In any ∆ ABC,

c) 9

d) 6

c) 3 abc

d)

∑ a 3 sin (B − C ) is equal to b) 3(a + b + c )

a) 0

ab + bc + ca

ANSWERS 1. 8. 15. 22. 29. 36.

a c b a c c

2. 9. 16. 23. 30. 37.

a b b d b c

3. 10. 17. 24. 31. 38.

b b c c d a

4. 11. 18. 25. 32. 39.

b a b b b d

5. 12. 19. 26. 33. 40.

d b c a d a


6. 13. 20. 27. 34.

d c a c c

7. 14. 21. 28. 35.

14

a c a a b

HEIGHTS & DISTANCES 1. A flag staff of 5 mt high stands on a building of 25 mt high. At an observer at a height of 30 mt, the flag staff and the building subtend equal angles. The distance of the observer from the top of the flag staff is a)

5 3 2

b)

5

3 2

c)

5

2 3

2. If a flag-staff of 6 metres high placed on the top of a tower throws a shadow of 2 that the sun makes with the ground is a) 60° b) 30° c) 45°

d) none of these

3 metres along the ground then the angle (in degrees) d) none of these

3. The top of a hill observed from the top and bottom of a building of height h is at angles of elevation p and q respectively. The height of the hill is a)

h cot q cot q − cot p

b)

h cot p cot p − cot q

c)

h tan p tan p − tan q

d) none of these

4. On the level ground the angle of elevation of the top of a tower is 30°. On moving 20 mt. nearer the tower, the angle of elevation is found to be 60°. The height of the tower is a) 10 mt

b) 20 mt

c)

10 3 mt

d) none of these

5. A person standing on the bank of a river observes that the angle subtended by a tree on the opposite bank is 60°, when he retires 40 metres from the bank he finds the angle to be 30°. Then the breadth of the river is a) 40 m b) 60 m c) 20 m d) 30 m. 6. A tree is broken by wind, its upper part touches the ground at a point 10 metres from the foot of the tree and makes an angle of 45° with the ground. The entire length of the tree is a) 15 metres

7. A vertical pole subtends an angle is a) tan −1 (1 / 4)

b) 20 metres

c) 10 (1 +

2 ) metres

F GH

d) 10 1 +

I JK

3 metres. 2

tan −1 1 / 2 at a point P on the ground. The angle subtended by the upper half of the pole at the point P b) tan −1 ( 2 / 9)

c)

tan −1 (1 / 8)

d) tan −1 ( 2 / 3)

8. An aeroplane flying at a height of 300 m above the ground es vertically above another plane at an instant when the angles of elevation of the two planes from the same point on the ground are 60° and 45° respectively. The height of the lower plane from the ground is a) 100 3 m

b)

100 / 3 m

c) 50 m

d) 150 ( 3 + 1) m

9. A pole 50 m high stands on a building 250 m high. To an observer at a height of 300 m, the building and the pole subtends equal angles. The horizontal distance of the observer from the pole is a) 25 m

b) 50 m

c)

25 6 m

d) 25 3 m

10. A man in a boat rowing away from a cliff 150 metres high observes that it takes 2 minutes to change the anlge of elevation of the top of the cliff from 60° to 45°. The speed of the boat is a) (1 / 2) ( 9 − 3 3 ) km / h

b)

(1 / 2) ( 9 + 3 3 ) km / h

c) (1 / 2) ( 9 3 ) km / h

d) none of these Quest Tutorials Head Office : 44C,Kalusarai New Delhi-16; Ph.(011) 46080363

15

11. When the length of the shadow of a pole is equal to the height of the pole then the elevation of source of light is a) 30° b) 45° c) 60° d) 75° 12. From the top of a light house 60 m high with its base at sea level the angle of depression of a boat is 15°. The distance of the boat from the light house is

 3 −1 m   3 +1

 3 +1 m   3 −1

 a) 60

 b) 60

c)

 3 −1 m 30  3 +1  

 3 +1

 m d) 30   3 −1

13. On the level ground, the angle of the top of the tower is 30°. On moving 20 metres nearer, the angle of elevation is 60°. Then the height of the tower is a)

20 3 metres

(

)

c) 10 3 − 1 metres

b) 10 3 metres

d) none of these

14. The angle of elevation of the top of a tower from a point 20 metres away from its base is 45°. The height of the tower is a) 10 m

b) 20m

c) 40 m

d) 20 3 m

15. A tree is broken by wind and its upper part touches the ground at a point 10 metres from the foot of the tree and makes an angle of 45° with the ground. The entire length of the tree is

(

)

(

)



3

  d) 101 + 2  m   16. Angle of depression from the top of a light house of two boats are 45° and 30° due east which are 60m apart. The height of the light house is c) 10 1 + 2 m

a) 15m

b) 20m

a) 60 3

b) 30 3 − 1

(

)

c)

30 3 + 1

d) none of these

17. The angle of elevation of the top of a tower at a point G on the ground is 30°. On walking 20m towards the tower the angle of elevation becomes 60°. The height of the tower is equal to a)

10

b) 20 3

3

c)

20

d) 10 3

3

18. Two posts are 25 metres and 15 metres high and the line ing their tops makes an angle of 45° with the horizontal. The distance between posts is a) 5m

b)

10 2

m

c) 10 m

d) 10 2 m

19. A portion of a 30m long tree is broken by tornado and the top struck up the ground making an angle of 30° with ground level. The height of the point where the tree is broken is equal to a)

30 3

m

b) 10m

c)

d) 60 m

30 3 m

ANSWERS 1. 8. 15.

b a c

2. 9. 16.

a c c

3. 10. 17.

b a d

4. 11. 18.

c b c

5. 12. 19.

c b b


6. 13.

c b

7. 14.

16

b b

SEQUENCE AND PROGRESSIONS 1. The value of 21 / 4 .41 / 8 .81 / 16 .161 / 32....................., is a) 1

b) 2

c) 3/2

d) 5/2

c) 44

d) none of these

2. The third term of a G.P. is 4. The product of first five is a) 43

b) 45

3. Sum of three numbers in GP be 14. If one is added to first and second and 1 is subtracted from the third, the new numbers are in AP. The smallest of them is a) 2

b) 4

c) 6

d) 8

b) a + c > b + d

c)

b) ac > bd

c) ad > bc

4. If a, b, c, d are in HP, then a)

a +b > c+d

a+d >b+c

d) none of these.

5. If a, b, c, d are in HP, then a) ab > cd

d) none of these

6. If x, y, z are in G.P. & a x = b y = c z then a)

log b a = log a c

7. If a,b, c are in HP then

b) log c b = log a c

c)

b) GP

c) HP

F 1I F 8. The sum of first 10 of the series G x + J + G x H xK H F x − 1IJ FG x + 1IJ + 20 a) G H x −1K H x K F x − 1IJ FG x − 1IJ + 20 c) G H x −1K H x K 2

2

1 + 2 x

IJ + FG x K H 2

22

2

18

d) none of these

a b c , , will be in b+c c+a a +b

a) AP

20

log b a = log c b

d) none of these

IJ +....... is K FG x − 1IJ FG x + 1IJ + 20 H x −1K H x K

3

1 + 3 x

2

18

20

b)

9

d) none of these

2

11

9

11

2

9. If n Arithmetic means are inserted between two quantities a and b, then their sum is equal to

n b) ba + b g c) 2nba + b g b g 2 10. If log 2, logc2 − 1h and logc2 + 3h are in AP, then the value of x is given by a)

n a +b

x

n a −b 2

d)

log 5 3

x

b) log 2 5

a) 5/2

b g

d)

c)

log 3 5

11. If the AM of the roots of a quadratic in x is A and GM is G, then the quadratic is a)

x 2 − Ax + G 2 = 0

12. The sum of n term of the series

a) -

b g

n n +1 2

b) x − 2 Ax + G = 0 2

c)

x 2 − 2 Ax + G 2 = 0

d) x 2

− Ax + G = 0

12 − 22 + 32 − 42 + 52 − 62 +...... is ( n is even no.) b)

b g

n n +1 2

c)

b g

−n n + 1


(d)

none of these 17

13. If a, b, c are in GP, then log a x ,logb x ,log c x are in a) AP

b) GP

c) HP

d) none of these

14. If the sum of n of a series is an2 + bn, then the series is ( a, b are constants ) a) an AP

b) a GP

c) AP

d) none of these

15. Let sn denote the sum of first n of an A.P. If s2 n = 3sn then the ratio s3n / sn is equal to a) 4

b) 6

16. If the roots of the equation a)

c) 8

d) 10

x 3 − 12 x 2 + 39 x − 28 = 0 are in A.P., then their common difference will be

±1

b) ±2

c)

±3

b) G.P.

c) H.P.

d) none of these

c) 2 ( f - d )

d) d - c.

c) n + 2-n - 1

d) 2n - 1.

c) 171

d) 180.

d)

±4

17. log32, log62 log122 are in a) A.P.

18. If a, b, c, d, e, f are in A.P., then e - c is equal to a) 2 ( c - a )

b) 2 ( d - c )

19. The sum of n of the series

1 3 7 15 + + + +...... is 2 4 8 16

a) 2n - n - 1

b) 1 - 2-n

20. The maximum value of the sum of the A.P. 30, 27, 24, 21, .... is a) 165

b) 168

21. The least value of n for which the sum of the series 3 + 6 + 9 + .... to n exceeds 1000 is a) 25

b) 26

c) 27

d) 28.

22. If (A1, A2) ( G1, G2 ) ( H1, H2 ) are two (arithmetic), ( geometric ), ( harmonic ) means between two positive real numbers a and b, then value of a)

G1G2 H1 + H2 . is H1 H2 A1 + A2

ab

23. Sum of the series 1 +

b)

a 2 + b2

c)

b a +a b

d) none of these.

1 1 1 is + +.....+ 1+ 2 1+ 2 + 3 1 + 2 +...+ n

2n a) n +1

n b) n +1

n c) 2(n + 1)

n2 . d) 2(n − 1)

24. Sum to n of the series 1 + ( 1 + 2 ) + ( 1 + 2 + 3 ) + ... is a)

1 n ( n + 1) ( n + 2) 2

25. The sum to n of the series

a) 1 − (1 / 3) n

b)

1 n ( n + 1) ( n + 2) 3

c)

1 n ( n + 1) ( 2n + 1) 4

d) none of these

n − n (1 / 3) n

d) none of these

2 8 26 80 + + + +.... is equal to 3 9 27 81 b) 2 −

1 ( 2 / 3) n 2

c)


18

26. If x, y, z are the pth, qth and rth respectively of an A.P. as well as that of a G.P., then x a) x y z

b) 0

y−z

y z − x z x − y is equal to

c) 1

d) none of these

c) a, b, c are in G.P.

d) a, b, c are in H.P.

27. If log ( a + c ), log ( c - a ), log ( a - 2b + c ) are in A.P., then a) a, b, c are in A.P.

b) a2, b2, c2 are in A.P.

28. The coefficient of x in the expansion of ( x - 1 ) ( x - 2 ) .... ( x - 100 ) is 99

a) 5050

b)

-5050

c) 3300

d) -3310

29. If x = 1 + y + y2 + .......to ∞ , then y is a)

x x −1

b)

x 1− x

c)

x −1 x

d)

1− x x

30. The first term of an A.P of consecutive integers is p 2 + 1 . The sum of (2 p + 1) of this series can be expressed as a)

( p + 1)2

b)

(2 p + 1) ( p + 1)2

c)

( p + 1)3

d)

p 3 + ( p + 1)3

31. If the sum of first n of an A.P is Pn + Qn 2 where P and Q are constants, then common difference of A.P will be a)

P+Q

b)

P −Q

c) 2P

d) 2Q

32. If a,b,c are in A.P, then 10 ax +10 , 10 bx +10 , 10 cx +10 , x ≠ 0 are in a) A.P b) G.P only when x >0 c) G.P for all x

d) G.P only when x < 0

33. Let the sequence a1 , a2 , a3 ⋅ ⋅ ⋅ ⋅ ⋅ an , form an A.P, then

a12 − a22 + a32 − a42 + .. + a2 n −12 − a2 n 2 is equal to

(

a)

n a12 − a22n 2n − 1

c)

n a12 + a22n n +1

(

34. The sum of the series a)

1 3

)

b)

)

(

2n 2 a2 n − a12 n −1

)

d) none of these

1 1 1 + + + .........t o ∞ is 3 × 7 7 × 11 11 × 15 b)

1 6

c)

1 9

d)

1 12

35. If the first, second and last of an arithmetic series are a, b and c respectively. Then the number of are a)

1 (a + b + c) 2

b)

1 (b + c − 2a ) 2

c)

( b + c − 2a ) /(b − a )

d) none of these

36. If the sum of n of two arithmetic series are in the ratio 7 n + 1 : 4n + 27 , then their 11th are in the ratio a) 3 : 4 b) 4 : 3 c) 78 : 61 d) 152 : 119 37. If b + c, c + a , a + b are in H.P., then a 2 , b 2 , c 2 are in a) A.P b) G.P

c) H.P


d) none of these

19

38. 8 + 88 + 888 + ....n = a)

39. If

80 n 8n (10 − 1) − 81 9 a n +1 + b n +1 a n + bn

b)

10 n (10 − 1) 81

c)

80 n 8n (10 − 1) + 81 9

d) none of these

is the H.M of a and b, then the value of n is

a) 1

b) - 1/2

c) - 1

d) 0

ANSWERS 1. 8. 15. 22. 29. 36.

b a b d c b

2. 9. 16. 23. 30. 37.

b b c a d a

3. 10. 17. 24. 31. 38.

a b c d d a

4. 11. 18. 25. 32. 39.

c c b d c c

5. 12. 19. 26. 33.

c a c c a


6. 13. 20. 27. 34.

c c a d d

7. 14. 21. 28. 35.

20

c a b b c

QUADRATIC EQUATIONS AND INEQUATIONS 1. The roots of the equation x − x − 6 = x + 2 are 2

a) -2, 1, 4

b) 0, 2, 4

c) 0, 1, 4

d) -2, 2, 4

2. If f ( x ) = ax + bx + c, g ( x ) = − ax + bx + c where ac ≠ 0, then f ( x ) g ( x ) = 0 has a) at least three real roots b) no real roots c) at least two real roots d) two real roots and two imaginary roots. 2

3. The equation 2 cos2

2

FG x IJ sin H 2K

a) no real solution

2

1 π , 0 ≤ x ≤ has 2 x 2

x = x2 +

b) one real solution

c) more than one real solution d) none of these.

4. The number of real roots of the equation ( x − 1) 2 + ( x − 2) 2 + ( x − 3) 2 = 0 is a) 1 b) 2 c) 3

d) none of these.

5. The roots of the equation log 2 ( x 2 − 4 x + 5) = ( x − 2) are a) 4, 5

b) 2, -3

c) 2, 3

d) 3, 5

6. The value of k for which the equation 3x 2 + 2 x ( k 2 + 1) + k 2 − 3k + 2 = 0 has roots of opposite sign, lies in the interval a)

( −∞, 0)

b)

( −∞, − 1)

c) (1, 2)

d) ( 3/2, 2 )

ax 2 + bx + c = 0 is c) cx 2 + ax + b = 0 d) bx 2 + ax + c = 0

7. The quadratic equation whose roots are reciprocal of the roots of the equation a)

cx 2 + bx + a = 0

8. If one root of the equation a) 0

b) bx 2 + cx + a = 0

5x 2 + 13x + k = 0 is reciprocal of other, then the value of k is b) 5

c) 1 / 6

d) 6

9. If the root of the equation x − px + q = 0 differ by unity, then 2

a)

p 2 = 4q

b) p 2 = 4q + 1

c) p 2 = 4q − 1

d) none of these

2

10. The number of real roots of the equation x − 3 x + 2 = 0 is a) 4

b) 3

b) -1

c) 0

c h = 5 +5

12. The number of solutions of the equation 2 sin e a) 0

d) 1

x 3 − ix + 1 − i = 0, then the other real root is

11. If x = 1+ i is a root of the equation a) 1

c) 2

x

b) 1

x

−x

is

c) 2

b

gb g

d) none of these

d) Infinitely many

13. If c and d are roots of the equation x − a x − b − k = 0 , then a , b are roots of the equation a) c)

b x − cgb x − d g − k = 0 b x − agb x − cg + k = 0

b) d)

b x − cgb x − d g + k = 0 b x − bgb x − d g + k = 0


21

14. If x 2 − 3x + 2 is a factor of x 4 − px 2 + q , then the values of p and q are a) 5, -4 b) 5, 4 c) -5 , 4

d) -5, -4,

15. If the expression x 2 − 11x + a and x 2 a) 0,24 b) 0, -24

d) -2, 1

− 14 x + 2a have a common factor, then the values of 'a' are c) 1, -1

b g b

g b

g

16. If the sum of the roots of the equation a + 1 x 2 + 2a + 3 x + 3a + 4 = 0 is -1, then the product of the roots is a) 0

b) 1

17. The values of x satisfying x = a) 3, -2

c) 2

d) 3

c) 3

d) none of these

6 + 6 + 6+... ∞ are

b) -2

2 + i 3 is a root of x 2 + px + q = 0 where p, q ∈R, then a) p = −4, q = 7 b) p = 4, q = 7 c) p = 4, q = −7

18. If

bg f b x g > 0 for all x

p = − 4 , q = −7

d)

19. Let f x = x 2 + 4 x + 1 . Then a)

bg

b) f x > 1 when x ≥ 0

c)

bg

bg b g

f x ≥ 1 when x ≤ −4 d) f x = f − x for all x

20. The ading figure shows the graph of y = ax 2 + bx + c. Then

a) a < 0

b)

b 2 < 4ac

c) c > 0

d) none of these

c) c > 0

d)

c) -1

d) 0

c) 1

d) 0

21. The diagram shows the graph of y = ax 2 + bx + c. Then,

a) a > 0 22. If

b) b < 0

b 2 − 4ac = 0

3n x 2 − x + 1 = 0, Then the value of x is

a) -1, 1

b) 1

23. The number of positive real roots of a) 3

b) 2

x 4 − 4 x − 1 = 0 is


22

x 4 − 4 x − 1 = 0 is

24. The number of negative real roots of a) 3

b) 2

c) 1

d) 0

25. The number of complex roots of the equation x 4 − 4 x − 1 = 0 is a) 3

b) 2

c) 1

d) 0

26. The number of values of k for which the equation x 3 − 3x + k = 0 has two distinct roots lying in the interval (0, 1) are a) 3

b) 2

27. If α, β are roots of

d) none of these

b g b g

ax 2 + bx + c = 0, then the equation ax 2 − bx x − 1 + c x − 1 2 = 0 has roots

α β , 1− α 1− β

a)

c) infinite

b)

1− α 1− β , α β

c)

α β , α +1 β +1

d)

α +1 β +1 , α β

7 6 x 28. If one root of the equation 2 x 2 = 0 is -9, the other roots are x 3 7 a) 2, 6

b) 3, 6

29. Given that

b) c > 0

c) c < 0

d)none of these

2 α and β be the roots of the equation x + x + 1 = 0. The equation whose roots are α19 , β 7 is

a) x 2 − x − 1 = 0

b)

x2 − x + 1 = 0

31. The value of 'k' for which one of the roots of a) 1

b) -2

a) Real

7 log7 ( x

c) x 2 + x − 1 = 0

c) 2

− 4 x +5)

b) 7

c) Rational

d) Equal

c) -2, -3

d) 2, -3

c) two solution

d) more than two solutions

c) two solutions

d) more than two solutions

x + 1 − x − 1 = 4 x − 1 has

a) no solution

b) one solution

d

35. The equation log 3 3 + a) no solution

d) none of these

= x − 1, x may have values

a) 2,3 34. The equation

x2 + x + 1 = 0

ax 2 + 2bx + c = 0 are

b) Imaginary 2

d)

2 x 2 − x + 3k = 0, is double of one of the roots of x − x + k = 0 is

32. If a, b, c are all positive and in H.P., then the roots of

33. If

d) 3, 7

ax 2 + bx + c = 0 has no real roots and a + b + c < 0, then

a) c = 0

30. Let

c) 2, 7

i

c

h

x + log 3 1 + x 2 = 0 has b) one solution


23

x 3 + x 2 + 2 x + sin x = 0 in [ −2 π, 2 π] is

36. The number of roots of the equation a) 1

b) 2

37. The line

c) 3

c

d) none of these

h

y + 14 = 0 cuts the curve whose equation is x x 2 + x + 1 + y = 0 at

a) three real points

b) One real point

38. If sin α ,cos α are the roots of the equajtion

c) at least one real point

d) no real point

ax 2 + bx + c = 0, then

a)

a 2 − b 2 + 2ac = 0

b)

c)

a 2 + b 2 − 2ac = 0

c)

)

(

ba + c g ba − c g

2

= b2 − c2

2

= b2 + c2

39. If the product of the roots of the equation x 3 + 3kx − 2e 2 ln k − 1 = 0 is 7, then the roots are real only if a) k = 2

b) k = −2

c) k = 4

40. If α, β are the roots of the equation 8 x 2 − 3 x + 27 = 0 then the value of a) 1/3

1. 8. 15. 22. 29. 36.

b) 1/4

d b a a c a

2. 9. 16. 23. 30. 37.

c b c c d b

d) k = −4

cα / βh + cβ / αh 2

1/ 3

c) 1/5

3. 10. 17. 24. 31. 38.

a a a c b a

ANSWERS 4. 11. 18. 25. 32. 39.

1/ 3

2

is:

d) 1/6

d b a b b a

5. 12. 19. 26. 33. 40.

c a c d a b


6. 13. 20. 27. 34.

c b a c a

7. 14. 21. 28. 35.

24

a b c c a

Complex Number 1. The product of cube roots of -1 is equal to a) 0 b) 1

c) -1

d) none of these.

2. Which of the following is correct ? a) 1 + i > 2 - i b) 2 + i > 1 + i

c) 2 - i > 1 + i

d) none of these

3. Let z be a purely imaginary number such that Im (z) > 0. Then arg (z) is equal to a) π b) π / 2 c) 0

d) - π / 2

4. Let z be any non-zero complex number. Then, arg (z) + arg ( z ) is equal to a) π b) - π c) 0

d) π / 2

5. The smallest positive integer n for which a) 3

FG 1 + i IJ H 1− i K

n

=1

b) 2

c) 4

6. Let z be a complex number. Then the angle between vectors z and iz is b) 0 c) π / 2 a) π 7. The locus of the points z satisfying the condition arg a) parabola 8. If z = x + iy and w =

d) none of these. d) none of these.

FG z − 1IJ = π is H z + 1K 3

b) circle

c) pair of straight lines

d) none of these.

1 − iz , then | w | = 1 implies that in the complex plane z −i

a) z lies on imaginary axis

b) z lies on real axis

c) z lies on unit circle

9. Let z be a complex number such that | z | = 4 and arg (z) = a) −2 3 + 2 i

b) 2 3 + 2 i

d) none of these

5π , then z = 6 c) 2 3 − 2 i

d)

− 3 +i

10. Let 3 - i and 2 + i be affixes of two points A and B in the argand plane and P represents the complex number z = x + i y. Then the locus of P if | z − 3 + i |=| z − 2 − i | is a) circle on AB as diameter c) the perpendicular bisector of AB d)

b) the line AB none of these

11. If the complex numbers sin x + i cos 2 x and cos x - i sin 2 x are conjugate to each other, then x is equal to b) ( n + 1/2 ) π c) 0 d) none of these. a) n π 12. Common roots of the equations z + 2 z + 2 z + 1 = 0 and 3

a) w , w2

2

b) 1, w, w2

z1985 + z100 + 1 = 0 are c) -1, w, w2

d) - w, - w2

c) Re ( z ) = 3

d) none of these

13. The inequality | z - 2 | < | z - 4 | represents the half plane a) Re ( z) ≥ 3

b) Re ( z ) ≤ 3

14. The cube roots of unity a) are collinear

b) lie on a circle of radius

3 c) from an equilateral triangle

d) none of these.


25

z − 5i = 1, then the locus of z is z + 5i

15. If z is a complex number such that a) x-axis

b) straight line y = 5

c) a circle ing through the origin

(d)none of these.

16. For any complex number z, the minimum value of | z | + | z - 1| is a) 1

b) 0

2

c) 1 / 2

d) 3 / 2.

c) a pair of straight line

d) none of these.

c) | z | > 5

d) none of these.

2

17. If z + | z | = 0 , then the locus of z is a) a circle

b) a straight line

FG | z | − | z | + 1IJ > 2 , then the locus of z is H 2 +| z | K 2

18. If log

3

a) | z | = 5

b) | z | < 5

19. Let z ( ≠ 2 ) be a complex number such that log1/2 | z - 2 | > log1/2 | z | , then a) Re (z) > 1

b) Im (z) > 1

c) Re (z) = 1

d) Im (z) = 1.

c) ( -1 )n - 1

d) ( -1 )n.

20. The product of n th roots of unity is a) 1

b) -1

21. If p is a multiple of n, then the sum of pth powers of nth roots of unity is a) p

b) n

22. If xn = cos

c) 0

d) none of these.

FG π IJ + i sinFG π IJ , n ∈ N , then x1.x2 .x3........x∞ is equal to H2 K H2 K n

a) 1

n

b) -1

c) 0

d) none of these

23. The region of the Argand diagram defined by | z - 1 | + | z + 1 | ≤ 4 is a) interior of an ellipse

b) exterior of a circle

c) interior and boundary of an ellipse

d) none of these.

iθ

24. If z = re , then | ei z | is equal to a)

e

− r sin θ

b) r e − r sin θ

d) r e − r cosθ

c) e − r cos θ

25. If | z - 25 i | ≤ 15, then | max arg(z) - min arg (z) | =

a) cos−1

FG 3IJ H 5K

−1

b) π − 2 cos

FG −3IJ H 5K

c)

FG IJ HK

π 3 + cos−1 2 5


d) sin

−1

FG 3IJ − cos FG 3IJ H 5K H 5K −1

26

26. The locus of the points representing the complex numbers z for which | z | - 2 = | z - i | - | z + 5 i | = 0 is a) a circle with centre at the origin

b) a straight line ing through the origin

c) the single point ( 0, -2 )

d) none of these.

27. If m, n, p, q are consecutive integers then the value of i m + i n + i p + i q is a) 1

b) 4

c) 0

d) none of these.

28. The value of z satisfying the equation log z + log z + ... + log z = 0 is 2

n

a) cos

4mπ 4mπ + i sin , m = 1, 2,.... n ( n + 1) n ( n + 1)

b)

c) sin

4mπ 4mπ + i cos , m = 1, 2,.... n n

d) 0

cos

4mπ 4mπ , m = 1, 2,.... − i sin n (n + 1) n (n + 1)

29. If z1 = z2 =.... = zn = 1, then the value of z1 + z2 +....+ zn is a) n

1 1 1 + +....+ z1 z2 zn

b)

c) 0

d) none of these.

30. The general value of θ which satisfies the equation

(cos θ + i sin θ) (cos 3θ + i sin 3θ) (cos 5θ + i sin 5θ) ...(cos(2n − 1)θ + i sin (2n − 1) θ) = 1 is a)

rπ n2

b)

(r − 1) π n2

c)

( 2r + 1) π n3

d)

2r π . n2

31. If the area of the ∆ on the complex plane formed by the complex numbers z, iz and z + iz is a | z |2, then the constant 'a' is equal to a) 1 / 2

b) 2 / 3

c) 3 / 4

d) none of these.

b) − ω 2

c) π / 2

d) none of these

32. The value of ii is a) ω 33. The value of the expression

FG1 + 1 IJ FG1 + 1 IJ + FG 2 + 1 IJ FG 2 + 1 IJ + FG 3 + 1 IJ FG 3 + 1 IJ +...+FG n + 1 IJ FG n + 1 IJ , where w is an imaginary cube H ωK H ω K H ωK H ω K H ωK H ω K H ωK H ω K 2

2

2

2

root of unity is

a)

n ( n 2 + 2) 3

b)

n ( n 2 − 2) 3

c)

n (n 2 + 1) 3

d) none of these.

34. The number of solutions of z + 2 z = 0 is 2

a) 2

b) 3

c) 4

d) 5


27

35. i + i + i + ...+ up to ( 2n + 1 ) = 2

4

6

a) i

b) -i

c) - 1

10

d) 1

23

36. If ω (≠ 1) is a cube root of unity, then the value of {(ω + ω )π − π / 4} is

− 3/2

a)

b)

−1 / 2

− 5π 4

c)

3/2

d)

37. Let z and ω be two complex numbers such that | z | = | ω | and Arg z + Arg ω = π . Then z equals a)

ω

b)

−ω

c)

ω

d)

−ω

ω2 1 1+ i + ω2 ω 2 − 1 equals −1 38. If ω(≠ 1) is a cube root of unity, then ∆ = 1 − i −i −i + ω − 1 −1 a) 0

b) 1

d) ω

c) i

39. Let zk ( k = 0, 1, 2, ..., 6 ) be the roots of the equation ( z + 1) 7 + z 7 = 0, then a) 3 - 2i

b) 0

40. If m and x are two real numbers, then e a) cos x + i sin x 10 41. The value of

6

∑ Re ( z

k

) is equal to

k =0

c) - 7/2 i 2 m cot

−1

xi + 1     xi − 1 

x

d) 3 + 2i

m

is equal to

b) m / 2

c) 1

d) ( m + 1 ) / 2.

c) 1

d) -1

 2kπ   2kπ   − i cos  is  11   11 

∑ sin

k =1

a) i

b) - i

42. If 1, α 1 , α 2 ,..., α n−1 are the n, nth roots of unity, then the value of (1 + α1 ) (1 + α 2 )......(1 + α n −1 ) is a) n

b) - n

c) - n / 2 + 1

d) none of these.

43. If z1 + z2 = z1 − z2 , then the difference of the arguments of z1 and z2 is b) π / 2

a) 0 44. If z lies on the circle a) π / 3

c) π

z − 2 i = 2 2 then arg

d) 2 π

FG z − 2 IJ is equal to H z + 2K

b) π / 4

c) π / 6

d) π / 8

45. If | z | = 2, then area of the triangle whose sides are magnitudes of the complex numbers z , w z and z+ wz is a)

3

b) 2

3

c) 4

3


d) 16

3 28

46. The curve represented by | z | = Re ( z ) + 2 is a) a straight line

b) a circle

c) an ellipse

d) none of these

47. If α(≠ 1) is a root of x5 = 1, then the value of (α − 1) (α 2 − 1) (α 3 − 1) (α 4 − 1) is a) 5

b) 10

c) - 5

d) - 10

48. If α is a root of x7 = 1, with α ≠ 1 , then value of α 101 + α 102 + ... + α 205 is a) 1

b) 0

c) - 104

d) 104

49. The value of (i 5 + i 6 + i 7 + i 8 + i 9 ) / (1 + i ) is

a)

1 (1 + i ) 2

b)

1 (1 − i ) 2

50. The system of equations | z + 1 − i | = a) no solution

c)

1 ( −1 − i ) 2

d)

1 ( −1 + i ) 2

2 and | z | = 3 has

b) one solution

c) two solutions

d) infinite number of solutions.

ANSWERS 1. 8. 15. 22. 29. 36. 43. 50.

c b a b b c b a

2. 9. 16. 23. 30. 37. 44.

d a a c d d b

3. 10. 17. 24. 31. 38. 45.

b c c a a a a

4. 11. 18. 25. 32. 39. 46.

c d c b d c d

5. 12. 19. 26. 33. 40. 47.

c a a c a c a


6. 13. 20. 27. 34. 41. 48.

c d c c c a b

7. 14. 21. 28. 35. 42. 49.

b c b a c d a

29

Permutation and Combination 1. A box contains two white balls, three black balls and four red balls. In how many ways can three balls be drawn from the box if at least one black ball is to be included in the draw ? All balls are dissimilar a) 129 b) 84 c) 64 d) none of these. 2. The exponent of 3 in 100! is a) 33 b) 44

c) 48

d) 52

3. A person wishes to make up as many different parties as he can out of his 20 friends such that each party consists of the same number of persons. The number of friends he should invite at a time is a) 5 b) 10 c) 8 d) none of these 4. If a denotes the number of permutations of x + 2 things taken all at a time, b the number of permutations of x things taken 11 at a time and c the number of permutations of x - 11 things taken all at a time such that a = 182 bc, then the value of x is a) 15 b) 12 c) 10 d) 18 5. The number of natural numbers smaller than 104, in the decimal notation of which all the digits are different is a) 5274 b) 5265 c) 4676 d) none of these 6. The sum of all the numbers that can be formed with the digits 2, 3, 4, 5 taken all at a time is a) 93324 b) 66666 c) 84844

d) none of these

7. All possible two-factor products are formed from the numbers 1,2,...., 100. The number of factors out of the total obtained which are multiple of 3 is. a) 2211 b) 4950 c) 2739 d) none of these 8. The number of ordered triplets of positive integers which are solutions of the equation x + y + z = 100 is a) 6005 b) 4851 c) 5081 d) none of these 9. The number of ways in which 52 cards can be divided into 4 sets, three of them having 17 cards each and the fourth one having just one card a)

52! 3 17!

b g

b)

52!

c)

(17!)3 3!

10. The total number of all proper factors of 75600 is a) 120 b) 119

51! 3 17!

b g

c) 118

d)

51! 3 17! 3!

b g

d) none of these

11. The number of ways in which 12 different balls can be divided between two friends, one receiving 8 and the other 4, is a)

12! 8!4!

b)

12! 2! 8!4!

12. The least positive integer n for which is a) 12

b) 13

c) n −1

12! 8! 4! 2!

d) none of these

C5 + n −1 C6 < n C7 is c) 14

d) 15

13. The number of five digit numbers (without repetition) which are divisible by 9 and which can be formed by using the digits 0, 1, 2, 3, 4, 7, and 8 is a) 212 b) 214 c) 412 d) none of these 14. A binary sequence is a sequence of 0's and 1's. The number of n - digit binary sequence which contain odd number of 0's is a) 2n −1

b)

2n − 2

c)

2 n−1 + 2

d) none of these


30

15. The tens digit of 1! + 2! + 3! +........+ 49! is a) 1 b) 2

c) 3

d) 4

16. If n C4 , n C5 and n C6 , are in A P, the value of n can be a) 14 b) 7

c) 7 and 14

d) none of these

17. A parallelogram is cut by two sets of m lines parallel to its sides. The number of parallelograms thus formed is a) ( m C2 ) 2

b) ( m+1 C2 ) 2

( m+2 C2 ) 2

c)

d) none of these

18. A code word consists of three letters of the English alphabet followed by two digits of the decimal system. If neither letter nor digit is repeated in any code word, the the total number of code words is a) 1404000 b) 16848000 c) 2808000 d) none of these 19. There are four balls of different colours and four boxes of colours, same as those of the balls. The number of ways in which the balls, one each in a box, could be placed such that a ball does not go to a box of its own colour is a) 9 b) 24 c) 12 d) none of these 20. The number of ways in which four persons be seated at a round table, so that all shall not have the same neighbours in any two arrangements is a) 24 b) 6 c) 3 d) 4 21. A library has a copies of one book, b copies of each of two books, c copies of each of three books, and single copy of d books. The total number of ways in which these books can be distributed is a)

ba + b + c + d g! a !b ! c !

b)

ba + 2b + 3c + d g! 2

a !(b !) ( c !)

3

c)

ba + 2b + 3c + d g! a !(b ! c !

d) none of these

22. The number of parallelograms that can be formed from a set of four parallel lines intersecting another set of three parallel lines is a) 6 b) 18 c) 12 d) 9 23. The number of 5 - digit even numbers that can be made with the digits 0, 1, 2 and 3 is a) 384 b) 192 c) 768

d) none of these

24. There are 10 points in a plane of wlhich no three points are collinear and 4 points are concyclic. The number of different circles that can be drawn through at least 3 points of these points is a) 116 b) 120 c) 117 d) none of these 25. In a polygon no three diagonals are concurrent. If the total number of points of intersection of diagonals interior to the polygon be 70 then the number of diagonals of the polygon is a) 20 b) 28 c) 9 d) none of these 26. The number of triangles that can be formed with 10 points as vertices, n of them being collinear, is 110. The n is a) 3 b) 4 c) 5 d) 6 27. The number of possible outcomes in a throw of n ordinary dice in which at least one of the dice shows an odd number is a)

6n − 1

b)

3n − 1

c)

6n − 3n

d) none of these

28. From 4 gentelmen and 6 ladies a committee of five is to be selected. The number of ways in which the committee can be formed so that gentlemen are in majority is a) 66 b) 156 c) 60 d) none of these 29. The number of ways in which the letters of the word ARTICLE can be rearranged so that the even places are always occupied by consonants is a) 576

b)

4

bg

C3 × 4!

c)

bg

2 4!

d) none of these


31

30. The number of ways in which a couple can sit around a table with 6 guests if the couple take consecutive seats is a) 1440 b) 720 c) 5040 d) none of these 31. There are 4 mangoes, 3 apples, 2 oranges and 1 each of 3 other varieties of fruits. The number of ways of selecting at least one fruit of each kind is a) 10 ! b) 9 ! c) 4 ! d) none of these 32. The number of proper divisors of 2 p.6 q.15r is a) c)

b p + q + 1gbq + r + 1gbr + 1g b p + qgbq + r gr − 2

b)

b p + q + 1gbq + r + 1gbr + 1g − 2

d) none of these

33. The number of ways to give 16 different things to three persons A, B, C so that B gets 1 more than A and C gets 2 more than B, is a)

34. If

16! 4 !5! 7 ! (a 2 −a )

C2 =

b) 4 !5! 7 ! (a 2 −a )

c)

16 3!5!8!

d) none of these

C 4 , then a =

a) 2

b) 3

c) 4

d) none of these

35. The number of diagonals for n sided polygon is a)

n (n − 1) / 2

b)

n (n − 1) (n − 2) / 6

c)

n (n − 1)

d)

n (n − 3) / 2

36. A bag contains unlimited number of black, blue, orange and red balls. The number of ways to select 10 balls so that the selection includes at least one ball of each colour is a) 1001 b) 286 c) 270 d) none of these 37. Six Xs have to be placed in the squares of the figure given below such that each row contains at least one X. The number of ways in which this can be done is

R1 R2 R3 a) 26

b) 27

c) 22

d) none of these

38. The number of ways to select 3 numbers in A.P from the first 2n + 1 natural numbers is given by a)

1 2 (n − 1) 4

b)

n2 5

c)

n

d) none of these

2

ANSWERS 1. 8. 15. 22. 29. 36.

c b a b a d

2. 9. 16. 23. 30. 37.

c b a a a a

3. 10. 17. 24. 31. 38.

b c c c c c

4. 11. 18. 25. 32.

b a a a b

5. 12. 19. 26. 33.

a c a c a


6. 13. 20. 27. 34.

a d c c b

7. 14. 21. 28. 35.

c a b a d

32

BINOMIAL THEOREM

b

1. The number of in the expansion of 2 x + 3 y − 4 z

g

n

is

b) n + 3

a) n + 1 2. Given positive integers

c)

2

d) none of these

b g

r > 1, n > 2 and the coefficients of (3r)th and (r + 2)th in the binomial expansion of 1 + x

Then a) n = 2r

b) n = 3 r 10

∑

3. The sum of the series

bn + 1gbn + 2g

20

c) n = 2r + 1

2n

are equal.

d) none of these

Cr is

r =0

a) 220

b) 219

c)

c

219 +

1 20 C10 2

h

4. If | x | < 1, then the coefficient of xn in the expansion of 1 + x + x + x ..... a) n

2

b) n - 1

2

b g

2n

b g

and 1 + x

b) 2 A + B

219 −

1 20 C10 2

is

c) n + 2

5. If A and B are coefficients of xn in the expansions of 1 + x a) A = B

3

d)

d) n +1 2 n −1

respectively, then

c) A = 2 B

d) none of these

18

3 a b  + 6. If the ( r + 1) term in the expansion of  contains a and b having the same power, then the value of r is  b 3a   th

a) 9

b) 10

c) 8

c

7. The coefficient of x4 in the expansion of 1 + x + x + x a) 900

2

b) 909

h

3 11

b g

8. If the coefficient of the middle term in the expansion of 1 + x are q and r , then a) p + q = r

c

a) 30

17

2n+2

4

d i

C5 x 6 − y

5

b)

17

C6

d xi

11

d y3

x− y

is p and the coeficients of middle in the expansion of (1 + x )2n +1

c) p = q + r

d) p + q + r = 0

c) 40

d) none of these

i

17

is c)

c

17

h

11. If the sum of the coefficients in the expansion of α 2 x 2 − 2αx + 1 a) 2

b) -1

d) 999

is

b) 60

10. The 14th term from the end in the expansion of a)

h b1 + xg

2 5

is c) 990

b) p + r = q

9. The coefficient of x5 in the expansion of 1 + x

d) 6

C4 x 13/ 2 y 2

51

vanishes, then the value of

c) 1


d) none of these

α is d) -2

33

12. If n is even, then the greatest coefficient in the expansion of a)

n

Cn 2 + 1

13. If there is a term containing

Cn2 − 1

b)

n

2r

in

x

b x + ag is n

c)

FG x + 1 IJ H xK

n

C

n

d) none of these

2

n−3

2

, then

a) n - 2r is an integral multiple of 3 c) n - 2r is odd

b) n - 2r is even d) none of these

14. The value of the sum of the series 3. n C 0 − 8 n C1 + 13 n C 2 − 18 n C 3 + ... , when n is an even natural no. a) 0

b g

15. Let 1 + x

b) n

n

= ∑ Cr x r and r =0

a) 1/2

b g

c) 1/3

d) 3

22 23 24 2 11 C1 + C2 + C 3 +... C 10 is 2 3 4 11

311 − 1 a) 11

113 − 1 c) 11

211 − 1 b) 11 n

d) none of these

5n

C1 C C C 1 + 2 2 + 3 3 +...+ n n = n n + 1 , then the value of k is C0 C1 C2 C n −1 k

b) 2

16. The value of 2 C 0 +

17. If x + y = 1, then

c)

3n

∑r

2 n

112 − 1 d) 11

Cr x r y n − r equals

r =0

a) n x y

b)

n x ( x + y n)

18. The coefficient of xn in the expansion of

a)

3n +1 − 1 2 . 3n +1

b)

c)

n x (n x + y )

d) none of these.

FG 3 − 1IJ H 3 K

d) none of these

1 is (1 − x ) (3 − x ) n +1

3n +1 − 1 3n +1

c) 2

n +1

19. If ( r + 1 )th term is the first negative term in the expansion of ( 1 + x )7/2 , then the value of r is a) 5 b) 6 c) 4

d) 7

20. The approximate value of ( 7.995)1/ 3 correct to four decimal places is a) 1.9995 b) 1.9996 c) 1.9990

d) 1.9991

21. The term independent of x in the expansion of (1 + x ) n (1 + 1 / x ) n is a) C02 + 2 C12 + 3 . C22 +.....+ ( n + 1) Cn2

b) ( C0 + C1 +....+ Cn ) 2

c) C02 + C12 +....+ Cn2

d) none of these.

22. The largest term in the expansion of ( 3 + 2 x )50 where x = 1/5 is a) 5th b) 51st c) 7th


d) none of these

34

23. If the coefficients of x7 and x8 in ( 2 + x / 3)n are equal, then n is equal to a) 56 b) 55 c) 45 24. The ratio of the coefficient of x

15

a) 1 / 4

F to the term independent of x in G x H

b) 1 / 16

25. For each n ∈ N, x 2 n −1 + y 2 n −1 is divisible by a) x + y b) ( x + y )2

2

d) 15

2 + x

IJ K

15

is

c) 1 / 32

d) 1 / 64.

c) x3 + y3

d) none of these

26. The middle term of ( 1 + x )2n is a)

1. 3 . 5 ........( 2n − 1) n n 2 x n!

b) 2n ! x n

c) n ! x n

d) none of these

27. Coefficient of x3 in the expansion of (1 − 2 x ) −1/ 2 is a) - 5/2 b) 5/2

c) 3/2

d) - 3/2

28. The greatest integer less than or equal to ( 2 + 1) 6 is a) 196 b) 197

c) 198

d) 199

29. The number of distinct in the expansion of ( x1 + x2 +.....+ xn ) 3 is a)

n +1

C3

b)

n +2

C3

c)

n +3

C3

d) none of these

30. If (1 + x + x ) = a0 + a1 x + a2 x +....+ a2 n x , then value of a02 − a12 + a22 − a32 +....+ a22n is 2 n

a) n!

2

b)

2n

2n

Cn

31. The remainder when 22000 is divided by 17 is a) 1 b) 2

c) n! / 2

d) none of these

c) 8

d) none of these 2n

32. If n is a natural number which is not a multiple of 3 and ( 1 + x + x ) = 2 n

∑

ar x r , then value of

r =0

a) - 1

b) 2

33. The coefficient of x3 in the expansion of (1 − x + x ) is a) 10 b) - 20

n

∑ ( −1)

r

(ar ) ( n Cr ) is

r =0

c) 0

d) ( - 1)n

c) - 50

d) - 30

2 5

34. The coefficient of x3 in the expansion of (1 − x ) 5 (1 + x + x 2 + x 3 ) 4 is a) 4 b) - 4 c) 0

d) none of these

35. 1.n C1 + 2 . n C2 + 3 . n C3 +....+ n . n Cn is equal to a)

n ( n + 1) n .2 4

b) 2 n + 1 - 3

c) n . 2 n - 1


d) none of these

35

n

10

∑ r.

36. The value of

r =1

a)

n

5 ( 2n − 9)

37. The sum a) 2

20

20

+

Cr Cr −1

is equal to

b) 10 n

c) 9 ( n - 4 )

d) none of these

C0 + 20C1 + 20C2 +.....+ 20 C10 is equal to

20! (10!) 2

b) 2

19

1 20! − . 2 (10!) 2

c) 219 +

20

C10

d) none of these

38. The sum of the last ten coefficients in the expansion of (1 + x)19 when expanded in ascending powers of x is b) 219 c) 2 18 - 19C10 d) none of these a) 218

ANSWERS 1. 8. 15. 22. 29. 36.

c c b c b a

2. 9. 16. 23. 30. 37.

a b a b a d

3. 10. 17. 24. 31. 38.

c c c c a a

4. 11. 18. 25. 32.

d c a a c

5. 12. 19. 26. 33.

c c a a d


6. 13. 20. 27. 34.

a a b b c

7. 14. 21. 28. 35.

c a c b c

36

Infinite Series 1. The product of the following series a)

b) 1 - e-1

e -1

2. The sum of the series 1 +

a)

e − e −1

ea − e a −1

4. The sum of the series

a)

e

c)

1 ( e − e −1 ) 2

d) none of these

1+ a 1+ a + a2 1+ a + a2 + a3 + + +.... is 2! 3! 4! ea − a c) a −1

b) e-1

2e

e/2

d) none of these

ea + e− a d) a −1

2 4 6 8 + + + +... is 3! 5! 7 ! 9 ! c)

1 ( e + e −1 ) 2

d) none of these

3 5 7 + + +.... is 1! 2 ! 3! b) e - 1

6. The sum of the series 1 + a)

1 ( e + e −1 ) 2

ea b) a −1

5. The sum of the series 1 + a)

c) e - 1

1 1 1 + + +..... is 2 ! 4 ! 6! b)

3. The sum of the series 1 +

a)

FG 1 + 1 + 1 +....IJ FG 1 − 1 + 1 − 1 +....IJ is H 1! 2! 3! K H 2! 3! 4! 5! K

c) 3e

d) 5e - 7

c) 3e / 2

d) 2e

c) e 2 - 1

d) none of these

2 2 32 4 2 + + +.... is 2 ! 3! 4 ! b) e

1 2 22 + + +..... is 7. The sum of the series 1 + 2 ! 3! 4 ! a)

e2


11 e

b) e 2 + 1

1.4 2 .5 3 .6 4 .7 5 .8 + + + + + .... is 0! 1! 2! 3! 4! b) 10 e

9. The sum of the series 1 . + 3 + a)

e


2 log ( a / b )

c) 9 e

d) 8 e

2 .4 3 .5 4 .6 + + +.... is 1.2 1.2 .3 1.2 .3 .4

b) 2e

FG H

a −b 1 a −b + 2 a a b) log 2 - 1

c) 3e

IJ K

2

FG H

1 a −b + 3 a

d)1+ 4e

IJ +.... is K 3

c) log ( b / a )


d) none of these 37

11. The sum of the series log4 2 - log8 2 + log16 2 - ..... is a)

e2

b) 1 + log2


log 2


log 2

c) log3 - 2

d) 1 - log2

c) log ( 3/2 )

d) none of these

1 1 1 1 − + − +... is 2 3 2 2 .2 3 .2 4 .2 4 b) log ( 1/2 )

11 1 1 1 1  1 1 1  +  −  2 − 2  +  3 + 3  + ....... is 2 2 3 4 2 3  62 3  b) log 3 / 2

c) log 3

d) log

20

14. ( 1 - x )3/2 can be expanded in ascending powers of x if a)

−1 < x < 1

b) x < −1

c)

x >1

d) none of these

c)

−5x 4

d) none of these

15. In the expansion of ( 1 + x ) -2, | x | < 1, the 5th term is a)

- 6x5

b)

5

C4 x 4

16. The coefficient of x5 in the expansion of a)

-1

1+ x2 ,| x | < 1, is 1+ x

b) 2

c) 0

d) - 2

e 2 . 3n c) n!

d) none of these

17. The coefficient of xn in the expansion of e 2 x + 3 is a)

2n n!

18. In the expansion of

a)

1 3 x 5!

e3 . 2n b) n!

ex − 1− x in ascending powers of x, the fourth term is x2 b)

1 4 x 4!

c)

19. If | x | < 1, the coefficient of x3 in the expansion of

a)

17 6

b) −

a) 21.

−

1 3

d) none of these

1 is e . (1 + x ) x

17 6

20. The constant term in the expansion of

1 3 x 3!

c)

−

11 6

d) none of these

c)

−

1 2

d)

x + log e (1 − x ) x3

b) 0

1 2 3 4 + + + + .... to ∞ is equal to 0! 1! 2 ! 3 ! a) 4e b) 3e

c) 2e Quest Tutorials


1 3

d) none of these 38

∞

22.

1 . x 2n 2n − 1

∑

n =1

x 1+ x  log  2 1− x 

a)

23. If x − a)

24. If

FG H

1 1+ x2 log 2 1− x2

IJ K

c) 1 − log e

1 2e

d) none of these

b) x

c) x + 1

d) none of these

e5 x + e x is expanded in a series of ascending powers of x and n is an odd natural number, then the coefficient of xn is e3x 2n n! ∞

n2 n!

∑

n =1

a)

b)

x2 x3 x4 y2 y3 + − +.... to ∞ = y then y + + +.... to ∞ is equal to 2 3 4 2! 3!

-x

a)

25.

is equal to

2 n +1 b) ( 2 n)!

22n c) ( 2 n )!

d) none of these

b) 3 e

c)

1 e 2

d) none of these

is equal to

2e

ex = B0 + B1 x + B2 x 2 +....+ Bn x n +....., then Bn − Bn −1 equals 26. If 1− x a)

27.

e a)

1 n!

b)

1 (n − 1)!

1 1 1 x − 1− ( x − 1) 2 + ( x − 1) 3 − ( x − 1) 4 + ..... 2 3 4

log ( x - 1 )

1 1 − n! (n − 1)!

c)

d) 1

is equal to

b) log x

c) x

d) none of these

28. The sum of the series log 4 2 − log 8 2 + log 16 2 − ...., is a)

b) log e 2 + 1

e2

c)

log e 3 − 2

d) 1 − log e 2.

29. The coefficient of xn in the expansion of log (1 + x + x 2 ), when n is not a multiple of 3, is a)

−

2 n

b)

1 n

c)

2 n

d) none of these.

30. The coefficient of xn in the expansion of log a (1+ x ) is a)

( − 1) n −1 n

( −1) n −1 log a e b) n

( −1) n −1 c) log e a n


( −1) n loga e . d) n 39

ANSWERS 1. 8. 15. 22. 29.

b a b a b

2. 9. 16. 23. 30.

b d d b b

3. 10. 17. 24.

a d b d

4. 11. 18. 25.

b d a a

5. 12. 19. 26.

c c d a


6. 13. 20. 27.

d b a c

7. 14. 21. 28.

d a c d

40

STRAIGHT LINES AND PAIR OF STRAIGHT LINES 1. Let P and Q be points on the line ing A ( -2, 5 ) and B ( 3, 1 ) such that AP = PQ = QB. Then the mid point of PQ is a) ( 1/2, 3 ) b)( -1/2, 4 ) c) ( 2, 3 ) d) ( -1, 4 ) 2. If O be the origin and if P1 ( x1 , y1 ) and P2 ( x2 , y2 ) be two points, then OP1.OP2 cos ( ∠POP 1 2 ) is equal to a) x1 y2 + x2 y1

b) ( x12 + y12 ) ( x22 + y22 )

d) x1 x2 + y1 y2

c) ( x1 − x2 ) 2 + ( y1 − y2 ) 2

3. The coordinates of the middle points of the sides of a triangle are ( 4, 2 ) ( 3, 3 ) and ( 2, 2 ), then the coordinates of its centroid are a) ( 3, 7/3 ) b) ( 3, 3 ) c) ( 4, 3 ) d) none of these 4. If A and B are two fixed pionts, then the locus of a point which moves in such a way that the angle APB is a right angle is a) a circle b) an ellipse c) a parabola d) none of these. 5. If a variable line es through the point of intersection of the lines x + 2y - 1 = 0 and 2x - y - 1= 0 and meets the coordinate axes in A and B, then the locus of the mid point of AB is a)

x + 3y = 0

b)

x + 3 y = 10

6. The image of the point ( -1, 3 ) by the line x - y = 0 is a) ( 3, -1 ) b) ( 1, -3 )

c)

x + 3 y = 10 xy

d) none of these.

c) ( -1, -1 )

d) ( 3, 3 )

7. If ( - 4, 0 ) and ( 1, -1 ) are two vertices of a triangle of area 4 square units, then its third vertex lies on a) y = x

b)

5x + y + 12 = 0

c)

x + 5y − 4 = 0

8. The equation of the line with gradient -3 / 2 which is concurrent with the lines a)

3x + 2 y − 2 = 0

b)

3x + 2 y − 63 = 0

c)

d) none of these

4 x + 3 y − 7 = 0 and 8 x + 5 y − 1 = 0 is

2 y − 3x − 2 = 0

d) none of these

9. The medians AD and BE of the triangle with vertices A ( 0, b ), B ( 0, 0 ) and C ( a, 0 ) are mutually perpendicular if a) b =

2a

b) a =

c) b = − 2 a

2b

d) a = −2 2 b

10. The point A ( 2, 1 ) is translated parallel to the line x - y = 3 by a distance 4 units, If the new position A' is in third quadrant, then the coordinates of A' are a) ( 2 + 2 2 ,1 + 2 2 ) b) ( −2 + 2 2 ,−1 − 2 2 ) c) ( 2 − 2 2 ,1 − 2 2 )

d) none of these

11. All points lying inside the triangle formed by the points ( 1, 3 ), ( 5, 0 ) and ( -1, 2 ) satisfy a) 3x + 2 y ≥ 0

b) 2 x + y − 13 ≥ 0

c) 2 x + 3 y + 12 ≤ 0

d) − 2 x + y ≥ 0

12. If a line ing two pionts A ( 2, 0 ) and B ( 3, 1 ) is rotated about A in anti-clockwise direction through an angle 15°, then the equation of the line in the new position is a)

3x− y=2 3

b)

3x+ y =2 3

13. Two sides of an isosceles triangle are given by the equation point ( 1, - 10 ), then its equations are

x − 3 y − 7 = 0 or 3x + y − 31 = 0 c) x − 3 y − 31 = 0 or 3x + y + 7 = 0

a)

c)

x + 3y = 2 3

d) none of these.

7 x − y + 3 = 0 and x + y − 3 = 0. If its third side es through the b)

x − 3 y − 31 = 0 or

3x + y − 7 = 0

d) none of these.


41

14. The number of lines that are parallel to a) 1 b) 2 15. The ratio in which the line a) 7 : 3

2 x + 6 y + 7 = 0 and have an intercept of length 10 between the coordinate axes is c) 4

d) infinitely many.

3x + 4 y + 2 = 0 divides the distance between 3x + 4 y + 5 = 0, and 3x + 4 y − 5 = 0, is b) 3 : 7

c) 2 : 3

d) none of these.

16. A ray of light coming from the point ( 1, 2 ) is reflected at a point A on the x - axis and then es through the point ( 5, 3 ). The coordinates of the point A are a)

FG 13 , 0IJ H5 K

b)

FG 5 , 0IJ H 13 K

c)

b−7, 0g

d) none of these.

17. The equation(s) of the bisector(s) of that angle between the lines x + 2 y − 11 = 0, 3x − 6 y − 5 = 0 which contains the point ( 1, -3 ) is a) 3x = 19 b) 3y = 7 c) 3x = 19 and 3y = 7 (d) none of these. 18. The orthocentre of the triangle formed by the lines xy = 0 and x + y = 1 is a) ( 1/2, 1/2 ) b) ( 1/3, 1/3 ) c) ( 0, 0 )

d) ( 1/4, 1/4 )

19. If the sum of the distances of a point from two perpendicular lines in a plane is 1, then its locus is a) square b) circle c) straight line d) two intersecting lines. 20. The equations to a pair of opposite sides of a parallelogram are are

x 2 − 5x + 6 = 0 and y 2 − 6 y + 5 = 0. The equations to its diagonals

a) x + 4 y = 13 and y = 4 x − 7

b) 4 x + y = 13 and 4 y = x − 7

c) 4 x + y = 13 and y = 4 x − 7

d) y − 4 x = 13 and y + 4 x = 7.

21. The line which is parallel to x-axis and crosses the curve y = x at an angle of 45° is a) x = 1/4 b) y = 1/4 c) y = 1/2

d) y = 1.

22. The point ( 4, 1 ) undergoes the following two successive transformations : (i) reflection about the line y = x (ii) translation through a distance 2 units along the positive x-axis. Then the final coordinates of the point are a) ( 4, 3 )

b) ( 3, 4 )

23. Area of the quadrilateral formed by the lines | x | + | y | = 1 is a) 4 b) 2 24. If a line is perpendicular to the line

a) ( 1, -1 )

d) (7/2, 7/2).

c) 8

d) none of these.

5x − y = 0 and forms a triangle with coordinate axes of area 5 sq. units, then its equation is

a) x + 5 y ± 5 2 = 0 b) x − 5 y ± 5 2 = 0 25. A point equidistant from the lines

c) ( 1, 4 )

c) 5x + y ± 5 2 = 0

d) 5x − y ± 5 2 = 0

4 x + 3 y + 10 = 0, 5x − 12 y + 26 = 0 and 7 x + 24 y − 50 = 0 is

b) ( 1, 1 )

c) ( 0, 0 )

d) ( 0, 1 ).

26. The points ( k − 1, k + 2 ), ( k , k + 1), ( k + 1, k ) are collinear for a) any value of k

b) k = - 1/2 only

c) no value of k

27. The straight lines x + y - 4 = 0, 3x + y - 4 = 0 and x + 3y - 4 = 0 from a triangle which is a) isosceles b) right angled c) equilateral Quest Tutorials Head Office : 44C,Kalusarai New Delhi-16; Ph.(011) 46080363

d) integral values of k only

d) none of these. 42

28. If the mid point of of ( x, y + 1 ) and ( x + 1, y + 2 ) is ( 3/2, 5/2 ); then the mid point of of ( x - 1, y + 1 ) and (x + 1, y - 1 ) is a) ( -1, -1 ) b) ( -1, 1 ) c) ( 1, -1 ) d) ( 1, 1 ) 29. If x cos α + y sin α = p where p = - sin α tan α be the equation of a line, then the length of the perpendiculars on the line from the points ( a2, 2a ), ( ab, a + b ) and ( b2, 2b ) form a) A.P.

b) G.P.

c) H.P.

d) none of these

30. The locus of the point equidistant from the points ( a + b, b - a ) and ( a - b, a + b ) is a) ax + by = 0 b) ax - by = 0 c) bx - ay = 0

d) ( a + b ) x + ( a - b ) y = 0

31. The number of lines that can be drawn through the point ( 4, -5 ) at a distance 12 from the point ( -2, 3 ) is a) 0 b) 1 c) 2 d) infinite 32. If a, b, c from an A.P. with common difference d (≠ 1) and x, y , z from a G.P. with common ratio r (≠ 1) then the area of the triangle with vertices; ( a, x ), ( b, y ) and ( c, z ) is independent of a) a b) d c) x d) r 33. If m1 , m2 are the roots of the equation x2 -ax - a - 1 = 0, then the area of the triangle formed by the three straight lines y = m1 x, y = m2 x and y = a ( a ≠ −1) is a)

a 2 (a + 2) if a > - 1 2 ( a + 1)

b)

− a 2 (a + 2) if a < - 1 2 ( a + 1)

c)

− a 2 (a + 2) if a ∈ R 2 ( a + 1)

d)

a 2 (a + 2) if a < - 1 2 ( a + 1)

34. If the product of the intercepts made by the line x tan a + y sec a = 1 on the coordinate axis is sin a, then a is equal to a) π / 4 b) π / 6 c) π / 3 d) 2 π / 3 35. Area of the rhombus enclosed by the lines ax ± by ± c = 0 is (a) 2a 2 / bc

b) 2b 2 / ca

c) 2c 2 / ab

d) none of these.

36. The distance between the orthocentre and circumcentre of the triangle with vertices ( 0, 0 ), ( 0, a ) and ( b, 0 ) is

a 2 − b2 / 2

a)

b) a + b

c) a- b

a 2 + b2 / 2

d)

37. If a, b, c from a G.P. with common ratio r, the sum of the ordinates of the points of intersection of the line ax + by + c = 0 and the curve x + 2y2 = 0 is a)

−r 2 / 2

b) −r / 2

c) r / 2

d)

r2 / 2

38. If a,b,c are in A.P, then the st. line ax + by + c = 0 will always thro’ a fixed point whose co-ordinates are b) (-1, 2) c) (1, 2) d) (-1, -2) a) (1, − 2 ) 39. The point (4,1) undergoes the following three transformations successively i) reflection about the line y = x ii) translation thro’ a distance of 2 units along the positive direction of x-axis iii) rotation thro’ an angle of π / 4 about the origin in the counter-clockwise direction. The final position of the point is given by

 1

a) 

 2

,

7   2

b)

(−

2, 7 2

)



c)  −



1 2

,

7   2

d)

(

2, 7 2

)

40. The equations of the sides of a triangle are x = 0, y = m1 x + c1 and y = m2 x + c2 . The area of the trinagle is

c1 − c2 a) m1 − m2

1 (c1 − c2 )2 b) 2 m1 − m2

1 c1 − c2 c) 2 (m1 − m2 )2


d)

(c1 − c2 )2 (m1 − m2 )2 43

ANSWERS 1. 8. 15. 22. 29. 36.

a a b b b d

2. 9. 16. 23. 30. 37.

d b a b c c

3. 10. 17. 24. 31. 38.

a c a a a a

4. 11. 18. 25. 32. 39.

a a c c a c

5. 12. 19. 26. 33. 40.

c a a a a b


6. 13. 20. 27. 34.

a c c a a

7. 14. 21. 28. 35.

c b c d c

44

CIRCLE 1. The centre of the circle ing through ( 0, 0 ) ( a, 0 ) and ( 0, b ) is a) ( a, b )

FG a , b IJ H 2 2K

b)

c)

FG − a , − b IJ H 2 2K

d) ( − a , − b )

2. The straight line y = mx + c cuts the circle x2 + y2 = a2 in real points if

a 2 (1 + m2 ) < c

a)

a 2 (1 − m2 ) < c

b)

c)

a 2 (1 + m2 ) > c

a 2 (1 − m2 ) > c

d)

3. The locus of the centre of a circle of radius 2 which rolls on the outside of the circle x 2 + y 2 + 3x − 6 y − 9 = 0 is a) x 2 + y 2 + 3x − 6 y + 5 = 0 c) x + y + 3x − 6 y + 2

2

b)

29 =0 4

x 2 + y 2 + 3x − 6 y − 31 = 0

d) none of these.

4. The circle described on the line ing the points ( 0, 1 ), ( a, b ) as diameter cuts the x-axis in points whose abscissae are roots of the equation a) x + ax + b = 0 2

x 2 − ax + b = 0

b)

(c) x + ax − b = 0 2

d) x − ax − b = 0 . 2

5. A line is drawn through a fixed point P ( α, β ) to cut the circle x 2 + y 2 = r 2 at A and B. Then PA . PB is equal to a) (α + β) 2 − r 2

b) α 2 + β 2 − r 2

c)

(α − β) 2 + r 2

d) none of these.

6. The centre of a circle ing through the points ( 0, 0 ), ( 1, 0 ) and touching the circle x 2 + y 2 = 9 is a)

FG 3 , 1 IJ H 2 2K

FG 1 , 3IJ H 2 2K

b)

c)

FG 1 , 1 IJ H 2 2K

d)

FG 1 , − 2 IJ H2 K

7. The equation x 2 + y 2 + 2 gx + 2 fy + c = 0 will represent a real circle if a) g 2 + f

2

−c<0

g2 + f 2 − c ≥ 0

b)

c) always

d) none of these.

8. The length of the chord cut off by y = 2x + 1 from the circle x 2 + y 2 = 2 is a)

5 6

b)

6 5

c)

6 5

d)

5 6

9. Equation of the circle with centre on the y-axis and ing through the origin and ( 2, 3 ) is a) x 2 + y 2 + 13 y = 0

b)

3x 2 + 3 y 2 − 13 y = 0 c) x 2 + y 2 + 13x + 3 = 0

d) 6 x 2 + 6 y 2 − 13x = 0.

10. ABCD is a square whose side is a. The equation of the circle circumscribing the square, taking AB and AD as axes of reference, is a) x 2 + y 2 + ax + ay = 0

b)

x 2 + y 2 + ax − ay = 0

c) x 2 + y 2 − ax − ay = 0

d)

x 2 + y 2 − ax + ay = 0

11. Two perpendicular tangents to the circle x 2 + y 2 = a 2 meet at P. Then the locus of P has the equation a) x 2 + y 2 = 2a 2

b) x 2 + y 2 = 3a 2

c)

x 2 + y 2 = 4a 2


d) none of these 45

12. If the equation of a given circle is x 2 + y 2 = 36, then the length of the chord which lies along the line

3 6

a)

b)

2 3

c)

6 3

3x + 4 y − 15 = 0 is d) none of these

13. The number of common tangents that can be drawn to the circles x 2 + y 2 − 4 x − 6 y − 3 = 0 and x 2 + y 2 + 2 x + 2 y + 1 = 0 is a) 1 b) 2 c) 3 d) 4 14. The condition that the chord x cos α + y sin α − p = 0 of x + y − a = 0 may subtend a right angle at the centre of the circle is 2

a) a 2 = 2 p 2

b)

p 2 = 2a 2

c)

2

2

a = 2p

d)

p = 2 a.

15. The coordinates of the point on the circle x 2 + y 2 − 12 x − 4 y + 30 = 0, which is farthest from the origin are a) ( 9, 3 ) b) ( 8, 5 ) c) ( 12, 4 ) d) none of these. 16. The circles x 2 + y 2 + x + y = 0 and x 2 + y 2 + x − y = 0 intersect at an angle of a) π / 6

b)

π/4

c)

π/3

d) π / 2

17. A circle of radius 5 units touches both the axes and lies in the first quadrant. If the circle makes one complete roll on x-axis along the positive direction of x-axis, then its equation in the new position is a) x 2 + y 2 + 20 π x − 10 y + 100 π 2 = 0

b)

x 2 + y 2 + 20 π x + 10 y + 100 π 2 = 0

c) x 2 + y 2 − 20 π x − 10 y + 100 π 2 = 0

d) none of these

18. The number of the tangents that can be drawn from ( 1, 2 ) to x 2 + y 2 = 5 is a) 1 b) 2 c) 3

d) 0

19. Equation of the chord of the circle x 2 + y 2 − 4 x = 0 whose mid point is ( 1, 0 ) is a) y = 2

b) y = 1

c) x = 2

d) x = 1

20. The two circles x 2 + y 2 − 5 = 0 and x 2 + y 2 − 2 x − 4 y − 15 = 0 a) touch each other externally b) touch each other internally c) cut each other orthogonally

d) do not intersect

21. If a circle es through ( 1, 2 ) and cuts the circle x 2 + y 2 = 4 orthogonally then the equation of the locus of its centre is

2x + 4 y − 9 = 0

a)

b) 2 x + 4 y + 9 = 0

c)

2x − 4 y + 9 = 0

d) none of these

22. The lines 3x − 4 y + 4 = 0 and 6 x − 8 y − 7 = 0 are tangents to the same circle. Then its radius is a) 1 / 4

FG H

23. If mi ,

IJ K

b) 1 / 2

c)

d) none of these

3/ 4

1 , i = 1, 2, 3, 4 are concyclic points, then the value of m1 m2 m3 m4 is mi

a) 1

b) -1

c) 0

d) none of these

24. The angle between the tangents drawn from the origin to the circle ( x − 7) + ( y + 1) = 25 is 2

a) π / 3

b)

π/6

c)

2

π/2

d) π / 8

25. The equation of a circle with origin as centre and ing through the vertices of an equilateral triangle whose median is of length 3 a is a) x 2 + y 2 = 9a 2

b)

x 2 + y 2 = 16 a 2

c)

x2 + y2 = 4 a2

26. The slope of the tangent at the point ( h, h ) of the circle x 2 + y 2 = a 2 is a) 0 b) 1 c) -1


d) x 2 + y 2 = a 2 .

d) depends on h

46

27. The circles x 2 + y 2 − 10 x + 16 = 0 and x 2 + y 2 = r 2 intersect each other in two distinct points if a) r < 2

b) r > 8

d) 2 ≤ r ≤ 8

c) 2 < r < 8

28. The number of common tangents to the circles x 2 + y 2 − x = 0, x 2 + y 2 + x = 0 is a) 2

b) 1

c) 4

d) 3

29. The equation ax + 2hxy + by + 2 gx + 2 fy + c = 0 represents a circle if a) a = h = 2, b = 0 b) b = h = 2, a = 0 c) a = b = 2, h = 0 2

2

d) none of these

30. The locus of the middle points of the chords of the circle x 2 + y 2 = 4a 2 which subtend a right angle at the centre of the circle is a)

x + y = 2a

b) x 2 + y 2 = a 2

c)

x 2 + y 2 = 2a 2

d) x 2 + y 2 = x + y

31. Tangents are drawn from the origin to a circle with centre at ( 2, -1 ). If the equation of one of the tangents is 3x + y = 0, the equation of the other tangent is a) 3x - y = 0

b) x + 3y = 0

c) x - 3y = 0

d) x + 2y = 0

32. A line meets the coordinate axes in A and B. A circle is circumscribed about the triangle OAB. If the distances from A and B of the tangent to the circle at the origin be m and n, then the diameter of the circle is a) m ( m + n )

b) m + n

c) n ( m + n )

d) m2 + n2

33. The circle ing through three distinct points ( 1, t ), ( t, 1 ) and ( t, t ) es through the point (for all values of t .) a) ( 1, 1 )

b) ( -1, -1 )

c) ( -1, 1 )

d) ( 1, -1 )

34. If OA and OB are tangents from the origin to the circle x 2 + y 2 + 2 gx + 2 fy + c = 0 and C is the centre of the circle, then area of the quadrilateral OACB is

c

1 c g2 + f 2 − c a) 2

h

b)

c

c g + f −c 2

2

h

c)

c

cg

2

+ f −c 2

h

d)

g2 + f 2 − c c

35. The locus of the point of intersection of the tangents to the circle x = r cos θ , y = r sin θ at points whose parametric angles differ by π /3 is a) x 2 + y 2 = 4 ( 2 − c) x 2 + y 2 = ( 2 −

3) r 2 3) r 2

b)

3( x2 + y2 ) = 1

d)

3( x2 + y2 ) = 4 r 2

36. If the two circles x 2 + y 2 + 2 gx + 2 fy = 0 and x 2 + y 2 + 2 g1 x + 2 f 1 y = 0 touch each other, then a)

f 1 g = fg1

b)

ff 1 = gg1

c)

f 2 + g 2 = f 12 + g12

d) none of these.

37. The equation of the chord of the circle x 2 + y 2 = a 2 ing through the point ( 2, 3 ) farthest from the centre is a) 2x + 3y = 13

b) 3x - y = 3

c) x - 2y + 4 = 0

d) x - y + 1 = 0

38. A circle is given by x 2 + y 2 + 4 x − 7 y + 12 = 0. The points P ( 0, 0 ) and Q ( -2, 4) are such that a) both lie inside the circle

b) both lie outside the circle

c) one lies inside and the other outside the circle

d) one lies on the circle and the other is outside the circle


47

39. If the equation of a circle is ax 2 + ( 2a − 3) y 2 − 4 x − 1 = 0 then its centre is a) ( 2, 0 )

b) ( 2/3, 0 )

c) ( -2/3, 0 )

d) none of these

c) a circle of non-zero radius

d) none of these

40. The equation x 2 + y 2 − 2 x + 4 y + 5 = 0 represents a) a point

b) a pair of straight lines

41. C1 is a circle of radius 1 touching the x-axis and the y-axis. C2 is another circle of radius > 1 and touching the axes as well as the circle C1. Then the radius of C2 is a)

3− 2 2

b)

3+ 2 2

c)

3+ 2 3

d) none of these

42. If p and q be the longest distance and the shortest distance respectively of the point ( -7, 2 ) from any point ( α, β ) on the curve whose equation is x 2 + y 2 − 10 x − 14 y − 51 = 0 then GM of p and q is equal to a)

b)

2 11

c) 13

5 5

43. A region in the x-y plane is bounded by the curve y =

d) none of these

25 − x 2 and the line y = 0. If the point ( a, a + 1 ) lies in the interior of the region then

a)

a ∈ ( −4, 3)

b)

a ∈ ( −∞, − 1) ∪ (3, + ∞)

c)

a ∈ ( −1, 3)

d) none of these

44. The equation of the diameter of the circle 3 ( x 2 + y 2 ) − 2 x + 6 y − 9 = 0 which is perpendicular to the line c)

3x − 2 y = 9

a) x 2 + y 2 + 6 x + 2 y = 0

b)

x + y − 6x + 2 y = 0

c) x 2 + y 2 − 6 x − 2 y = 0

d)

x 2 + y 2 + 8x − 2 y = 0

a)

3x − 2 y = 3

b) 3x − 2 y + 1 = 0

2 x + 3 y = 12 is

d) none of these

45. The line x + 3 y = 0 is the diameter of the circle 2

2

46. The distance between the chords of of the tangent to the circle x 2 + y 2 + 2 gx + 2 fy + c = 0 from the origin and the point (g, f) is a) g 2 + f 2

b)

1 2 2 ( g + f + c) 2

c)

g2 + f 2 + c

d)

2 (g 2 + f 2 )

g2 + f 2 − c 2 (g 2 + f 2 )

47. The condition for the two circles x 2 + y 2 + 2k1 x + k 2 = 0 and x 2 + y 2 + 2k 2 y + k 2 = 0 to touch each other externally is a) k12 + k 22 = k 2

b ) k12 − k 22 = k 2

c) k 2 ( k12 − k 22 ) = k12 k 22

d)

2

2

2

2

2

k ( k1 + k 2 ) = k1 k 2

ANSWERS 1. 8. 15. 22. 29. 36. 43.

b c a c c a c

2. 9. 16. 23. 30. 37. 44.

c b d a c a a

3. 10. 17. 24. 31. 38. 45.

b c d c c c b

4. 11. 18. 25. 32. 39. 46.

b a a c b b d

5. 12. 19. 26. 33. 40. 47.

b c d c a a d


6. 13. 20. 27. 34. 41.

d c b c b b

7. 14. 21. 28. 35. 42.

b a a d d a

48

CONIC SECTIONS - PARABOLA 1. If the focus of a parabola is at ( 0, -3 ) and its directrix is y = 3, then its equation is a) x 2 = −12 y

b) x 2 = 12 y

c) y 2 = −12 x

d) y 2 = 12 x

2. The equation of the directrix of the parabola x 2 − 4 x − 3 y + 10 = 0 is a) y = −

5 4

b) y =

5 4

c) y = −

3 4

d) y =

3. The angle made by a double ordinate of length 8a at the vertex of the parabola y 2 = 4 ax is a) π / 3 b) π / 2 c) π / 4

5 . 4

d) π / 6

4. The coordinates of a point on the parabola y 2 = 8 x whose focal distance is 4, are a) (1 / 2 , ± 2)

b) (1, ± 2 2 )

c)

( 2, ± 4)

d) none of these

5. If the parabola y 2 = 4 ax es through ( 3, 2 ), then the length of its latus rectum is a) 2 / 3 b) 4 / 3 c) 1 / 3

d) 4.

6. The coordinates of the focus of the parabola x 2 − 4 x − 8 y − 4 = 0 are a) ( 0, 2 ) b) ( 2, 1 ) c) ( 1, 2 )

d) ( -2, -1 ).

7. The area of the triangle inscribed in the parabola y 2 = 4 x the ordinates of whose vertices are 1, 2 and 4 is a) 7 / 2 sq. units

b) 5 / 2 sq. units

c) 3 / 2 sq. units

8. The length of the latusrectum of the parabola whose focus is ( 3, 3 ) and directrix is a) 2 b) 1 c) 4

d) 3 / 4 sq. units

3x − 4 y − 2 = 0, is d) none of these.

9. The locus of the points of trisection of the double ordinates of the parabola y 2 = 4 ax is a) y 2 = ax 10. If

b) 9 y 2 = 4 ax

c) 9 y 2 = ax

d) y 2 = 9 ax.

y = mx + c touches the parabola y 2 = 4 a ( x + a ), then

a) c =

a m

b) c = am +

a m

c) c = a +

a m

d) none of these.

11. The parametric representation ( 2 + t2 , 2 t + 1 ) represents a) a parabola with focus at ( 2, 1 ) b) a parabola with vertex at ( 2, 1 ) c) an ellipse with centre at ( 2, 1 )

d) none of these.

2

12. If ( at , 2 at ) are the coordinates of one end of a focal chord of the parabola y 2 = 4 ax , then the coordinates of the other end are a) ( at 2 , − 2 at )

b) ( − at 2 , − 2 at )

c)

FG a , 2 a IJ Ht t K 2

d)

FG a , − 2 a IJ Ht t K 2

13. If the vertex and focus of a parabola are ( 3, 3 ) and ( -3, 3 ) respectively, then its equation is a) x 2 + 6 x − 24 y + 63 = 0

b) x 2 − 6 x + 24 y − 63 = 0

c) y 2 − 6 y + 24 x − 63 = 0

d) y 2 + 6 y − 24 x + 63 = 0 .


49

14. The locus of the foot of the perpendicular from the focus upon a tangent to the parabola y 2 = 4 ax is a) the directrix b) tangent to the vertex c) x = a

d) none of these.

15. The slope of the normal at the point ( at 2 , 2 at ) of the parabola, y 2 = 4 ax is, a)

1 t

b) t

c) - t

16. If 2x + y + k = 0 is a normal to the parabola y 2 = −8 x , then the value of k is a) - 16 b) - 8 c) - 24 17. The equation of the normal at the point of of a tangent a) y = mx − 2 am − am 3

d) -

1 t

d) 24.

FG a , 2 a IJ is Hm m K 2

b) m 3 y = m 2 x − 2 am 2 − a

c) m3 y = 2 am2 − m2 x + a d)

none of these.

18. If the vertex of the parabola y = x 2 − 8 x + c lies on x-axis, then the value of c is a) - 16 b) - 4 c) 4

d) 16.

19. The locus of the point of intersection of the ⊥ tangents to the parabola x 2 = 4 ay is a) y = a b) y = - a c) x = a

d) x = - a

20. If PSQ is the focal chord of the parabola y = 8 x such that SP = 6. Then the length SQ is a) 6 b) 4 c) 3

d) none of these

21. The curve represented by x = 3 ( cos t + sin t ), y = 4 ( cos t - sin t ) is a) ellipse b) parabola

c) hyperbola

d) circle

22. The graph represented by the equations x = sin2 t y = 2 cos t is a) a portion of a parabola b) a parabola

c) a part of sine graph

d) a part of hyperbola.

2

23. The tangents at the points ( at12 , 2at1 ), ( at 22 , 2at 2 ) on the parabola y2 = 4ax are at right angles if a) t1 t 2 = −1

b) t1 t 2 = 1

c) t1 t 2 = 2

d) t1 t 2 = −2

24. The two ends of latusrectum of a parabola are the points ( 3, 6 ) and ( -5, 6 ). The focus is a) ( 1, 6 ) b) ( -1, 6 ) c) ( 1, -6 )

d) ( -1, -6 ).

25. Three normals to the parabola y 2 = x are drawn through a point ( C, 0 ), then a) C = 1 / 4 b) C = 1 / 2 c) C > 1 / 2

d) none of these

26. The circles on focal radii of a parabola as diameter touch a) the tangent at the vertex b) the axis

c) the directrix

d) none of these.

27. The focus of the parabola 4 y 2 + 12 x − 20 y + 67 = 0 is a) ( -7/2, 5/2 ) b) (-3/4, 5/2 )

c) (-17/4, 5/2 )

d) (5/2, -3/4 )

28. The line x + y = 6 is a normal to the parabola. y2 = 8x at the point a) ( 18, -12 ) b) ( 4, 2 )

c) ( 2, 4 )

d) ( 3, 3 )

29. The straight line x + y = k + 1 touches the parabola y = x ( 1 - x ). If a) k = -1 b) k = 0

c) k = 1

d) k takes any value


50

30. A line bisecting the ordinate PN of a point P ( at2, 2at ), t > 0, on the parabola y2 = 4ax is drawn parallel to the axis to meet the curve at Q. If NQ meets the tangent at the vertex at the point T, then the coordinates of T are a) ( 0, (4/3) at ) b) ( 0, 2 at ) c) ( (1/4) at2, at ) d) ( 0, at ) 31. If perpendiculars are drawn on any tangent to a parabola y2 = 4ax from the points ( a ± k , 0) on the axis. The difference of their squares is a) 4 b) 4a c) 4k d) 4ak 32. For the parabola y 2 + 8 x − 12 y + 20 = 0, which of the following are correct a) vertex ( 2, 6 ) b) focus ( 1 , 6 ) c) Length of the latusrectum = 4 d) axis is y = 0 33. Equation of a tangent to the parabola y2 = 7x which is inclined at an angle of 45° to the axis of the parabola is a)

4x − 4 y + 7 = 0

b) 4x + 4y - 7 = 0

c)

7x − 7 y + 4 = 0

d)

7x + 7 y + 4 = 0

34. The point on the parabola y2 = 36 x, whose ordinate is three times its abcissa are a) ( 0, 1) b) ( 3, 9 ) c) ( 4, 12 )

d) ( 6, 18 )

35. The slope of tangent to the parabola y2 = 9x which es through ( 4, 10 ) is a) 9 / 4 b) 5 / 4 c) 3 / 4

d) 1 / 3

36. The point of of the tangent in the above example is a) ( 4, 2 ) b) ( 36, 18 )

d) ( 1/4, 3/2 )

c) ( 4, 6 )

37. If the focus of a parabola is ( -2, 1 ) and the directrix has the equation x + y = 3 then the vertex is a) ( 0, 3 ) b) ( -1, 1/2 ) c) ( -1, 2 )

d) ( 2, -1 )

38. If the vertex and the focus of a parabola are ( -1, 1 ) and ( 2, 3 ) respectively then the equation of the directrix is a)

3x + 2 y + 14 = 0

b)

3x + 2 y − 25 = 0

c) 2 x − 3 y + 10 = 0

d) none of these.

39. If the vertex = ( 2, 0 ) and extremities of the latus rectum are ( 3, 2 ) and ( 3, -2 ) then the equation of the parabola is a) y 2 = 2 x − 4

b) x 2 = 4 y − 8

c) y 2 = 4 x − 8

d) none of these.

40. The equation of the parabola whose vertex and focus are on the positive side of the x-axis at distance a and b respectively from the origin is a) y 2 = 4 (b − a ) ( x − a )

b) y 2 = 4 ( a − b) ( x − b)

c) x 2 = 4 (b − a ) ( y − a )

d) none of these.

41. The equation λx 2 + 4 xy + y 2 + λx + 3 y + 2 = 0 represents a parabola if λ is a) - 4 b) 4 c) 0

d) none of these.

42. The equation of the axis of the parabola 9 y 2 − 16 x − 12 y − 57 = 0 is a) 2x = 3

b) y = 3

c) 3y = 2

d) x + 3y = 3

43. The parametric equation of a parabola is x = t 2 + 1, y = 2t + 1. The Cartesian equation of its directrix is a) x = 0

b) x + 1 = 0

c) y = 0

d) none of these.

44. If two tangents drawn from the point ( α, β ) to the parabola y2 = 4x be such that the slope of one tangent is double of the other then a) β =

2 2 α 9

b) α =

2 2 β 9

c) 2α = 9 β

2


d) none of these.

51

ANSWERS 1. 8. 15. 22. 29. 36. 43.

a a c b b b a

2. 9. 16. 23. 30. 37. 44.

b b d a a c b

3. 10. 17. 24. 31. 38.

b b c b d a

4. 11. 18. 25. 32. 39.

c b d c a c

5. 12. 19. 26. 33. 40.

b d b c a a


6. 13. 20. 27. 34. 41.

b c c c c b

7. 14. 21. 28. 35. 42.

d b a c a c

52

ELLIPSE AND HYPERBOLA 1. The equation ax 2 + 2hxy + by 2 + 2 gx + 2 fy + c = 0 represents an ellipse if a) ∆ = 0, h 2 < ab

b) ∆ ≠ 0, h 2 < ab

c) ∆ ≠ 0, h 2 > ab

2. The equation to the ellipse (referred to its axes as the axes of x and y respectively ) whose foci are

x2 y2 + =1 a) 12 16

x2 y2 + =1 b) 16 12

3. The eccentricity of the ellipse 9 x 2 + 5 y 2 − 30 y = 0 is a) 1 / 3 b) 2 / 3

d) ∆ ≠ 0, h 2 = ab

( ±2, 0) and eccentricity 1/2, is

x2 y2 + =1 c) 16 8

d) none of these.

c) 3 / 4

d) none of these.

4. Equations x = a cos θ, y = b sin θ(a > b) represent a conic section whose eccentricity e is given by a) e 2 =

a 2 + b2 a2

b) e = 2

a 2 + b2 b2

c) e = 2

a 2 − b2 a2

5. If A and B are two fixed points and P is a variable point such that PA + PB = 4, the locus of P is a) a parabola b) an ellipse c) a hyperbola

d) e = 2

a 2 − b2 . b2

d) none of these.

6. If the focal distance of an end of the minor axis of an ellipse ( referred to its axes as the axes of x and y respectively ) is k and the distance between its foci is 2h, then its equation is a)

x2 y2 + =1 k 2 h2

7. The equation a) a < 4

b)

x2 y2 + =1 k 2 k 2 − h2

c)

x2 y2 + =1 k 2 h2 − k 2

d)

x2 y2 + =1 k 2 k 2 + h2

x2 y2 + = 1 represents an ellipse if 10 − a 4 − a b) a > 4

c) 4 < a < 10

d) a > 10.

8. The curve with parametric equations x = 1 + 4 cos θ, y = 2 + 3 sin θ is a) an ellipse

b) a parabola

c) a hyperbola

d) a circle

x2 y2 9. If S and S ' are two foci of an ellipse 2 + 2 = 1 ( a < b) and P ( x1 , y1 ) a point on it, then SP + S ' P is equal to a b a) 2a

b) 2b

c) a + e x1

d) b + e y1

10. The length of the latus-rectum of the ellipse 5x 2 + 9 y 2 = 45 is (a)

5 3

b)

10 3

c)

2 5 5

d)

5 3

11. A set of points is such that each point is three times as far away from the y-axis as it is from the point ( 4, 0 ). Then the locus of the points is a) hyperbola b) parabola c) ellipse d) circle. 12. The product of the perpendiculars drawn from the two foci of an ellipse to the tangent at any point of the ellipse is a) a2 b) b2 c) 4a2 d) 4b2 . Quest Tutorials Head Office : 44C,Kalusarai New Delhi-16; Ph.(011) 46080363

53

13. If the normal at one end of the latusrectum of an ellipse a) e4 - e2 + 1 = 0

x2 y2 + = 1 es through one end of the minor axis, then a 2 b2

b) e2 - e + 1 = 0

14. If a tangent having slope of −

c) e2 + e + 1 = 0

d) e4 + e2 - 1 = 0

x2 y2 4 + = 1 intersects the major and minor axes in points A and B respectively, then the to the ellipse 18 32 3

area of ∆ OAB is equal to a) 12 sq. units

b) 48 sq. units

15. If e is the eccentricity of the ellipse a) b 2 = a 2 (1 − e 2 )

c) 64 sq. units

d) 24 sq. units

c) a 2 = b 2 ( e 2 − 1)

d) b 2 = a 2 (e 2 − 1)

x2 y2 + = 1 ( a < b ) , then a 2 b2 b) a 2 = b 2 (1 − e 2 )

16. The number of normals that can be drawn from a point to a given ellipse is a) 2 b) 3 c) 4 17. The radius of the circle ing through the foci of the ellipse a) 4

x2 y2 + = 1, and having its centre ( 0, 3 ) is 16 9

b) 3

18. The centre of the ellipse a) ( 0, 0 )

d) 1

c)

12

d) 7/2.

( x + y − 2) 2 ( x − y ) 2 + = 1 is 9 16 b) ( 1, 1 )

c) ( 1, 0 )

d) ( 0, 1 )

x2 y2 + = 1 intersect on the curve 19. Two perpendicular tangents drawn to the ellipse 25 16 a) x =

a e

20. The ellipse a)

b) x 2 + y 2 = 41

c) x 2 + y 2 = 9

d) x 2 − y 2 = 41

x2 y2 + = 1 and the straight line y = mx + c intersect in real points only if a 2 b2

a 2 m2 < c 2 − b 2

b) a 2 m2 > c 2 − b 2

c) a 2 m2 ≥ c 2 − b 2

d) c ≥ b .

21. The eccentricity of the hyperbola whose latusrectum is 8 and conjugate axis is equal to half the distance between the foci, is a) 4 /3

b) 4 /

3

c) 2 /

22. If e and e' be the eccentricities of a hyperbola and its conjugate, then a) 0

b) 1

3

d) none of these.

1 1 + 2 = 2 e e′

c) 2

d) none of these

23. The eccentiricity of the hyperbola 3x 2 − 4 y 2 = −12 is a)

7 3

b)

7 2

c) −

7 3


d) −

7 2 54

24. The locus of the point of intersection of the straight lines a) a circle

x y 1 x y + = λ and − = ( λ is a variable )is a b λ a b

b) a parabola

c) an ellipse

d) a hyperbola.

25. The equation of the chord of the hyperbola x 2 − y 2 = 9 which is bisected at ( 5, -3 ) is

5x + 3 y = 16

d)

5x − 3 y = 9

x2 y2 − = 1 is 26. The equation of a tangent parallel to y = x drawn to 3 2 a) x − y + 1 = 0 b) x − y + 2 = 0 c) x + y − 1 = 0

d)

x− y+2 = 0

a)

5x + 3 y = 9

b)

5x − 3 y = 16

c)

27. If the line y = 3 x + λ touches the hyperbola 9 x 2 − 5 y 2 = 45, then the value of λ is a) 36

b) 45

c) 6

d) 15.

28. A point moves in a plane so that its distance PA and PB from two fixed points A and B in the plane satisfy the relation PA - PB = k (k ≠ 0) , then the focus of P is a) a parabola

b) an ellipse

c) a hyperbola

29. The equation of the conic with focus at ( 1, -1 ), directrix along x - y + 1 = 0 and with eccentricity a) x 2 − y 2 = 1 c)

d) a branch of a hyperbola

2 is

b) xy = 1

2 xy − 4 x + 4 y + 1 = 0

d)

2 xy + 4 x − 4 y − 1 = 0 .

30. The centre of the hyperbola 9 x 2 − 16 y 2 − 36 x + 96 y − 252 = 0 are a) (2,3) b) ( -2, -3 ) c) ( -2, 3 )

31. The eccentricity of the hyperbola with latusrectum 12 and semi-conjugate axis 2 a) 2

b) 3

c)

d) none of these.

3 , is

3 2

d) 2 3.

32. The equation of the hyperbola with vertices ( 3, 0 ) and ( -3, 0 ) and semi - latusrectum 4, is given by a) 4 x 2 − 3 y 2 + 36 = 0

b) 4 x 2 − 3 y 2 + 12 = 0

c) 4 x 2 − 3 y 2 − 36 = 0

33. The equation of the tangent to the conic x 2 − y 2 − 8 x + 2 y + 11 = 0 at ( 2, 1 ) is a) x + 2 =0 b) 2x + 1 = 0 c) x - 2 = 0 34. The value of m for which y = mx + 6 is a tangent to the hyperbola

a)

17 20

b)

20 17

d) none of these.

d) x + y + 1 = 0.

x2 y2 − = 1 is 100 49 c)

3 20


d)

20 3 55

35. The equation of the tangent to the hyperbola 4 y 2 = x 2 − 1 at the point ( 1, 0 ) is a) x = 1 b) y = 1 c) y = 4 36. The number of normals to the hyperbola a) 2

d) x = 4.

x2 y2 − = 1 from an external point is a 2 b2

b) 4

37. If the eccentricity of a conic is 3/5, the conic is a) a parabola b) an ellipse

c) 6

d) 5

c) a hyperbola

d) none of these.

38. The point ( at2, 2bt ) lies on the hyperbola, x / a − y / b = 1 for 2

a) all values of t

b)

2

2

2

t2 = 2 + 5

t2 = 2 − 5

c)

d) no real value of t

39. y = mx + c is a normal to the ellipse x 2 / a 2 + y 2 / b 2 = 1 if c2 is equal to a)

(a 2 − b 2 ) 2 a 2 m2 + b 2

b)

(a 2 − b 2 ) 2 a 2 m2

40. If e,e' are the eccentricities of hyperbolas

a) e = e'

x2 a2

−

y2 b2

( a 2 − b 2 ) 2 m2 a 2 + b 2 m2

c)

= 1 and

x2 b2

−

y2 a2

b) e = −e'

d)

( a 2 − b 2 ) 2 m2 a 2 m2 + b 2

= 1 , then

c) ee'= 1

d)

1 e

2

+

1 e' 2

=1

ANSWERS 1.

b

2.

b

3.

b

4.

c

5.

b

6.

b

7.

a

8.

a

9.

b

10.

b

11.

c

12.

b

13.

d

14.

d

15.

b

16.

c

17.

a

18.

b

19.

b

20.

c

21.

c

22.

b

23.

a

24.

d

25.

c

26.

a

27.

c

28.

c

29.

c

30.

a

31.

a

32.

c

33.

c

34.

a

35.

a

36.

b

37.

b

38.

b

39.

c

40.

d


56

FUNCTIONS, LIMITS AND CONTINUITY 1. Let A and B be two finite sets having m and n elements, respectively. Then the total number of mappings from A to B is a) mn b) 2mn c) mn d) nm 2. The total number of injective mappings from a set with m elements to a set with n elements, m ≤ n , is a) mn

b) nm

n! (n − m)!

c)

d) n !

3. The total number of injective mappings from a finite set with m elements to a set with n elements for m > n is a)

n! (n − m)!

b)

m! (m − n)!

4. If f (x) = cos (loge x ), then f ( x ) f ( y ) − a) 0

b)

c) nm

1 [ f ( x / y ) + f ( xy )] is 2

1 f ( x) f ( y) 2

5. If f ( x ) = (a − x n )1/ n , where a > 0 and a) a b) x

d) none of these

f ( x + y)

c)

d) none of these

n ∈ N , then fof (x) is equal to d) an

c) xn

6. Which of the following functions from Z to itself are bijections ? a) f ( x ) = x 3

f ( x) = x + 2

b)

7. If f : R → R be a mapping defined by f ( x ) = x 3 + 5, then f a) ( x + 5)1/ 3 8. Let f : R → R ,

F x + 7 IJ a) G H 2 K

b) ( x − 5)1/ 3

f ( x) = 2 x + 1

c)

c)

−1

d) f ( x ) = x 2 + x.

( x ) is equal to

(5 − x )1/ 3

d) 5 - x

g : R → R be two functions given by f ( x ) = 2 x − 3, g ( x ) = x 3 + 5. Then ( fog ) −1 ( x )

1/ 3

b)

FG x − 7 IJ H 2K

1/ 3

9. If the function f : R → A given by f ( x ) = a) R

b) [ 0, 1 ]

c)

FG x − 2 IJ H 7K

1/ 3

d)

x2 is a surjection, then A = x2 + 1 c) ( 0, 1 ]

FG x − 7 IJ H 2K

is equal to

1/ 3

d) [ 0, 1 )

10. Which of the following functions is inverse of itself a) f ( x ) =

1− x 1+ x

b)

f ( x ) = 5log x

c)

f ( x) = 2

x ( x −1)

d) none of these

11. If f (x) is defined on [0, 1] by the rule

f ( x) =

RS x, if x is rational T1- x, if x is irrational.

Then for all x ∈[ 0, 1], f ( f ( x )) is a) constant

b) 1 + x

c) x

d) none of these

12. The function f : R → R defined by f ( x ) = ( x − 1) ( x − 2) ( x − 3) is a) one-one but not onto

b) onto but not one-one

c) both one -one and onto


d) niether one-one nor onto. 57

13. The composite mapping fog of the maps f : R → R , f ( x ) = sin x and a) x2 sin x

b) ( sin x )2

g : R → R , g ( x ) = x 2 , is

c) sin x2

d)

sin x . x2

14. The number of surjections from A = { 1, 2, ...., n }, n ³ 2 onto B = { a, b } is a)

n

P2

b) 2n - 2

c) 2n - 1

d) none of these

15. Let f :( e, ∞ ) → R be defined by f (x) = log [ log ( log x )], then a) f is one-one but not onto c) f is both one-one and onto 16. The value of lim x→0

b) f is onto but not one-one d) the range of f is the subset of its codomain.

x2 + 1 − 1 x2 + 9 − 3

a) 3

is

b) 4

c) 1

x + x 2 +....+ x n − n 17. The value of lim is x →1 x −1 n +1 a) n b) 2

c)

d) 2

n(n + 1) 2

d)

n( n − 1) 2

2

e x − cos x 18. lim is equal to x→0 x2 a) 3 / 2

b) 1 / 2

c) 2 / 3

d) none of these

[ x ] + [ 2 x ] + [ 3x ]+.....+[ nx ] equals n →∞ n2

19. If [x] denotes the greatest integer less than or equal to x, then lim a) x / 2

b) x / 3

20. The value of lim x→0

x →1

b) 3 / 5

c) 3 / 2

d) 3 / 4

1 − cos 2 ( x − 1) x −1

a) exists and it equals

2

c) does not exist because

b) exists and it equals –

( x − 1) → 0

F 1 + 5x IJ 22. The value of lim G H 1 + 3x K 2

x →0

a) e2

d) 0

1 − cos3 x is x sin x cos x

a) 2 / 5

21. lim

c) x / 6

2

2

d) does not exist because left hand limit is not equal to right hand limit

1/ x 2

is b) e

c) e-1


d) none of these

58

F xI 23. The value of lim G cos J H mK

m

is

m→∞

a) 1

b

24. The value of lim log 5 5x x →1

b) e

g

x →0

d) none of these

c) e-1

d) none of these

c) n / m

d) none of these

c) -1

d) none of these

c) b

d) none of these

b) 0

c) e

d) none of these

b) e

c) e2

d) e3.

b) 0

c) -1

d) none of these

b) 1

c)

1 2

d) none of these

b) 1 / 2

c) 1 / 3

d) none of these

log x 5

a) 1 25. lim

c) e-1

is

b) e

sin x n ,(m < n) is equal to (sin x ) m

a) 1

b) 0

F x sinFG 1 IJ − x I G H x K JJ is 26. The value of lim G GH 1 − | x | JK a) 0 b) 1 2

x →∞

x

27. The value of lim a sin x →∞

FG b IJ is ( a > 1 ) Ha K

a) b log a

28. If lim x→a

b) a log b

a x − xa = −1, then the value of a is x x − aa

a) 1

F x + 2 IJ 29. lim G H x + 1K

x +3

x →∞

is equal to

a) 1 30. The value of lim x →∞

sin x is x

a) 1 31. lim x →0

sin x is equal to x

a) 0 32. lim x→0

a) 1

x

−

tan x − x equals x 2 tan x


59

33. If f ( x ) = a) -2

2( 2 x + 1) 2 x−3 , then lim [ f ( x ) + g ( x ) + h( x )] is and h( x ) = − 2 , g( x) = x→3 x + x − 12 x−3 x+4 2 b) -1 c) − d) 0 7

e x − (1 + x ) is x2


a) 0

b) 1 / 2

c) 2

sec −1 x , where [x ] denotes the greatest integer less than or equal to x is defined for all x belonging to x − [ x]

35. The function f ( x ) =

b) R − {( −1, 1) ({n | n . Z })} c)

a) R

n →∞

x →0

-(0, 1)

d) R + − {n | n . N }.

n

a) 1

x

+

R

FG x IJ cosFG x IJ cosFG x IJ .....cosFG x IJ is H 2K H 4K H 8K H 2 K

36. The value of lim cos

37. lim

d) e

b)

sin x x

c)

x sin x

b)

1 2

c) 1

d) ∞

b) -1

c) e2

d) e

b) 0

c) 2

d) ∝

b) -1

c) 0

d) none of these

d) none of these

is equal to

−1

tan 2 x

a) 0

R F π IU 38. lim StanG + xJ V T H 4 KW

1/ x

is

x →0

a) 1

2 x 2 − 4 f ' ( x) 39. If f ' ( 2) = 2, f ' ' ( 2) = 1, then lim is x→2 x−2 a) 4 40. The value of lim x→0

sin x x2

a) 1

4− x is 4x − x3 2

41. The function f (x) =

a) discontinuous at only one point c) discontinuous exactly at three points

R| 36 − 9 − 4 + 1 , 42. If f ( x ) = S 2 − 1 + cos x |T k , x

x

a) 16 2 log 2 log 3

x

b) discontinuous exactly at two points d) none of these

x≠0

is continuous at x = 0, then k equals

x=0

b) 16 2 ln 6

c) 16 2 ln 2 ln 3 Quest Tutorials


(d) none of these 60

R| sin x 43. The function f defined by S x |T 0,

2

,x ≠ 0

is

x=0

a) continuous and derivable at x = 0 c) continuous but not derivable at x = 0 44. If f ( x ) = ( x + 1) a) 0

cot x

be continuous at x = 0, then f (0) is equal to b) 1 / e c) e

R| x − 5x + 4 , |( x − 1) ( x − 2)| | 45. Let f ( x ) = S || 126,, T 4

b) neither continuous nor derivable at x = 0 d) none of these.

2

a) R

x ≠ 1, 2 x =1 x=2

Then f (x) is continuous on the set

b) R - {1}

R| mx + 1, 46. If f ( x ) = S |Tsin x + n,

c) R - {2}

π 2 is continuous at x = π , then 2 π x> 2 b) m =

nπ +1 2

c)

n=m

π 2

d) m = n =

π . 2

tan{π[ x − π ]} , where [x] denotes the greatest integer less than or equal to x, is 1 + [ x ]2

a) discontinuous at some x c) f ' (x) exists for all x.

b) continuous at all x, but f ' (x) does not exist for some x d) f ' (x) exists for all x, but f '' (x) does not exist

48. If f (x) = [ x sin p x ], then f (x) is a) discontinuous at x = 0 b) continuous in ( -1, 0 )

49. f ( x) =

d) R - { 1, 2 }

x≤

a) m = 1, n = 0 47. The function f ( x ) =

d) none of these

c) differentiable at x = 1

d) nondifferentiable in ( -1, 1 )

2x + 1 1 + px − 1 − px ,0 ≤ x ≤ 1 is continuous in the interval [-1,1], then p equals ,−1 ≤ x < 0 = x−2 x b) −

a) -1

1 2

c)

1 2

d) 1

 | x | −3  −1  + [log(4 − x )] is defined for  2 

50. The function f ( x ) = cos −1  a) [−1, 0] ∪ [1, 5]

b) [−5, − 1] ∪ [1, 4]

c) [−5, − 1] ∪ [1, 4) − {3} ]

d) [1, 4) − {3}

51. The value of the parameter α , for which the function f ( x ) = 1 + αx , α ≠ 0 is the inverse of itself, is a) - 2 b) - 1 c) 1 d) 2 Quest Tutorials Head Office : 44C,Kalusarai New Delhi-16; Ph.(011) 46080363

61

 x + sin x  =  x − cos x 

52. lim  x →∞

a) 0

b) 1

c) - 1

53. The values of x where the function f ( x ) = a) (−∞, 2) ∪ {3}

tan x log ( x − 2) x 2 − 4x + 3

is not defined are given by

(−∞, 2) ∪ {3, nπ + π / 2 : n ≥ 1}

b)

c) (−∞, 2)

d) none of these

d) none of these

54. The domain of definition of the function f ( x ) = 3e

x 2 −1

log(x − 1) is

a) (1, ∞)

b) [1, ∞) d)

(−∞, − 1) ∪ (1, ∞)

c)

abc

d)

c)

a = 1, b = −2

d) none of these

c) set of all reals different from 1 2/ x

 a x + bx + c x   ; a > b > c > 0 is equal to 55. Lt   x→∞  3   a) 0

b) a 2

a bc

 x3 + 1   , then  x 2 + 1 − (ax + b)  = 2  

56. If Lt  x →∞

a) a = 1, b = 1

57. Let f ( x ) = a) 1/2

58.

Lt

x→ − ∞

b) a = 1, b = 2

1 − sin x (π − 2x ) 2

, when x ≠ π / 2 and f (π / 2) = k . The value of k which makes f continuous at x = π / 2 is c) 1/8

d) none of these

b) 1

c) zero

d) ∞

b) - 1

c) 0

d) - 2

1 x 4 sin   + x 2 x equals 1+ | x |3

a) - 1

59. Lt x →0

b) 1/4

sin −1 x 1− x +1

=

a) 1

ANSWERS 1. 8. 15. 22. 29. 36. 43. 50. 57.

d d c a b b a c c

2. 9. 16. 23. 30. 37. 44. 51. 58.

c d a a b b c b a

3. 10. 17. 24. 31. 38. 45. 52. 59.

d a c b a c d b d

4. 11. 18. 25. 32. 39. 46. 53.

a c a b c a c b

5. 12. 19. 26. 33. 40. 47. 54.

b b a a c d c a


6. 13. 20. 27. 34. 41. 48. 55.

b c c c b c b b

7. 14. 21. 28. 35. 42. 49. 56.

b b a a b c b c

62

DIFFERENTIABILITY AND DIFFERENTIATION 1. The set of points where the function f (x) = x | x | is differentiable is (a)

( −∞, ∞)

( −∞,0) ∪ (0, ∞)

(b)

(c)

(0, ∞)

(d)

[0, ∞)

2. Let f ( x ) = ( x +| x|) | x|. Then for all x (a) f is continuous but f ' is not continuous (c) f ' is continuous

(b) f is differentiable for some x (d) none of these

3. If f ( x ) = a |sin x| + b e + c | x| and if f (x) is differentiable at x = 0, then |x |

3

(b) a + b = 0 ; c ∈ R

(a) a = b = c = 0

4. Let h (x) = min { x, x2 }, for every real number x. Then (a) h is continuous for all x (c) h' (x) = -1, for all x > 1 5. If f ( x ) =

RS T(x

3

(c) b = c = 0, a ∈ R

(d) c = 0, a = 0, b ∈ R

(b) h is differentiable for all x (d) h is not differentiable at 3 values of x

| x − 4|, for x ≥ 1 , then / 2) − x + 3x + (1 / 2), for x < 1 2

(a) f ( x ) is continuous at x = 1 and at x = 4

(b)

f ( x ) is differentiable at x = 4

(c) f ( x ) is continuous and differentiable at x = 1

(d)

f ( x ) is only continuous at x = 1.

6. If f ( x ) =

R|x sinFG 1 IJ , S| H x K T 0, 2

x≠0

, then

x=0

(a) f and f ' are continuous at x = 0 (c) f is continuous at x = 0 and f ' is not continuous at x = 0

(b) f is derivable at x = 0 (d) f ' is derivable at x = 0.

7. Let [ x ] denotes the greatest integer less than or equal to x and f (x) = [ tan2 x ]. Then, (a) lim f ( x ) does not exist

(b) f (x) is continuous at x = 0

(c) f (x) is not differentiable at x = 0

(d) f ' (0) = 1.

x →0

8. If f ( x + y ) = f ( x ) f ( y ) for all x, y ∈ R , f ' (0) = 3 Then f '(5) equals (a) 3e15

(b) 3

(b) 44

1 1 + ( g ( x )) 3

(b)

(d) log 3

(c) 28

10. Let g (x) be the inverse of the function f (x) and f ' ( x ) = (a)

(d) none of these

x , y ∈ R. Suppose that f (3) = 3 , then f ' (3) is equal to

9. Let f ( x + y ) = f ( x ) f ( y ) for all (a) 22

(c) 5

1 1 + ( f ( x )) 3

1 . Then g ' ( x ) is equal to 1+ x3 (c) 1 + ( g ( x )) 3

(d) 1 + ( f ( x )) 3

11. If f ( x ) = min {tan x ,cot x}, then (a) f ( x ) is not differentiable at x = 0, π / 4, 5π / 4

(b)

z

π/2

f ( x ) dx = ln 2

0

(c) f ( x ) is periodic with period π / 2 .

(d) none of these Quest Tutorials


63

12. If y = cos

(a)

−1

FG 2 x IJ , then dy is H 1 + x K dx 2

−2 for all x 1+ x2

(b)

−2 for all | x | > 1 1+ x2

(c)

2 for all | x | < 1 1+ x2

(d) none of these

13. If f ( x ) = x − 5x + 6 , then f '(x) equals 2

(a) 2x - 5 for 2 < x < 3

(b) 5 − 2 x for 2 < x < 3

(c) 2 x − 5 for 2 ≤ x ≤ 3

(d) 5 − 2 x for 2 ≤ x ≤ 3

14. If

y = cos−1

FG 2 cos x − 3sin x IJ , then dy is H 13 K dx

(a) zero

15. If

(b) constant = 1

2 x + 2 y = 2 x + y , then the value of

(a) 0

(d) none of these.

(c) 1

(d) 2.

dy at x = y = 1 is dx

(b) -1

16. The expression of

(c) constant ≠ 1

.......∞ ax dy of the function y = a x is dx

y2 (a) x (1 − y log x )

y 2 log y (b) x (1 − y log x )

y 2 log y (c) x (1 − y log x log y )

y 2 log y . (d) x (1 + y log x log y )

17. If y = log cos x sin x , then

dy is equal to dx

(a) (cot x log cos x + tan x log sin x ) / (log cos x ) (c) (cot x log cos x + tan x log sin x ) / (log sin x ) 18. If

y = e sin

(a) e sin

−1

x

−1

x

and u = log x , then

/ 1− x

2

(b)

2

2

(b) (tan x log cos x + cot x log sin x ) / (log cos x ) (d) none of these

dy is du

xe

sin −1 x

(c)

x e sin

−1

x

(d) none of these

1− x2

19. The differential coefficeint of f (x) = log ( log x ) with respect to x is (a)

x log x

20. If x y = ( x + y ) p

q

(b) p+q

, then

log x x

(c) ( x log x )

−1

(d) x log x

dy is equal to dx


64

2

(a)

y x

21. If x = φ( t ), y = ψ ( t ), then

(a)

φ ' ψ ' '− ψ ' φ' ' ( φ' ) 2

22. If f ( x ) = x

1/ x

(c)

x y

(d)

qy . px

(c)

φ'' ψ' '

(d)

ψ' ' φ' '

(c)

e1/( e − 2 )

(d) none of these.

(b)

F 3 tan x + 3IJ fG H 5 tan x + 6 K

d2y is equal to dx 2 (b)

φ ' ψ ' '− ψ ' φ ' ' ( φ' ) 3

then f ' ' ( e) is equal to

(a) e1/( e − 3)

23. If y = f

px qx

(b)

(b) e1/ e

FG 3x + 4 IJ and f ' ( x) = tan x H 5x + 6 K

(a) − 2 tan FG 3 x + 4 IJ H 5x + 6 K

2

2

then

dy is equal to dx 2

1 × (5 x + 6) 2

(c) tan x2

2

2

tan x 2

(d) none of these.

24. If f ' ( x ) = g ( x ) and g ' ( x ) = − f ( x ) for all x and f (2) = 4 = f ' (2), then f 2 (16) + g2 (16) is (a) 16 (b) 32 (c) 64

(d) none of these

d2y π 25. If x = 2 sin t - sin 2t, y = 2 cos t - cos 2t, then the value of is 2 at t = dx 2 (a) 2

26.

(b) -1/2

(c) -3/4

(d) -3/2.

(c) 1/a

(d) log (ax) + 1

d (log( ax ) x ), where a is a constant is equal to dx (a) 1

27. If y = tan −1

(b) log ax

1 1 1 + tan −1 2 + tan −1 2 +.....+ upto n , then y' (0) is equal to 2 1+ x + x x + 3x + 3 x + 5x + 7

(a) −1 / ( n 2 + 1)

(b) − n 2 / ( n 2 + 1)

(c) n 2 / (n 2 + 1)

(d) none of these

28. The number of points at which the function f ( x ) = x − 0.5 + x − 1 + tan x does not have a derivative in the interval (0, 2) is (a) 1 29. The function f ( x ) =

(b) 2

(c) 3

(d) 4

1 , where [x] is equal to the greatest integer not exceeding x, is x − [ x]

(a) continuous at all points (c) discontinuous when x is an integer or zero

(b) continuous at x = 0 (d) discontinuous at all points


65

30. The set of all points where the function (a)

( −∞, ∞)

f ( x) =

(b)

x is differentiable is 1+ x

(0, ∞)

(c)

31. Which of the following statements is (are) necessarily true ? 1. If a function is differentiable at x, then it is continuous at x 2. If a function is continuous at x, then it is differentiable at x 3. If a function is integrable on [ x - a, x + a ], then it is continuous at x (a) only 1 (b) only 2 32. If y = 10

4 log10 x

+ 56 log5 x + 5, then

(a) 0

F dy I F dx I 33. If x = f (t) cost - f ' (t) sin t, y = f (t) sin t + f ' (t) cos t, then G J + G J H dt K H dt K 2

34. If

x y = e x − y , then

(a)

log x (1 + log x ) 2

35. If y = e 3log x , then

36. If

(a)

1 1+ x2

(d) none of these

2

is equal to

(c)

1+ x 1 + log x

(d)

(c)

3y x

(d) 3 xy

(d) none of these

1 − log x 1 + log x

dy is equal to dx

dy at x = 0 is dx

(b) -1

(c) 1

lim

z

f ( x)

x→2

(b) 3 f ' (2)

−1

(c) 10

(b) not defined

37. If f (x) is a differentiable function on [0, 3] and f (2) = 3, then

38. If y = tan

(d) 1 and 3 only

dy is dx

n

(a) 5 f ' (2)

(c) 1 and 2 only

(c) [ f ( t )]2 + [ f ' ' ( t )]2

y = (1 + x ) (1 + x 2 ) (1 + x 4 ).....(1 + x 2 ) then

(a) 0

( −∞,−1) ∪ ( −1, ∞)

(b) [ f ( t ) + f ' ' ( t )]2

(b) 2 log x

(a) 3 x2

(d)

dy at x = 1 is equal to dx

(b) -2

(a) [ f ( t ) − f ' ' ( t )]2

( −∞, 0) ∪ (0, ∞)

3

(d) none of these

5t dt x−2 (c) 15 f ' (2)

(d) none of these

x +1 1− x dy + tan −1 , then is 1− x 1+ x dx (b)

1 1− x2

(c)

2x 1+ x2


(d) 0

66

1 1 − x 2 at x = 2 is

 1   w.r.t  2x2 −1 

39. The derivative of sec −1  (a) 2

(b) 4

(c) 1

(d) -2

40. If f (x ) = log a log a (x ) , then f ' ( x ) is (a)

log a e x log e x

(b)

log e a x log a x

(c)

log e a x

(d)

x log e a

ANSWERS 1. 8. 15. 22. 29. 36.

a a b d c c

2. 9. 16. 23. 30. 37.

c d c a a c

3. 10. 17. 24. 31. 38.

b c a b a d

4. 11. 18. 25. 32. 39.

a a c d c b

5. 12. 19. 26. 33. 40.

a a c d b a


6. 13. 20. 27. 34.

c b a b a

7. 14. 21. 28. 35.

b b b c a

67

APPLICATION OF DERIVATIVES 1. If there is an error of k% in measuring the edge of a cube then the percentage error in estimating its volume is a) k b) 3k c) k / 3 d) none of these 2. If the line ax + by + c = 0 is a normal to the rectangular hyperbola xy = 1 then a) ab > 0

b) ab < 0

c)

a > 0, b > 0

d) none of these

3. Let f (x) = ( x - p )2 + ( x - q )2 + ( x - r )2. Then f (x) has a minimum at x = λ where λ is equal to a) ( p + q + r ) / 3

b) ( p q r )1/ 3

c)

3 1 1 1 + + p q r

d) none of these

2− x is (are) x2

4. The critical point (s) of f ( x ) = a) x = 0

b) x = 2

c)

x = 0 and x = 2

d) none of these

5. If f ( x ) = x / sin x and g ( x ) = x / tan x where 0 < x ≤ 1 , then in the interval a) both f (x) and g(x) are increasing function

b)

f ( x ) is an increasing function

c) both f (x) and g(x) are decreasing function

d)

g ( x ) is an increasing function

6. If 4a + 2b + c = 0 then the equation 3ax2 + 2bx + c = 0 has at least one real root lying between a) 0 and 1 b) 1 and 2 c) 0 and 2

d) none of these

7. The function f ( x ) = x + λx + 5x + sin 2 x will be an invertible function if λ belongs to 3

a)

( −∞,−3)

2

b)

( −3, 3)

c)

(3,+∞)

d) none of these

8. The difference between the greatest and the least values of the function f ( x ) = sin 2 x − x on [ − π / 2, π / 2] is

3+ 2 2

a)

b)

3+ 2 π + 2 6

3+ 2 π − 2 3

c) π

d)

c) 8

d) 10

c) k < 1

d) k < -1

9. The minimum value of f ( x ) = 3 − x + 2 + x + 5 − x is a) 0

b) 7

10. If f ( x ) = kx − sin x is monotonically increasing, then a) k > 1 11. If ax + a)

b) k > -1

b ≥ c for all positive value of x and a, b, c are positive constants, then x

ab ≥ c 2 / 4

b) ab ≤ c / 4 2

c)

bc ≥ a 2 / 4

12. The value of b for which the function f (x) = sin x - bx + c is decreasing in the internal a) b < 1

b) b ≥ 1

d) ac ≥ b / 4 2

( −∞, ∞) is given by

c) b > 1 Quest Tutorials


d) b ≤ 1 68

x + 2 y − 10 = 0 are

13. The equation of the tangents to 2 x 2 − 3 y 2 = 36 which are parallel to the straight line a) x + 2y = 0

b) x + 2 y +

288 =0 15

c)

x + 2y +

1 =0 15

d) none of these

14. The slope of the tangent to the curve represented by x = t 2 + 3t − 8 and y = 2t − 2t − 5 at the point M (2, − 1) is a) 7 / 6 b) 2 / 3 c) 3 / 2 d) 6 / 7 2

15. The number of real roots of the equation e x −1 + x − 2 = 0 is a) 1 b) 2

c) 3

d) 4

16. If the two curves y = a x and y = b x intersect at an angle α , then tan α equals a)

log a − log b 1 + log a log b

b)

log a + log b 1 − log a log b

c)

log a − log b 1 − log a log b

d) none of these.

17. For the curve x = t 2 − 1, y = t − 1 the tangent line is perpendicular to x-axis where a) t = 0

c)

b) t = ∞

t=

1 3

d) t = −

1 . 3

18. The normal to the curve x = a (cos θ + θ sin θ), y = a (sin θ − θ cos θ) at any θ is such that a) it makes a constant angle with x-axis b) it es through the origin c) it is at a constant distance from the origin d) none of these 19. The tangent to the curve y = e 2 x at the point (0, 1) meets x-axis at a) (0, 2)

b) (2, 0)

c)

FG − 1 ,0IJ H 2 K

d) none of these

c)

y = ±2 x

d) none of these

c)

p = −2, q = −7

d)

20. The equation of the tangents at the origin to the curve y 2 = x 2 (1 + x ) are a) 21. If a)

y = ±x

b)

x = ±y

y = 4 x − 5 is a tangent to the curve y 2 = px 3 + q at (2, 3), then p = 2 , q = −7

b)

p = −2 , q = 7

p = 2, q = 7.

22. Tangents are drawn from the origin to the curve y = cos x. Their points of lie on a) x 2 y 2 = y 2 − x 2

b) x 2 y 2 = y 2 + x 2

c)

x2 y2 = x2 − y2

d) none of these

23. The minimum value of ( x + a ) ( x + b) is a)

(a + b) 2 4

b)

( a − b) 2 4

c) a b

d) none of these

24. If f is an increasing function and g is a decreasing function such that gof is defined, then a) gof is increasing b) gof is decreasing c) gof may not be monotomic d) none of these


69

25. The function f ( x ) = x 3 + 24 x 2 + ax − 10 attains its relative minimum value at x = 1. Then the value of a is a) 51 b) -51 c) -45 d) none of these

26. If

a0 a a a + 1 + 2 +....+ n −1 + an = 0. Then the function f (x ) = ao x n + a1x n −1 + a2 x n − 2 + .......... + an has in ( 0, 1) n +1 n n −1 2

a) at least one zero

b) at most one zero

c) only 3 zeros

d) only 2 zeros.

27. If x 2 + y 2 = u − 1 / u and x 4 + y 4 = u 2 + 1 / u 2 then a) x 3 y y ' = 1

b) xy' = y

x 2 y y' = 1

c)

28. The equation of the horizontal tangent to the graph of the function y = e x + e − x is a) y = 2 b) y = -2 c) y = 0

d) none of these

d) none of these

29. The image of the interval [ -1, 3] under f ( x ) = 4 x 3 − 12 x is a)

[ −2, 0]

b)

30. The maximum value of x1 / x is a) 1 / e

[ −8, 72]

c)

b) e

[ −8, 0]

c) e1 / e

d)

[8, 72]

d) (1 / e )e

31. The number of solutions of the equation a f ( x ) + g ( x ) = 0, where a > 0, g (x) ≠ 0 and g (x) has minimum value 1/4, is a) one b) two c) infinitely many d) zero 32. Let x and y be two real numbers such that x > 0 and xy = 1. The minimum value of a) 1

b) 1 / 2

33. The maximum value of a) 1 / 2

x2 − x + 1 for all real values of x is x2 + x + 1 b) 1

x + y is

c) 2

d) 1 / 4

c) 2

d) 3

34. The point of intersection of the tangents drawn to the curve x2y = 1 - y at the points where it is met by the curve xy = 1 - y is given by a) (0, -1) b) (1, 1) c) (0, 1) d) none of these. 35. If y = 1 + x + a) x

2 3 dy x x + + ...... ∞ then is equal to dx 2! 3!

b) 1

36. If f (a ) = 2, f ' (a ) = 1, g (a ) = −1, g ' (a ) = 2, then the value of Lt a) -5

37. If

a)

x →a

b) 1/5

1 − x 2 + 1 − y 2 = a (x − y ), then

1− y2 1 − x2

b)

d) ∞

c) y

g (x ) f (a ) − g (a ) f (x ) is x−a

c) 5

d) 0

dy is dx

1 − x2 1− y

2

c)

1 − x2


d)

1− y2

70

38. Let f (x + y ) = f ( x ) f ( y ) for all x and y. Suppose that f (3) = 3 and f ' (0 ) = 11 then f ' (3) is given by a) 22 b) 33 c) 28 d) none of these 39. If y 2 = p ( x ), a polynomial of degree 3, then 2 a) p' ' ' (x) + p' (x)

(

b) p ' ' ( x ) p ' ' ' (x )

)

40. If y = cos −1 4 x 3 − 3 x , then

1− x

c)

p ( x) p ' ' ' (x )

d) a constant

dy is equal to dx

3

a)

d  3 d 2 y  is equal to y dx  dx 2 

b) −

2

3

c)

1 − x2

1 − x2

d) none of these

ANSWERS 1. 8. 15. 22. 29. 36.

b c a c b c

2. 9. 16. 23. 30. 37.

b b a d c a

3. 10. 17. 24. 31. 38.

a a a b d b

4. 11. 18. 25. 32. 39.

a a c b c c

5. 12. 19. 26. 33. 40.

b c c a d b


6. 13. 20. 27. 34.

c d a d c

7. 14. 21. 28. 35.

b d a a c

71

INDEFINITE INTEGRATION 1. The integral

z

e tan

−1

x

FG 1 + x + x IJ dx is equal to H 1+ x K 2

2

−1

−1

e tan x a) +C 1+ x2 2. If

b)

z z

K = -1/3

c)

K = -1/5

d) none of these

b)

e x tan x / 2 + C

c)

e x cot x + C

d)

( (a / b) tan x ) + C

b)

tan −1 ( (b / a ) tan x ) + C

1 tan −1 ( (b / a ) tan x ) + C ab

d)

none of these

e x (1 + sin x ) dx is equal to 1 + cos x

2

−1

∫

sin α

dα is

1 + cosα

a)

− 2 2 cos α / 2 + C

−

2 cos α / 2 + C

6. The value of

z

b)

2 2 cos α / 2 + C

c)

2 cos α / 2 + C

d)

dx 5 + 4 cos x is

FG IJ H K FG 1 tan xIJ + C H3 K

FG IJ H K FG 1 tan ( x / 2)IJ + C H3 K

a)

1 −1 tan x tan +C 3 3

b)

1 −1 tan ( x / 2) tan +C 3 3

c)

2 tan −1 3

d)

2 tan −1 3

z

dx sin x + cos x is equal to

b

g

a) log tan π / 4 + x / 8 + C

c)

sin log x + C

dx is equal to a cos x + b 2 sin 2 x 2

5. The value of

7.

d) none of these

b)

a) sin c)

x e tan x +C 1+ x2

cos 4 x + 1

a) log tan x + C

4.

+C

c)

∫ cot x − tan x dx = K cos 4x + C then

a) K = -1/2

3.

xe

tan −1 x

b

g

1 log tan x / 2 + π / 8 + C 2

b

g

b

g

b)

2 log tan x / 2 + π / 8 + C

d)

2 log tan x / 4 − π / 8 + C


72

z

8. The value of

cos x dx is sin x + cos x

b

g

a)

x2 + log sin x + cos x + C 2

b)

1 (x + log(sin x + cos x )) + C 2

c)

2 log (sin x + cos x ) + x + C

d)

none of these

9. If

z

dx = log f ( x ) + C then sin x cos x f ( x ) = sin x + cos x

a)

10. If f ( x) = lim

n →∞

x n − x −n n

x +x

−n

f ( x ) = tan x

b)

, 0 < x < 1, n∈ N then

z

c)

c) constant 11. If

f ( x ) = lim[2 x + 4 x 3 + ... 2nx 2 n −1 ](0 < x < 1) then n→∞

− 1− x2

a)

12.

zc

1

13.

zb

x ex

1+ x

z

g

d)

none of these

f ( x ) dx is equal to c)

2

2

1 tan −1 2

b)

1 x −1

d)

2

1 1− x2

F GH

2x 1+ x2

I JK

c)

1 tan −1 2

F GH

2x 1− x2

I JK

d) none of these

dx is equal to

ex +C x +1

b g

ex

b)

ex x +1 + C

c)

−

b)

( ae) x log ae

c)

ex 1+ log a

b x + 1g

2

+C

d)

ex +C 1+ x2

e x log a . e x dx is equal to

a) (ae)x

15. If

x sin −1 x + 1 − x 2 + C

2

2

a)

14.

2

−1

a)

1− x

b)

dx is equal to

h 1− x F 2x I 1 tan G 2 H 1− x JK

1+ x

z

1

b)

d) none of these

(sin −1 x ) f ( x ) dx is equal to

−[ x sin −1 x + 1 − x 2 ] + C

a)

f ( x ) = sec 2 x

z

g ( x ) dx = g ( x ), then

z

b g

l

d) none of these

q

g ( x ) f ( x ) + f ' ( x ) dx is equal to

a)

g( x) f ( x) − g( x) f ' ( x) + C

b)

g( x) f ' ( x) + C

c)

g( x) f ( x) + C

d)

g( x) f 2 ( x) + C


73

16.

z

cos3 x e log(sin x ) dx is equal to

a) −

17.

z

2 tan x + C

∫

sin x − cos x

a)

e sin x + C

19. If

1 − sin 2 x

z

c

b) 2

z z

b) e

h

e 3log x x 4 + 1

−1

23.

d) none of these

5x

cot x + C

c)

tan x +C 2

d) none of these

sin x − cos x

+C

c)

e sin x + cos x + C

c)

− log x 4 + 1

c

h

1 log x 4 + 1 + C 4

b)

55

c

x

blog 5 g

3

+C

55

b)

h

d) none of these

5

x

blog 5g

5x

3

+C

55

+C

c)

blog 5g

b)

f ' x g x − f x g' x

3

d) none of these

bg

[ f x g"( x ) − f " ( x ) g ( x )] dx is equal to

c)

f x g' x − f ' x g x

bg bg bgbg

z

d)

bg bg bg bg f b x g g ' b x g + f ' b x g gb x g A=

4 e x + 6 e− x dx = A x + B log 9 e 2 x − 4 + C , then x −x 9e −4e

c

h

a)

3 35 A = − , B = ,C = 0 2 36

b)

c)

3 35 A= − , B= , C ∈R 2 36

d)

none of these.

b)

1 f ax + b n +1

a)

cos x − sin x

x

f ( x) g '( x )

z

d) e

dx is equal to

a)

22. If

e sin x +C 4

55 .55 .5x dx is equal to

a)

21.

c)

e sin x cos x dx is equal to

a) log ( x 4 + 1) + C 20.

cos4 x +C 4

−

b)

tan x dx is equal to sin x cos x

a)

18.

sin 4 x +C 4

b

f ' ax + b

35 3 , C = − , C ∈R 36 2

g m f bax + bgr dx is equal to

mb

1 f ax + b n +1

n

gr

n +1

+ C , ∀ n except n = -1

mb


gr

n +1

+ C , ∀n

74

+C

mb

1 f ax + b a (n + 1)

c)

24.

1

∫

sin x cos 3 x

+ C , ∀n except n = -1

tan

a)

−1

1

c)

2 2

z

log

Fx GH x

b)

2 2

I JK

+ 2 x +1 +C − 2 x +1

2x2 + 3

)(

(

n +1

+ C , ∀n

2 +C tan x

d)

−2 tan x + C

d)

F GH

I JK

1 x2 −1 tan −1 2 2x none of these

)

8

2

x cos2 x

( x − 1) dx

∫x

(1 − x 3 )

(1/2, 1/2)

c)

(-1 , 1)

d) (1, - 1)

b)

1 − sin 2 x 2

c)

1 − sin x 2

d) − sin 2 x

b)

log{(x − 3) /( x − 2)}

c)

log{( x − 2 ( x − 3)}

d) none of these

c)

- 1/3

d) - 2/3

b)

1 [ 3 tan −1 (3 tan x ) + x ] 9

d)

none of these

is equal to

a) log{( x − 3) 2 /( x − 2)}

dx

b)

dx =

1 sin 2 x 2

∫ (x − 2) (x − 3)

= a log

(1 − x 3 ) + 1 b)

30. The value of the integral

2

(1 − x 3 ) − 1

a) 1/3

a)

gr

 x +1 −1 x , dx = a log   + b tan 2 then (a, b) is  x −1 x2 −1 x2 + 4

∫ 1 − 2 sin

29. If

mb

1 f ax + b a (n + 1)

1+ x 2 dx is equal to 1+ x 4

x +C

8

28.

c)

2

sin x − cos x

a)

2 tan x + C

b)

a) (- 1/2, 1/2)

27.

d)

dx is equal to


∫

n +1

−2 +C tan x

a)

26.

gr

+ b, then a =

2/3

2

cos x dx

∫ cos

2

2

x + 9 sin x

is

1 [3 tan −1 (3 tan x ) − x ] 8

1

−1



c) 8  tan (3 tan x ) − x   27  Quest Tutorials


75

31.

∫

log(x + 2) − log x dx is equal to x ( x + 2) 2

2

  x + 2   + c a) − log   x 

b)

  x + 2  − 2 log  + c   x 

d)

none of the above

c)

2 −1 3/ 2 cos cos x + c 3

d) none of the above

c)

a periodic function


b)

x x − + x − log( x + 1) + c 3 2

2

1   x + 2   + c c) − log 4   x  32.

3

cos x − cos x

∫

dx is equal to

3

1 − cos x

(

)

2 −1 3/ 2 sin cos x + c 3

a)

b)

(

)

3 −1 3/ 2 sin cos x + c 2

(

)

x

∫

33. The function f ( x ) = log( t + 1 + t 2 ) dx is 0

a) an even function 34.

∫

2

3

a)

x x − +c 3 2

c)

x x − + x + log( x + 1) + c 3 2

3

e (1 + x )

∫ cos (x e 2

x

x

∫a

x

2

+c

x

b)

sec ( x e ) + c

x

d) tan ( x + e x ) + c

c)

tan ( xe ) + c

b)

a x ex +c log a

c)

(a e) +c x +1

a x ex +c d) 1 + log a

b)

1 tan 2 x + c 2

c)

1 3 tan x + c 3

d)

b)

1 −1 3 tan x 6

c)

1 −1 3 tan x 6

x

x

a) a e + c 2

sin x

∫ cos a)

2( x + 1)

e dx = x

37.

4

dx =

)

2 log cos ( x e ) + c

a)

d)

2

x

2

x

36.

an odd function

x3 dx = x +1 3

35.

b)

4

x

x

dx =

1 2 tan x + c 3

3sin 2x − 4 cosx + c 38. The value of a)

∫

x 2 tan −1 x 3

1 −1 3 tan x 6

1+ x

6

will be

(

)

2

(

)

3


d)

(

1 −1 3 tan x 3 76

)

4

39. The value of a)

x sin

x + sin x

∫ 1 + cos x dx is

x 2

40. The value of

b)

log x

∫ (x + 1)

2

x cos

x 2

c)

x tan

x 2

d) x sin x + cos x

dx

a)

− log x + log x − log ( x + 1) x +1

b)

log x + log x − log ( x + 1) x +1

c)

log x + log x + log( x + 1) x +1

d)

− log x − log x − log ( x + 1) x +1

ANSWERS 1. 8. 15. 22. 29. 36.

b b c c a d

2. 9. 16. 23. 30. 37.

d b b c a c

3. 10. 17. 24. 31. 38.

b a a b c b

4. 11. 18. 25. 32. 39.

c d a b c c

5. 12. 19. 26. 33. 40.

a c b a a a


6. 13. 20. 27. 34.

d a c b b

7. 14. 21. 28. 35.

c b c a c

77

DEFINITE INTEGRATION

z n

1. The value

[ x ] dx (where [ x ] is the greatest integer function) is

0

a)

b g

n n +1 2

b g

n n −1 2

b)

c)

n (n −1)

d) none of these

b g z FGH 2 + dtd cos tIJK dt then F ' (x) is equal to x

2. If the F x =

3

a) 2 − sin x

b) 2 + sin t

z k

3. Let f be a positive function and l1 =

c b gh

x f x 1 − x dx , l2 =

1− k

then l1 / l2 is a) 2 4. The value of

z

0

z

z k

d) 0

c b gh

f x 1 − x dx

1− k

b) k

c) 1/2

d) 1

b) 1

c)

π/4

66 log 3

c)

66 5 13 − log 3 log 3

11 10100

c)

1 1010

dx is 1 + tan 3 x

π/ 2

a) 0 4

5. The value of

c) sin x

2 x +1

3

d)

π/2

dx is

0

a)

FG H

4 6 13 − log 3 log 3

z 1

6. The value of

IJ K

b g

x 1− x

b)

99

FG H

IJ K

d) none of these

dx is

0

a)

1 10100

b)

zm 3

7. The value of

r

x − 2 + [ x ] dx , where [x] denotes the greatest integer less than or equal to x is

−1

a) 7

b) 5 1000

8. The value of

∑ n =1

e1000 − 1 a) 1000

z

d) none of these

n

n −1

c) 4

d) 3

e x − [ x ] dx is ( [ x ] is the greatest integer function ) e1000 − 1 b) e −1

c)

e −1 1000


b g

d) 1000 e − 1

78

9. If f : R → R , g : R → R are continuous functions then the value of the integral

z

π/2

( f ( x ) + f ( − x )) ( g ( x ) − g ( − x )) is

− π/2

π

a)

10. The value of

z

b) 1 π

0

b)

2

zc ∞

11. If

0

z

3/ 2

−1

b2m + 1g!

c)

n!

hc

b) 1/80

b) 2 π + 1

bg bg bg

2

b) 2e


z

x

0

z

1

0

2

c) 1/40

d) 1/20

c) 1

d) 4

f ( x ) φ ( x ) dx is c)

2e

d) 2e −3/ 2

c)

-1

d) 2

c)

−π / 2

d) none of these

is

x

b) 1


z

FG H

IJ K

d 1 tan −1 dx is −1 dx x 1

π/2

b)

π/4

16. The equation of the tangent to the curve y =

2 y +1= x

z

d) none of these

cos t 2 dt

a) 0

a)

cos2 m−1 x dx

x sin πx dx = kπ 2 , then the value of k is

13. If f ( 0) = 2, f ' x = f x , φ x = x + f (x) , then

a)

0

h

a) 3π + 1

a) e

z

π

dx = kπ then the value of k is x + 4 x2 + 9 2

a) 1/60 12. If

d) 0

sin n x cos2 m+1 x dx is

b2m + 1g! bn!g

a)

c) -1

b)

z

x3

x

2

dt 1+ t2

at x = 1 is

3 x +1= y

c)

3 x + 1+ 3 = y

d) none of these

π2 / 4

17.

sin x dx =

0

a) 0

b) 1

c) 2

d) 4

z

π/3

18. The value of the integral a) c)

x sin x dx is 2 x cos − π/3

bπ / 3 − log tan 3π / 2g 3b π / 2 − log sin π / 12g

b

b) 2 2 π / 3 − log tan 5π / 12

g

d) none of these Quest Tutorials Head Office : 44C,Kalusarai New Delhi-16; Ph.(011) 46080363

79

zc 3

19. The value of

h

x − 2 + [ x ] dx is ([x] stands for greatest integer less than or equal to x)

−1

a) 7

b) 5

z

100


b

c) 4

d) 3

c) 100 π

d) 200 π

c) n/2

d) none of thes

c) 4/3

d) none of these

c) 0

d) -3

g

sin x − [ x ] π dx is

0

100 / π

a)

z

zc

nπ

21. If

200 / π

b) π

h

f (cos x )dx = k f cos2 x dx , then the value of k is 2

0

0

a) 1

b) n

z

π/2


cos x − cos3 x dx is

− π/2

a) 0

b) 2/3


z

1 + cos 2 x dx is 2

π

0

a) -2 24. Let I1=

b) 2

z

2

1

1 1+ x

2

dx and I 2 =

I1 > I2

a)

z

b)


z

1/ 2

−1/ 2

a) 0

1

1 dx . Then x

I2 > I1

cos x log b)

zc

2

1 2

z

0

a)

d) none of these

h

x 3 + x cos x + tan5 x + 1 dx is equal to

a) 0 π/2

d) I1 > 2I2

1 2

c) −

− π/2

27.

I1 = I2

FG 1 + x IJ dx is H 1− xK

π/2

26. The value of

c)

b) 2

c)

π

d) none of these

sin 2 x dx is equal to sin x + cos x

π 2

28. The value of the integral a) b - a

b)

z

b

a

d

2 log

2 +1

x dx , 0 < a < b x

i

c)

1 log 2

d

i

2 +1

d) none of these

is

b) a - b

c) b + a Quest Tutorials


d) none of these 80


z

π/ 2

sin x − cos x dx is

0

a) 0

b) 2

30. The value of the integral a)

(0,1)

31. The function f ( x ) =

z z

1

0

b) x

0

log

bg f bb − x g dx

z

b

a

2a

0

z

−2

( −1,0)

b) an odd function

c)

(1, e)

d) none of these

c) a periodic function

d) none of these

xf ( x) dx is equal to a +b 2

z

b

a

bg

f x dx

c)

b−a 2

z

b

a

bg

f x dx

d) none of these

h

px 3 + qx + s dx it is necessary to know the values of the constants: b) q

bg bg b

c) s

d) p and s.

c) a

d) none of these

f x f x + f 2a − x dx is equal to

a) 0

g

b) 2a

bg

d) 2

b

a

zc 2

a) p

z

bg

bg b g

bg b g

35. If f x and g x are continous functions satisfying f x = f a − x and g x + g a − x = 2, then equal to a)

z

a

0

g ( x ) dx

36. The value of a) 1 37. The value of a) 1

z z

b)

π/2

−π/2

2π

0

a

f ( x ) dx

a

0

bgbg

f x g x dx is

c) 0

d) none of these

b) -1

c) 0

d) none of these

b) -1

c) 99

d) 0

0

{ e

j}

cos99 x dx is

z

2

0

x[ x ] dx is

7 2

z

z

z

sin log x + x 2 + 1 dx is

38. The value of the integral a)

i

2 +1

2 2

FG 1 − x IJ dx is H1+ xK

b)

33. To find the numerical value of


d

c)

2

32. If f ( a + b − x ) = f x , then

a +b 2

)

2 −1

e x dx lies in the interval

a) an even function

a)

(

b)

3 2

c)

5 2

d) none of these

b)

π 8

c)

π 16

d) none of these

x3 dx is 39. The value of 0 1+ x 8 a)

π 4

1


81

40. If f ( x ) =

RS x, T x − 1,

for x < 1 for x ≥ 1

a) 1

b)


z

π/2

42. nlim →∞

b)

43. The value of

π

0

z

1

−1

4 3

c)

5 3

d)

5 2

5 π 32

c)

3 π 16

d) none of these

c) log e 2

d) 1

c)

1 2

d) 2

c)

−π

d) −2π

sin 3 x cos2 x dx is b) 1

2 sin x dx =

2π / 3

a)

x 2 f ( x ) dx is equal to

b) 0

a) 0

z

0

LM 1 + 1 + 1 +...+ 1 OP is equal to Nn + 1 n + 2 n + 3 n + n Q

a) log e 3

44.

2

sin 6 x dx is

0

3π 4

a)

z

, then

−5π / 3

b) n +1

45. If for every integer n,

∫

4

2 f ( x) dx = n , then the value of

∫ f ( x) dx is

−2

n

a) 16

b) 14

c) 19

d) none of these

n 1

∑ ∫ f (k − 1 + x) dx is


k =1 0

1

∫

a)

2

f ( x) dx

b)

0

0

47. The value of

∫

tan x

1/ e

t dt 1+ t

a) 1

48.

1

∫

−1

log

∫ a)

π/3 0

2

+

∫

cot x

1/ e

dt t (1 + t 2 )

c)

∫ f ( x) dx 0

d) n

∫ f ( x) dx 0

is

b) 2e

c) 2/e

d) none of these

b) 1

c) 2

d) none of these

2− x dx = 2+ x

a) 0

49.

∫ f ( x) dx

1

n

| tan x − 1 | dx =

π 1 + log 2 2 2

b)

π + log 8 6

c) log 2 −

π 3


d) none of these 82


∫

dx

π/2

π/6

a) 1 4

51.

dx

∫

{( x − 2) ( 4 − x)} a) π / 2

b) 2

c) 2(1 + 3−1/ 4 )

d) none of these

b) π

c) 0

d) none of these

=

2

1 c

52. Assuming that f is continuous everywhere,

∫

a)

b

a

is given by

3

(cos x sin x

f ( x) dx

b)

1 c

∫

b

a

∫

bc

ac

x f   dx = c

f ( x) dx

c) c

∫

b

a

f ( x) dx

d) none of these

2 n   1 + + ..... = 2 2 2 1 − n  1 − n 1 − n

53. nlim →∞  a) 0

b) - 1/2 π/ 2

54. If A =

∫

c) 1/2

π/ 2

2 sin x dx and B =

0

∫

π/ 2 2 cos x dx and C =

0

a) A − B = 0

d) none of these

∫ sin

3

x dx , then

0

b) A + B = 0

c)

A+C = 0

d) A − C = 0

π/ 2

55. If u10 =

∫x

10

sin x dx , then the value of u10 + 90u8 is

0

π  2

a) 9  36

56. If

8

π b)  

9

c) 10  π 

2

9

π  2

d) 9 

2

9

1

∫ 2 x + 9 dx = log k , then k is equal to 0

a) 3 [ x]

57. The value of

2x

∫2

[ x]

b) 9/2

c) 9

b) [x] log 2

c)

d) 81

dx is

0

[x] a) log 2

1 [ x] 2 log 2

d) none of these

ANSWERS 1. 8. 15. 22. 29. 36. 43. 50. 57.

b d a c b c a d a

2. 9. 16. 23. 30. 37. 44. 51.

a d a c c d a b

3. 10. 17. 24. 31. 38. 45. 52.

c c c b a b c a

4. 11. 18. 25. 32. 39. 46. 53.

c a b a b c c b

5. 12. 19. 26. 33. 40. 47. 54.

d a a c c c a a


6. 13. 20. 27. 34. 41. 48. 55.

a b b c c b a c

7. 14. 21. 28. 35. 42. 49. 56.

a b b a b c d a 83

AREA UNDER THE CURVENMENT 1. The area of the figure bounded by the curves y = x − 1 and y = 3 − x a) 2

b) 3

c)

is 4

2. Area lying in the first quadrant and bounded by the circle x 2 + y 2 = 4 the line x = a)

π

b)

π 2

c)

d) 1

3 y and x-axis is

π 3

3. If A is the area lying between the curve y = sin x and x-axis between x = 0 and x =

d) none of these

π / 2 . Area of the region between the curve

y = sin 2 x and x- axis in the same interval is given by a) A/2

b) A

c)

2A

d) none of these

c)

1

d) 7/3

c)

2

d) none of these

c)

7/8

d) 9/8

c)

1 ab 4

d)

4. The area bounded by y = x 2 , y = x + 1 , x ≤ 1 and the y-axis is a) 1/3

b) 2/3

5. The area enclosed within the curve x + y = 1 is a) 1

b) 1.5

6. The area bounded by the curve x a) 3/8

2

= 4 y and the straight line b) 5/8

x = 4 y − 2 is

x y x2 y2 7. The area between 2 + 2 = 1 and the straight line + = 1 is a b a b a)

1 ab 2

b)

1 π ab 2

1 1 π ab − ab 4 2

8. The area cut off a parabola by any double ordinate is k times the corresponding rectangle contained by that double ordinate and its distance from the vertex. The value of k is a) 2/3 b) 3/2 c) 1/3 d) 3 9. Area bounded by the curve y = x sin x and x-axis between x = 0 and a) 2π b) 3π 10. The area of the region bounded by y = x − 1 and a) 1

x = 2 π is c) 4π

d) none of these

c)

d) none of these

y = 1 is

b) 2

11. The area bounded by the curve y = x x , x − axis and the ordinates a) 0

b) 1/3

b

a) 2 π − 2

g

b)

x = 1, x = −1 is given by c)

12. The smaller area enclosed by the circle x 2 + y 2 = 4 and the line

π−2

1/2

2/3

d) none of these

x + y = 2 is equal to c)

2π − 1

13. Area lying in the first quadrant and bounded by the curve y = x 3 and the line y = 4 x is a) 2 b) 3 c) 4


d) none of these

d) 5

84

14. In the interval 0, π / 2 , area lying between the curves a) log 2

b)

y = tan x , y = cot x and x- axis is

1 log 2 2

c)

2 log

FG 1 IJ H 2K

d) none of these

b gb gb x − 3g and x-axis lying between the ordinates x = 0 and x = 3 is equal to

15. Area bounded by the curve y = x − 1 x − 2 a) 9/4

b) 11/4

16. The area bounded by the curves y = e x , y = e − x and

b

a) log 16 / e

g

b)

c)

11/2

d) none of these

c)

2 ( 2e2 - 1 )

d) none of these

=2

d) none of these

3

d) 6

y = 2 is

b g

log 4 / e

17. The area of the figure bounded by y = e x −1 , y = 0, x = 0 and x = 2 is a) < 2 b) > 2 c) 18. Area bounded by the curves y = x − 1 , y = 0 and x = 2 is a) 4

b) 5

c)

19. The ratio of the areas between the curves y = cos x and y = cos 2 x and x-axis from x = 0 to x = π / 3 is a) 1 : 2

b) 2 : 1

c)

d) none of these

3 :1

20. For 0 ≤ x ≤ π , the area bounded by y = x and y = x + sin x , is a) 2

b) 4

c)

4π

d)

2π

21. Area of the region bounded by the parabola ( y − 2) 2 = x − 1 , the tangent to it at the point with the ordinate 3, and the x-axis is given by a) 9/2 sq. units b) 9 sq. units c) 18 sq. units d) none of these

ANSWERS 1. 8. 15.

c a b

2. 9. 16.

c c b

3. 10. 17.

b a b

4. 11. 18.

b c b

5. 12. 19.

c b c


6. 13. 20.

d c a

7. 14. 21.

d a b

85

DIFFERENTIAL EQUATIONS 1. The general solution of the differential equation a) x 3 − y 3 = C

b)

dy x 2 = IS dx y 2

x3 + y3 = C

c)

x2 + y2 = C

d)

x2 − y2 = C

2. The order of the differential equation of all circles of radius r , having centre on y -axis and ing through the origin is a) 1 b) 2 c) 3 d) 4 3. The order of d the differential equation whose solution is y = a cos x + b sin x + Ce a) 3 b) 2 c) 1

−x

is d) none of these

4. The slope of a curve at any point is the reciprocal of twice the ordinate at the point and it es through the point ( 4 , 3 ). The equation of the curve is a) x 2 = y + 5

b)

y2 = x − 5

c)

y2 = x + 5

d)

x2 = y − 5

5. The curve in which the slope of the tangent at any point equals the ratio of the abscissa to the ordinate of the point is a) an ellipse b) a parabola c) a rectangular hyperbola d) a circle. 6. The equation of the curve whose subnormal is constant is a) y = ax + b b) y2 = 2ax + b

c)

ay2 - x2 = a

d) none of these

7. A differential equation associated to the primitive y = a + be5 x + c e −7 x is a)

y3 + 2 y2 − y1 = 0

8. The solution of a)

b)

4 y3 + 5 y2 − 20 y1 = 0

y3 + 2 y2 − 35 y1 = 0 d) none of these

c)

a = 0, b ≠ 0

dy ax + h represents a parabola when = dx by + k

a = 0, b = 0

b)

a = 1, b = 2

9. The solution of the differential equation y ≠ a)

c)

y2 = x2 − 2x + 2

b)

bg

y2 = 2x2 − x − 1

FG IJ H K

2

= x log

c)

y = x2 − 2x + 2

d) none of these

c)

3

d) none of these

d2y is dx 2

b) 2

11. The order of the differential equation of all tangent lines to the parabola y = x 2 is a) 1 d) 2 c) 3

b

g

a = 2, b = 1

dy = x − 1 satisfying y 1 = 1 is dx

d2y dy + 10. The degree of the differential equation 2 dx dx a) 1

d)

d) 4

bg

12. The particular solution of log dy / dx = 3x + 4 y , y 0 = 0 is a) e 3 x + 3e −4 y = 4

b)

4e 3 x − 3e −4 y = 3

c)

3e 3 x + 4e +4 y = 7


d)

4e 3 x + 3e −4 y = 7 86

bg 2x f b xg = 2− x

b

g

13. If y = f x ing through (1,2) satisfies the differential equation y 1 + xy dx − x dy = 0 , then a)

b)

2

bg

f x =

x +1 x2 + 1

c)

bg

f x =

x −1 4 − x2

d)

bg

f x =

4x 1− 2x2

14. The equation of the curve satisfying the differential equation y2 ( x 2 + 1) = 2 xy1 ing through the point ( 0, 1 ) and having slope of tangent at x = 0 as 3 is a)

y = x 2 + 3x + 2

b)

y 2 = x 2 + 3x + 1

b) 1 and 2

( a + x)

16. Solution of the differential equation : a)

y = Ae

( 2 / 3) ( 2 a − x ) ( a + x )

c)

y = Ae

( 2 / 3) ( 2 a + x ) ( a + x )

17. The integrating factor of x a) x

y = x 3 + 3x + 1

d) none of these

c)

1 and 4

d) 2 and 4

b)

y = Ae

− ( 2 / 3) ( a − x ) ( a + x )

d)

y = Ae

− ( 2 / 3) ( 2 a − x ) ( a + x )

c)

−x

c)

cos x. tan y = c

1/ 4

  dy  2  = 15. Order and degree of differential equation y +    dx 2   dx   d2y

a) 4 and 2

c)

are

dy + xy = 0 is dx

dy − 2 y = x 2 + sin(1 / x 2 ) is dx

2

b) 1 / x 2

2

d) none of these

18. The solution of the differential equation dy − sin x. sin y dx = 0 , is a) e cos x tan y / 2 = c

b)

e cos x tan y = c

d) cos x. sin y = c

19. Equation of the curve ing through (3, 9) which satisfies the differential equation dy / dx = x + (1 / x 2 ) is a) 6 xy = 3 x 2 − 6 x + 29

b)

2

6 xy = 3 x − 29 x + 6

c)

3

6 xy = 3 x + 29 x − 6

d) none of these

20. Solution of the differential equation 2 y sin x dy / dx = 2 sin x cos x − y 2 cos x and at x = π / 2, y = 1 , is given by 2

y = sin x

2

y = cos x + 1

c)

y + 2 xy + x = 0

a) ( a + m) y = e mx + c

b)

ye ax = me mx + c

y = e mx + ce−ax

d)

b)

y = sin x

2

c)

a)

d) none of these

21. The solution of differential equation ( x + y ) dy − ( x − y ) dx = 0 a)

2

2

y + 2 xy + x = c

b)

22. The solution of differential equation

c)

2

2

y + 2 xy − x = c

2

2

d)

2

2

y − 2 xy + x = c

dy + ay = e mx is dx ( a + m) y = e mx + ce − ax


87

23. The solution of the differential equation 2 x a) Straight line

dy − y = 3 represent dx

b) circles

dy 1 + cos 2 y + = 0 is dx 1 − cos 2 x a) tan y + cot x = c b) tan y − cot x = c

c)

parabolas

d) ellipses

c)

cot y + tan x = c

d) none of these

b)

y +1 1 x −1 1 +c = log log y−2 4 x+3 3

d)

none of these

24. Solution of the equation

2

25. The solution of the equation

a)

dy ( y − y − 2) = is dx x 2 + 2 x − 3

y−2 1 x+3 1 +c = log log y +1 4 x −1 3

c) 4 log

y−2 x −1 = 3 log +c y +1 x+3

ANSWERS 1. 8. 15. 22.

a c d d

2. 9. 16. 23.

a a a c

3. 10. 17. 24.

a d b b

4. 11. 18. 25.

c a a c

5. 12. 19.

c d c


6. 13. 20.

b a a

7. 14. 21.

c c b

88

DETERMINANTS

x2 − 2x + 3 7x + 2 x+4 2 2x + 7 x − x+2 3x 1. If = ax 6 + bx 5 + cx 4 + dx 3 + ex 2 + fx + g the value of g is 3 2x − 1 x2 − 4x + 7 a) 2

b) 1

c)

b

sin A + B + C 2. If A + B + C = π then value of ∆ =

− sin B

b

cos A + B a)

tan 2 A

b

sin B

tan A + B + C

g

x k +3 y k +3 z k +3 b)

k = −3

b

cos2 C

gb gb gFGH 1x + 1y + 1z IJK , then c)

k = −1

γ b)

x +1 x + 2 5. If ∆ = x + 2

x+3 x+3 x+4

2p

x +1 ω 6. If ω ≠ 1 is a cube root of unity and ω x + ω2 ω2 1

b g

a) x = 1

b)

7. If U n = n

n3 N

a) 2

∑n n =1

1

d) k = 3

γ

γ α is equal to α β c)

3q

d) none of these

c)

H.P

d) none of these

x + b = 0 , then a, b, c are in x+c b) G.P

2

k =1

x+a

a) A.P

n

d) none of these

= x− y y−z z−x

4. If α, β, γ are the roots of the equation x 3 + px + q = 0 , then ∆ = β

− pq

tan A is

c)

α β

a)

cos C 0

2

x k +2 y k +2 z k +2

g

d) none of these

− tan A

b) sin B

xk 3. If x , y , z are distinct and y k zk a)

g

-2

5

ω2 1 = 0, then x+ω

x=ω

c)

x=ω

c)

1 N 2 ∑n 2 n =1

2

d) none of these

N

2 N + 1 2 N + 1 , then ∑ U n is equal to n =1 3N 2 3N + 1 N

b) 2

∑n

2

n =1


d) 0 89

0 −b − c 8. If a, b, c are complex number, then z = b 0 − a is equal to c a) 0

a

0

b) a b c

1/ a a2 b2 1 / c c2

∆ is independent of a, b, c

3abc

d) 4 a b c

c)

a 2b 2 c 2

d) none of these

bc

9. If a, b, c are non-zero real numbers and ∆ = 1 / b

a)

c)

ca , then ab

b) a b c

10. The system of equations x + y + z = 4, x + ky + z = 3, x + 2 ky + z = 4 is consistent if a)

k =1

k =2

b)

11. If a > 0, b > 0, c > 0, are respectively the p , q , r th

a) 1

th

th

1 12. The value of the determinant

m

C1 C2

a) 1

1 m +1 m +1

C1 C2

d) none of these

of a GP, then the value of determinant

b) 0

m

k = −2

c)

log a p 1 logb q 1 is log c r 1

c)

-1

d) none of these

c)

0

d) none of these

1 m+ 2 m+ 2

C1 is equal to C2

b) -1

13. If A is a square matrix of order n such that its elements are polynomial in x and its r - rows become identical for x = k, then a) c)

bx − k g bx − k g

r

is a factor of A

r +1

is a factor of A

d)

4 14. If x, y, z are in AP then the value of the determinant A is, where A =

a) 0

bx − k g bx − k g

b)

b) 1

r −1

r

5 6

x

5 6 7

y

6 7 8

z

x

0

c)

y

z

is a factor of A

is a factor of A

2

d) none of these.

1 ω 3 ω5 3 15. The value of the determinant ω 1 ω4 , where ω is an imaginary cube root of unity, is ω5 ω 4 1 a)

b1 − ω g

2

b) 3

c)

-3


d) none of these 90

x b b x b , then 16. If ∆1 = a x b and ∆ 2 = a x a a x a)

b g

∆1 = 3 ∆ 2

2

b)

b g

d ∆ 1 = 3∆ 2 dx

b g

c)

d ∆ 1 = 3∆22 dx

b g

d) ∆ 1 = 3 ∆ 2

3/ 2

x a b 17. The factors of a

x b are 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

c)

ω2

1 ω ω2 18. If ω is a cube root of unity, then ω ω 2 1 = 2 ω 1 ω a) 1

b)

ω

d) 0

19. Let a, b, c be positive real numbers. The following system of equations in x, y and z

x2 y2 z2 x2 y2 z2 x2 y2 z2 + − = 1 , − + = 1 , − + + = 1 has a 2 b2 c2 a 2 b2 c2 a 2 b2 c2 a) no solution

b) unique solution

c)

infinitely many solutions

d) finitely many solutions.

20. If every element of a third order determinant of value ∆ is multiplied by 5, then the value of new determinant is a)

∆

b) 5∆

c)

25∆

d) 125∆

c)

x =ω2

d) x = 0

c)

1

d) 0

21. If ω is a cube root of unity, then a root of the following equation

x +1

ω ω2 a)

ω x +ω2

ω2

1

x +ω

1

x =1

1 22. 4 42 a) 2

1 3 32

= 0 is

b)

x =ω

1 2 equals 22 b) -2 Quest Tutorials Head Office : 44C,Kalusarai New Delhi-16; Ph.(011) 46080363

91

1

2

3

1

6

3

23. If ∆ = 2

− 1 0 , then 4 − 6 0 is equal to 3 4 5 3 12 5

a)

b)

∆

2∆

sec x sin x 24. Value of the determinant ∆ = 0 1

c)

6∆

d) none of these

c)

1

d) none of these

tan x 0 is given by

tan x cot x sec x a) 0

b) - 1

25. If f (x) is a polynomial satisfying f ( x ) = a) 624

b) - 124

x+ y

x 26. If 2 x

f (1 / x ) − f ( x ) and f (2) = 17 , then the value of f (5) is f (1 / x )

1 f ( x) 2 1

c)

626

d) 126

c)

4

d) 6

x+ y+z

3x + 2 y

4 x + 3 y + 2 z = 64 , then the real value of x is

3 x 6 x + 3 y 10 x + 6 y + 3 z a) 2

b) 3

1 27. The parameter, on which the value of the determinant cos( p − d ) x

sin( p − d ) x a) a

b) p

28. For the equations : x + 2 y + 3z = 1, a) there is only one solution c) there is no solution

a)

b 3 − 1 = 0 then c3 − 1

30. The value of a) 0

b)

c

c

a b a+b

d

d) x

b) there exists infinitely many solution d) none of these.

b2 c c2

b+c a b c+a

2

2 x + y + 3z = 2, 5x + 5 y + 9 z = 4,

a3 − 1

a+b+c =0

a

cos px cos( p + d ) x does not depend upon, is sin px sin( p + d ) x c)

a a2 29. If a,b,c, are different and b

a

abc = 1

c) a + b + c = 1

d) ab + bc + ca = 0

a+b+c

c) 4abc

d) abc

is b)


92

ANSWERS 1. 8. 15. 22. 29.

d a b b b

2. 9. 16. 23. 30.

d a b c c

3. 10. 17. 24.

b d a c

4. 11. 18. 25.

d b d c

5. 12. 19. 26.

a a b c


6. 13. 20. 27.

d b d b

7. 14. 21. 28.

b a d a

93

MATRICES 1. If A is a square matrix, then A + A' is a) symmetric matrix

b) skew symmetric matrix

2. If A = [ 2 -3 5 ], then A' A is a matrix of order a) 1 × 1 b) 3 × 3 3. If A =

LMcos θ N sin θ

OP, A cos θ Q

− sin θ

−1

c) scalar matrix

d) diagonal matrix

c) 1 × 3

d) 3 × 1

c) - A'

d) A

c) AB = 0

d) BA = 0

c) B 3

d) A + B.

is given by

a) - A

b) A'

4. If A and B are square matrices, then ( A + B ) 2 = A 2 + 2 AB + B 2 ifff a) A 2 = 0 or B 2 = 0

5.

LM3 If A = 0 MM N0

b) AB = BA

OP PP Q

LM MM N

0 0 a1 a2 3 0 and B = b1 b2 0 3 c1 c2

a) B

3

b) 3B

LM i 0OP then A is equal to N0 i Q L1 0 OP L−1 0 OP b) M a) M N0 −1Q N 0 −1Q L cos θ sin θ OP, then A = I if If A = M N− sin θ cos θQ

c)

a) θ = 0

c) θ = π / 2

6. If A =

7.

OP P c PQ

a3 b3 then AB is equal to

2

LM1 0OP N0 1Q

d)

LM−1 0OP N 0 1Q

2

b)

θ = π/4

FG 1 aIJ , then A (where n ∈N) equals H 0 1K F 1 n aI FG 1 naIJ b) G H0 1 K H 0 1 JK

8. If A =

n

2

a)

d) none of these.

9. A matrix A = [ aij ] is an upper triangular matrix if a) it is a square matrix and aij = 0, i < j c) it is not a square matrix and aij = 0, i > j

c)

FG 1 naIJ H0 0 K

d)

FG n naIJ H0 n K

b) it is a square matrix and aij = 0, i > j d) it is not a square matrix and aij = 0, i < j

10. If A is any m × n matrix such that AB and BA are both defined, then B is an a) m × n matrix b) n × m matrix c) n × n matrix

d) m × m matrix

11. If A is a square matrix of order n × n and k is a scalar, then adj ( kA ) is equal to a) k adj A b) kn adj A c) k n - 1 adj A

d) kn + 1 adj A


94

12. If A = [ aij ] is a square matrix of order n × n and k is a scalar, then | kA | = a)

kn A

b)

c) k n−1 A

k A

d) none of these

13. If A and B are matrices such that AB and A + B both are defined, then a) A and B can be any two matrices b) A and B are square matrices not necessarily of the same order c) A, B are square matrices of the same order d) number of columns of A is same as the number of rows of B

LM5 2OP, then A N3 1 Q LM 1 −2OP N −3 5 Q

14. If A =

a)

−1

=

LM−1 2 OP N 3 −5Q 0O −1 1 P1Q and A - 2B = LMN 0 −1OPQ , then A = L2 / 3 1 / 3OP b) M N1 / 3 2 / 3Q b)

LM1 N1 L1 1OP a) M N2 1Q L3 −1OP LM x OP = LM4 OP 16. M N2 5 Q N y Q N−3Q 15. If A + B =

a) x = 3, y = - 1

b) x = 2, y = 5

LM−1 −2OP N − 3 − 5Q

d)

LM1 2OP N3 5 Q

c) none of these

d)

LM1 / 3 N2 / 3

c) x = 1, y = - 1

d) x = - 1, y = 1

c)

OP 1 / 3Q 1/ 3

2

17. With 1, ω,ω as cube roots of unity, inverse of which of the following matrices exists ? a)

LM 1 Nω

OP ω Q ω

2

b)

2

18. Matrix theory was introduced by a) Cauchy - Riemann

LMa 0 0OP 19. If A = 0 b 0 then A MM P N0 0 Q LMa 0 0OP a) MM00 b0 0P N Q

LMω 1 OP N 1 ωQ

−1

c)

b) Caley - Hamilton

LM ω Nω

2

ω2 1

OP Q

d) none of these

c) Newton

d) Cacuchy - Schwarz

=

LMa MM 0 N0

2

b)

0 ab 0

OP 0P aQ 0

c)

LM1 / a MM 00 N

0 1/ b 0


OP 0 P d) 1 / Q 0

LM−a MM 00 N

0 −b 0

OP 0P − Q 0

95

LM−1 20. If A = 2 MM N2

OP P 1 PQ

−2 −2 1 −2 then adj A = −2

a) A

b) AT

21. If In is the identity matrix of order n, then ( In ) -1 = a) does not exist b) In

c) 3 A

d) 3 AT

c) 0

d) n In

22. If A and B are square matrices of order 3 such that | A | = −1, | B | = 3 , then | 3AB| = a) -9 b) -81 c) -27

d) 81

cos α − sin α 0 23. Let F (α) =  sin α cos α 0 , then F (α) F (α' ) is equal to    0 0 1 a)

F (α α ' )

b)

α F   α' 

c) F (α + α' )

d) F (α − α' )

c) 3

d) - 3

1 0 2 1   24. The value of x for which [1 1 x] 0 2 1 1 = 0 is   2 1 0 1 a) 2

b) - 2

a b  α β  2 and A =    , then b a  β α

25. If A = 

a) α = a 2 − b 2 , β = 2ab c)

2

2

2

α = a +b , β = a −b

b) α = a 2 + b 2 , β = 2ab 2

d) α = 2ab, β = a 2 + b 2

1 2  1 0  and B =    and X be a matrix such that A = BX . Then X = 3 − 5 0 2 

26. Let A = 

a)

2 4   3 − 5  

b)

1 2 4  .  2  3 − 5

c)

1  − 2 4 .  2  3 5


1 1 n  and n∈ N , then A is equal to 1 1  

27. If A =  a)

2n A

b)

2 n −1 A

c) n A


d) none of these

96

a b   such that ad − bc ≠ 0 , then A −1 is c d  

28. If A = 

a)

1  d b   ad − bc  − c a 

b)

 d − b   − c a 

c)

1  d − b   ad − bc  − c a 

d)

none of these

ANSWERS 1. 8. 15. 22.

a a d b

2. 9. 16. 23.

b b c c

3. 10. 17. 24.

b b d b

4. 11. 18. 25.

b c b b

5. 12. 19. 26.

b a c b


6. 13. 20. 27.

b c d b

7. 14. 21. 28.

97

a b b c

VECTORS 1. If AO + OB = BO + OC then A, B, C are a) coplanar

G

G

b) collinear

c) Non-collinear

d) none of these.

G

G

2. If | a + b | =| a − b |, then

G

G

G

a) a is parallel to b

G

c) | a | = | b |

G

G

G

G

b) a ⊥ b

G

d) none of these.

G

3. If a and b are two unit vectors inclined at an angle θ such that a + b is a unit vector, then θ is equal to

π 3

a)

b)

π 4

c)

π 2

d)

2π . 3

9 19

d)

6 19

4. The pojection of the vector i − 2 j + k on the vector 4i − 4 j + 7 k is

5 6 10

a)

b)

G G

G

G

19 9

c)

G

G

G

5. If a and b are two vectors such that a . b = 0 and a × b = 0 , then

G

G

G

G

G

b) a ⊥ b

a) a || b

G

c) either a or b is a null vector

d) none of these.

G

G 6. The number of vectors of unit length pependicular to vectors a = i + j and b = j + k is a) one

b) two

c) three

d) none of these.

7. The unit vector perpendicular to vector i − j and i + j forming a right handed system is b) − k

a) k

c)

1 (i − j ) 2

d)

1 (i + j ) 2

8. If the constant forces 2 i − 5 j + 6 k and − i + 2 j − k act on a particle due to which it is displaced from a point A ( 4, -3, -2 ) to a point B ( 6, 1, -3 ), then the work done by the forces is a) 15 units

b) -15 units

c) 9 units

d) -9 units.

G

9. The work done by the force F = 2 i − j − k in moving an object along the vector 3 i + 2 j − 5k is a) -9 units

b) 15 units

c) 9 units

d) none of these.

10. If forces of magnitudes 6 and 7 units acting in the directions i − 2 j + 2 k and 2 i − 3 j − 6k respectively act on a particle which is displaced from the point P ( 2, -1, -3 ) to Q ( 5, -1, 1), then the work done by the forces is a) 4 units b) -4 units c) 7 units d) -7 units.


98

11. If n1 , n2 are two unit vectors and θ is the angle between them, then cos θ / 2 = a)

1 | n1 + n2 | 2

b)

1 | n1 − n2 | 2

c)

1 | n1 . n2 | 2

d)

| n1 × n2 | 2 | n1 | | n2 |

G

12. The projection of the vector a = 4i − 3 j + 2 k on the axis making equal acute angles with the coodinate axes is a) 3

G G

3

b)

G

G

G

d) none of these.

3

G

13. a .(b + c ) × ( a + b + c ) = a) 0

1

c)

G G G

G G G

b) 2[ a b c ]

c) [ a b c ]

d) none of these.

14. A unit vector in xy-plane making an angle of 45° with the vector i + j and an angle of 60° with the vector 3i − 4 j is

i + j 2

i − j d) none of these. 2 G G G G G G 15. If the unit vectors a = xi + y j + z k and b = j are such that a , c and b form a right handed system, then c is G a) z i − x k b) 0 c) y j d) − z i + x k . a)

i

b)

c)

16. If C is the middle point of AB and P is any point outside AB, then a)

PA + PB = PC

b)

PA + PB = 2PC

c)

G PA+ PB + PC = O

d)

G PA + PB + 2PC = O

17. The value of b such that the scalar product of the vector i + j + k with the unit vector parallel to the sum of the vectors 2i + 4 j − 5k and bi + 2 j + 3k is one, is a) -2 b) -1

c) 0

d) 1

18. Volume of the parallelopiped whose coterminal edges are 2i − 3 j + 4 k , i + 2 j − 2 k , 3i − j + k , is a) 5 units b) 6 units c) 7 units d) 8 units

19. i .( j × k ) + j .( k × i ) + k .(i × j ) = a) 1 b) 3

G G

c) -3

G

d) 0

G

G

G

G

G

G

20. Each of the angle between vectors a , b and c is equal to 60°. If | a | = 4,| b | = 2 and | c | = 6, then the modulus of a + b + c is a) 10 b) 15 c) 12 d) none of these 21. If a = i + 2 j + 3k , b = −i + 2 j + k and c = 3i + j , then t such that a + t b is at right angle to c will be equal to a) 5 b) 4 c) 6 d) 2 22. If | a | = 2, | b | = 5 and | a × b | = 8 then a . b can be equal to a) 4 b) 6

c) 5

d) none of these.

23. If a . b = b . c = c . a = 0, then [ a, b, c ] is equal to a) 0 b) 1

c) -1

d) | a | | b | |c |.


99

24. Let a, b and c be three non-coplanar vectors, and let p, q and r be the vectors defined by the relations

b×c c×a , q= [ a b c] [abc]

p=

and r =

a ×b [abc]

Then the value of the expression ( a + b ) . p + (b + c ) . q + ( c + a ) . r is equal to a) 0

b) 1

c) 2

d) 3

25. The volume of the parallelopiped whose sides are given by OA = 2i − 3 j , OB = i + j − k , OC = 3i − k is a) 4 / 13

b) 4

c) 2 / 7

d) none of these.

26. Let a, b, c be distinct non negative numbers. If the vectos ai + aj + ck , i + k and ci + cj + bk lie in a plane, then c is a) the arithmetic mean of a and b c) the harmonic mean of a and b

b) d)

the geometric mean of a and b equal to zero

27. Given a = i + j − k , b = −i + 2 j + k and c = −i + 2 j − k . A unit vector perpendicular to both a + b and b + c is a)

2 i + j + k 6

b) j

c) k

28. A unit tangent vector at t = 2 on the curve x = t 2 + 2,

1 (i + j + k ) 3

a)

b)

y = 4t - 5,

d)

i + j + k 3

z = 2t 2 - 6t is

1 ( 2i + 2 j + k) 3

c)

1 (2i + j + k ) 6

d) none of these.

1 2 1 3 t , z = t . The acceleation at t = 1 is 2 3 c) 2 j + k d) none of these.

29. A particle moves along a curve so that its coordinates at time t are x = t, y = a)

j + 2 k

b) j + k

30. If a and b are two unit vectors and φ is the angle between them, then b) π / 2

a) 0

1 | a − b | is equal to 2 c) | sin φ / 2 |

d) | cos φ / 2 |

c) 0

d) 3a

c) [ a b c ]2

d) 0

31. The value of i × (a × i ) + j × ( a × j ) + k × ( a × k ) is a) a

b) 2a

32. The value of [ a × b, b × c, c × a ] is a) 2 [ a b c ]

b) [ a b c ]

33. If a = xi + 5 j + 7 k , b = i + j − k , c = i + 2 j + 2 k are coplanar then the value of x is a) 1

b) -2

c) -1

d) none of these.

G G 34. If a and b are position vector of A and B respectively, then the position vector of a point C in AB produced such that AC = 3 AB is G G G G G G G G b) 3 b − a c) 3 a − 2b d) 3 b − 2a a) 3 a − b

ANSWERS 1. 8. 15. 22. 29.

b b a b a

2. 9. 16. 23. 30.

b c b d c

3. 10. 17. 24. 31.

d a d d b

4. 11. 18. 25. 32.

b a c b c

5. 12. 19. 26. 33.

c b b b d


6. 13. 20. 27. 34.

b a a c d

7. 14. 21. 28.

a d a b

100

THREE DIMENSIONAL GEOMETRY 1. The distance of the point P ( a, b, c ) from the x-axis is

b2 + c2

a)

a 2 + c2

b)

a 2 + b2

c)

d) none of these.

2. If α, β, γ are the angles which a directed line makes with the positive directions of the coordinate axes, then sin2 α + sin2 β + sin2 γ is equal to a) 1 b) 2 c) 3 d) none of these. 3. Equation of the line ing thro’ (1,1,1) and perpendicular to 2 x + 3 y + z + 5 = 0 is a)

x −1 y −1 z −1 = = −1 1 1

b)

4. The value of λ for which the lines a) 0

x −1 y −1 z −1 = = 2 3 1

c)

x −1 y −1 z −1 = = 3 3 1

d)

x −1 y −1 z −1 = = 1 3 2

x −1 y − 2 z +1 x +1 y +1 z − 2 = = = = and are perpendicular to each other is 2 1 1 λ −1 −λ b) 1

5. The number of spheres of radius r touching the coordinate axes is a) 4 b) 6

c) -1

d) none of these.

c) 8

d) none of these.

6. The equation of the sphere which es through the points ( 1, 0, 0 ), ( 0, 1, 0 ), ( 0, 0, 1 ) and having radius as small possible, is a) 3 ( x 2 + y 2 + z 2 ) − 2 ( x + y + z ) − 1 = 0 c)

3 ( x 2 + y 2 + z 2 ) − 2 ( x + y + z) + 1 = 0

7. The angle between the lines a)

π/2

b)

x2 + y2 + z2 − x − y − z − 1 = 0

d) none of these.

x −1 2y + 3 z + 5 x − 2 y +1 = = is = , z = 2 and 1 3 2 3 −2 b) π / 3

c)

π/6

d) none of these.

8. The DCs of the line 6x - 2 = 3y + 1 = 2z - 2 are a)

1 1 1 , , 3 3 3

b)

1 2 3 , , 14 14 14

c) 1, 2, 3

d) none of these.

9. A line es through two points A ( 2, -3, -1 ) and B ( 8, -1, 2 ). The coordinates of a point on this line at a distance of 14 units from A are a) ( 14, 1, 5 )

b) ( -10, -7, -7 )

c) ( 86, 25, 41 )

d) none of these.

10. The position vector of the point in which the line ing the points i − 2 j + k and 3i − 2 j cuts the plane through the origin and the points 4 j and 2i + k , is a) 6i − 10 j + 3k

b)

3ˆ 3 i − 2 ˆj + kˆ 2 4

c)

−6i + 10 j − 3k

11. The coordinate of the middle point of the line ing the points ( -1, -1, 1 ) and ( -1, 1, -1 ) are a) ( 0, 0, 0 ) b) ( -1, 0, 0 ) c) ( 0, -1, 1 )


d) none of these.

d) ( 0, 1, -1 ).

101

12. The direction cosines of the line ing the points ( 1, 2, -3 ) and ( -2, 3, 1 ) are a) -3, 1, 4

−3 1 4 , , 26 26 26

−1 5 −2 , , 30 30 30

b) -1, 5, -2

c)

b) y = 0

c) z = 0

d) z = c, c ≠ 0

b) z = 0, y = 0

c) x = 0, y = 0

d) x = k, y = -k., ( k ≠ 0)

d)

13. The equation of the XOY plane is a) x = 0 14. The equation of z-axis is a) z = 0, x = 0

15. The ratio in which the yz plane divides the line ing the points ( -2, 4, 7 ) and ( 3, -5, 8 ) is a) 2 : 3 b) 3 : 2 c) 4 : 5

d) -7 : 8

16. The coordinates of the point equidistant from the points ( a, 0, 0 ), ( 0, a, 0 ), ( 0, 0, a ) and ( 0, 0, 0 ) are a) ( a/3, a/3, a/3 ) b) ( a/2, a/2, a/2 ) c) ( a, a, a ) d) ( 2a, 2a, 2a, ) 17. The coordinate of the foot of the perpendicular from the point ( a, b, c ) on z-axis is a) ( a, 0, 0 ) b) ( 0, b, 0 ) c) ( 0, 0, c )

d) ( a, b, 0 )

18. l = m = n = 1 represents the direction cosines of a) x-axis b) y-axis

d) none of these.

c) z-axis

19. The direction cosines of a line equally inclined with the co-ordinate axes are a)

1 1 1 , , 2 2 2

b)

1 1 1 , , 2 2 2

c)

1 1 1 , , 3 3 3

d)

3 3 3 , , 2 2 2

20. A line makes an angle of 60° with each of x and y axis, the angle which it makes with z axis is a) 30° b) 45° c) 60° d) none of these. 21. If l1 , m1 , n1 and l2 , m2 , n2 be direction cosines of two perpendicular lines, then a) l1 l2 + m1 m2 + n1 n2 = 0 b) l1 l2 + m1 m2 + n1 n2 = 1 c)

l1 m1 n1 + + =1 l2 m2 n2

22. Algebraic sum of the intercepts made by the plane x + 3y - 4z + 6 = 0 on the axes is a) - 13/2 b) 19 / 2 c) - 22/3

d)

l1 m1 n1 + + =0 l2 m2 n2

d) 26/3

23. The equation of the plane ing through the point ( 1, -1, 2 ) and parallel to the plane 3x + 4y - 5z = 0 is a)

3x + 4 y − 5z + 11 = 0

b)

3x + 4 y − 5z = 11

24. The reflection of the point A ( 1, 0, 0 ) in the line a) ( 3, -4, -2 ) 25. The centre of the sphere a) ( -1, 2, 3 )

c)

6 x + 8 y − 10z = 1

d)

3x + 4 y − 5z = 2

x − 1 y + 1 z + 10 is = = 2 8 −3

b) ( 5, -8, -4 )

c) ( 1, -1, -10 )

d) ( 2, -3, 8 )

( x + 1) ( x − 1) + ( y − 2) ( y + 2) + ( z − 3) ( z + 3) = 0 is b) ( 1, -2, -3 )

c) ( 0, 0, 0 )


d) ( 1, 2, 3 )

102

26. A sphere of constant radius k es through the origin and meets coordinate axes in A, B, C. The centroid of the triangle ABC lies on a)

x2 + y2 + z2 = k 2

b) 4 ( x 2 + y 2 + z 2 ) = 9 k 2

c)

2 ( x2 + y2 + z2 ) = 3 k 2

d) 9 ( x 2 + y 2 + z 2 ) = 4 k 2

27. The ratio in which the plane

(

)

G G G G G G G G G G r . i − 2 j + 3k = 17 divides the line ing the points − 2i + 4 j + 7 k and 3i − 5 j + 8k is

a) 1 : 5

b) 1: 10

c) 3 : 5

d) 3 : 10

28. The direction ratios of the diagonals of a cube which s the origin to the opposite corner are (when the 3 concurrent edges of the cube are co-ordinate axes a)

−2 2 2 , , 3 3 3

b) 1,1,1

c) 2, -2, 1

d) 1, 2, 3

ANSWERS 1. 8. 15. 22.

a b a a

2. 9. 16. 23.

b a, b b a

3. 10. 17. 24.

b b c b

4. 11. 18. 25.

b b d c

5. 12. 19. 26.

c c c d


6. 13. 20. 27.

a c b d

7. 14. 21. 28.

103

a c a b

PROBABILITY 1. Three of the six vertices of a regular hexagon are chosen at random. The probability that the triangle with these vertices is equilateral, equals a) 1/2 b) 1/5 c) 1/10 d) 1/20 2. The probability of India winning a test match against West Indies is 1/2. Assuming independence from match to match the probability that in a 5 match series, India's second win occurs at the third test is a) 1/8 b) 1/4 c) 1/2 d) 2/3 3. If two events A and B are such that a) 3/4

P( A') = 0.3, P( B) = 0.4 and P( A ∩ B') = 0.5, then P ( B / A ∪ B ' ) equals

b) 5/6

c) 1/4

d) 3/7

4. If the probabilities that A and B will die within a year are x and y respectively then the probability that exactly one of them will be alive at the end of the year is a)

x + y + 2 xy

b)

x + y + xy

c)

x + y − xy

5. The probability of three persons having the same date and month for the birthday is c) 1/(365)3 a) 1/365 b) 1/(365)2

d)

x + y − 2 xy

d) none of these

6. A speaks truth in 70 percent cases and B speaks the truth in 80 percent cases. The probability that they will say the same thing while describing a single event is a) 0.56 b) 0.62 c) 0.38 d) 0.94 7. One mapping is selected at random from all the mappings from the set S = {1,2,3,..,n} into itself. The probability that the selected mapping is oneto-one is a)

l / nn

b)

l / n!

c)

bn − 1g!/ n

n −1

8. Two non-negative integers are chosen at random. The probability that their product is divisible by 5 is a) 7/25 b) 8/25 c) 9/25

d) none of these

d) 2/5

9. A man is known to speak the truth 3 out of 4 times. He throws a die and reports that it is a six. The probability that it is actually a six is a) 3/8 b) 1/5 c) 3/4 d) none of these

b

g

10. Three six-faced fair dice are thrown together. The probability that the sum of the numbers appearing on the dice is k 3 ≤ k ≤ 8 is a) ( k − 1) ( k − 2) / 432

b) k ( k − 1) / 432

c) k 2 / 432

d) none of these

11. A determinant is chosen at random from the set of all determinants of order 2 with elements 0 or 1 only. The probability that the determinant chosen is non-zero is a) 3 / 16 b) 3 / 8 c) 1 / 4 d) none of these 12. Three persons A, B and C are to speak at a function along with 5 other persons. If the persons speak in random order, the probability that A speaks before B and B speaks before C is a) 3/8

b) 1/6

c) 3/5

d) none of these

13. Three identical dice are rolled. The probability that the same number will appear on each of them is Quest Tutorials Head Office : 44C,Kalusarai New Delhi-16; Ph.(011) 46080363

104

a)

1 6

b)

1 36

c)

1 18

d)

3 28

14. If there are 6 girls and 5 boys who sit in a row, then the probability that no two boys sit together is

a)

6!6! 2 !11!

b)

7 !5! 2 !11!

c)

6!7 ! 2 !11!

d) none of these

15. Two dice are rolled one after the other. The probability that the number on the first is smaller than the number on the second is a) 1 / 2 b) 7 / 18 c) 3 / 4 d) 5 / 12 16. A four figure number is formed of the figures 1, 2, 3, 5 with no repetitions. The probability that the number is divisible by 5 is a) 3/4 b) 1/4 c) 1/8 d) none of these 17. In the above question the probability that the number is odd is a) 3/4 b) 1/4 18. Suppose a) 1 −

c) 1/8

d) none of these

n (≥ 3) persons are sitting in a row. Two of them are selected at random. The probability that they are not together is

2 n

b)

2 n −1

c) 1 −

1 n

d) none of these

19. Two events A and B have probabilities 0.25 and 0.50 respectively. The probability that both A and B occur is 0.14. Then the probability that neither A nor B occurs a) 0.39

b) 0.25

c) 0.11

d) none of these

20. An integer is chosen at random from the first 200 positive integers. The probability that the integer chosen is divisible by 6 or 8 is a) 1 / 3 b) 1 / 4 c) 1 / 5 d) none of these 21. If two squares are chosen at random on a chess board, the probability that they have a side in common is a) 1 / 9 b) 1 / 18 c) 2 / 7 d) none of these 22. The probability that the 13th day of a randomly chosen month is a Friday, is a) 1/12 b) 1/7 c) 1/84

d) none of these.

23. If the letters of the word 'REGULATION' be arranged at random, the probability that there will be exactly 4 letters between R and E is a) 1/10 b) 1/9 c) 1/5 d) 1/2 24. A coin is tossed three times. The probability of getting head and tail alternately is a) 1/8 b) 1/2 c) 1/4

d) none of these

25. A box contains 100 bulbs out of which 10 are defective. A sample of 5 bulbs is drawn. The probability that none is defective, is a)

FG 1 IJ H 10K

5

F 1I b) G J H 2K

F 9I c) G J H 10K

5

5

d)

9 10

26. A carton contains 20 bulbs, 5 of which are defective. The probability that, if a sample of 3 bulbs is chosen at random from the carton, 2 will be Quest Tutorials Head Office : 44C,Kalusarai New Delhi-16; Ph.(011) 46080363

105

defective, is a) 1 / 16

b) 3 / 64

c) 9 / 64

d) 2 / 3

27. In a class of 125 students 70 ed in Mathematics, 55 in Statistics and 30 in both. The probability that a student selected at random from the class, has ed in only one subject is a) 13 / 25 b) 3 / 25 c) 17 / 25 d) 8 / 25. 28. 3 mangoes and 3 apples are in a box. If 2 fruits are chosen at random, the probability that one is a mango and the other is an apple is a) 3 / 5 b) 5 / 6 c) 1 / 36 d) none of these 29. There are 4 white and 4 black balls in a bag and 3 balls are drawn at random. If balls of same colour are identical, the probability that none of them is black, is a) 1 / 4 b) 1 / 14 c) 1 / 2 d) none of these 30. A father has 3 children with at least one boy. The probability that he has 2 boys and one girl is a) 1 / 4 b) 3 /7 c) 2 / 3

d) none of these

31. If x ∈[0, 5], then what is the probability that x − 3x + 2 ≥ 0 ? a) 4/5 b) 1/5 c) 2/5

d) none of these

2

32. An almirah stores 5 black and 4 white socks well mixed. A boy pulls out 2 socks at random. The probability that 2 are of the same colour is a) 4/9 b) 5/8 c) 5/9 d) 7/12 33. If x ∈[1,5], then the probability that x 2 − 5 x + 4 < 0 is a) 3/4 b) 4/4

c) 2/4

d) 1/4

ANSWERS 1. 8. 15. 22. 29.

c c d c b

2. 9. 16. 23. 30.

b a b b b

3. 10. 17. 24. 31.

c a a c a

4. 11. 18. 25. 32.

d b a c a

5. 12. 19. 26. 33.

b b a c a


6. 13. 20. 27.

b b b a

7. 14. 21. 28.

c c b a

106

STATISTICS 1. Which of the following is not a measure of central tendency a) Mean b) Median

c) Mode

d) Range

2. The mean of first n natural numbers is a)

n(n + 1) 2

b) n(n + 1)

c)

n +1 2

d)

(n + 1)

3. For a continuous series the mean is computed by the following formula a) Mean = A +

∑f n

b) Mean = A +

∑d ∑f

c) Mean = A +

∑f ∑d

d) Mean = A +

∑ fd ∑f

4. The weighted mean of first n natural numbers whose weights are equal to the squares of corresponding number is a)

n +1 2

3n(n + 1) 2(2n + 1)

b)

c)

(n + 1) (2n + 1) 6

d)

n(n + 1) 2

5. The mean of a set of numbers is x . If each number is increased by λ , the mean of the new set is a) x

x+λ

b)

c) λ x

d) none of these

6. The relationship between mean, median and mode for a moderately skewed distribution is a) Mode = Median -2 Mean b) Mode = 2 Median - Mean c) Mode = 3 Median -2 Mean d) Mode = 2 Median -3 Mean 7. If in a moderately asymmetrical distribution mode and mean of the data are 6λ and 9λ respectively, then median is b) 7λ c) 6λ d) 5λ a) 8λ 8. If a variable takes the discrete values

7 2

5 2

1 2

1 2

α + 4, α − , α − , α − 3, α − 2, α + , α − , α + 5(α > 0) , then the median is a) α −

5 4

b) α −

1 2

c) α − 2

d) α +

5 4

2 3 n 9. Geometric mean of 2,2 ,2 ,⋅ ⋅ ⋅ ⋅ ⋅2 is

a)

2 2n

b)

n 22

c)

n −1 2 2

d) 2

n +1 2

10. The harmonic mean of 3,7,8,10,14 is

3 + 7 + 8 + 10 + 14 a) 5

1 1 1 1 1 + d) + + + 3 7 8 10 14

1 1 1 1 1 + + + + 5 c) 3 7 8 10 14 d) 1 1 1 1 1 4 + + + + 3 7 8 10 14

2 11. If each observation of a raw data, whose variance is σ , is multiplied by λ , then the variance of the new set is

a) σ 2

b) λ2σ 2

c) λ + σ 2 Quest Tutorials

Head Office : 44C,Kalusarai,New Delhi-16; Ph.(011) 46080363

d) λ2 + σ 2 107

12. The harmonic mean of the numbers 2,3,4, is 1

b) (24)3

a) 3

c)

36 13

d)

13 36

13. Let s be the standard deviation of n observations. Each of the n observations is multiplied by a constant c. Then the standard deviation of the resulting numbers is a) s

b) cs

d) s c

d) none of these

14. The S.D of the first n natural numbers is a)

n +1 2

b)

n(n + 1) 2

c)

n2 − 1 12

d) none of these

15. Which of the following, in case of a discrete data, is not equal to the median ? a) 50th percentile b) 5th decile c) 2nd quartile

d) lower quartile

16. For a frequency distribution, 7th decile is computed by the formula

n   − c  ×i a) D7 = l +  7 f

n   − c  ×i b) D7 = l +  10 f

 7n  − c   ×i c) D7 = l +  10 f

 10n  − c   ×i d) D7 = l +  7 f

17. For a continuous series, the mode is computed by the formula a) l +

  f m − f m−1 f m − f1 f m −1 f1  × i b) l + × C or l +  × C or l + ×i f m − f m −1 − f m +1 f m − f1 − f 2 f m − f m−1 − f m+1  f m − f1 − f 2 

c) l +

2 f m − f m−1 2 f m − f1 f m − f m−1 f m − f1 × C or l + × i d) l + × C or l + ×i 2 f m − f m−1 − f m+1 2 f m − f1 − f 2 f m − f m −1 − f m +1 f m − f1 − f 2

18. The standard deviation for the set of numbers 1,4,5,7,8 is 2.45 nearly. If 10 are added to each number, then the new standard deviation will be a) 2.45 nearly b) 24.45 nearly c) 0.245 nearly d) 12.45 nearly 19. For a frequency distribution, lower quartile is computed by

N   −C  ×i a) Q1 = l +  4 f

N   −C  ×i b) Q1 = l +  2 f

 3N  −C  4   ×i c) Q1 = l + f

d) Q1 = l +

(N − C ) × i f

Quest Tutorials Head Office : 44C,Kalusarai,New Delhi-16; Ph.(011) 46080363

108

20. The upper quartile for the following distribution

th

 31 + 1   item a)   4 

Size of items Frequency

  31 + 1   b)  2   4 

th

item

1 2

2 4

3 5

4 8

  31 + 1   c) 3   4 

5 7

6 3

th

item

7 2

is given by the size of

  31 + 1   d)  4   4 

th

item

21. The A.M of a set of 50 numbers is 38. If two numbers of the set, namely 55 and 45 are discarded, the A.M of the remaining set of numbers is a) 38.5 b) 37.5 c) 36.5 d) 36. 22. An automobile driver travels from plane to a hill station 120 km distant at an average speed of 30 km per hour. He then makes the return trip at an average speed of 25 km per hour. He covers another 120 km distance on plane at an average sped of 50 km per hour. His average speed over the entire distance will be a) c)

30 + 25 + 50 km/hour 3

b)

3 km/hour 1 1 1 + + 30 25 50

(30,25,50 )1 / 3

d) none of these

23. If x1 , x2 , x3 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ xn be n observations, then the quantity (x1 , x2 , x3 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ xn )1 / n is called a) H.M b) A.M c) G.M

24. If the mean of 1,2,3,.........n is a) 10

d) none of these

6n then n is 11 b) 12

c) 11

d) 13

25. If the standard deviation of 0,1,2,3,........9 is K, then the standard deviation of 10,11,12,13.......19 is a) K

c) K + 10

b) K + 10

d) 10K

26. If in a moderately asymmetrical distribution mode and mean of the data are 6λ and 9λ respectively, then median is b) 7λ c) 6λ d) 5λ a) 8λ

27. If a variable takes the discrete values α + 4, α − 7 / 2, α − 5 / 2, α − 3, α − 2, α + 1 / 2, α − 1 / 2, α + 5(α > 0) , then the median is a) α − 5 / 4

b) α − 1 / 2

c) α − 2

d) α + 5 / 4

28. For a symmetrical distribution P25 and P75 are 40 and 60 respectively. The value of median will be a) 50 b) 40 c) 60 d) none of these 29. The standard deviation of first n natural numbers is

n(n + 1) (2n + 1) a) 6

2

n −1 b) 12

c)

 n2 − 1     12   


d) none of these 109

30. If S.D of a variate x is σ , then S.D of

c σ a) a

b)

ax + b where a, b, c are constants is c

c2 a

2

σ

c)

b σ c

d)

a σ c

31. If the variance of observations x1 , x2 ,......... xn is σ 2 , then the variance of px1 , px2 ,....... pxn . p ≠ 0 is given by a) pσ 2

b) p 2 σ 2

c) σ 2 / p 2

d) none of these

ANSWERS 1. 8. 15. 22. 29.

d a d c c

2. 9. 16. 23. 30.

c d c c d

3. 10. 17. 24. 31.

d d c c b

4. 11. 18. 25.

b b a a

5. 12. 19. 26.

b c a a


6. 13. 20. 27.

c b c a

7. 14. 21. 28.

110

a c b a

CORRELATION AND REGRESSION ANALYSIS 1. If a linear relation aX + bY + c = 0 exists between the variables X and Y and ab < 0, then the coefficients of correlation between X and Y is a) 1 b) -1 c) 0 d) any number between -1 and 1. 2. Let X and Y be two variables with the same variance and U and V be two variables such that equal to a) Cov ( X, Y ) b) 0 c) 1

U = X + Y, V = X - Y. Then Cov ( U, V ) is

3. If X and Y are independent variables, then the two lines of regression are a) x = 0, y = 0 b) x = 0, y = const c) x = const, y = 0

d) -1

d) x = const, y = const.

4. If bYX and bXY are both positive, then a)

1 1 2 + ≤ bYX bXY r

b)

1 1 2 + ≥ bYX bXY r

c)

1 1 r + ≤ bYX bXY 2

d) none of these.

5. The line of regression of X on Y referred to the means of X and Y as the origin is a)

x− X =r

σX σX ( y − Y ) b) x = r (y −Y ) σY σY

6. If the regression of Y on X is a) is

3 4

c) x = r

σX y σY

d) none of these.

4 , then the regression coefficient of X on Y 3 b) is less than or equal to

3 4

c) is less than 1

d) can take any value

7. Let X and Y be two variables with the same mean. If the lines of regressions of Y on X and X on Y respectively y = α x + b and x = α y + b, then the value of the common mean is a)

b 1− a

b)

1− a b

c)

β 1− a

d)

b 1− α

8. The arithmetic mean of the first n odd natural numbers is a) n b) ( n + 1 ) /2

c) ( n - 1 )

d) none of these.

9. The arithmetic mean of the squares of the first n natural numbers is a) ( n + 1 ) / 6 b) ( n + 1 ) ( 2n + 1 ) / 6

c) ( n2 - 1 ) / 6

d) none of these.

c) ( 2 n + 1) / n

d) none of these.

2 3

... n

n

n

10. The arithmetic mean of the series 1, 2, 22 , ....., 2n-1 is a)

2n / n

b) ( 2 n − 1) / n

11. The arithmetic mean of the data given by Variate (x) 0 1 Frequency (f )

n

a) ( n + 1 ) / 2

b) n / 2

C0

n

C1

C2

c) 2n / n


C3

...

n

Cn

d) none of these. 111

12. The mean height of 25 male workers in factory is 61 cms, and the mean height of 35 female workers in the same factory is 58 cms. The combined mean height of 60 workers in the factory is a) 59.25

b) 59.5

c) 59.75

d) 58.75.

13. The mean weight of 9 items is 15. If one more item is added to the series the mean becomes 16. The value of 10th item is a) 35

b) 30

c) 25

d) 20.

14. If X and Y are two random variables with the same standard deviation and coefficient of correlation r, then the coefficient of correlation between X and X + Y is a)

1+ r 2

1+ r 2

b)

c)

1− r 2

15. Suppose ∑ x = 15, ∑ y = 36, ∑ xy = 110, n = 5, then cov ( x, y ) is a) 0.4 b) 0.6 c) 0.8 16. If the correlation coefficient between X and Y is 0.6 and if U =

d) 0.9.

X −5 Y −3 and V = , 2 3

then the coefficient of correlation between U and V is a) 0.8 b) 0.6

(c) 0.3

17. Suppose the correlation coefficient between X and Y is -0.7 and if U = between U and V is a) 0.7

1− r . 2

d)

b) - 0.7

d) 0.2

X −3 Y −4 , then the correlation coeff. and V = −2 5

c) 0.8

d) none of these.

18. If cov (X, Y ) = -13.5, Var ( X ) = 2.25 and Var (Y ) = 100, then the coefficient of correlation between X and Y is a) 0.8

b) - 0.8

c) -0.9

d) 0.9.

19. If two lines of regressions are 3x + 12y = 19 and 9x + 3y = 46, the correlation coefficient is a)

b)

1/ 2 3

−1 / 2 3

c) 0.52

d) - 0.52.

c) ( x / σ x , y / σ y )

d) none of these.

20. The two lines of regressions intersect in a)

(σ x , σ y )

b)

( x , y)

ANSWERS 1. 8. 15.

a a a

2. 9. 16.

b b b

3. 10. 17.

d b a

4. 11. 18.

b b c

5. 12. 19.

c a b


6. 13. 20.

b c b

7. 14.

112

a b

SETS, RELATIONS AND MAPPINGS FUNCTIONS 1. A survey shows that 63% of the Americans like cheese whereas 76% like apples. If x% of the Americans like both cheese and apples, then a) x = 39

c) 39 ≤ x ≤ 63

b) x = 63

2. If X = {4 n − 3n − 1 | n ∈ N } and Y = {9 ( n − 1) | n ∈ N }, then X ∪ Y is equal to a) X b) Y c) N

RS T

3. If sets A and B are defined as A = ( x , y ) | y = a)

A∩ B = A

UV W

l

d)

x ≤ 39

d) none of these

q

1 , 0 ≠ x ∈ R , B = ( x , y ) | y = − x , x ∈ R , then x

b) A ∩ B = B

c)

A∩ B = φ

d) none of these

4. If a N = { a x | x ∈ N } and b N ∩ c N = d N , where b, c ∈ N then a) d = bc 5. If A and B are two sets, then

b) c = bd

c) b = cd

d) none of these

c) φ

d) none of these

A ∩ ( A ∪ B) equals

a) A

b) B

6. Which of the following is the empty set ? a) { x | x is a real number and x2- 1 = 0 } c) { x | x is a real number and x2- 9 = 0 }

b) { x | x is a real number and x2 + 1 = 0 } d) { x | x is a real number and x2 = x + 2}.

7. A set contains n elements. The power set contains. a) n elements b) 2n elements

c) n2 elements

d) none of these

8. Let A and B be two non-empty subsets of a set X such that A is not a subset of B, then a) A is a subset of complement of B b) B is a subset of A c) A and B are dist d) A and the complement of B are non-dist. 9. If A = { 1, 2, 3 } and B = { 3, 8 }, then a) {( 3, 1 ), ( 3, 2 ), ( 3, 3 ), ( 3, 8 )} c) {( 1, 2 ), ( 2, 2 ), ( 3, 3 ), ( 8, 8 )}

( A ∪ B) × ( A ∩ B) is b) {( 1, 3 ), ( 2, 3 ), ( 3, 3 ), ( 8, 3 )} d) {( 8, 3 ), ( 8, 2 ), ( 8, 1 ), ( 8, 8 )}

10. For any set A, (A' )' is equal to a) A'

b) A

11. The symmetric difference of A = { 1, 2, 3 } and B = ( 3, 4, 5 } is a) { 1, 2 } b) { 1, 2, 4, 5 }

c) φ

d) none of these

c) { 4, 3 }

d) { 2, 5, 1, 4, 3 }.

12. Let U be the universal set containing 700 elements. If A, B are sub-sets of U such that n (A) = 200, n (B) = 300 and Then

n( A ∩ B) = 100.

n( A'∩ B') =

a) 400

b) 600

c) 300

d) none of these

13. If A = {x ∈ R | 0 < x < 1} and B = {x ∈ R | − 1 < x < 1}, then A × B is the set a) of all points lying inside the rectangle having vertices at (1, 1), (0, 1), (0, -1) and (1, -1) b) of all points lying inside the rectangle having vertices at (1, 0), (1, 1), (0, 1) and (0, 0) Quest Tutorials Head Office : 44C,Kalusarai,New Delhi-16; Ph.(011) 46080363

113

c) of all points lying on the sides of the rectangle whose vertices are at (1,1), (0,1), (0,-1) and (1,-1) d) none of these 14. If A = { a, b, c }, B = { c, d, e }, C = { a, d, f }, then a) {( a, d ), ( a, e ), ( a, c )}

A × ( B ∪ C) is

b) {( a, d ), ( b, d ), ( c, d )}

c) {( d, a ), ( d, b ), ( d, c )} d) none of these

15. If R is a relation from a finite set A having m elements to a finite set B having n elements, then the number of relations from A to B is a) 2mn

b) 2mn -1

c) 2mn

d) mn

16. Let R be a reflexive relation on a set A and I be the identity relation on A. Then a)

R⊂I

b) I ⊂ R

c) R = I

d) none of these

17. Let A be the non-void set of the children in a family. The relation 'x is a brother of y' on A is a) reflexive

b) symmetric

c) transitive

d) none of these

18. The relation R defined in N as a R b ⇔ b is divisible by a is a) reflexive but not symmetric

c) symmetric but not transitive

c) symmetric and transitive d) none of these

19. Given the relation R = {( 1, 2 ), ( 2, 3 )} on the set A = { 1, 2, 3 }, the minimum number of ordered pairs which when added to R make it an equivalence relation is a) 5

b) 6

c) 7

d) 8

20. Let R1 be a relation defined by R1 = {( a, b ) | a ³ b, a, b ∈ R }. Then R1 is a) an equivalence relation on R

b) reflexive, transitive but not symmetric

c) symmetric, transitive but not reflexive symmetric

d)

neither transitive nor reflexive but

21. Let P = {( x, y ) | x2 + y2 = 1, x, y ∈ R }. Then P is a) reflexive

b) symmetric

22. For real numbers x and y, we write x Ry ⇔ x − y + a) reflexive

c) transitive

d) antisymmetric

2 is an irrational number. Then the relation R is

b) symmetric

c) transitive

d) none of these.

23. Let n be a fixed positive integer. Define relation R on the set Z of integers by, a Rb ⇔ n | a − b | Then R is a) reflexive

b) symmetric

c) transitive

d) equivalence.

24. Let L denote the set of all straight lines in a plane. Let a relation R be defined by α R β ⇔ α ⊥ β, α , β ∈ L. Then R is a) reflexive

b) symmetric

c) transitive

d) none of these.

25. If R = {( x, y ) | x, y ∈ Z , x 2 + y 2 ≤ 4} is a relation in Z, then domain of R is a) { 0, 1, 2 }

b) { 0, -1, -2 }

c) {-2, -1, 0, 1, 2}


d) none of these 114

26. Let R be a relation defined in the set of real numbers by a R b ⇔ 1 + ab > 0. Then R is a) equivalence relation

b) transitive

c) symmetric

d) anti-symmetric.

27. Which of the following is not an equivalence relation on Z ? a) a R b ⇔ a + b is an even integer

b) a R b ⇔ a − b is an even integer

c) a R b ⇔ a < b

d) a R b ⇔ a = b

28. The relation R defined on the set A = { 1, 2, 3, 4, 5 } by R = {( x , y ):| x 2 − y 2 | < 16} is given by a) R1 = {( 1, 1 ), ( 2, 1 ), ( 3, 1 ), ( 4, 1 ), ( 2, 3 )} c) R3 = {( 3, 3 ), ( 4, 3 ), ( 5, 4 ), ( 3, 4 )}

b) R2 = {( 2, 2 ), ( 3, 2 ), ( 4, 2 ), ( 5, 2 )} d) none of these

29. Sets A and B have 3 and 6 elements each. What can be minimum number of elements in A ∪ B ? a) 3 b) 6 c) 9

d) 18

30. Two finite sets have m and n elements respectively. The total number of subsets of first set is 56 more than the total number of subsets of the second set. The values of m and n respectively are a) 7, 6 b) 6, 3 c) 5, 1 d) 8, 7 31. Let S be the set of integers. For a , b ∈ S , a Rb if and only if | a - b | < 1, then a) R is not reflexive

b) R is not symmetric

c) R = {( a, a );

a ∈ I}

d) R is not an equivalence relation.

32. A and B are two sets having 3 and 4 elements respectively and having 2 elements in common. The number of relations which can be defined from A to B is b) 210 - 1 c) 212 - 1 d) none of these a) 25 33. If f (x) is a polynomial satisfying f (x ) f (1 / x ) = f (x ) + f (1 / x ) and f ( 3) = 28, then f ( 4 ) is given by a) 63

b) 65

c) 67 −1

d) 68

34. The function f : [ -1/2, 1/2 ] → [ − π / 2, π / 2] defined by f ( x ) = sin ( 3x − 4 x ) is a) both one-one and onto

35. Given

f ( x) =

b) neither one-one nor onto

1 and g ( x ) = | x|− x

3

c) onto but not one-one

d) one-one but not onto

1 . Then x −| x|

a) dom f ≠ φ and dom g = φ

b) dom f = φ and dom g ≠ φ

c) dom f ≠ φ and dom g ≠ φ

d) dom f = φ and dom g = φ

36. Which of the following functions is not onto a)

f : R → R , f ( x ) = 3x + 4

b)

c)

f : R + → R + , f ( x) = x

d) none of these

f : R → R + , f ( x) = x 2 + 2

37. Which of the following functions is not one - one Quest Tutorials Head Office : 44C,Kalusarai,New Delhi-16; Ph.(011) 46080363

115

a)

f : R → R, f ( x) = 2 x + 5

b)

f :[ 0, π ] → [ −1, 1], f ( x ) = cos x

c)

f :[ − π / 2, π / 2] → [ −1, 1], f ( x ) = 2 sin x + 3

d)

f : R → [ −1, 1], f ( x ) = sin x

38. Let f ( x ) = x 2 and g ( x ) = 2 x then the solution set of fog(x) = gof (x) is a) R b) { 0 } c) { 0, 2 }

d) none of these

39. The range of the function f ( x) = cos[ x] for - π / 2 < x < π / 2 contains a) { -1, 1, 0 }

b)

{ cos 1, 1, cos 2 }

c) { cos 1, -cos 1, 1 }

1− x , the domain of f -1 (x) contains 1+ x b) ( −∞, − 1) a) ( −∞, ∞)

d) [ -1, 1 ].

40. If f ( x ) =

c)

( −1, ∞)

d)

(− ∞,−1)U (− 1, ∞ )

41. Let g ( x ) = x − 4 x − 5 then 2

a) g is one - one on R

b) g is one - one on

c) g is not one - one on 42. Let

f ( x) =

( −∞,4)

d) none of these

sin x . Then dom f does not contain 1 + sin x 3

a) ( 0, π )

b)

43. The inverse of the function y =

( −2π, − π)

b)

1 1+ x log10 2 1− x

44. Let the function f : R − {−b} → R − {1} be defined by f ( x ) =

(2 π, 3π)

d)

( 4 π , 6π )

c)

1 log10 (2 x − 1) 2

d)

1 2x log 4 2− x

x+a ( a ≠ b). Then x +b

a) f is one - one but not onto c) f is both one - one and onto

b) f is onto but not one - one d) none of these

sin([ x ]π ) , where [x] denotes the integral part of x, them x2 + x + 1

a) f is one-one c) f is a constant function

1. 8. 15. 22. 29. 36. 43.

c)

10 x − 10− x is 10 x + 10− x

a) log10 ( 2 − x )

45. If f ( x ) =

( −∞,2)

c d a a b b b

2. 9. 16. 23. 30. 37. 44.

b) f is not one-one and non-constant function d) none of these

b b b d b d c

3. 10. 17. 24. 31. 38. 45.

c b c b c c c

ANSWERS 4. 11. 18. 25. 32. 39.

a b a c d b

5. 12. 19. 26. 33. 40.

a c c c b d


6. 13. 20. 27. 34. 41.

b a b c a b

7. 14. 21. 28. 35. 42.

116

b d b d a d

LINEAR PROGRAMMING 1. The solution set of the inequation 2x + y > 5 is a) half plane that contains the origin c) whole xy-plane except the points lying on the line 2x + y = 5

b) open half not containing the origin d) none of these

2. If a point ( h, k ) satisfies an inequation ax + by ≥ 4 , then the half plane represented by the inequation is a) b) c) d)

the halfplane containing the point ( h, k ) but excluding the points on ax + by = 4 the halfplane containing the point ( h, k ) and the points on ax + by = 4 whole xy-plane none of these.

3. In equation y - x ≤ 0 represents a) the half plane that contains the positive x-axis y-axis c) half plane that contains the negative x-axis 4. In equations 3x − y ≥ 3 and 4x - y > 4 a) have solution for positive x and y positive x and y c) have no solution for negative x

b) closed half plane above the line y = x which contains positive d) none of these.

b)

have no solution to

d) have no solution for negative y.

5. The maximum value of P = 3x + 5y, subject to a) 15 b) 16

x ≤ 2, y ≤ 3, x + y ≤ 4, x ≥ 0 and y ≥ 0, is c) 18

6. The maximum value of P = x + y, subject to 2 x + y ≤ 4, x + 2 y ≤ 4, x ≥ 0 and y ≥ 0 is a) 2 b) 3 c) 4/3

d) 20

d) 8/3

7. The minimum value of P = x + 3y , subject to 2 x + y ≥ 6, x + y ≥ 4, x ≤ y, x ≥ 0 and y ≥ 0 , is a) 8

b) 7

c) 6

8. If S is a convex subset of the plane bounded by lines in the plane, then a linear function z = cx + dy, ( x, y ) attains its extreme value at the a) boundary points

b) points inside S

c) vertices of S

d) none of these.

∈S, where c, d are scalars d) none of these.

9. A firm manufacturing two types of electrical items A and B can make a profit of Rs 160 per unit of A and Rs. 240 per unit of B. Both A and B make use of two essential components, a motor and a transformer. Each unit of A requires 3 motors and 2 transformers and each unit of B requires 2 motors and 4 transformers and total supply of components per month is restricted to 210 motors and 300 transformers. If x and y denotes the number of items A and B which the firm should produce in order to maximize the profit then the objective function is a) z = 3x + 2y b) z = 2x + 4y c) z = 210x + 300y d) none of these. 10. The minimum value of z = 2x + y subject to the contraints 5 x + 10 y ≤ 50, x + y ≥ 1, y ≤ 4 and x, y ≥ 0 is a) 1/3

b) 1

c) 1/2

d) 5/2

11. A firm manufactures three products A, B and C. The profits are Rs 3, Rs 2 and Rs 4 respectively. The firm has 2 machines and below is the required processing time in minutes for each machine Products Machine C D

A 4 2

B 3 2

C 5 4 Quest Tutorials

Head Office : 44C,Kalusarai,New Delhi-16; Ph.(011) 46080363

117

Machines C and D have 2000 and 2500 machine minutes. The firm must manufacture 100 A's, 200 B's and 50 C 's but not more than 150 A's. The number of constraints the firm has is a) 6

b) 7

c) 5

d) 8

12. A manufacturer of a line of patent medicines is preparing a production plan on medicines A and B, there are sufficient ingredients available to make 20,000 bottles of A and 40,000 bottles of B but there are only 45,000 bottles into which either of the medicines can be put. Further more it takes 3 hours to prepare enough material to fill 1000 bottles of A. It takes one hour to prepare enough material to fill 1000 bottles of B and there are 66 hours available for this operation. The number of constraints the manufacturer has is a) 4 b) 5 c) 6 d) 7 13. The maximum value of z = 5x + 7y subject to x + y ≤ 4, 3 x + 8 y ≤ 24, 10 x + 7 y ≤ 35, x, y ≥ 0 is a) 22 b) 24.8 c) 26.4

d) 25

14. The maximum value of z = 3x + 5y subject to x + 2 y ≤ 2000, x + y ≤ 1500, x ≤ 600, x, y ≥ 0 is a) 5000

b) 5300

c) 6000

d) 6500

15. A firm manufacturer trucks and buses. Inputs available are 720 man year 900 machines weeks and 1900 tons of steel production of a truck requires 1 man - year, 3 machine-week and 5 tons of steel. Production of a bus requires 2 man-years, 1 machine-week and 4 tons of steel. The price of a truck is Rs. 600,000 and the price of a bus is Rs. 800,000. If a manufacturer produces x trucks and y buses then his objective function is a) z = x + 3y b) z = 3x + 5y c) z = 600000 x + 800000 y

d) z = 720 x + 900 y

16. Maximum value of 80 x + 120 y subject to x + y ≤ 9, x ≥ 2, y ≥ 3, 20 x + 50 y ≤ 360 is a) 928 b) 840 c) 960

d) 1140

17. The maximum value of P = 4x + 2y subject to 4 x + 2 y ≤ 46, x + 3 y ≤ 24, x ≥ 0, y ≥ 0 is a) 42 b) 48 c) 52

d) 46

18. Maximum Z = 6 x1 + 11x2

2 x1 + x2 ≤ 104 , x1 + 2 x2 ≤ 76 and x1 , ≥ 0, x2 ≥ 0 , is

Subject to a) 240

b) 540

c) 440

d) none of these

c) 34.8

d) none of these

c) 4

d) none of these

19. Maximum Z = 5 x1 + 7 x2

x1 + x2 ≤ 4, 3 x1 + 8 x2 ≤ 24

Subject to

10 x1 + 7 x2 ≤ 35 and x1 , x2 ≥ 0 is a) 14.8 20. Minimum Subject to a) 5

b) 24.8

Z = x+ y 3x + 2 y ≥ 12 , x + 3 y ≥ 11 and x ≥ 0, y ≥ 0 is b) 6

21. For Maximum value of Z = 5 x + 2 y Subject to a) 18/7, 2/7

2x + 3y ≥ 6 , x − 2 y ≤ 2 ,

6 x + 4 y ≤ 24 , − 3x + 2y ≤ 3 and x ≥ 0, y ≥ 0 the values of x and y are

b) 7/2, 3/4

c) 3/2, 15/4


d) none of these

118

22. For Maximum value of Z = 6 x1 − 2 x2

2 x1 − x2 ≤ 2, x1 ≤ 3 and x1 , x2 ≥ 0 , the values of x1 and x2 are

Subject to a) 3, 4

b) 2, 3

c) 1, 2

d) none of these

23. Minimum Z = − x1 + 2x2

− x1 + 3 x2 ≤ 10 ,

Subject to a) - 4

x1 + x2 ≤ 6 ,

x1 − x2 ≤ 2 and x1 , x2 ≥ 0 , is

b) - 2

c) 2

d) none of these

24. For x + y ≤ 5, 2 x + y ≥ 4, x + 5 y ≥ 5, x ≥ 3, y ≤ 3 , then minimum value of z = x + 3 y is a) 19/2 b) 11 c) 11/3

d) none of these

ANSWERS 1. 8. 15. 22.

b c c a

2. 9. 16. 23.

b d c b

3. 10. 17. 24.

a b d c

4. 11. 18.

a a c

5. 12. 19.

c a b


6. 13. 20.

d b a

7. 14. 21.

a b b

119

NUMERICAL METHODS 1. In decimal system, the number (3C 2)16 is equivalent to a)

(962)10

b)

(862)10

c)

(864)16

c)

xn +1 = xn −

d)

(860)10

2. Newton’s method for finding the root of the equation, f (x ) = 0 , is a) xn +1 = xn −

f ' ( xn ) f ( xn )

b) xn +1 = xn +

f ' ( xn ) f ( xn )

f ( xn ) f ' ( xn )

d) none of these

3. The root of the equation f ( x ) = 0 lying in the interval (a, b ) is a)

af (b ) − bf (a ) b−a

b)

bf (a ) − af (b ) b−a

4. In decimal form, the number (362)16 is equivalent to a) 866 b) 867

c)

af (b ) − bf (a ) f (b) − f (a)

c) 868

d) none of these

d) 869

5. In normalized floating point respesentation, 0 ⋅ 8642 E 02 ÷ 0 ⋅ 2562 E 02 gives a) 3⋅ 3731E 0

c) 3⋅ 3731 E 04

d) none of these

6. Gauss-Elimination method is used for solving a) algebraic equations b) exponential equations

c) trignometric equations

d) Linear simultaneous equations

7. The number of significant digits in the number 0 ⋅ 00452000 is a) 3 b) 5

c) 8

d) none of these

8. In decimal systems, the number (11011101) 2 is equivalent to a) 221 b) 222

c) 2021

d) none of these

b) 3 ⋅ 3373

9. Using the successsive bisection method, the approximate value of a root of the equation x 3 − x − 4 = 0 lying between 1 and 2 at the end of third iteration is a) 1.875 b) 1.796 c) 1.8125 d) none of these 10. If the equation x 3 − 3x + k = 0 has all real roots, then b) − 1 < k < 1 a) − 2 < k < 2

c) 0 < k < ∞

11. The round off error when the number 8.987652 is rounded to five significant digits is a) 0.00048 b) -0.00048 c) -0.000048

d) none of these d) none of these

12. By false positioning, the second approximation of a root of equation f ( x ) = 0 is (where x0 , x1 are initial and first approximations respectively) a) x0 −

f (x0 ) f (x1 ) − f (x0 )

b)

x0 f (x1 ) − x1 f ( x0 ) f ( x1 ) − f (x0 )

c)

x0 f (x0 ) − x1 f (x1 ) f (x1 ) − f (x0 )

d) x1 −

f (x0 ) f (x1 ) − f (x0 )

13. Let f (x ) be a polynomial. Then if f (x1 ) f (x2 ) < 0 , then f ( x ) = 0 has a) an odd number of roots in ( x1 , x2 )

c) no root or an even number of roots in ( x1 , x2 )

b) any number of roots in ( x1 , x2 ) d) only one root in ( x1 , x2 )


120

b

14. Trapezoidal rule for evaluation of

∫ f (x )dx requires the interval (a, b ) to be divided into a

a) 2n sub-intervals of equal width c) any number of sub-intervals of equal width

b) 2 n + 1 sub-intervals of equal width d) 3n sub-intervals of equal width 1

15. By Simpson rule, taking n = 4 , the value of the integral

1

∫ 1 + x 2 dx is equal to 0

a) .785

b) .788

c) .781

d) none of these

16. By the method of successive bisection, root of the equation x 3 − x − 1 = 0 ; lying between 1 and 2, correct to 3 places of decimal is a) 1.432 b) 1.323 c) 1.324 d) 1.325 17. If a and a + b are two consecutive approximate roots of the equation f (x ) = 0 as obtained by Newton’s Method, then b is equal to a)

f (a ) f ' (a )

f ' (a ) f (a )

b)

18. Let f (0 ) = 1,

c) −

f ' (a ) f (a )

f (1) = 2.72 , then the trapezoidal rule gives approximate value of

d) −

f (a ) f ' (a )

1

∫ f (x )dx as 0

a) 3.72

b) 1.86

c) 1.72

d) 0.86

ANSWERS 1. 8. 15.

a a a

2. 9. 16.

c c c

3. 10. 17.

c a d

4. 11. 18.

a c b

5. 12.

b b


6. 13.

d a

7. 14.

121

d c

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