complex no. arihant

In each of the questions below, four choices are given of which only one is correct. You have to select the correct answer which is the

most appropriate.

1. Let a b c, , be three cube roots of unity, the value of

e

e

e

e

e

e

e

e

e

a

b

c

a

b

c

a

b

c

2

2

2

3

3

3

1

1

1

−

−

−

is

(a) ( )1 3+ a

(b) ( )1 3+ b

(c) ( ) , ( )a b c n Nn+ + ∈3

(d) ( ) , ( )a b c n In+ + ∈2 3 2

2. If Imaz z

bz z

++

=

1

2

1 (where z z, 1 and z2 are complex

numbers and a b, are real numbers). Then z lies on

(a) a straight line(b) a circle(c) a parabola(d) an ellipse

3. If m and x are two real numbers, then

exi

xiim i x

m

2 1 1

11cot , ( )

− +−

= −where is equal to

(a) cos sinx i x+ (b) m/2(c) 1 (d) ( )/m + 1 2

4. If x i= +2 5 (where i2 1= − ) and 21

1 9

1

3 7

1

5 5! ! ! ! ! !+

+

= 2a

b !, then the value of ( )x x x3 25 33 19− + − is equal to

(a) a (b) b(c) a b− (d) a b+

5. If| |z i− + ≤3 2 4, (where i = −1 ) then the difference of

greatest and least values of| |z is(a) 2 11 (b) 3 11

(c) 2 13 (d) 3 13

6. The value of ( )z z+ 4 and ( )z z− 4 are respectively

(where z i i= + = −4 3 20 1, )

(a) 1296,400 (b) 72,16

(c) 36,9 (d) 216,25

7. If z lies on the circle centred at origin. If area of the triangle

whose vertices are z z, ω and z z+ ω where ω is the cube

root of unity, is 4 3 sq unit. Then radius of the circle is

(a) 1 unit (b) 2 unit

(c) 3 unit (d) 4 unit

8. If ω is a complex cube root of unity, then

{ }cos ( ) ( ) ( ) ( )1 1 10 10900

2 2− − + + − −

ω ω ω ω πK

(a) − 1 (b) 0

(c) 1 (d) 3 2/

9. If | ( )| | Im ( )|,z i z z z− = −Re (where i = −1) , then z

lies on

(a) Re( )z = 2 (b) Im( )z = 2

(c) Re( ) Im ( )z z+ = 2 (d) none of these

10. For a complex number z, the minimum value of

| | | cos sin |z z i+ − −α α (where i = −1) is

(a) 0 (b) 1

(c) 2 (d) none of these

11. If a b c p q r, , , , , are six complex numbers, such thatp

a

q

b

r

ci+ + = +1 and

a

p

b

q

c

r+ + = 0, wherei = −1 then

value ofp

a

q

b

c

r

2

2

2

2

2

2+ + is

(a) 0 (b) − 1

(c) 2i (d) − 2i

12. If z be complex number such that equation

| | | |z a z a− + − =2 2 3 always represents an ellipse, then

range of a R( )∈ + is

(a) ( , )1 2 (b) [ , ]1 3

(c) ( , )− 1 3 (d) ( , )0 3

1Complex Numbers

Objective Questions Type I [Only one correct answer]

13. If A z B z( ) , ( )1 2 and C z( )3 are three points in the argand

plane where | | || | | ||z z z z1 2 1 2+ = − and

|( ) | | | | |,1 1 3 1 3 1− + = + −i z i z z z z where i = −1 then

(a) A B, and C lie on a fixed circle with centre( )z z2 3

2

+

(b) A B, and C are collinear points(c) ABC form an equilateral triangle(d) ABC form an obtuse angle triangle

14. The least positive integer n for which

1

1

2 1

21

2+−

= +−i

i

x

x

n

πsin , where x > 0 and

i = −1 is

(a) 2 (b) 4(c) 8 (d) 12

15. If ω ≠ 1 is a cube root of unity and x y z+ + ≠ 0, then

x

y

z

y

z

x

z

x

y

1

1

1

1

1

12

2

2

2

2

2

+

+

+

+

+

+

+

+

+

ω

ω ω

ω

ω ω

ω

ω

ω

ω

ω ω

= 0 if

(a) x y z2 2 2 0+ + =(b) x y z+ + =ω ω2 0 or x y z= =(c) x y z≠ ≠ = 0

(d) x y z= =2 3

16. One of the values of i i is ( )i = − 1

(a) e− π/2 (b) eπ/2

(c) eπ (d) e− π

17. If x irr r= −cos ( / ) sin ( / )π π3 3 , ( where i = − 1), then

value of x x1 2⋅ ⋅ ∞K , is

(a) 1 (b) − 1(c) − i (d) i

18. The area of the triangle on the argand plane formed by thecomplex numbers − −z iz z iz, , , is (where i = − 1)

(a)1

2

2| |z (b) | |z 2

(c)3

22| |z (d) none of these

19. Let z i1 6= + and z i2 4 3= − (where i = − 1) . Let z be a

complex number such that

argz z

z z

−−

=1

2 2

π, then z satisfies

(a) | ( )|z i− − =5 5 (b) | ( )|z i− − =5 5

(c) | ( )|z i− + =5 5 (d) | ( )|z i− + =5 5

20. The number of solutions of the equation z z2 2 0+ =| | ,

where z C∈ is

(a) one (b) two

(c) three (d) infinitely many

21. If z i= + + −( ) ( )λ λ3 5 2 ; then the locus of z is (where

i = − 1)

(a) a straight line (b) a circle(c) an ellipse (d) a parabola

22. The locus of z which satisfies the inequalitylog | | log | |0.3 0.3z z i− > −1 is given by

(a) x y+ < 0 (b) x y+ > 0

(c) x y− > 0 (d) x y− < 0

23. If z1 and z2 are any two complex numbers, then

| ( )| | ( )|z z z z z z1 12

22

1 12

22+ − + − − is equal to

(a) | |z1 (b) | |z2

(c) | |z z1 2+ (d) none of these

24. If z ≠ 0, thenx

z=∫ 0

100[arg| |] dx is

(where [.] denotes the greatest integer function)(a) 0 (b) 10(c) 100 (d) not defined

25. The centre of square ABCD is at z = 0. A is z1. Then the

centroid of triangle ABC is

(a) z i1 (cos sin )π π± (b)z

i1

3(cos sin )π π±

(c) z i1 2 2(cos / sin / )π π± (d)z

i1

32 2(cos / sin / )π π±

26. The point of intersection of the curves arg ( ) /z i− =3 3 4πand arg ( ) /2 1 2 4z i+ − = π (where i = − 1) is

(a) 1 4 3 9/ ( )+ i (b) 1 4 3 9/ ( )− i

(c) 1 2 3 2/ ( )+ i (d) no point

27. If S n i in n( ) ,= + − where i = − 1 and n is an integer, then

the total number of distinct values of S n( ) is(a) 1 (b) 2(c) 3 (d) 4

28. Consider the following statements :

S i i1 8 2 4 4 16: ( ) ( )− = × = − × −S2 4 16 4 16: ( ) ( ) ( ) ( )− × − = − × −S3 4 16 64: ( ) ( )− × − =S4 64 8: =

of these statements, the incorrect one is

(a) S1 only (b) S2 only

(c) S3 only (d) none of these

29. If the multiplicative inverse of a complex number is( )/3 4 19+ i , (where i = − 1) then the complex number

itself is

(a) 3 4− i (b) 4 3+ i

(c) 3 4+ i (d) 4 3− i

30. If z1 and z1 represent adjacent vertices of a regular polygon

of n sides whose centre is origin and ifIm( )

Re( )

z

z1

1

= −2 1,

then n is equal to

(a) 8 (b) 16

(c) 24 (d) 32

2

Complex Numbers

31. If| | ,z = 1 then1

1

++

z

zequals

(a) z (b) z(c) z−1 (d) none of these

32. For x x y y R1 2 1 2, , , .∈ If 0 1 2 1 2< < =x x y y, and

z x iy1 1 1= + , z x i y2 2 2= + , (where i = − 1) and

z z z3 1 2

1

2= +( ), then z z1 2, and z3 satisfy

(a) | | | | | |z z z1 2 3= = (b) | | | | | |z z z1 2 3< <(c) | | | | | |z z z1 2 3> > (d) | | | | | |z z z1 3 2< <

33. If z is any non-zero complex number, thenarg ( ) arg ( )z z+ is equal to(a) 0 (b) π /2(c) π (d) 3 2π /

34. If the following regions in the complex plane, the only onethat does not represent a circle is

(a) z z i z z+ − =( ) 0 (b) Re1

10

+−

=

z

z

(c) argz i

z i

−+

= π2

(d)z i

z

−+

=

11

35. If xi= +

1

2, (where i = − 1) then the expression

2 2 34 2x x x− + + equals

(a) 3 2− ( / )i (b) 3 2+ ( / )i

(c) ( )/3 2+ i (d) ( )/3 2− i

36. If 1 2, ,ω ω are the three cube roots of unity, then for

α β γ δ, , , ∈ R, the expressionα βω γω δωβ αω γω δω

+ + ++ + +

2 2

2is

equal to(a) 1 (b) ω(c) − ω (d) ω−1

37. Ifω is a complex cube root of unity and ( ) ,1 7+ = +ω ωA B

then A and B are respectively equal to

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

(c) 1, 0 (d) − 1 1,

38. If 1, ω and ω2 are the three cube roots of unity, then the

roots of the equation ( )x − − =1 8 03 are

(a) − − − − +1 1 2 1 2 2, ,ω ω (b) 3 2 2 2, ,ω ω(c) 3 1 2 1 2 2, ,+ +ω ω (d) none of these

39. If 1 2 1, , , . . . ,ω ω ωn− are n n, th roots of unity, the value of

( ) ( ) ( )9 9 92 1− − − −ω ω ωK

n will be

(a) n (b) 0

(c)9 1

8

n −(d)

9 1

8

n +

40. If 8 12 18 27 03 2i z z z i+ − + = , (where i = − 1) then

(a) | | /z = 3 2 (b) | | /z = 2 3(c) | |z = 1 (d) | | /z = 3 4

41. If z rei= θ, then| |eiz is equal to

(a) e r− sin θ (b) re r− sin θ

(c) e r− cos θ (d) re r− cos θ

42. If z z z1 2 3, , are three distinct complex numbers and a b c, ,are three positive real numbers such that

a

z z

b

z z

c

z z| | | | | |2 3 3 1 1 2−=

−=

−, then

a

z z

b

z z

c

z z

2

2 3

2

3 1

2

1 2( ) ( ) ( )−+

−+

−is equal to

(a) 0 (b) abc

(c) 3abc (d) a b c+ +43. For all complex numbers z z1 2, satisfying | |z1 12= and

| | ,z i2 3 4 5− − = the minimum value of| |z z1 2− is

(a) 0 (b) 2

(c) 7 (d) 17

44. If z z z1 2 3, , are the vertices of an equilateral triangle in theargand plane, then ( ) ( )z z z k z z z z z z1

222

32

1 2 2 3 3 1+ + = + +is true for

(a) k = 1 (b) k = 2

(c) k = 3 (d) k = 4

45. The value of i i+ −( ) (where i = − 1) is

(a) 0 (b) 2

(c) − i (d) i

46. If z z z1 2 3, , are the vertices of an equilateral triangle with

centroid z0, then z z z12

22

32+ + equals

(a) z02

(b) 2 02

z

(c) 3 02

z (d) 9 02z

47. If a complex number z lies on a circle of radius 1/2, then

the complex number ( )− +1 4z lies on a circle of radius

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

48. If n is a positive integer but not a multiple of 3 and

z i= − +1 3, (where i = − 1) then( )z zn n n n2 22 2+ + is

equal to

(a) 0 (b) − 1(c) 1 (d) 3 2× n

49. If the vertices of a triangle are 8 5 3 2 3+ − + − −i i i, , , themodulus and the argument of the complex numberrepresenting the centroid of this triangle respectively are

(a) 24

,π

(b) 24

,π

(c) 2 24

,π

(d) 2 22

,π

50. The value of Σk

ki

k

=−

1

10 2

11

2

11sin cos

π πis

(a) − 1 (b) 0(c) − i (d) i

51. The set of points in an argand diagram which satisfy both

| |z ≤ 4 and arg z = π/3 is

(a) a circle and a line (b) a radius of a circle(c) a sector of a circle (d) an infinite part line

52. If the points represented by complex numbersz a ib z c id1 2= + = +, (where i = − 1) and z z1 2− are

collinear, then

3

Complex Numbers

(a) ad bc+ = 0 (b) ad bc− = 0

(c) ab cd+ = 0 (d) ab cd− = 0

53. Let A B, and C represent the complex numbers z z z1 2 3, ,respectively on the complex plane. If the circumcentre ofthe triangle ABC lies at the origin, then the nine pointcentre is represented by the complex number

(a)z z

z1 23

2

+ − (b)z z z1 2 3

2

+ −

(c)z z z1 2 3

2

+ +(d)

z z z1 2 3

2

− −

54. Let α and β be two distinct complex numbers such that

| | | |.α β= If real part ofα is positive and imaginary part ofβ is

negative, then the complex number ( )/( )α β α β+ − may be

(a) zero (b) real and negative(c) real and positive (d) purely imaginary

55. The complex number z satisfies the condition

zz

−

=25

24. The maximum distance from the origin of

co-ordinates to the point z is(a) 25 (b) 30(c) 32 (d) none of these

56. The points A B, and C represent the complex numbersz z i z iz1 2 1 21, , ( )− + (where i = − 1) respectively on the

complex plane. The triangle ABC is(a) isosceles but not right angled(b) right angled but not isosceles(c) isosceles and right angled(d) none of the above

57. The system of equations| |

| |

z i

z

+ + ==

1 2

3, (where i = − 1)

has(a) no solution (b) one solution(c) two solutions (d) none of these

58. The centre of the circle represented by| | | |z z+ = −1 2 1on the complex plane is(a) 0 (b) 5/3(c) 1/3 (d) none of these

59. The value of the expression

2 1 1 3 2 1 2 12 2( ) ( ) ( ) ( )+ + + + +ω ω ω ω+ + + + + + +4 3 1 3 1 1 12( ) ( ) ( ) ( )ω ω ωK n n ( )nω2 1+is

(ω is the cube root of unity)

(a)n n2 21

4

( )+(b)

n nn

( )+

+1

2

2

(c)n n

n( )+

−1

2

2

(d) none of these

60. If Σ Πk

k

p

pi i x iy= =

+ = +0

200

1

50

, (where i = − 1) then ( , )x y is

(a) ( , )0 1 (b) ( , )1 1−(c) ( , )2 3 (d) ( , )4 8

61. If| | ,| | ,| |z z z1 2 31 1 2 2 3 3− < − < − < , then| |z z z1 2 3+ +(a) is less than 6 (b) is more than 3

(c) is less than 12 (d) lies between 6 and 12

62. If| | max | |,| |z z z= − +{ }1 1 , then

(a) | |z z+ = 1

2(b) z z+ = 1

(c) | |z z+ = 1 (d) z z− = 5

63. The equation | | | |z i z i k+ − − = (where i = − 1)

represents a hyperbola if

(a) − < <2 2k (b) k > 2

(c) 0 2< <k (d) none of these

64. If | |z i− ≤ 2 and z i1 5 3= + , (where i = − 1) then the

maximum value of| |iz z+ 1 is

(a) 2 31+ (b) 7

(c) 31 2− (d) 31 2+

65. If | | , ,a k nk < ≤ ≤3 1 then all the complex numbers z

satisfying the equation 1 01 22+ + + + =a z a z a zn

nK

(a) lie outside the circle| |z = 1

4

(b) lie inside the circle| |z = 1

4

(c) lie on the circle| |z = 1

4

(d) lie in1

3

1

2< <| |z

66. If X be the set of all complex numbers z such that| |z = 1and define relation R on X by z R z1 2 is

|arg argz z1 2

2

3− =| ,

πthen R is

(a) reflexive (b) symmetric

(c) transitive (d) anti-symmetric

67. The number of points in the complex plane that satisfying

the conditions| | , ( ) ( )z z i z i− = − + + =2 2 1 1 4, (where

i = − 1) is

(a) 0 (b) 1

(c) 2 (d) more than 2

68. If arg ( )z < 0, then arg ( ) arg ( )− −z z is equal to

(a) π (b) − π

(c) − π2

(d)π2

69. If α α α α0 1 2 1, , , . . . , n − be the n, nth roots of the unity, then

the value of Σi

ni

i=

−

−0

1

3

αα( )

is equal to

(a)n

n3 1−(b)

nn

−−

1

3 1

(c)nn

+−

1

3 1(d)

nn

+−

2

3 1

70. If| | | | ,z z− + + ≤1 3 8 then the range of values of| |,z − 4(where i = − 1) is

(a) ( , )0 7 (b) ( , )1 8

(c) [ , ]1 9 (d) [ , ]2 5

4

Complex Numbers

71. sin ( ) ,− −

1 11

iz where z is non real and i = −1, can be

the angle of a triangle if

(a) Re ( ) , ( )z z= =1 2Im

(b) Re ( ) , ( )z z= − ≤ ≤1 1 1Im

(c) Re ( ) ( )z z+ =Im 0

(d) none of the above

72. If z is a complex number and a a a b b b1 2 3 1 2 3, , , , , , all are

real, then

a z b z

b z a z

b z a

a z b z

b z a z

b z a

a z b z

b z

1 1

1 1

1 1

2 2

2 2

2 2

3 3

3

+++

+++

+++

a z

b z a

3

3 3

(a) ( ) | |a a a b b b z1 2 3 1 2 32 2+ (b)| |z 2

(c) 3 (d) none of these

73. Let A e B e C ei i i= = =− −2

3

2

3

2

3

2 6 5 6π π π/ / /, , (where

i = − 1) be three points forming a triangle ABC in the

argand plane. Then ∆ ABC is

(a) equilateral (b) isosceles

(c) scalene (d) none of these

74. If| | ,z = 1 then1

1

1

1

++

+ ++

z

z

z

z

n n

is equal to

(a) 2 cos ( ( ))n zarg (b) 2 sin ( ( ))arg z

(c) 2 2cos ( ( / ))n zarg (d) 2 2sin ( ( / ))n zarg

75. If all the roots of z az bz c3 2 0+ + + = are of unit modulus,

then

(a) | |a ≤ 3 (b) | |b > 3

(c) | |c ≤ 3 (d) none of these

76. The trigonometric form of z i= −( cot )1 8 3 (where

i = − 1) is

(a) cosec3 24 3 28 ⋅ −ei( / )π (b) cosec3 24 3 28 ⋅ − −e i( / )π

(c) cosec3 36 28 ⋅ −ei( / )π (d) cosec2 24 28 ⋅ − +e i π/

77. If | | ,ai i< ≥1 0λ for i n= 1 2 3, , , ,K and

λ λ λ λ1 2 3 1+ + + + =K n , then the value of

| |λ λ λ1 1 2 2a a an n+ + +K is

(a) = 1 (b) < 1

(c) > 1 (d) none of these

78. Perimeter of the locus represented by argz i

z i

+−

= π4

,

(where i = − 1) is equal to

(a)3

2

π(b)

3

2

π

(c)π2

(d) none of these

79. The digit in the unit’s place in the value of ( )739 49 is

(a) 3 (b) 4

(c) 9 (d) 2

80. If z z1 2≠ − and| |z zz z

1 21 2

1 1+ = +

, then

(a) at least one of z z1 2, is unimodular(b) both z z1 2, are unimodular(c) z z1 2⋅ is unimodular(d) none of the above

81. If z x iy= + and ω = −−

1 iz

z i, (where i = − 1) then| |ω =1

implies that in the complex plane(a) z lies on imaginary axis (b) z lies on real axis(c) z lies on unit circle (d) none of these

82. If S x d i xrr= ∫ sin ( ), where ( )i = − 1 , then Σ

r

n

rS=

−

1

4 1

is

( )n N∈(a) − +cos x c (b) cos x c+(c) 0 (d) not defined

83. For complex numbers z x iy1 1 1= + and z x iy2 2 2= + ,

( )where i = −1 we write z z1 2∩ of x x1 2≤ and y y1 2≤ , then

for all complex numbers z with 1 ∩ z, we have1

1

−+

∩z

zK is

(a) 0 (b) 1(c) 2 (d) 3

84. Let 3 − i and 2 + i, (where i = − 1)be affixes of two points

A and B in the argand plane and P represents of thecomplex number z x iy= + , then the locus of P if| | | |z i z i− + = − −3 2 is(a) circle on AB as diameter(b) the line AB(c) the perpendicular bisector of AB(d) none of the above

85. The distances of the roots of the equation|sin | |sin | |sin | |sin | ,θ θ θ θ1

32

23 4 3z z z+ + + = from

z = 0, are(a) greater than 2/3(b) less than 2/3(c) greater than|sin | |sin | |sin | |sin |θ θ θ θ1 2 3 4+ + +(d) less than|sin | |sin | |sin | |sin |θ θ θ θ1 2 3 4+ + +

86. Let S be the set of complex number z which satisfy

log log (| | | | )/ /1 3 1 22 4 3 0{ }z z+ + < , then S is

(where i = − 1)

(a) 1 ± i (b) 3 − i

(c)5

24+ i (d) empty set

87. Let f z z( ) sin= and g z z( ) cos .= If * denotes acomposition of functions, then the value of( )* ( )f ig f ig+ − (where i = − 1) is

(a) ieeiz

(b) ie eiz−

(c) iee iz−(d) ie e iz− −

88. Let f e e e epi p i p i p i p( ) / / / /α α α α α= ⋅ ⋅

2 2 22 3K , (where

i = − 1 and p N∈ ), then lim ( )n

nf→ ∞π is

(a) 1 (b) − 1(c) i (d) − i

5

Complex Numbers

In each of the questions below four choices of which one or more than one are correct. You have to select the correct answer(s)

accordingly.

1. If z satisfies | | | |z z− < +1 3 then ω = + −2 3z i, (wherei = − 1) satisfies

(a) | | | |ω ω− − < + +5 3i i (b) | | | |ω ω− < +5 3(c) Im ( )iω >1 (d) | ( )| /arg ω π− <1 2

2. ω is a cube root of unity and n is a positive integersatisfying 1 02+ + =ω ωn n ; then n is of the type

(a) 3m (b) 3 1m +(c) 3 2m + (d) none of these

3. Ifz

z i

++

1is a purely imaginary number, (where i = − 1)

then z lies on a(a) straight line(b) circle(c) circle with radius = 1 2/

(d) circle passing through the origin

4. The equation whose roots are nth power of the roots of theequation, x x2 2 1 0− + =cos ,θ is given by

(a) ( cos ) sinx n n+ + =θ θ2 2 0

(b) ( cos ) sinx n n− + =θ θ2 2 0

(c) x x n2 2 1 0+ + =cos θ(d) x x n2 2 1 0− + =cos θ

5. If ω( )≠ 1 is a cube root of unity, then the value ofω ω1994 1995+ is

(a) − ω (b) − ω2

(c) − ω3 (d) − ω4

6. If| | | | | | ,z z z z1 22

12

22+ = + then

(a)z

z1

2

is purely real (b)z

z1

2

is purely imaginary

(c) z z z z1 22 10+ = (d) amp

z

z1

2 2

= π

7. If z z z z1 2 3 4, , , are the four complex numbers representedby the vertices of a quadrilateral taken in order such that

z z z z1 4 2 3− = − and ampz z

z z4 1

2 1 2

−−

= π

, then the

quadrilateral is a(a) rhombus (b) square(c) rectangle (d) cyclic quadrilateral

8. Let z z1 2, be two complex numbers represented by pointson the circle| |z = 1 and| |z = 2 respectively, then(a) max| |2 41 2z z+ = (b) min| |z z1 2 1− =

(c) zz

21

13+

≤ (d) none of these

9. If α is a complex constant such that α αz z2 0+ + = has a

real root, then(a) α α+ = 1(b) α α+ = 0(c) α α+ = − 1(d) the absolute value of the real root is 1

10. If z z z z1 2 3 4, , , are roots of the equationa z a z a z a z a0

41

32

23 4 0+ + + + = where a a a a0 1 2 3, , , and

a4 are real, then(a) z z z z1 2 3 4, , , are also roots of the equation(b) z1 is equal to at least one of z z z z1 2 3 4, , ,(c) − − − −z z z z1 2 3 4, , , are also roots of the equation(d) none of the above

11. The reflection of the complex number2

3

−+

i

i, (where

i = − 1) in the straight line z i z i( ) ( )1 1+ = − is

(a)− −1

2

i(b)

− +1

2

i

(c)i i( )+ 1

2(d)

−+1

1 i

12. The common roots of the equations

z i z i z i3 21 1 0+ + + + + =( ) ( ) , (where i = − 1) and

z z1993 1994 1 0+ + = are

(a) 1 (b) ω(c) ω2 (d) ω981

13. The argument and the principal argument of the complex

number2

4 1 2

++ +

i

i i( ), (where i = − 1) are

(a) tan ( )− −1 2 (b) − −tan 1 2

(c) tan−

1 1

2(d) −

−tan 1 1

2

14. If z z1 2, are two complex numbers satisfying the equationz z

z z1 2

1 2

1+−

= , then z z1 2/ is a number which is

(a) positive real (b) negative real(c) zero (d) purely imaginary

15. If z a ib1 = + and z c id2 = + (where i = − 1) are two

complex numbers such that | | | |z z1 2 1= = andRe ( )z z1 2 0= , then the pair of complex numbers,ω1 = +a ic and ω2 = +b id satisfy(a) | |ω1 1= (b) | |ω2 1=(c) | |ω ω1 2 1= (d) Re ( )ω ω1 2 0=

16. The real value of θ for which the expression1

1 2

+−

i

i

cos

cos

θθ

(where i = − 1) is a real number is

(a) 22

n n Iπ π+ ∈, (b) 22

n n Iπ π− ∈,

(c) 22

n n Iπ π± ∈, (d) 24

n n Iπ π± ∈,

17. If cos cos cos sin sin sin ,α β γ α β γ+ + = + + =0 then(a) cos ( ) cos ( ) cos ( )2 2 2 0α β γ+ + =(b) sin ( ) sin ( ) sin ( )2 2 2 0α β γ+ + =(c) cos ( ) cos ( ) cos ( )β γ γ α α β+ + + + + =0(d) sin ( ) sin ( ) sin ( )β γ γ α α β+ + + + + =0

6

Complex Numbers

Objective Questions Type II [One or more than one correct answer(s)]

18. If 1 2 1, , , . . . . ,ω ω ωn − are the n, nth roots of unity, then

( ) ( ) ( )2 2 22 1− − − −ω ω ωK

n equals

(a) 2 1n −(b) n n n

nC C C1 2+ + +. . . .

(c) [ . . . . ] /20

2 11

2 12

2 1 1 2 1n n n nnC C C C+ + + + −+ + +

(d) 2 1n +19. If z z z z1 2 3 4, , , are the vertices of a square in that order,

then

(a) z z z z1 3 2 4+ = +(b) | | | | | | | |z z z z z z z z1 2 2 3 3 4 4 1− = − = − = −(c) | | | |z z z z1 3 2 4− = −(d) ( )/( )z z z z1 3 2 4− − is purely imaginary

20. If 21

cosθ = +xx

and 21

cos φ = +yy

, then

(a)x

y

y

x+ = − φ2 cos ( )θ

(b) x yx y

m nm n

m n+ = + φ1

2 cos ( )θ

(c)x

y

y

xm n

m

n

n

m+ = − φ2 cos ( )θ

(d) xyxy

+ = + φ12 cos ( )θ

21. If ( ) ,1 0 1+ = + + +x C C x C xnn

nK where n is a positive

integer, then

(a) C C Cnn

0 2 422

4− + − =

. . . cos/ π

(b) C C Cnn

1 3 522

4− + − =

. . . sin/ π

(c) C C Cnn n

0 4 82 2 22 2

4+ + + = +

− −K

( )/ cosπ

(d) C C Cnn

0 3 6

1

32 2

3+ + + = +

K cosπ

In these questions, a passage (paragraph) has been given followed by questions based on each of the passages. You have to answer the

questions based on the passage given.

PASSAGE 1

If x2 1 0+ = ⇒ x2 1= − or x i iota= ± − = ±1 ( ) is called the imaginary unit.

Also, i i i i i i2 3 21 1= − = ⋅ = − = −, ( ) and i i4 2 2 21 1= = − =( ) ( ) .

ie, i i i i n In n n n+ + + = ∀ ∈+ + +1 2 3 0 (Integer) and x3 1 0− = ⇒ ( ) ( )x x x− + + =1 1 02

⇒ ( ) ( ) ( )x x x− − − =1 02ω ω

∴ x = 1 2, ,ω ω are the cube roots of unity. ie, ω ω ωn n n n I+ + = ∀ ∈+ +1 2 0 (Integer)

Now let z a ib= + if | : | :a b = 3 1 or 1 3:

Then, convert z in terms of ω or ω2. Also, | | | |1 1 32− = − =ω ω

On the basis of above information, answer the following questions :

1. The value of the sum Σn

n ni i=

++1

161( ), where i = − 1,

equals

(a) i (b) i − 1

(c) − i (d) 0

2. For positive integers n n1 2, ; the value of the expression,

( ) ( ) ( ) ( ) ,1 1 1 11 1 2 23 5 7+ + + + + + +i i i in n n n

where i = − 1, is real if

(a) n n1 2 1= + (b) n n1 2 1= −(c) n n1 2= (d) n n1 20 0> >,

3. If ( )ω ≠ 1 is a cube root of unity and i = − 1, then

1

1

1

1

1

1

1

2 2

2−−

+ +−

− + −−

−

i

i

i

i

ω

ω

ω

ω is equal to

(a) 0 (b) 1(c) i (d) ω

4. Let ω = − +1

2

3

2i , where i = −1, then the value of the

determinant

1 1 1

1 1

1

2 2

2 4

− − ω ωω ω

is

(a) 3 ω (b) 3 1ω ω( )−(c) 3 2ω (d) 3 1ω ω( )−

5. If i = − 1, then

4 51

2

3

23

1

2

3

2

334 365

+ − +

+ − +

i

iis equal to

(a) 1 3− i (b) − +1 3i

(c) i 3 (d) − i 3

7

Complex Numbers

Linked-Comprehension Type

6. The complex numbers z z1 2, and z3 satisfyingz z

z z

i1 3

2 3

1 3

2

−−

= −(where i = −1) are the vertices of a

triangle which is(a) of area zero (b) right angled isosceles(c) equilateral (d) obtuse angled isosceles

7. The value of

Σr

n

r r r=

− − −2

21( ) ( ) ( )ω ω , where ω is an imaginary cube

root of unity, is(a) ( )Σn n2 2− (b) ( )Σn n2 −(c) ( )Σn n2 + (d) ( )Σn n2 2+

8. The smallest positive integer n for which1

11

+−

=i

i

n

,

where i = − 1, is

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

9. If α β, and γ are the roots of x x x3 23 3 7 0− + + = , then

Σ (( )/( ))α β− −1 1 is(a) 0 (b) 3 ω(c) 3/ω (d) 2 2ω(where ω is cube root of unity)

10. If3

2

3

23

50

25+

= −i

x iy( ) where x y, are real and

i = −1 , then the ordered pair ( , )x y is given by

(a) ( , )0 3 (b)1

2

3

2, −

(c) ( , )− 3 0 (d) −

1

2

3

2,

11. The roots of the cubic equation ( ) ( )z + = ≠αβ α α3 3 0 ,

represent the vertices of a triangle of sides of length

(a)1

3| |α β (b) 3| |α

(c) 3| |β (d)1

3| |α

12. If zi= +3

2where i = − 1, then ( )z i101 103 105+ is equal

to

(a) z (b) z2

(c) z3 (d) z4

PASSAGE 2

Let z a ib rei= + = θ where a b R, , θ ∈ and i = − 1

Then, r a b z= + =( ) | |2 2 and θ =

=−tan ( )1 b

azarg

Now, | | ( ) ( )z a b a ib a ib zz2 2 2= + = + − = ⇒ 12z

z

z=

| |

and | . . . . | | || || |. . . .| |z z z z z z z zn n1 2 3 1 2 3=

If| ( )|f z = 1, then f z( ) is called unimodular. In this case f z( ) can always be expressed as f z e Ri( ) ,= ∈α α

Also, e e ei ii

α βα β

α β+ = −

+

2 2

2cos and e e e ii i

iα β

α βα β− = −

+

2 2

2sin where α β, ∈ R.


1. If z z z1 2 3, , are complex numbers such that

| | | | | | | | ,z z z z z z1 2 3 1 2 3 1= = = + + = then1 1 1

1 2 3z z z+ +

is

(a) equal to 1 (b) less than 1(c) greater than 3 (d) equal to 3

2. If | | ,| | ,| |z z z1 2 31 2 3= = = and | |z z z1 2 3 1+ + = , then| |9 41 2 3 1 2 3z z z z z z+ + is equal to(a) 6 (b) 36(c) 216 (d) 1296

3. If | | | | . . . . | |z z zn1 2 1= = = = , then the value of| |z z z zn1 2 3+ + + +K is equal to(a) 1(b) | | | | | | | |z z z zn1 2 3+ + + +K

(c)1 1 1 1

1 2 3z z z zn

+ + + +

. .

(d) n

4. If z x iy= + is a complex number with rationals x and

y iand = −1 and| |z = 1, then| |z n2 1− is ( )n N∈(a) an irrational number

(b) a rational number

(c) non terminating non recurring

(d) a positive real number

5. The value of tan lnia ib

a ib

−+

, (where i = −1 )is equal

to

8

Complex Numbers

(a)2

2 2

ab

b a−(b)

22 2

ab

a b−

(c)a b

a b

2 2

2 2

+−

(d)a b

a b

2 2

2 2

−+

6. If | |z i− =3 3, (where i = −1) and arg z ∈ ( , / )0 2π , then

cot ( )arg zz

− 6is equal to

(a) 0 (b) − i(c) i (d) π

7. If z1 and z2 are complex numbers satisfyingz z

z z1 2

1 2

1+−

=

and argz z

z zm m I1 2

1 2

−+

≠ ∈π( ), then

z

z1

2

is

(a) zero (b) a rational number(c) a positive real number (d) a purely imaginary

PASSAGE 3

Let a quadratic equation az bz c2 0+ + = where a b c R, , ∈ and a ≠ 0. If one root of this equation is p iq+ , then other must be the

conjugate p iq− and vice-versa. ( ,p q R∈ and i = −1). But if a b c, , are not real, then roots of az bz c2 0+ + = are not conjugate

to each other.

ie, if one root is real, then other may be non real, Now, combining both cases we can say that az bz c2 0+ + = where a b c C, , ∈and a ≠ 0.


1. If one root of the quadratic equation

( ) ( ) ( ) ,1 7 3 6 8 02+ − + + + =i x i x i where i = −1 is

4 3− i, then the other root must be(a) 4 3+ i (b) 1 − i(c) 1 + i (d) i i( )− 1

2. The condition that the equation z az b2 0+ + = has a

purely imaginary root where a and b are complexconstants, is

(a) ( ) ( ) ( )a a ab ab b b+ − + + =2 0

(b) ( ) ( ) ( )a a ab ab b b+ + + − =2 0

(c) ( ) ( ) ( )a a ab ab b b− + + − =2 0

(d) ( ) ( ) ( )a a ab a b b b+ + + + =2 0

3. The condition that the equation z az b2 0+ + = has a

purely real root where a and b are complex constants, is

(a)ab ba

b b

b b

a a

+−

= +−

(b)ab ba

b b

b b

a a

++

= ++

(c)ab ba

b b

b b

a a

−+

= −+

(d)ab ba

b b

b b

a a

−−

= −−

4. The condition that the equation az bz c2 0+ + = has both

real roots where a b c, , are complex constants is

(a)a

a

b

b

c

c= = (b)

a

a

b

b

c

c= = −

(c) − = =a

a

b

b

c

c(d)

a

a

b

b

c

c= − =

5. The condition that the equation az bz c2 0+ + = has both

purely imaginary roots where a b c, , are complex constantsis

(a)a

a

b

b

c

c= =

(b)a

a

b

b

c

c= − = −

(c)a

a

b

b

c

c= = −

(d)a

a

b

b

c

c= − =

6. If equations az bz c2 0+ + = and z z2 2 3 0+ + = have a

common root where a b c R, , ∈ , then a b c: : is

(a) 2 3 1: : (b) 1 : 2 : 3

(c) 3 : 1 : 2 (d) 3 : 2 : 17. If α is a non real complex number and x x2 0+ + =α α has

a real root γ, then

(a) γ α α= +(b) γ α α= −(c) γ =1

(d) γ α α= −| |

8. If the quadratic equation z a ib z c id2 0+ + + + =( ) , where

a b c d, , , are non-zero real and i = − 1, has a real root,

then

(a) abd b c d= +2 2

(b) abc bc d= +2 2

(c) abd bc ad= +2 2

(d) abc bc ad= +2 2

9

Complex Numbers

PASSAGE 4

Let z a ib a b= + = ( , ) be any complex number ∀ ∈a b R, and i = − 1. If ( , ) ( , )a b ≠ 0 0 , then arg ( ) tanzb

a=

−1

where − < ≤π πarg( )z and arg argif arg

if arg( ) ( )

, ( )

, ( )z z

z

z+ − =

<− >

ππ

0

0


1. If arg ( ) ,z < 0 then arg arg( ) ( )− −z z is equal to(a) π (b) − π(c) − π

2(d)

π2

2. Let z and w be two non zero complex numbers such that| | | |z w= and arg arg( ) ( ) ,z w+ = π then z equals

(a) w (b) − w

(c) w (d) − w

3. If arg arg( ) ( )2 31 2z z− = π, then the value ofz

z1

2

is equal

to

(a) 0.5 (b) 1.0(c) 1.5 (d) 2.0

4. If arg( )z > 0, then arg( ) arg( )− − =z z λ 1 and if arg ( )z < 0then arg arg( ) ( )z z− − = λ2, then(a) λ λ1 2 0+ = (b) λ λ1 2 0− =(c) 3 2 01 2λ λ− = (d) 2 3 01 2λ λ− =

5. The value of

{arg } { }( ) arg( ) arg( ) arg( )z z z z+ − − − +2π∀ = + = − >z x iy i x y, ( ) ,where 1 0 is

(a) π (b) − π(c) 0 (d) not defined

PASSAGE 5

Let z a ib a b1 1 1 1 1= + ≡ ( , ) and z a ib a b2 2 2 2 2= + ≡ ( , ); where i = − 1 , be two complex numbers

If ∠ =POQ θ, From Rotation theorem

z

z

z

zei2

1

2

1

0

0

−−

=| |

| |θ ⇒ z z

z z

z

zei2 1

1 1

2

1

=| |

| |θ

⇒ z z

z

z

zei2 1

12

2

1| |

| |

| |= θ ⇒ z z z z ei

2 1 1 2=| || | θ

⇒ z z z z i2 1 1 2= +| || |( sin )cos θ θ

∴ Re ( ) | || |cosz z z z2 1 1 2= θ ...(i)

and Im ( ) | || |sinz z z z2 1 1 2= θ ...(ii)

The dot product of z1 and z2 is defined by z oz z z z z1 2 1 2 2 1= =| || |cos Re ( )θ [from (i)] and cross product of z1 and z2 is defined

by

z z z z z z1 2 1 2 2 1× = =| || |sin ( )θ Im [from Eq. (ii)]


1. If z i z i1 22 5 3= + = −, , then the value of

( )z z z z1 2 2 1⋅ + × is equal to

(a) 2 (b) 3(c) 2 3 (d) 3 2

2. If z i1 3 4= + and z i2 4 3= + , then the value of

sin θ π θ π< <

3

2is equal to

(a) − 1

7(b) − 7

25

(c) − 24

25(d) − 1

25

3. If z i1 5 12= + and z i2 3 4= + , then (the projection of z1 onz2 + projection of z2 on z1) is equal to

(a)4131

65(b)

3411

65

(c)1134

65(d)

1341

65

4. If a and b are real numbers between 0 and 1 such that thepoints z a i z bi1 2 1= + = +, and z3 0= form an equilateraltriangle, then ( , )a b is(a) ( , )2 3 3 1− − (b) ( , )3 1 3 1− −(c) ( , )2 3 2 3− − (d) ( , )3 1 2 3− −

5. Let z1 and z2 be roots of the equation z z2 1 0+ + = . If

∠ = ≠ < <POQ α α π0 0( ) and OP OQ= where O is the

origin, then α is equal to(a) π/4 (b) π/2(c) π/3 (d) 2 3π/

10

Complex Numbers

θP z( )1

O

Q z( )2

PASSAGE 6

The equation zn − =1 0has n roots which are called the nth roots of unity. The n n, th roots of unity are1 2 1, , , . . . . ,α α α n − which

are in GP, where α π π=

+

= −cos sin ;2 2

1n

in

i

then we have following results :

(i) Σr

nr

=

−=

0

1

0α or Σr

n r

n=

−

=0

1 20cos

πand Σ

r

n r

n=

−

=0

1 20sin

π(ii) z zn

r

nr− = −

=

−1

0

1

Π ( )α

(iii) Πr

nr n

=

−−= −

0

111α ( ) (iv) Σ

r

nkr n k n

k n=

−=

0

1

0α

,

,

if is multiple of

if is not multiple of


1. The value of Σr

n

r=

−

−1

1 1

2( )αis equal to

(a) ( )n n− 2 2 (b)( )n n

n

− +−

−2 2 1

2 1

1

(c)( )n n

n

−−

−2 2

2 1

1

(d)( )n n

n

−−

−1 2

2 1

1

2. If ω be non real complex cube root of unity, then the value

of

Π

Π

p

p

q

q

=

=

+

−

1

4

1

72

( )

( )

ω α

ω αis equal to

(a)− +

1 3

2

i(b)

1 3

2

−

i

(c)− −

1 3

2

i(d)

1 3

2

+

i

3. The algebraic sum of perpendicular distances from thepoints 1 2 3 1, , , ,α α α αK

n − to the line a z az b+ + = 0,

(where a is complex number and b is real) is equal to

(a)n

a2| |(b)

n b

a

| |

2

(c)nb

a| |(d)

nb

a2| |

4. If α π π=

+

cos sin2

7

2

7i , then equation whose roots

are α α α+ +2 4 and α α α3 5 6+ + is

(a) z z2 2 0− − = (b) z z2 2 0− + =(c) z z2 2 0+ − = (d) z z2 2 0+ + =

5. If n N∈ , then the value of

Πr

n r

n=

−

1

1

sinπ

is equal to

(a)n

n2 1− (b) n n⋅ −2 1

(c)n

n2 1+ (d) n n⋅ +2 1

6. If n I n∈ ≥, 2, then the value of Σr

n

n rr

n=

−−

1

1 2( ) cos

πis

equal to(a) − n (b) n

(c) − n

2(d)

n

2

7. The expression Σ Σm

n

k

m k

mi

k

m=

+

=

−

−

1

4 1

1

1 2 2sin cos

π π

m

(a) has real part equal to 1

(b) purely real(c) purely imaginary

(d) has imaginary part equal to 2

PASSAGE 7

Let A z B z C z( ), ( ), ( )1 2 3 be the vertices of an equilateral triangle ABC such that| | | | | |z z z1 2 3 2= = = . A circle is inscribed in the

triangle ABC which touches the sides AB BC, and CA at D z E z( ), ( )4 5 and F z( )6 respectively. P z( ) be any point on its incircle

other than D E F, , .


1. The value of ( ) ( ) ( )AB BC CA2 2 2+ + is equal to

(a) 9 (b) 18

(c) 27 (d) 36

2. The value of ( ) ( ) ( )PA PB PC2 2 2+ + is equal to

(a) 9 (b) 12

(c) 15 (d) 18

11

Complex Numbers

3. The value of ( ) ( ) ( )DE EF FD2 2 2+ + is equal to

(a) 12 (b) 3

(c) 3 3 (d) 9

4. The value of Re ( )z z z z z z1 2 2 3 3 1+ + is equal to(a) 0 (b) 6(c) −6 (d) −3

5. If z i1 3= + , i = −1, then the value of

| | | |z z z z2 32

2 32− + + is equal to

(a) 0 (b) 2(c) 4 (d) 6

6.z

z1

3

is equal to

(a) 1 3− i (b) 1 3+ i

(c)− +1 3

2

i(d)

1 3

2

+ i

PASSAGE 8

The general equation of straight line is az a z b+ + = 0 …(i)

where a is complex number and b is real number. The real and complex slopes of the line are − +−

ia a

a aand − a

a,

( )where i = −1 respectively. If adding z z in LHS (i), then (i) convert in general equation of circle

ie z z a z az b, + + + = 0

with centre − a and radius | |a b2 − if a = 0, then circle | |z b2 0+ =

which is defined only when b < 0

or | |z b r2 2= − = (say)

∴ | |z r= (radius of circle)


1. If lines a z az b+ + = 0 and a z a z b1 1 1 0+ + = where a a, 1

are complex numbers and b b R, ;1 ∈ are perpendicularthen(a) a a a a1 1 0− =(b) a a a a1 1 0+ =(c) aa a a− =1 1 0(d) aa aa1 1 0+ =

2. Reflection of the line ( ) ( ) , ,2 3 2 3 0 1+ + − = = −i z i z i

on the real axis is(a) ( ) ( )2 3 2 3 0− − + =i z i z(b) ( ) ( )2 3 2 3 0+ − − =i z i z(c) ( ) ( )2 3 2 3 0+ + − =i z i z(d) ( ) ( )2 3 2 3 0− + + =i z i z

3. Two different non parallel lines meet the circle| |z r= inthe points a b, and c d, ; where a b c d, , , are complexnumbers respectively, then these lines meet in the point zis given by

(a)a c b d

a c b d

− − − −

− − − −+ − −

−

1 1 1 1

1 1 1 1

(b)a b c d

a b c d

− − − −

− − − −+ − −

−

1 1 1 1

1 1 1 1

(c)a d b c

a d b c

− − − −

− − − −+ − −

−

1 1 1 1

1 1 1 1

(d)b c a d

b c a d

− − − −

− − − −+ − −

−

1 1 1 1

1 1 1 1

4. If | |z i− ≤ 2 and z i0 5 3= + , the maximum value of

| |i z z+ 0 , where i = − 1, is

(a) 2 31+ (b) 31 2−(c) 7 (d) − 7

5. The number of values of z which satisfies both the

equations | |z i− − =1 2 and | |z i+ + =1 2, where

i = −1, is

(a) 0 (b) 1

(c) 2 (d) infinitely many

6. If z z1 2, and z3 be three points on| |z = 1. If θ θ1 2, , and θ3 be

the arguments of z z1 2, and z3 respectively, then

Σ cos ( )θ θ1 2− is

(a) ≥ − 3

2(b) ≤ − 3

2

(c) ≥ 3

2(d) ≥ 1

7. If the circles zz a z a z b+ + + =1 1 1 0 and

zz a z a z b+ + + =2 2 2 0 (where b b R1 2, , )∈ intersect

orthogonally, then

(a) a a a a b b1 2 1 2 1 2+ = +(b) a a a a b b1 2 1 2 1 2− = −(c) a a a a b b1 2 1 2 1 2+ = +(d) a a a a b b1 2 1 2 1 2− = −

12

Complex Numbers

PASSAGE 9

If a b ccos cos cosα β γ+ + = 0 = + +a b csin sin sinα β γ; where a b c R, , ∈ and − <π α β γ π, , ≤

Then, let A e B ei i= =α β, and C ei= γ where i = −1

∴ aA bB cC+ + = 0 anda

A

b

B

c

C+ + = 0

we get ( ) ( ) ( ) ,aA bB cC abc ABC3 3 3 3+ + =

a

A

b

B

c

C

abc

ABC

+

+

=3 3 3

3and aBC bCA cAB+ + = 0


1. IfA

B

B

C

C

A+ + =1, then Σ cos ( )α β− is equal to

(a) − 3

2(b) 0

(c) 1 (d)3

2

2. If cos cos cos cos( )3 3 3α λ β µ γ α β γ+ + = + +v ,then the value of| | | |µ λ µ− + −3 v is equal to(a) 0 (b) 6(c) 12 (d) 16

3. If sin sin sinn n nα β γ+ + = 3

2

and cos ( ) cos ( ) cos ( )α β β γ γ α λ+ + + + + = ,

then the value of ( , )n λ is

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

(c) ( , )2 0 (d) ( , )3 0

4. If sin sin sin ,α β γ+ + =2 3 0 then the value of α β γ+ − is

equal to

(a) −2 α (b) −2β(c) −2 γ (d) 0

5. If sin ( ) sin ( )2 8 2α β γ β γ α− − + − −+ − − =27 2sin ( )γ α β N1

and sin ( ) sin ( ) sin ( )α β β γ γ α+ + + + + =N2,

then the value of N N1 2+ is equal to

(a) 36 (b) 18

(c) 9 (d) 0

PASSAGE 10

Let A z B z( ), ( )1 2 and C z( )3 be the vertices of a triangle ABC on the complex plane which is circumscribed by a circle| |z = 1. If

the altitude of the triangle through the vertex A z( )1 meets BC at D and circle| |z = 1 at P.


1. The complex number associated with the point D is equalto

(a)1

21 2 3

1 2

3

z z zz z

z+ + −

(b)1

21 2 3

2 3

1

z z zz z

z+ + −

(c)1

21 2 3

3 1

2

z z zz z

z+ + −

(d)1

21 2 3

1 2

3

z z zz z

z− + −

2. The complex number associated with the point P is equalto

(a) − z z

z1 2

3

(b)z z

z1 2

3

(c) − z z

z2 3

1

(d)z z

z2 3

1

3. If Q be the image of P about the line BC , then complexnumber associated with the point Q is equal to

(a) z z z1 2 3+ + (b)z z z1 2 3

2

+ +

(c)z z z1 2 3

3

+ +(d)

2

31 2 3( )z z z+ +

4. If R be the image of P about the origin ( )O , then the

distance between the points B and R is equal to

(a) | |z z1 2− (b) | |z z2 3−(c) | |z z3 1− (d) | |z z1 3+

5. The orthocentre of the triangle ABC associated with the

point O ′ is equal to

(a)2

31 2 3z z z− +

(b)z z z1 2 32

3

+ −

(c) z z z1 2 3− + (d) z z z1 2 3+ +

13

Complex Numbers

Solve the following problems and mark your response against their respective grids. Write your answer in the top row of the grid and darken

the concerned numbers in the respective columns.

For example. If answer of a question is 0247, then

1. z1 and z2 are two complexnumbers such that

z z

z z1 2

1 2

2

2

−−

is unimodular,

while z2 is not

unimodular, then | |z1

must be equal to

2. If| | ,| | ,z z1 22 3= = | |z3 4=and| |2 3 4 91 2 3z z z+ + = , thenabsolute value of8 272 3 3 1z z z z+ + 64 1 2z zmust be equal to

3. If a b c, , are distinctintegers and ω ≠1 is acube root of unity and ifminimum value of| | |a b c a+ + +ω ω2

+ +b cω ω2 | = n 1 4/ , then

the value of n must beequal to

4. If ω ≠1 is a cube root ofunity, and a b+ = 21,a b3 3 8001+ = , then the

value of( )( )a b a bω ω ω ω2 2+ +must be equal to

5. If the equation of all thecircles which areorthogonal to | |z = 1 and| |z − =1 4 is

| |z ib+ − =7 (λ + b i2),

= − ∈1 and b R, then thevalue of λ must be equalto

6. If the area of the triangleon the complex planeformed by the pointsz z i z, + and iz is 200,then the value of | |3zmust be equal to (wherei i= − )

7. If

Σ Σp q

pq

= =+

1

32

1

10

3 22

11( ) sin

π

−

iq

p

cos2

11

4π = λ

(where i = −1), then the

value of λ must be equal

to

8. If απ

= e

i2

7 and f x( ) =

A A xk

kk

01

20

+=Σ , then the

value of Σr

rf x= 0

6

( )α

= +n A A xnn( 0 + A xn

n2

2 )

then n must be equal to

9. The sum of maximum andminimum modulus of acomplex number zsatisfying| | ,z i i− ≤ = −25 15 1must be equal to

14

Complex Numbers

0 000

1 111

2 222

3 333

4 444

5 555

6 666

7 777

8 888

9 999

0 000

1 111

2 222

3 333

4 444

5 555

6 666

7 777

8 888

9 999

0 000

1 111

2 222

3 333

4 444

5 555

6 666

7 777

8 888

9 999

0 000

1 111

2 222

3 333

4 444

5 555

6 666

7 777

8 888

9 999

0 000

1 111

2 222

3 333

4 444

5 555

6 666

7 777

8 888

9 999

0 000

1 111

2 222

3 333

4 444

5 555

6 666

7 777

8 888

9 999

0 000

1 111

2 222

3 333

4 444

5 555

6 666

7 777

8 888

9 999

0 000

1 111

2 222

3 333

4 444

5 555

6 666

7 777

8 888

9 999

0 000

1 111

2 222

3 333

4 444

5 555

6 666

7 777

8 888

9 999

Numerical Grid-Based Problems

0 00

0 2 4 7

0

1 111

2 222

3 333

4 444

5 555

6 666

7 777

8 888

9 999

10. If z1 and z2 are complexnumbers, such that| |15 131 2

2z z−+ +| |13 151 2

2z z

= +λ (| | | | )z z12

22 , then

the value of

λ λ λ λ ...... ∞ must

be equal to

11. If 27 3 5cos sinθ θ= a sin 8θ− +b csin sin6 4θ θ+ d sin 2θ and θ is realthen the value ofa b4 4+ + +c d4 4must

be equal to

12. If z is a complex numberand the minimum valueof| | |z z+− + −1 2 3| | |z is λ and ify x x= + = −2 3 3[ ] [ ]λ ,then the value of[ ]x y+must be equal to

(where [.] denotes thegreatest integerfunction)

13. Let A A An1 2, , . . . , bevertices of n sidesregular polygon such

that1

1 2A A= 1

1 3A A

+ 1

1 4A A, then the value

of n must be equal to

14. If ω( )≠ 1 is a cube root of

unity, then the value of( )( )1 1 2+ +ω ω

( )( )1 13 4+ +ω ω( )1 5+ ω K ( )1 30+ ω

must be equal to

15. The value of

2199

199 sinπ

sin

2

199

π

sin3

199

π

K sin

198

199

π

must be equal to

16. If

Σ2008

12009

rr

=−( )

cos2

2009

πr

= − n

2,then the digit in the

unit’s place of( )9417709487 n must be

equal to

15

Complex Numbers

0 000

1 111

2 222

3 333

4 444

5 555

6 666

7 777

8 888

9 999

0 000

1 111

2 222

3 333

4 444

5 555

6 666

7 777

8 888

9 999

0 000

1 111

2 222

3 333

4 444

5 555

6 666

7 777

8 888

9 999

0 000

1 111

2 222

3 333

4 444

5 555

6 666

7 777

8 888

9 999

0 000

1 111

2 222

3 333

4 444

5 555

6 666

7 777

8 888

9 999

Given below are Matching Type Questions, with two columns (each having some items) each . Each item of Column I has to be

matched with the items of Column II, by encircling the correct match(es).

NOTE An item of Column I can be matched with more than one items of Column II. All the items of Column II have to be matched.

1. Observe the following columns :

Column I Column II

(A) Locus of the point z satisfying the equation

Re ( ) Re ( )z z z2 = +(P) A hyperbola with eccentricity 2

(B) Locus of the point z satisfying the equation

| | | | ,z z z z R− + − = ∈ +1 2 λ λ and λ < −| | |z z1 2

(Q) A straight line

(C) Locus of the point z satisfying the equation2

1

z i

zm

−+

=

where i = −1 and m R∈ +

(R) An ellipse

(S) A rectangular hyperbola

(T) A circle

0 000

1 111

2 222

3 333

4 444

5 555

6 666

7 777

8 888

9 999

Matrix-Match Type

T T TT T T(A) (B) (C)P P PP P PQ Q QQ Q QR R RR R RS S SS S S

0 000

1 111

2 222

3 333

4 444

5 555

6 666

7 777

8 888

9 999

16

Complex Numbers


Column I Column II

(A) If | | ;ai i< ≥1 0λ for i n=1 2 3, , , ..., and λ λ λ1 2 1+ + + =K n

and ω is a complex cube root of unity, then|λ ω λ ω1 1 2 2

2a a+ + +K λ ωn nna |cannot exceed

(P) | || |

zz

n + 1

(B) If Re ( )z < 0, then the value of ( )1 2+ + + +z z znK can not

exceed(Q) 2

(C) If ω( )≠ 1 is a cube root of unity, then1

3| |1 2 3 32 3 1+ + + + −ω ω ωK n n ( )n N∈ cannot exceed

(R) n

(S) 1

(T) | | | | . . .| |a a an1 2+ +


Column I Column II

(A) If1 2 1, , , ...,ω ω ωn − are the n, nth roots of unity, then

( )( )2 2 2− −ω ω K( )2 1− −ωn equals

(P) 2 1n −

(B) If z z zn1 2, , ,K lie on a circle | |z = 2 , then the value of

| |z z zn1 2 4+ + + −K

1 1 1

1 2z z zn

+ + ... is

(Q) n n n nC C C C0 1 2 3− + − +...

(C) If p is a multiple of n, then the sum of the pth power of nth

roots of unity is

(R) nn

nn

nnC C C− − −+ + +1 2 3 ...+ nC0

(S) nC1

(T) ( ... )( ) ( ) ( )2 10

2 11

2 1 1n n nnC C C+ + ++ + + −


Column I Column II

(A) If z z z1 2 3, , are the vertices of an equilateral triangle with z0 asits nine point centre, then z z z1

222

32+ + is

(P) ( ) ( )z z z z1 22

1 32− = − ( )z z3 2−

(B) If z z z1 2 3, , are the vertices A B C, , respectively of an isoscelesright angled triangle with right angle at C, then

(Q) 3 02z

(C) If z z z1 2 3, , are the vertices A B C, , respectively of an isoscelestriangle and the angles at Band Care each equal to π/ 6, then

(R) ( )z z2 32− = − −3 3 1 1 2( )( )z z z z

(S) z z z z z z1 2 2 3 3 1+ +(T) z z

z zei2 3

1 3

2−−

= π/




1. If a ib i+ = −( )1 3 100, where i = −1, then

a b= =K , . . . .

2. If the real part of ( )/( )z z i+ +4 2 is equal to 1/2, wherei = −1, then the point z lies on a .....

3. If z z i+ + + =2 1 0| | , where i = −1, then z = . . . . . . . .

4. If z x iy i z a ib a ba b= + = − = − ≠ ± ≠, , , , ,/where 1 01 3

thenx

a

y

bk a b− = −( ),2 2 where k = . . . . . . .

5. If α ≠ 1 is a nth root of unity, thenS = + + +1 3 5 2α α . . . . . . . up to n terms is equal to......

6. If the expression

sin cos tan ( )

sin

x xi x

ix

2 2

1 22

+

−

+

is real,

where i = −1 then the set of all possible values of x is ....

7. For any two complex numbers z z1 2, and any real numbersa and b az bz bz az.| | | | . . . . .1 2

21 2

2− + + =8. If a and b are real numbers between 0 and 1 such that the

points z a i z bi1 2 1= + = +, and z3 0= where i = −1,

form an equilateral triangle, then a = . . . . and b = . . . . .

9. ABCD is a rhombus. Its diagonals AC and BD intersect atthe point M and satisfy BD AC= 2 . If the points D and Mrepresent the complex numbers1 + i and 2 − i respectively,where i = −1, then A represents the complex number......

or .....

10. Suppose z z z1 2 3, , are the vertices of an equilateral triangleinscribed in the circle | | .z = 2 If z i1 1 3= + , where

i = −1, then z z2 3= =. . . . . , . . . . ..

11. The value of expression

1 2 2 2 3 32 2⋅ − − + − −( ) ( ) ( ) ( )ω ω ω ω+ + − − −. . ( ) ( ) ( ),n n n1 2ω ω

where ω is an imaginary cube root of unity, is ....

12. The equation | | | |z z z z k− + − =12

22 (where k is a real

number) will represent a circle if ....

13. If| | | | ,z z2 21 1− = + then z lies on a .....

14. If A and B have affixes z1 and z2, then we define the

complex slope of the line AB as µ = − −( )/( )z z z z1 2 1 2 .

Two lines with complex slopes µ1 and µ 2 will be

perpendicular if µ1 and µ 2 are connected by .....

15. Length of perpendicular from w to the line az az c+ + = 0

(where c is real) is ....

16. If z z1 2, be two non-zero complex numbers satisfying the

equationz z

z z1 2

1 2

1+−

= , then

z

z

z

z1

2

1

2

+

is ....

17. If z i z i1 22 5 3= − = −, , where i = −1, then magnitude of

projection of z1 on z2 is ....

18. The locus of z in the argand diagram is the circle | | ,z = 2then the locus of z + 1 is .......

19. The maximum amplitude of z such that| |z i− − ≤1 3 1

where i = −1, is ....

20. The principal value of arg z where

z i= + +16

5

6

5cos sin

π πwhere i = −1 is given by ....

21. If the real part ofz

z

++4

2 1is equal to the imaginary part of

1

3z +, then the point z lies on .....

22. Let γ be the imaginary part of ( ) ( )z e z ei i− + −− −1 1 1α α

where z is complex and i = −1 and α is real, then γ = 0

implies that z lies on a circle of centre .... and radius ......

23. If ω is an imaginary cube root of unity, then the value of

17

Complex Numbers


Column I Column II

(A) If G be the greatest and L be the least values of | |,z −1 if| |z i+ + ≤2 1 where i = −1, then

(P) LG = 9

(B) If G be the greatest and L be the least values of| |,z i+ 2 wherei = −1, if1 1 3≤ − ≤| |z , then

(Q) L G+ = 6

(C) If G be the greatest and L be the least values of | |z − 2 , if

| |z i+ ≤1 where i = −1 , then

(R) ( )2 2 42G L− =

(S) LG = 4

(T) G L− = 2


Fill in the Blanks

cos ( )ω ω π π17 7

3+ +

is ......

1. If a b c, , are three real numbers such that a b c+ + = 0 (at

least one of a b c, , is different from zero) and

az bz cz1 2 3 0+ + = , then z z z1 2 3, , are collinear.

2. If three complex numbers are in arithmetic progression,

then they lie on a circle in the complex plane.

3. If the complex numbers z z1 2, and z3 represent the vertices

of an equilateral triangle such that| | | | | |,z z z1 2 3= = then

z z z1 2 3 0+ + =4. If

( ) ,1 20 1 2

22

2+ + = + + + + + +x x a a x a x a x a xnr

rn

nK K

then a a a n0 3 6

13+ + + = −. . . . .

5. The cube roots of unity when represented on arganddiagram form the vertices of an equilateral triangle.

6. If z z2 1/( )− is always real, then z always lies on a circle.

7. If| / | ,z z− =4 2 then the greatest value of| |z is 5.

8. If ω π πk k n i k n= +cos ( / ) sin ( / ),2 2 where i = −1

for k n= −0 1 2 1, , , . . . . , ,

then ( ) ( ) ( )a b a b a b a bnn n+ + + = +−ω ω ω0 1 1K

9. If α β γ δ, , , are four complex numbers such that γ δ/ is real

and αδ βγ− ≠ 0, then

zt

t= +

+α βγ δ

, t R∈ represents a circle.

10. If | |z < 1 and z z z zkk= + + + + −1 2 1. . . . for

k n= 1 2, , . . . . , , then the point z = 0 lies within the polygonwith vertices, z z zn1 2, , . . . . , .

11. If z z z1 2 3, , are three non-zero and distinct complex

numbers such that1 1 1

01 2 3z z z

+ + = , then origin lies within

the triangle with vertices z z1 2, and z3.

12. If a b c ii i i, , , ( , , )= 1 2 3 are complex numbers and

z

a

a

a

b

b

b

c

c

c

=

1

2

3

1

2

3

1

2

3

, then z

a

a

a

b

b

b

c

c

c

=

1

2

3

1

2

3

1

2

3

13. | | | ( )| | ( )| | |z R z I z z≤ + ≤ 2 for all complex numbers z.

14. If z a b ic( )1 + = + , where i = −1 and a b c2 2 2 1+ + = ,

then1

1 1

+−

= ++

iz

iz

a ib

c.

15. If the triangles whose vertices are z z z1 2 3, , and a b c, , aresimilar, then a z z b z z c z z( ) ( ) ( )2 3 3 1 1 2 0− + − + − =

16. If z is a complex number, then zeiα is equivalent to rotatingthe line oz through an angle α in the anticlockwisedirection about o.

17. The inequality a ib c id+ < + , where i = −1, holds if a c<and b d< .

18. There exists a non-zero integral solution of the equation

( )1 2 5+ =i x x , where i = −1 .

The following questions consist of two statements, one labelled as ‘Assertion (A)’ and the other labelled as ‘Reason (R)’. You are toexamine these two statements carefully and decide if the Assertion (A) and the Reason (R) are individually true and if so, whether theReason (R) is the correct explanation for the given Assertion (A). Select your answer to these items using the codes given below andthen select the correct option.

Codes :

(A) Both A and R are individually true and R is the correct explanation of A

(B) Both A and R are individually true but R is not the correct explanation of A

(C) A is true but R is false

(D) A is false but R is true

1. Assertion (A) : If z is a complex number ( )z ≠ 1 , thenz

zz

| ||arg |− ≤1

Reason (R) : In a unit radius circle chord (AP) ≤ arc AP( )

(a) A (b) B

(c) C (d) D

2. Assertion (A) : If| |z ≥ 2, then the least value of z + 1

2is

3

2.

Reason (R) :| | | | | |z z z z1 2 1 2+ ≤ +(a) A (b) B

(c) C (d) D

3. Assertion (A) : 7 4 5 3+ > +i i, where i = −1

Reason (R) : 7 5> and 4 3>(a) A (b) B

(c) C (d) D

4. Assertion (A) : If| |z < −2 1, then| sin |z z2 2 1+ <α

Reason (R) :| | | | | |z z z z1 2 1 2+ < + and|sin |α ≤ 1

(a) A

(b) B

18

Complex Numbers

Assertion & Reason

True / False

(c) C

(d) D

5. Assertion (A) : ( ) ( ) ( )( )− − = − − =2 3 2 3 6

Reason (R) : If a and b both negative, then a b ab≠(a) A (b) B(c) C (d) D

6. Assertion (A) : Σr

nri i i

=

+= − = −

1

4 11

1,

Reason (R) : Sum of four consecutive powers of i is zero.(a) A (b) B(c) C (d) D

7. Assertion (A) : If5

112

1

z

zis purely imaginary then

2 3

2 311 2

1 2

z z

z z

+−

=

Reason (R) :| | | |z z=ie, | | | |,a ib a ib i+ = − = −1

(a) A (b) B(c) C (d) D

8. Assertion (A) : If A z B z C z( ), ( ), ( )1 2 3 are the vertices of anequilateral triangle ABC, then

argz z z

z z2 3 1

3 2

2

4

+ −−

= π

Reason (R) : If ∠ =B α, thenz z

z z

AB

BCei1 2

3 2

−−

= α

or argz z

z z1 2

3 2

−−

= α

(a) A (b) B(c) C (d) D

9. Assertion (A) : If | | ,| | ,| |z z z1 2 31 2 3= = = and| |z z z1 2 32 3 6+ + = , then the value of| |z z z z z z2 3 3 1 1 28 27+ + is 36Reason (R) :| | | | | | | |z z z z z z1 2 3 1 2 3+ + ≤ + +(a) A (b) B(c) C (d) D

10. Assertion (A) : If z i i= + + −( ) ( ),5 12 12 5 then the

principal values of arg ( )z are ± ±π π4

3

4, , where i = −1.

Reason (R) : If z a ib= + , then

zz a

iz a= ± +

+ −

| | | |

2 2for b > 0

and zz a

iz a= ± +

− −

| | | |

2 2for b < 0

(a) A (b) B(c) C (d) D

11. Assertion (A) : If| | ,z i− + ≤3 2 4 then the sum of least andgreatest value of| |z is 8Reason (R) :|| | | || | | | | | |z z z z z z1 2 1 2 1 2− ≤ + ≤ +(a) A (b) B(c) C (d) D

12. Assertion (A) : If z z z z1 2 1 2* | || |cos= θ whenA z B z( ), ( )1 2 and ∠ =AOB θ, thenz z1 2* is equal to Re ( )z z1 2 or Re( )z z1 2

Reason (R) : z z ei2 1= θ (Rotation theorem) and

cos ( )θ θ= Re ei and sin Im ( )θ θ= ei , where i = −1.

(a) A (b) B

(c) C (d) D

13. Assertion (A) : If cos( )1 − = +i a i b where a b R, ∈ and

i = −1, then a ee

b ee

= +

= −

1

2

11

1

2

11cos , sin

Reason (R) : e iiθ θ θ= +cos sin

(a) A (b) B

(c) C (d) D

14. Assertion (A) : If xx

+ =11 and p x

x= +4000

4000

1and q be

the digit at unit place in the number 2 12n

n N+ ∈, and

n > 1, then the value of p q+ = 6.

Reason (R) : ω ω, 2 are the roots of xx

+ = −11, then

xx

2

2

11+ = − , x

x

3

3

12+ =

(a) A (b) B

(c) C (d) D

15. Assertion (A) : Let z be a complex number satisfying| | | |,z z− ≤ −3 1 | | | |,| |z z z i− ≤ − −3 5 ≤ +| |z i and| | | |.z i z i− ≤ − 5 Then the area of region in which z lies is

12 sq unit.

Reason (R) : Area of trapezium

= ×1

2(Sum of parallel sides)

× ( )Distance between parallel sides

(a) A (b) B(c) C (d) D

16. Assertion (A) : If z i1 9 5= + and z i2 3 5= + and

if argz z

z z

−−

=1

2 4

π, then the value of | |z i− − =6 8 3 2,

where i = −1.

Reason (R) : The three points are non-collinear, then theymust lie on a circle and the angle at centre by a chord isdouble the angle on circumference by that chord.

19

Complex Numbers

A z( )1

C z( )3B z( )2

α

θA z( )2

B z( )1

O

(a) A (b) B (c) C (d) D

Objective Questions Type I [Only one correct answer]

Objective Questions Type II [One or more than one correct answer(s)]

Linked-Comprehension Type

Passage 1 1. (d) 2. (d) 3. (a) 4. (b) 5. (c) 6. (c) 7. (b) 8. (d) 9. (c) 10. (b) 11. (b) 12. (c)

Passage 2 1. (a) 2. (a) 3. (c) 4. (b) 5. (b) 6. (c) 7. (d)

Passage 3 1. (c) 2. (b) 3. (d) 4. (a) 5. (d) 6. (b) 7. (c) 8. (a)

Passage 4 1. (a) 2. (d) 3. (c) 4. (b) 5. (a)

Passage 5 1. (d) 2. (b) 3. (c) 4. (c) 5. (d)

Passage 6 1. (b) 2. (b) 3. (d) 4. (d) 5. (a) 6. (c) 7. (c)

Passage 7 1. (d) 2. (c) 3. (d) 4. (b) 5. (c) 6. (c)

Passage 8 1. (b) 2. (d) 3. (b) 4. (c) 5. (c) 6. (a) 7. (c)

Passage 9 1. (c) 2. (c) 3. (c) 4. (c) 5. (d)

Passage 10 1. (b) 2. (c) 3. (a) 4. (c) 5. (d)

Numerical Grid-Based Problems

Matrix-Match Type

1. A → (P, S); B → (Q, R); C → (Q,T) 2. A → (Q, S); B → (P, T); C → (R)

3. A → (Q, S, T); B → (P); C → (R) 4. A → (P, R T); B → (P); C → (S)

5. A → (P, T); B → (Q, R, S); C → (S, T)

1. (c) 2. (b) 3. (c) 4. (b) 5. (c) 6. (a) 7. (d) 8. (b) 9. (d) 10. (b)

11. (c) 12. (d) 13. (a) 14. (b) 15. (b) 16. (a) 17. (c) 18. (c) 19. (b) 20. (d)

21. (b) 22. (c) 23. (d) 24. (a) 25. (d) 26. (d) 27. (c) 28. (b) 29. (a) 30. (a)

31. (a) 32. (d) 33. (a) 34. (d) 35. (a) 36. (b) 37. (b) 38. (c) 39. (c) 40. (a)

41. (a) 42. (a) 43. (b) 44. (a) 45. (b) 46. (c) 47. (c) 48. (a) 49. (b) 50. (d)

51. (c) 52. (b) 53. (c) 54. (d) 55. (a) 56. (c) 57. (a) 58. (b) 59. (b) 60. (b)

61. (c) 62. (c) 63. (a) 64. (b) 65. (a) 66. (b) 67. (c) 68. (a) 69. (a) 70. (c)

71. (b) 72. (d) 73. (a) 74. (a) 75. (a) 76. (a) 77. (b) 78. (b) 79. (c) 80. (c)

81. (b) 82. (b) 83. (a) 84. (c) 85. (a) 86. (d) 87. (b) 88. (c)

1. (b, c, d) 2. (b, c) 3. (b, c, d) 4. (b, d) 5. (a, d)

6. (b, c, d) 7. ( c, d) 8. (a, b, c) 9. (a, c, d) 10. (a, b)

11. (b, c, d) 12. (b, c) 13. (a, b) 14. (c, d) 15. (a, b, c, d)

16. (a, b, c) 17. (a, b, c, d) 18. (a, b, c) 19. (a, b, c, d) 20. (a, b, c, d)

21. (a, b, c, d)

1. 0 0 0 2 2. 0 2 1 6 3. 0 1 4 4 4. 0 3 8 1

5. 0 0 4 8 6. 0 0 6 0 7. 1 6 4 8 8. 0 0 0 7

9. 0 0 5 0 10. 0 3 9 4 11. 1 3 2 9 12. 0 0 3 0

13. 0 0 0 7 14. 1 0 2 4 15. 0 3 9 8 16. 0 0 0 7

Answers

Fill in the Blanks

1. a b= − =2 2 399 99, 2. straight line 3. − −2 i 4. k = 4

5. ( )/ ( )− −2 1n α 6. x n n n I= + ∈2 4π π π, / , 7. ( )(| | | | )a b z z2 21

22

2+ +

8. a b= = −2 3 9. 32

− ior 1

3

2− i 10. z z i2 32 1 3= − = −, 11.

1

41 3 42n n n n( )( )− + +

12. k z z≥ −1

21 2

2| | 13. an imaginary axis 14. µ µ1 2 0+ = 15.| |

| |

aw aw c

a

+ +2

16. zero 17.11

1018. a circle 19. π / 2

20. − 3

5

π21. { ( ) }4 9 8zz z z+ + + { ( ) }zz z z+ + +3 9 + − + + + =i z z zz z z( )( ( ) )4 2 1 0

22. centre (1, 0); radius 1 23. − 1

2

True / False

Assertion & Reason

21

Complex Numbers

1. T 2. F 3. T 4. T 5. T 6. F 7. F 8. F 9. F 10. F

11. T 12. T 13. T 14. T 15. T 16. T 17. F 18. F

1. (a) 2. (b) 3. (d) 4. (a) 5. (d) 6. (d) 7. (a) 8. (a) 9. (b) 10. (b)

11. (a) 12. (b) 13. (a) 14. (b) 15. (d) 16. (a)

complex no. arihant

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