MCQ QUESTIONS FOR COMPLEX NUMBERS

101
1. If a + ib =  i k , then ( a , b ) equals to
k =1

(a) ( 0,1 ) (b) ( 0, 0 ) (c) ( 0, −1) (d) ( 1,1)

n
 1+ i 
2. If   = −1, n  , then least value of n is :
 1− i 
(a) 1 (b) 2 (c) 3 (d) 4

2
3. The conjugate of a complex number z is z = . Then Re ( ) equals to
1− i
(a) −1 (b) 0 (c) 1 (d) 2

4. The number of complex number z such that z − i = z + i = z + 1 is

(a) 0 (b) 1 (c) 2 (d) inf inite

5. If z + 2 z = π + 4i , then Im ( z ) equals

(a) π (b) 4 (c) π 2 + 16 (d) None of these

6. If z = z + 3 − 2i , then z equals

7 7 5 5
(a) + i (b) − + 2i (c) − + 2i (d) + i
6 6 6 6

( )
11
7. If ω (  1) is a cube root of unity and 1 + ω 2 = a + bω + cω 2 , then ( a , b, c ) equals

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

1 + y + ix
8. If x 2 + y 2 = 1 and x  −1 then equals
1 + y − ix

(a) 1 (b) 2 (c) x + iy (d) y + ix

9. If z is non-zero complex number, then arg( z ) + arg( z ) equals

(a) 0 (b) π (c) 2π (d) None of these

10. If z  and 2 z = z + i , then z equals to

3 1 3 1 3 1 3 1
(a) + i (b) + i (c) + i (d) + i
6 2 6 3 6 4 6 6

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 7 7
 1 1   1 1 
11. If z =  + i + − i  , then
 3 2   3 2 

(a) Re ( z ) = 0 (b) Im ( z ) = 0 (c) Re( z )  0, Im ( z )  0 (d) Re( z )  0, Im ( z )  0

( ) = (1 + ω )
n n
12. If ω (  1) is a complex cube root of unity and 1 + ω 4 8
, then the least positive
integral value of n is

(a) 2 (b) 3 (c) 6 (d) 12

1 + cos θ + i sin θ  π
13. If z =  0  θ   then z equals
sin θ + i ( 1 + cos θ )  2

θ
(a) 2 sin θ (b) 2 cos θ (c) 1 (d) cot  
 2

14. All the roots of ( z + 1) = z lie on

4 4

(a) a straight line parallel to x -axis

(b) a straight line parallel to y -axis

(c) a circle with centre at −1 + 0i

(d) a circle with centre at 1+ i

15. If α (  1 ) is a fifth root of unity and β (  1 ) is the fourth root of unity then
z = ( 1 + α )( 1 + β ) ( 1 + α 2 )( 1 + β 2 )( 1 + α 3 )( 1 + β 3 ) equals

(a) α (b) β (c) αβ (d) 0

16. Suppose z1 , z2 , z3 are vertices of an equilateral triangle whose circum −3 + 4i , then

z1 + z2 + z3 is equal to

(a) 5 (b) 10 3 (c) 15 (d) 15 3

5
17. If z  0 lies on the circle z − 1 = 1 and ω = , then ω lies on
z
(a) a circle (b) an ellipse (c) a straight line (d) a parabola

25
18. If z = 3i + , then z cannot exceed
z + 3i
(a) 3 (b) 8 (c) 16 (d) 18

19. If z − 1 = z + 1 = z − 2i , then value of z is

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 5 3
(a) 1 (b) 1 (c) (d)
4 4

20. Ther number of complex numbers satisfying z = iz 2 is

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

21. If z  , z  , and a = z 2 + 3z + 5, then a cannot take value

−2 5 11 11
(a) (b) (c) (d) −
5 2 4 5

22. Suppose a, b, c  , and a = b = c = 1 and abc = a + b + c, then ab + bc + ca is equal to

(a) 0 (b) −1 (c) 1 (d) None of these

2
23. The Number of complex numbers z which satisfy z 2 + 2 z = 2 is

(a) 0 (b) 2 (c) 3 (d) 4

24. Suppose a  and the equation z + a z + 2i = 0 has no solution in , then a satisfies the
relation.

(a) a  1 (b) a  1 (c) a  2 (d) a  2

be such that z = ( 1 + z ) = 1, then the least value

n n
25. Suppose z is a complex number and n
of n is

(a) 3 (b) 6 (c) 9 (d) 18

z−i 1
26. Let z   i be a complex number such that is purely imaginary number, then z + is
z+i z

(a) a non-zero real number other than 1

(b) a purely imaginary number

(c) a non-zero real number

(d) 0

27. The points z1 , z2 , z3 , z4 are in the complex plane are the vertices of a parallelogram taken in
order if and only if

(a) z1 + z4 = z2 + z3 (b) z1 + z3 = z2 + z4 (c) z1 + z2 = z3 + z4 (d) None of these

28. If the complex numbers z1 , z2 and z 3 represent the vertices of an equilateral triangle such that

z1 = z2 = z3 , then

(a) z1 + z2 + z3 = 0 (b) z1 + z2 − z3 = 0 (c) z1 − z2 + z3 = 0 (d) z1 + z2 + z3  0

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 6
 2πk 2πk 
29. The value of S =   sin
k =1 7
− i cos
7 
is

(a) −1 (b) 0 (c) −i (d) i

30. The complex numbers sin x + i cos 2 x and sin x − i cos 2 x are conjugate to each other for

 1
(a) x = nπ, n  I (b) x =  n + π, n  I (c) x = 0 (d) no value of x
 2 

31. If z1 and z 2 are two complex numbers and a, b are two real numbers, then
2 2
az1 − bz2 + bz1 + az2 equals

( )
(a) a 2 + b 2 z1 z2 (b) a 2 + b 2 ( )( z1
2
(
+ z2 2 ) (c) ( a 2 + b 2 ) z1 + z2
2 2
) (d) 2ab z1 z 2

32. If a and b are real numbers between 0 and 1 such that the points z1 = a + i , z2 = 1 + ib and
z3 = 0 form an equilateral triangle, then

(a) a = b = 2 − 3 should look like this

(b) a = 2 − 3, b= 3 −1

(c) a = 3 − 1, b = 2 − 3

(d) None of these

π
33. If z  0 is a complex number such that arg( z ) = , then
4
2
( ) 2 2
( )
(a) Re z = 0 (b) Im z = 0 (c) Re( z ) = Im z
2
( ) (d) None of these

34. Let z and w be two non-zero complex numbers such that z = w and arg ( z ) + arg ( w ) = π.
Then z equals

(a) w (b) − w (c) w (d) − w

z −1
35. If z = 1 and w = ( where z  −1 ) . Then Re( w ) equals
z +1

1 z 1 2
(a) 0 (b) − (c) (d)
z +1
2
z +1 z +1 2
z +1
2

36. Let z and w be two complex numbers such that z = w = 1 and z + iw = z − iw = 2. Then z
equals

(a) 1 or i (b) i or -i (c) 1 or -1 (d) i or -1

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 z − 5i
37. The complex numbers z = x + iy which satisfy thew equation = 1 , lie on
z + 5i

(a) the x -axis

(b) the straight line y = 5.

(c) a circle passing through origin

(d) None of these

38. The inequality z − 4  z − 2 represents the region given by

(a) Re ( z )  0 (b) Re ( z )  3 (c) Re ( z )  0 (d) Re ( z )  3

z1 − z2
39. If z1 and z 2 are two complex numbers such that = 1, then
z1 + z2

(a) z 2 = kz1 , k  (b) z2 = ikz1 , k  (c) z2 = z1 (d) None of these

40. For any complex number z , the minimum value of z + z − 2i is

(a) 0 (b) 1 (c) 2 (d) None of these

41. If x = 2 + 5i , then the value of x 3 − 5 x 2 + 33 x − 19 is equal to

(a) −5 (b) −7 (c) 7 (d) 10

1 − iz
42. If z = x + iy & w = , then w = 1 implies, that in the complex plane
z−i
(a) z lies on the imaginary axis

(b) z lies on real axis

(c) z lies on the unit circle

(d) None of these

1
43. The real part of z = is
1 − cos θ + i sin θ

1 1 1
(a) (b) (c) tan θ (d) 2
1 − cos θ 2 2

2z + 1
44. If the imaginary part of is −4 , then the locus of the point representing z in the complex
iz + 1
plane is

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

45. The area of the triangle whose vertices are the points represented by the complex number z , iz
and z + iz is

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 1 2 1 2 1 2 1
(a) z (b) z (c) z (d) z
4 8 2 2

x +1 w w2
46. If w is a complex cube root of unity, then a root of the equation w x + w2 1 = 0 is
w2 1 x+w

(a) x = 1 (b) x = w (c) x = w 2 (d) x = 0

z1 z2
47. Let z1 & z2 be two complex numbers such that + = 1, then the origin and points
z2 z1
represented by z1 & z2

(a) lie on a straight line (b) form a right triangle (c) form an equilateral triangle (d) None of these

( )
n
48. If 1 + x + x 2 = a0 + a1 x + a2 x 2 + ... + a2 n x 2 n , then value of a0 + a3 + a6 + ... is

(a) 1 (b) 2n (c) 2n−1 (d) 3n−1

1 1 − 2i 3 + 5i
49. Let z = 1 + 2i −5 10i , then
3 − 5i −10i 11

(a) z is purely imaginary (b) z is purely real (c) z = 0 (d) None of these

1
x y
50. If ( x + iy ) 3 = a + ib, then + equals
a b

(
(a) 4 a + b
2 2
) 2
(
(b) 2 a − b
2
) (
(c) 2 a + b
2 2
) (d) None of these

51. If z  , the minimum value of z + z − i is attained at

(a) exactly one point (b) exactly two points (c) infinite number of points (d) None of these

52. For all complex numbers z1 , z2 satisfying z1 = 12 and z2 − 3 − 4i = 5 , the minimum value of
z1 − z2 is

(a) 0 (b) 2 (c) 7 (d) 17

z−2
53. If z lies on the circle z − 1 = 1, then equals to
z

(a) 0 (b) 2 (c) −1 (d) None of these

1 1 1
54. If 1, w , ...., w n −1 are the nth roots of unity , then value of + + .... + is
2− w 2− w 2
2 − w n −1

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(a) n
1
(b)
n 2n − 1 ( ) (c)
( n − 2 ) 2 n −1
(d) None of these
2 −1 2n + 1 2n − 1
π π
55. If w = cos + i sin , then value of 1 + w + w 2 + ... + w n−1 is
n n

 π   π
(a) 1 + i cot   (b) 1 + i cot   (c) 1+ i (d) None of these
 2n   n

z
56. If z = 1& z  1, then all the values of lie on :
1 − z2
(a) a line not passing through the origin

(b) z = 2

(c) the x -axis

(d) the y -axis,

57. The locus of the centre of a circle which touches the circle z − z1 = a & z − z2 = b  a
externally is

(a) an ellipse (b) a hyperbola (c) a circle (d) a pair of straight lines.
2
58. If z 2 − 1 = z + 1, then z lies on

(a) a circle (b) the imaginary axis (c) the real axis (d) an ellipse

59. If z 2 + z + 1 = 0, where z is a complex number, then values of

2 2 2 2
 1  1  1  1
S =  z +  +  z 2 + 2  +  z 3 + 3  + ... +  z 6 + 6  is
 z  z   z   z 

(a) 12 (b) 18 (c) 54 (d) 6

60. If z + 4  3, then maximum value of z + 1 is

(a) 4 (b) 10 (c) 6 (d) 0

61. If z, w be two complex numbers such that z + iw = 0 and arg ( zw ) = ππ then arg z equals

3π π π 5π
(a) (b) (c) (d)
4 2 4 4

62. If z1 + z2 + z3 = 0 & z1 = z2 = z3 = 1, then value of z1 + z2 + z3 equals

2 2 2

(a) −1 (b) 0 (c) 1 (d) 3

63. If z satisfies the relation z − i z = z + i z , then

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(a) Im ( z ) = 0 (b) z = 1 (c) Re ( z ) = 0 (d) None of these

β−α
64. If α &β are distinct complex numbers with β = 1 , then value of equals
1 − αβ

(a) 1 (b) α (c) 2 (d) None of these

1 − z + z2
65. Suppose z  ,&z  . If w = is a real number, then z equals
1 + z + z2

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

4
66. If z − = 2, then the minimum value of z is
z

(a) 1 (b) 2 + 2 (c) 3 + 1 (d) 5 +1

1
67. If ω = 2, then the set of points x + iy = ω − lie on
ω
(a) circle (b) ellipse (c) parabola (d) hyperbola

 1 
68. If z = 1, z  1, then value of arg   cannot exceed
 1− z 

π 3π
(a) (b) π (c) (d) 2π
2 2

z2
69. If z  1, is real , then point represented by the complex z lies
z −1
(a) on circle with centre at the origin

(b) either on the real axis or on a circle not passing through the origin

(c) on the imaginary axis

(d) either on the real axis or on a circle passing through the origin
100
3 3
70. If 3 ( x + iy ) =  + i
49
 , y & x = ky , then value of k is
2 2 


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

( ) ( )
71. If ( 4 + i ) z + z − ( 3 + i ) z − z + 26i = 0, then the value of z is
2

(a) 13 (b) 17 (c) 19 (d) 11

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  π π
72. Let z = a  cos + i sin  , a  , a  1, then S = z 2015 + z 2016 + z 2017 + .... equals
 5 5

a 2015 a 2015 a 2015 a 2015

(a) (b) (c) (d)
z −1 1− z 1− a a −1

73. If z = 20i − 21 + 20i + 21, then one of the possible value of arg ( z ) equals

π π 3π
(a) (b) (c) (d) π
4 2 8

74. If ( a + ib ) = x + iy , ( x , y , a , b  ) , then ( b + ia )
11 11
equals

(a) y + ix (b) − y − ix (c) − x − iy (d) x + iy

, ω  1, is a cube root of unity and ( a + bω ) = x + yω, then ( b + aω )

7 7
75. If a, b, x, y 
equals

(a) y + xω (b) − y − xω (c) x + yω (d) − x − yω

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