1.

Complex numbers with 0 as real part are called:

A. imaginary numbers
B. pure non real numbers
C. pure imaginary numbers
D. pure complex numbers

2. The argument of which of the following number is not defined?

A. 0
B. 1
C. 1/0
D. i

3. If θ is the principal argument Arg(z) of a complex number z, then:

A. 0 ≤ θ ≤ 2π
B. −π ≤ θ ≤ π
C. −π ≤ θ < π
D. −π < θ ≤ π
4. For k ∈ Z, the relationship between arg(z) and Arg(z) is:
A. arg(z)= Arg(z)+2kπ
B. Arg(z)= arg(z)+2kπ
C. arg(z)= Arg(z)−2kπ
D. All of these

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BSc Multiple Choice Questions Page 2 of 12

5. Which of the following is unique?

A. Arg(z)
B. arg(z)
C. Both A and B
D. None of these

6. We can write r(cos θ + i sin θ) as:

A. rsicθ
B. rcsiθ
C. rcisθ
D. r cos θ

7. The value of arg(5) is:

A. 0◦
B. 90◦
C. 180◦
D. 270◦
8. The value of arg(−5) is:
A. 0◦
B. 90◦
C. 180◦
D. 270◦

9. The value of arg(5i) is:

A. 0◦
B. 90◦
C. 180◦
D. 270◦
10. The value of arg(−5i) is:
A. 0◦
B. −90◦
C. 180◦
D. 270◦

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BSc Multiple Choice Questions Page 3 of 12
11. The value of Arg(−5i) is:
A. 0◦
B. 90◦
C. 180◦
D. 270◦
12. The value of Arg(−5) is:
A. 0◦
B. 90◦
C. 180◦
D. 270◦

13. The equation of a circle with center at origin and radius 2 is:
A. |z| = 2
B. |z| = 4
√
C. |z| = 2
D. None of these

14. Which of the following is not true?

A. arg(z1z2)= arg(z1) + arg(z2)
B. Arg(z1z2)= Arg(z1) + Arg(z2)
C. zz = |z|2
D. arg( zz1 )= arg(z1) - arg(z2)
2

15. The least value of |z1 + z2| is:

A. ||z1| + |z2||
B. ||z1||z2||
C. ||z1|/|z2||
D. ||z1| − |z2||
16. The inequality ||z1| − |z2|| ≤ |z1 + z2| ≤ |z1| + |z2| is called:
A. Triangle Inequality
B. Minkowski Inequality
C. Cauchy-Schwarz Inequality
D. Holder’s Inequality

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BSc Multiple Choice Questions Page 4 of 12

17. The principal argument of any complex number can not be:
A. 7π
8
B. 7π
6
C. π2
π
D. −
2
19. If |z| = 2i(1 − i)(2 − 4i)(3 + i), then |z| equals:
A. 20
B. −20
C. 40
D. −40
20. z = a + ib is pure imaginary if and only if:
A. z = −z
B. z = z
C. z = −z
D. z = z−1
21. If z1 = 24 + 7i and |z2| = 6, then the least value of |z1 + z2| is:
A. 31
B. 19
C. −19
D. −13
|az+b|
22. =1, for z =?
|bz+a| | |
A. 1
B. 0
C. 2
D. −1
23. Locus of the points satisfying Re(iz) = 3 is:
A. a line parallel to x-axis
B. a line parallel to y-axis
C. a circle
D. a parabola

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BSc Multiple Choice Questions Page 5 of 12

24. For all integers n, we have:

A. (cos θ + i sin θ)n = cos nθ + i sin nθ
B. (cos θ + i sin θ)n = cos nθ − i sin nθ
C. (cos θ − i sin θ)n = cos nθ + i sin nθ
D. (cos θ√+ i sin θ)−n = cos nθ + i sin nθ
3− i 6
25. The value of ( ) is:
√
3+i
A. 0
1
B. 2
C. 1
D. −1
26. For any integers n, we have (sin x + i cos x)n =
A. sin n(π − x) + i cos n(π − x)
2 2
B. cos n(2π − x) + i sin n(2π − x)
C. sin n(π + x) + i cos n(π + x)
2 2
D. sin n(π + x) + i cos n(π + x)
2 2

1
27. If x = cos θ + i sin θ, then the value of x =
A. cos θ + i sin θ
B. sin θ + i cos θ
C. cos θ − i sin θ
D. sin θ − i cos θ
1
28. If x = cos θ + i sin θ, then the value of n =
x
A. cos nθ + i sin nθ
B. sin nθ + i cos nθ
C. cos nθ − i sin nθ
D. sin nθ − i cos nθ

29. If x = cos θ + i sin θ, then the value of xn +xn1 =

A. 2i sin nθ
B. 2i cos nθ
C. 2 cos nθ
D. 2 sin nθ

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BSc Multiple Choice Questions Page 6 of 12

30. If x = cos θ + i sin θ, then the value of xn − xn1 =

A. 2i sin nx
B. 2i cos nx
C. 2 cos nx
D. 2 sin nx
31. If |z| = r and arg(z) = θ, then all the nth roots of z are:
1
A. r n 2kπ+θ
n )
cis( 2 π+θ
1
kn )
B. r n
2π+kθ
cis( n )
1
2kπ+θ
C. r n
kn )
cis(
1
D. r n
cis(
32. 1, ω, ω2, ..., ωn−1 are nth roots of:
A. zero
B. unity
C. 2i
D. None of these

33. If z is a root of w, then which of following is also a root of w?

A. 1
B. −z
C. z
D. z−1

34. Three cube roots of 8i are:

A. 2, 2ω, 2ω2
B. 2i, 2iω, 2iω2
C. −2, −2ω, −2ω2
D. −2i, −2iω, −2iω2
35. Sum of four fourth roots of unity is:
A. 0
B. 1
C. i
D. −1

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 BSc Multiple Choice Questions Page 7 of 12
(cos θ+i sin θ)n
(cos φ+i sin φ)m equals:
36.

A. cos(mθ + nφ) + i sin(mθ + nφ)

B. cos(nθ + mφ) + i sin(nθ + mφ)
C. cos(mθ − nφ) + i sin(mθ − nφ)
D. cos(nθ − mφ) + i sin(nθ − mφ)
(cos α—i sin α)11
37.
(cos β+i sin β)9
equals:
A. cos(11α + 9β) + i sin(11α + 9β)
B. cos(11α − 9β) + i sin(11α − 9β)
C. cos(−11α + 9β) + i sin(−11α + 9β)
D. cos(−11α − 9β) + i sin(−11α − 9β)
−
eiz−e iz
38. For a complex number z, iz −iz
i(e
=
A. cot z +e )
B. tan z
C. coth z
D. tanh z

39. sin2 z + cos2 z=

A. 1
B. −1
C. 0
D. 2 sin z cos z

40. sin iz=

A. sinh z
B. sinh iz
C. i sin z
D. i sinh z

41. cos iz=

A. cosh z
B. cosh iz
C. i cos z
D. i cosh z

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BSc Multiple Choice Questions Page 8 of 12

42. an iz=
A. tanh z
B. tanh iz
C. i tan z
D. i tanh z

43. sinh iz=

A. sin z
B. i sin z
C. sinh z
D. i sinh z

44. cosh iz=

A. cos z
B. i cos z
C. cosh z
D. i cosh z

45. tanh iz=

A. tan z
B. i tan z
C. tanh z
D. i tanh z

Important Points
(i). ez is never zero.
(ii). For z = x + iy, |ez| = ex.
(iii). |eiθ| = 1, where θ ∈ R.
(iv). ez = 1 if and only if z = 2kπi, where k ∈ Z.
(v). ez1 = ez2 if and only if z1 − z2 = 2kπi, where k ∈ Z.

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BSc Multiple Choice Questions Page 9 of 12

46. Multiplication of a vector z by ... rotates the vector z counterclockwise through an angle
of measure α.
A. eα
B. e−α
C. eiα
D. e−iα

47. −3 − 4i=
A. 5 i tan−1 43

e
−1 4
B.
5ei(− tan −1 34)
C.
5ei(π−tan
i(π+tan−1 4 )
3)

D. 5e 3

48. For any complex number z, log z=

A. ln |z| + i arg z
B. ln z + i arg |z|
C. ln |z| + i arg |z|
D. All of these

49. Which number(s) has(have) no complex logarithm?

A. 0
B. Negative real numbers
C. Non positive real numbers
D. None of these

50. For any complex number z, Logz=

A. ln |z| + i Arg z
B. ln z + i Arg |z|
C. ln |z| + i Arg |z|
D. All of these
51. The value of Log(−i) is:
A. π2 i
3π
B. 2
i
C. − 2π i
3π
D. − 2 i

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BSc Multiple Choice Questions Page 10 of 12

52. If x is any negative real number, then Logx is:

A. ln x + iπ
B. ln x − iπ
C. ln(−x) + iπ
D. ln(−x) − iπ
53. log(ez)=
A. z
B. z + 2nπ
C. z + 2nπi
D. ez

54. If z is a positive real number, then

A. Log(z)=log(z)
B. Log(z)=log(z)+ 2nπ
C. log(z)=Log(z)+ 2nπ
D. None of these
55. sinh−1 z= √
A. log(z + z2 + 1)
√
B. log(z − z2 + 1)
√
C. log(z + z2 − 1)
√
D. log(z − z2 − 1)
56. cosh z=
−1
√
A. log(z + z2 + 1)
√
B. log(z − z2 + 1)
√
C. log(z + z2 − 1)
√
D. log(z − z2 − 1)
57. sin−1 z= √
A. i log(iz + 1 + z2)
√
B. −i log(iz − 1 − z 2 )
√
C. −i log(iz + 1 + z 2 )
√
D. −i log(iz + 1 − z 2 )

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BSc Multiple Choice Questions Page 11 of 12

58. If z and w are complex numbers, then zw=

A. exp(z log w)
B. z exp(log w)
C. exp(w log z)
D. w exp(log z)

59. If z and w are complex numbers, then the principal value of zw is:
A. exp(zLogw)
B. z exp(Logw)
C. exp(wLogz)
D. w exp(Logz)
i
60. The principal value of i is:
π
A. e2 π
B. −e 2
−π
C. e2
π
−
D. −e 2
61. The principal value of (−1)i is:
π
A. e
B. e−π
C. −eπ
D. −e−π
62. The principal value of −( i)−i is:
π
A. e π
2

B.π −e 2
−
C. e2
π
− 2
D. −e
63. If a is a positive real number, then the principal value of ai is:
A. cos(ln a) + i sin(ln a)
B. cos(a) + i sin(a)
C. sin(a) + i cos(a)
D. sin(ln a) + i cos(ln a)

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BSc Multiple Choice Questions Page 12 of 12
64. Log(1 − i)=
A. 1 ln 2 + πi
2 4
1 πi
B. 2 ln 2 − 4
1
C. ln 2 + 3πi
2 4
1 3πi
D. ln 2 −
2 4
√
i+ 3
65. (−1 + i) =
√
A. exp[(i − √ 3) log(−1 − i)]
B. exp[(i
C. exp[(−1 log(i + √
+ +3)i)log(−1 3)]
+ i)]
√
D. exp[(i + 3) log(−1 − i)]

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