Complex Numbers MCQ Quiz - Objective Question with Answer for Complex Numbers - Download Free PDF

Last updated on Jun 30, 2025

Latest Complex Numbers MCQ Objective Questions

Complex Numbers Question 1:

If α is the fifth root of unity, then 

  1. |1 + 2α + 3α2 + 4α3 + 5α4| = 0

  2. |1 + α + α2 + α3| = 1

  3. |1 + α + α2| = 2 cos (17π/5)

  4. |1 + α| = 2 cos (19π/10)

Answer (Detailed Solution Below)

Option 2 :

|1 + α + α2 + α3| = 1

Complex Numbers Question 1 Detailed Solution

Concept:

  • nth Root of Unity: The complex numbers satisfying αn = 1 are called the nth roots of unity.
  • For n = 5, the fifth roots of unity are: 1, α, α2, α3, α4, where α = e2πi/5.
  • The sum of all five roots of unity is always zero: 1 + α + α2 + α3 + α4 = 0
  • We can use symmetry of these roots on the unit circle in the complex plane to evaluate modulus expressions involving them.

 

Calculation:

Given that α is a fifth root of unity,

⇒ α5 = 1

The 5 roots are: 1, α, α2, α3, α4

Sum of all roots: 1 + α + α2 + α3 + α4 = 0

⇒ 1 + α + α2 + α3 = −α4

Take modulus on both sides:

|1 + α + α2 + α3| = |−α4|

Since α4 lies on the unit circle, |α4| = 1

⇒ |1 + α + α2 + α3| = 1

∴ The correct answer is Option (2): |1 + α + α2 + α3| = 1

Complex Numbers Question 2:

If \(z\ne0\) is a complex number, then what is  equal to?

  1. 0
  2. π/2
  3. π
  4. 2π

Answer (Detailed Solution Below)

Option 1 : 0

Complex Numbers Question 2 Detailed Solution

Concept:

1. The amplitude (or argument) of a complex number \( z = r(\cos\theta + i\sin\theta) \) is given by \( \text{amp}(z) = \theta \).

2. The conjugate of \( z \), denoted as \( \overline{z} \), has an amplitude \( \text{amp}(\overline{z}) = -\theta \), because conjugating a complex number reflects it across the real axis in the Argand plane.

Formula Used:

\( \text{amp}(z) + \text{amp}(\overline{z}) = \theta + (-\theta) = 0 \).

Calculation:

\( z = r(\cos\theta + i\sin\theta) \)

\( \overline{z} = r(\cos\theta - i\sin\theta) \)

\( \text{amp}(z) + \text{amp}(\overline{z}) = \theta + (-\theta) = 0 \)

Conclusion:

\( \therefore \text{amp}(z) + \text{amp}(\overline{z}) = 0 \).

Hence, the correct answer is Option 1.

Complex Numbers Question 3:

What is \( \left( \frac{\sqrt{3}+i}{\sqrt{3}-i} \right)^3 \) equal to?

  1. -1
  2. 0
  3. 1
  4. 3

Answer (Detailed Solution Below)

Option 1 : -1

Complex Numbers Question 3 Detailed Solution

Concept:

The key concept involves simplifying complex numbers in the polar form. We use the argument (angle) of the complex number and De Moivre's theorem, which states:

  (r(cosθ + i sinθ))n = rn(cos(nθ) + i sin(nθ))

Calculation:

⇒ \(\left( \frac{\sqrt{3} + i}{\sqrt{3} - i} \right)^3 \)

Multiply the numerator and the denominator by the conjugate of the denominator

⇒ \(\frac{\sqrt{3} + i}{\sqrt{3} - i} \times \frac{\sqrt{3} + i}{\sqrt{3} + i} \)

⇒ \(\frac{(\sqrt{3} + i)^2}{(\sqrt{3} - i)(\sqrt{3} + i)} \)

 \(\frac{2 + 2\sqrt{3}i}{4} = \frac{1 + \sqrt{3}i}{2} \)

Convert to polar form. The modulus r is 

\(r = \sqrt{\left( \frac{1}{2} \right)^2 + \left( \frac{\sqrt{3}}{2} \right)^2} = \sqrt{\frac{1}{4} + \frac{3}{4}} = \sqrt{1} = 1 \)

The argument θ is 

\(\theta = \tan^{-1}\left( \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} \right) = \tan^{-1}(\sqrt{3}) = \frac{\pi}{3} \)

So the polar form of the complex number is

\(1 \left( \cos \frac{\pi}{3} + i \sin \frac{\pi}{3} \right) \)

To cube the complex number, we use De Moivre’s Theorem

In our case, r = 0 so,

\(\left( \cos \frac{\pi}{3} + i \sin \frac{\pi}{3} \right)^3 = \cos \pi + i \sin \pi = -1 + 0i = -1 \)

Thus, the result of cubing the complex number is \(\boxed{-1} \)

Hence, the correct answer is Option 1.

Complex Numbers Question 4:

Let z1 = 2 + 3i and z2 = 3 + 4i. The set  

\(\rm S=\left\{z \in C:\left|z-z_{1}\right|^{2}-\left|z-z_{2}\right|^{2}=\left|z_{1}-z_{2}\right|^{2}\right\}\) represents a

  1. straight line with sum of its intercepts on the coordinate axes equals 14
  2. hyperbola with the length of the transverse axis 7
  3. straight line with the sum of its intercepts on the coordinate axes equals –18 
  4. hyperbola with eccentricity 2

Answer (Detailed Solution Below)

Option 1 : straight line with sum of its intercepts on the coordinate axes equals 14

Complex Numbers Question 4 Detailed Solution

Concept:

Locus of Points Defined by Difference of Squares of Distances:

  • In complex geometry, a point \( z \) such that \( |z - z_1|^2 - |z - z_2|^2 \) is constant represents a geometric locus.
  • If the difference is constant, it can represent a straight line.
  • The equation \( |z - z_1|^2 - |z - z_2|^2 = c \) simplifies to linear form in many cases.
  • Here, it simplifies to a straight line equation.

Complex Number:

  • Definition: A complex number is of the form \( z = x + iy \), where \( x \) is the real part and \( y \) is the imaginary part.
  • SI Unit: Dimensionless
  • Modulus: \( |z| = \sqrt{x^2 + y^2} \)

Distance in Complex Plane:

  • Definition: Distance between two complex numbers \( z_1 \) and \( z_2 \) is \( |z_1 - z_2| \)
  • Formula: \( |z_1 - z_2|^2 = (x_1 - x_2)^2 + (y_1 - y_2)^2 \)

 

Calculation:

Given,

\( z_1 = 2 + 3i,\quad z_2 = 3 + 4i \)

\( z \in \mathbb{C} \) such that \( |z - z_1|^2 - |z - z_2|^2 = |z_1 - z_2|^2 \)

Let \( z = x + iy \)

\( |z - z_1|^2 = (x - 2)^2 + (y - 3)^2 \)

\( |z - z_2|^2 = (x - 3)^2 + (y - 4)^2 \)

\( |z_1 - z_2|^2 = (-1)^2 + (-1)^2 = 2 \)

\( (x - 2)^2 + (y - 3)^2 - (x - 3)^2 - (y - 4)^2 = 2 \)

\( [x^2 - 4x + 4 + y^2 - 6y + 9] - [x^2 - 6x + 9 + y^2 - 8y + 16] = 2 \)

\( x^2 - 4x + 4 + y^2 - 6y + 9 - x^2 + 6x - 9 - y^2 + 8y - 16 = 2 \)

\( 2x + 2y - 12 = 2 \)

\( 2x + 2y = 14 \Rightarrow x + y = 7 \)

⇒ Equation of line: \( x + y = 7 \)

⇒ x-intercept = 7 (when y = 0), y-intercept = 7 (when x = 0)

∴ The locus represents a straight line with sum of intercepts = 14.

Complex Numbers Question 5:

Find the value of (1 - i)4, Where i = \(\sqrt {-1}\)

  1. -4i
  2. 4
  3. -4
  4. -1
  5. None of the above

Answer (Detailed Solution Below)

Option 3 : -4

Complex Numbers Question 5 Detailed Solution

Concept:

Power of i:

  • i = \(\sqrt{-1}\)
  • i2 = -1
  • i3 = -i × i2 = -i
  • i4 = (i2)2 = (-1)2 = 1
  • i4n = 1

 

Calculation:

To Find: Value of (1 - i)4

(1 - i)4

= [(1 - i)2]2

= [12 + i2 - 2i]2         (∵ (a - b)= a2 + b2 - 2ab)

= [1 - 1 - 2i]2            (∵ i2 = -1)

= [-2i]2

= 4i2

= 4 × -1

= -4

Top Complex Numbers MCQ Objective Questions

Find the conjugate of (1 + i) 3

  1. -2 + 2i
  2. -2 – 2i
  3. 1 - i
  4. 1 – 3i

Answer (Detailed Solution Below)

Option 2 : -2 – 2i

Complex Numbers Question 6 Detailed Solution

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Concept:

Let z = x + iy be a complex number.

  • Modulus of z = \(\left| {\rm{z}} \right| = {\rm{}}\sqrt {{{\rm{x}}^2} + {{\rm{y}}^2}} = {\rm{}}\sqrt {{\rm{Re}}{{\left( {\rm{z}} \right)}^2} + {\rm{Im\;}}{{\left( {\rm{z}} \right)}^2}}\)
  • arg (z) = arg (x + iy) = \({\tan ^{ - 1}}\left( {\frac{y}{x}} \right)\)
  • Conjugate of z =  = x – iy

 

Calculation:

Let z = (1 + i) 3

Using (a + b) 3 = a3 + b3 + 3a2b + 3ab2

⇒ z = 13 + i3 + 3 × 12 × i + 3 × 1 × i2

= 1 – i + 3i – 3

= -2 + 2i

So, conjugate of (1 + i) 3 is -2 – 2i

NOTE:

The conjugate of a complex number is the other complex number having the same real part and opposite sign of the imaginary part.

What is the value of (i2 + i4 + i6 +... + i2n), Where n is even number.

  1. 1
  2. 0
  3. -1
  4. None of the above

Answer (Detailed Solution Below)

Option 2 : 0

Complex Numbers Question 7 Detailed Solution

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Concept:

i2 = -1

i3 = - i

i4 = 1

i4n = 1

Calculation:

We have to find the value of (i2 + i4 + i6 +... + i2n)

(i2 + i4 + i6 +... + i2n) = (i2 + i4) + (i6 + i8) + …. + (i2n-2 + i2n)

= (-1 + 1) + (-1 + 1) + …. (-1 + 1)

= 0 + 0 + …. + 0

= 0

If (1 + i) (x + iy) = 2 + 4i then "5x" is

  1. 11
  2. 13
  3. 14
  4. 15

Answer (Detailed Solution Below)

Option 4 : 15

Complex Numbers Question 8 Detailed Solution

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Concept:

Equality of complex numbers.

Two complex numbers z1 = x1 + iy1 and z2 = x2 + iy2 are equal if and only if x1 = x2 and y1 = y2

Or Re (z1) = Re (z2) and Im (z1) = Im (z2).

Calculation:

Given: (1 + i) (x + iy) = 2 + 4i

⇒ x + iy + ix + i2y = 2 + 4i

⇒ (x – y) + i(x + y) = 2 + 4i

Equating real and imaginary part,

x - y = 2         …. (1)

x + y = 4        …. (2)

Adding equation 1 and 2, we get

x = 3

Now,

5x = 5 × 3 = 15

The value of ω6 +  ω7 + ω5 is

  1. ω5
  2. 1
  3. 0
  4. ω 

Answer (Detailed Solution Below)

Option 3 : 0

Complex Numbers Question 9 Detailed Solution

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Concept:

Cube Roots of unity are 1, ω and ω2

Here, ω = \(\frac{{ - {\rm{\;}}1{\rm{\;}} + {\rm{\;i}}\sqrt 3 }}{2}\) and ω2 = \(\frac{{ - {\rm{\;}}1{\rm{\;}} - {\rm{\;i}}\sqrt 3 }}{2}\)

 

Property of cube roots of unity

  • ω3 = 1
  • 1 + ω + ω2 = 0
  • ω3n = 1

 

Calculation:

ω6 +  ω7 + ω5

= ω5 (ω + ω2 + 1)

= ω5 × (1 + ω + ω2)

= ω5 × 0

= 0

What is the modulus of \(\rm \dfrac{4+2i}{1-2i}\) where \(\rm i=\sqrt{-1} ?\)

  1. 2√5 
  2. 4
  3. 3
  4. 2

Answer (Detailed Solution Below)

Option 4 : 2

Complex Numbers Question 10 Detailed Solution

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Concept:

Let z = x + iy be a complex number, Where x is called real part of the complex number or Re (z) and y is called Imaginary part of the complex number or Im (z)

Modulus of z = |z| = \(\rm \sqrt {x^2+y^2} = \sqrt {Re (z)^2+Im (z)^2}\)

Calculations:

Let \(\rm z= x + iy = \dfrac{4+2i}{1-2i}\)

\(\rm = \dfrac{4+2i}{1-2i}\times\dfrac{1+2i}{1+2i}\)

\(\rm= \dfrac{4+10i+4i^2}{1-4i^2}\)   

As we know i2 = -1 

\(\rm = \dfrac{4+10i-4}{1+4}\)

\(\rm x + iy =\dfrac{10i}{5} = 0 + 2i\)

As we know that if z = x + iy be any complex number, then its modulus is given by,|z| = \(\rm \sqrt{x^2+y^2}\)

∴ |z| = \(\rm \sqrt{0^2+2^2} = 2\)

Find the conjugate of (i - i2)3

  1. -2 - 2i
  2. -2 + 2i
  3. i - 1
  4. 2 + 2i

Answer (Detailed Solution Below)

Option 1 : -2 - 2i

Complex Numbers Question 11 Detailed Solution

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1Concept:

Let z = x + iy be a complex number.

  • Modulus of z = \(\left| {\rm{z}} \right| = {\rm{}}\sqrt {{{\rm{x}}^2} + {{\rm{y}}^2}} = {\rm{}}\sqrt {{\rm{Re}}{{\left( {\rm{z}} \right)}^2} + {\rm{Im\;}}{{\left( {\rm{z}} \right)}^2}}\)
  • arg (z) = arg (x + iy) = \({\tan ^{ - 1}}\left( {\frac{y}{x}} \right)\)
  • For calculating the conjugate, replace i with -i.
  • Conjugate of z = x – iy

Calculation:

Let z = (i - i2)3

⇒ z = i3 (1 - i) 3  = - i (1 - i)3

For calculating the conjugate, replace i with -i.

⇒ z̅  =  -(- i) (1 - (- i))3

⇒ z̅  =  i(1 + i)3

Using (a + b) 3 = a3 + b3 + 3a2b + 3ab2

⇒ z̅  =  i(1 + i3 +3 ×12 × i + 3 × i2 × 1 ) 

⇒ z̅  =  i(1 - i + 3i - 3

⇒ z̅  =  i(-2 + 2i)

⇒ z̅  = -2i + 2i2

⇒ z̅  = -2 - 2 i

So, the conjugate of  (i - i2)3 is -2 - 2i

The value of ω3n + ω3n+1 + ω3n+2, where ω is cube roots of unity, is

  1. -1
  2. \(\frac{{ - {\rm{\;}}1{\rm{\;}} + {\rm{\;\;i}}\sqrt 3 }}{2}\)
  3. 1
  4. 0

Answer (Detailed Solution Below)

Option 4 : 0

Complex Numbers Question 12 Detailed Solution

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Concept:

Cube Roots of unity are 1, ω and ω2

Here, ω =  \(\frac{{ - {\rm{\;}}1{\rm{\;}} + {\rm{\;\;i}}\sqrt 3 }}{2}\) and ω2\(\frac{{ - {\rm{\;}}1{\rm{\;}} - {\rm{\;\;i}}\sqrt 3 }}{2}\)

 

Property of cube roots of unity

  • ω3 = 1
  • 1 + ω + ω2 = 0
  • ω = 1 / ω 2 and ω2 = 1 / ω
  • ω3n = 1

 

Calculation:

We have to find the value of ω3n + ω3n+1 + ω3n+2

⇒ ω3n + ω3n+1 + ω3n+2 

=  ω3n (1 + ω + ω2)           (∵ 1 + ω + ω2 = 0)

= 1 × 0 = 0

If 1, ω,  ω2 are the cube roots of unity then the roots of the equation (x - 1)+ 8 = 0 are

  1. -1, 1 + 2ω1 + 2ω2, 
  2. -1, 1 - 2ω1 - 2ω2
  3. -1, 1, 2
  4. -2, -2ω-2ω2

Answer (Detailed Solution Below)

Option 2 : -1, 1 - 2ω1 - 2ω2

Complex Numbers Question 13 Detailed Solution

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Concept 

Cube Roots of unity are 1, ω and ω2

Here, ω = \(\frac{{ - \;1\; + \;i\sqrt 3 }}{2}\) and ω2 = \(\frac{{ - \;1\; - \;i\sqrt 3 }}{2}\)

Property of cube roots of unity:

  • ω3 = 1
  • 1 + ω + ω2 = 0
  • ω = 1 / ω 2 and ω2 = 1 / ω
  • ω3n = 1

 

Calculation:

Given that,

(x - 1)+ 8 = 0

⇒ (x - 1)3 = (-2)3

⇒ (x - 1) = -2(1)1/3

(x - 1) = -2(1, ω,  ω2)

⇒ x = -1, 1 - 2ω, 1 - 2ω2  

The smallest positive integer for which \(\rm \left(\frac{1-i}{1+i}\right)^{n}=-1\), where i = √-1, is:

  1. 1
  2. 2
  3. 3
  4. 4

Answer (Detailed Solution Below)

Option 2 : 2

Complex Numbers Question 14 Detailed Solution

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Concept:

Complex Numbers:

  • A complex number is a number of the form a + ib, where a and b are real numbers and i is the complex unit defined by i = √-1.
  • i2 = -1, i3 = -i, i4 = 1 etc.
  • In general, i4n + 1 = i, i4n + 2 = -1, i4n + 3 = -i, i4n = 1.
  • For a complex number z = a + ib, conjugate of z is z̅ = a - ib.

Calculation:

Rationalizing the complex number \(\rm \frac{1-i}{1+i}\), by multiplying and dividing by the conjugate of the denominator, we get:

\(\rm \frac{1-i}{1+i}=\frac{(1-i)(1-i)}{(1+i)(1-i)}=\frac{1^2-2i+i^2}{1^2-i^2}=\frac{-2i}{1+1}\) = -i.

Now, \(\rm \left(\frac{1-i}{1+i}\right)^{n}=-1\).

⇒ \(\rm (-i)^{n}=-1\)

⇒ (-i)n for n = 2.

(-i)2 = (-1)2 × (i)2 = 1 × -1 = -1.

∴  n = 2

The conjugate of the complex number \(\rm 3i+4\over2-3i\) is:

  1. \(\rm {-1\over13}-{18\over13}i\)
  2. \(\rm {18\over13}i+{1\over13}\)
  3. \(\rm {18\over13}i-{1\over13}\)
  4. \(\rm {1\over13}-{18\over13}i\)

Answer (Detailed Solution Below)

Option 1 : \(\rm {-1\over13}-{18\over13}i\)

Complex Numbers Question 15 Detailed Solution

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Concept: 

Let z = x + iy be a complex number.

  • Modulus of z = \(\left| {\rm{z}} \right| = {\rm{}}\sqrt {{{\rm{x}}^2} + {{\rm{y}}^2}} = {\rm{}}\sqrt {{\rm{Re}}{{\left( {\rm{z}} \right)}^2} + {\rm{Im\;}}{{\left( {\rm{z}} \right)}^2}}\)
  • arg (z) = arg (x + iy) = \({\tan ^{ - 1}}\left( {\frac{y}{x}} \right)\)
  • Conjugate of z = z̅ = x – iy


Calculation:

Given complex number is z = \(\rm 3i+4\over2-3i\)

z = \(\rm {3i+4\over2-3i}\times{2+3i\over2+3i}\)

z = \(\rm 6i+8-9+12i\over2^2-(3i)^2\)

z = \(\rm 18i-1\over13\)

z = \(\rm {-1\over13}+{18\over13}i\)

Conjugate of z = (z̅) = \(\rm {-1\over13}-{18\over13}i\)

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