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

Concept:

• n^th Root of Unity: The complex numbers satisfying αⁿ = 1 are called the n^th roots of unity.
• For n = 5, the fifth roots of unity are: 1, α, α², α³, α⁴, where α = e^2πi/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

Concept:

1. The amplitude (or argument) of a complex number is given by .

2. The conjugate of , denoted as , has an amplitude , because conjugating a complex number reflects it across the real axis in the Argand plane.

Formula Used:

.

Calculation:

Conclusion:

.

Hence, the correct answer is Option 1.

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:

⇒ 

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

⇒ 

⇒ 

Convert to polar form. The modulus r is 

The argument θ is 

So the polar form of the complex number is

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

In our case, r = 0 so,

Thus, the result of cubing the complex number is 

Hence, the correct answer is Option 1.

Calculation:

Given,

The condition is .

We need to find the locus of all complex numbers satisfying this.

Step 1: Write in Cartesian form.

Let Then

Step 2: Translate the modulus equation.

Squaring both sides:

Step 3: Expand and simplify.

Left:
Right:

Bring all terms together and divide by 3:

Step 4: Complete the square.

This is a circle of radius centered at

Hence, the correct answer is Option 4. 

Concept:

Principal Argument and Cube Roots of Unity:

• Principal Argument (arg(z)): The principal argument of a complex number , denoted as , is the angle formed by the complex number in the complex plane, typically in the range .
• Cube Root of Unity: The cube roots of unity are the solutions to the equation , which are:
  • , the primitive cube root of unity.
  • , the other cube root of unity.
• The cube roots of unity have the property that and .

Calculation:

Given,

• is the cube root of unity, and the sum involves alternating powers of .

Let's express the sum:

We notice that every pair of terms cancels out:

This simplifies the sum to a few remaining terms:

We now compute the exponents modulo 3, since :

The sum becomes:

The remaining terms give:

Next, we find using the argument of :

The argument of is .

Conclusion:

Hence, the value of is .

Concept:

Let z = x + iy be a complex number.

• Modulus of z = 
• arg (z) = arg (x + iy) = 
• 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.

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) + …. + (i^2n-2 + i²ⁿ)

= (-1 + 1) + (-1 + 1) + …. (-1 + 1)

= 0 + 0 + …. + 0

= 0

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

Concept:

Cube Roots of unity are 1, ω and ω2

Here, ω =  and ω2 = 

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

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| = 

Calculations:

Let 

As we know i2 = -1 

As we know that if z = x + iy be any complex number, then its modulus is given by,|z| = 

∴ |z| = 

1Concept:

Let z = x + iy be a complex number.

• Modulus of z = 
• arg (z) = arg (x + iy) = 
• 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

Concept:

Cube Roots of unity are 1, ω and ω²

Here, ω =   and ω² = 

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

Concept 

Cube Roots of unity are 1, ω and ω2

Here, ω =  and ω2 = 

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)3 + 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  

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, i^{4n + 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 , by multiplying and dividing by the conjugate of the denominator, we get:

 = -i.

Now, .

⇒ 

⇒ (-i)ⁿ for n = 2.

(-i)² = (-1)2 × (i)2 = 1 × -1 = -1.

∴  n = 2

Concept: 

Let z = x + iy be a complex number.

• Modulus of z = 
• arg (z) = arg (x + iy) = 
• Conjugate of z = z̅ = x – iy

Calculation:

Given complex number is z = 

z = 

z = 

z = 

z = 

Conjugate of z = (z̅) =