De Moivre's Theorem MCQ Quiz - Objective Question with Answer for De Moivre's Theorem - Download Free PDF

Calculation

Using De Moivre's Theorem:

Comparing with -i, we have:

This implies:

⇒ 

⇒ 

⇒ 

⇒ 

For the least positive integer value of k, let n = 0:

Let n = 1:

Let n = 2:

So, m = 3.

For the greatest negative integer value of k, we can analyze the values of k for negative n.

For n = -1:

So, n = -1.

Formula Used:

1. Euler's formula:

2. De Moivre's theorem:

3. and

Calculation:

⇒

⇒

⇒

⇒ (Since )

⇒

⇒ Real part =

∴ The real part of the given expression is .

Hence option 4 is correct

Calculation

Using De Moivre's Theorem:

Comparing with -i, we have:

This implies:

⇒ 

⇒ 

⇒ 

⇒ 

For the least positive integer value of k, let n = 0:

Let n = 1:

Let n = 2:

So, m = 3.

For the greatest negative integer value of k, we can analyze the values of k for negative n.

For n = -1:

So, n = -1.

Hence option 1 is correct

Concept Used:

Factoring the equation and expressing roots in cis form. Using De Moivre's Theorem.

cis(θ) = cos(θ) + i sin(θ)

Calculation

⇒ 

⇒ 

Thus, or

Let's analyze the options:

Option 1:

Option 3:

Option 4:

Option 2:

Since option 4 satisfies , it is a root of the equation.

∴ The root is

Hence option 4 is correct

Concept Used:

Complex numbers in polar form: z = |z|(cos(θ) + i sin(θ))

De Moivre's Theorem: (cos(θ) + i sin(θ))^n = cos(nθ) + i sin(nθ)

Calculation

Given:

|x| = |y| = 1

Arg(x) = 2α

Arg(y) = 3β

α + β = π/36

⇒ 

⇒ 

⇒ 

⇒ 

⇒ 

⇒ 

⇒ 

⇒ 

⇒ 

⇒  

Hence option 3 is correct

Concept:

Euler's formula:

A complex number z = cos θ + i sin θ can also be written as eiθ.

Calculation:

From the Euler's formula, we know that:

 = e^2iθ = cos 2θ + i sin 2θ.

Concept:

1. Euler's Formula on Complex Numbers:

• e^ix = cos x + i sin x
• e^-ix = cos x - i sin x

2. Trigonometry formulas:

• 1 – cos θ = 2 sin² (θ/2)
• sin θ = 2 sin (θ/2) cos (θ/2

Calculation:

We have to find the value of 

                          (∵e^ix = cos x + i sin x and e^-ix = cos x - i sin x)

= cos (π/2) + i sin (π/2) = 0 + i  = i

Concept:

Calculation:

Given: 

x = (cos π/14 + i sin π/14), y = (cos 9π/14 + i sin 9π/14)

As we know that, 

We can re-write x and y as:

⇒ x = e^{i ⋅ π/14} and y = e^{i ⋅ 9π/14}

⇒ x⁵ = e^{i ⋅ 5π/14} and y¹⁵ = e^{i ⋅ 135π/14}

⇒  x5 ⋅ y¹⁵ = e^{i ⋅ 10π} 

Concept:

Calculation:

Given: x =  (cos π/9 + i sin π/9 )18 and y = (cos π/16 + i sin π/16 )⁸

As we know that, 

⇒ y = (cos π/16 + i sin π/16 )8 = [ei ⋅ π/16]8 = ei ⋅ π/2

⇒ ei ⋅ π/2 = cos π/2 + i sin π/2 = i

⇒ y = i 

So, y^-2 = i^-2 = - 1  ----(1)

Similarly, 

As we know that, 

⇒ x = (cos π/9 + i sin π/9 )18 = [ei ⋅ π/9]18 = ei ⋅ 2π

⇒ ei ⋅ 2π = cos 2π + i sin 2π = 1

⇒ x = 1     -------(2)

So, from (1) and (2), we get

⇒ x ⋅ y^-2 = 1 × - 1 = - 1

Concept:

Calculation:

As we know that, 

⇒ (cos π/9 + i sin π/9 )18 

= [e^i ⋅ π/9]¹⁸ = e^{i ⋅ 2π}

⇒ e^{i ⋅ 2π} 

= cos 2π + i sin 2π = 1

Concept:

Calculation:

As we know that, 

⇒ (cos π/16 + i sin π/16 )8 = [e^{i ⋅ π/16}]⁸ = e^{i ⋅ π/2}

⇒ e^{i ⋅ π/2} = cos π/2 + i sin π/2 = i

Concept:

Relation between modulus and argument of a complex number:

For any complex number  if the modulus is given by  and argument is then the following relation always holds:

Calculation:

Let the given complex number be  then we have  and arg . 

Therefore, .

Now using the above formulae,

Therefore, the required complex number is .

Concept:

Let z = x + iy be any complex number then its polar form is , where r =  and 

De Moivre’s Theorem 

Given any complex number cos θ + i sin θ and any integer n,

(cos θ + i sin θ )n = cos nθ + i sin nθ 

Calculations:

To Find 

First write the complex numbers  and  in polar form and apply De - Moivers theorem.

Let z = x + iy be any complex number then its polar form is  , where r =  and 

The polar form of  is 

The polar form of  is 

Consider, 

 =  + 

Apply De Moivre’s Theorem 

Given any complex number cos θ + i sin θ and any integer n,

(cos θ + i sin θ )n = cos nθ + i sin nθ .

=  + 

= 

Hence, = 

Concept:

Calculation:

The given expression: can re-written as

 =  

 = 

As we know that sin π/4 = 1/√2 = cos π/4

So, we can write the given expression as

 = (cos π/4 + i sin π/4)⁶⁴

As we know that, 

⇒ (cos π/4 + i sin π/4)⁶⁴ = (e^i ⋅ π/4 )⁶⁴

⇒ (cos π/4 + i sin π/4)64 = e^{i ⋅ 16π}

Concept:

De Moivre's formula:

If z = re^iθ = r(cos θ + i sin θ) , then zⁿ = rⁿe^inθ(cos nθ + i sin nθ)

Calculation:

Given, z = 

⇒ z = 

⇒ z = 

⇒ z = 

⇒ z = 

⇒ z = 

⇒ z = 

⇒ z = 

⇒ z = 2 cos  = √3

∴ Im (z) = 0