### Complex Numbers and Quadratic Equations Exercise 02

## Question:

Convert each of the complex numbers given in the Exercise 3 to 8 in the polar form: 8. i

## Answer:

Given: 8. i

Step 1: Convert 8. i to its algebraic form.

Answer: 8. i = 0 + 8i

Step 2: Find the magnitude of 8i.

Answer: Magnitude of 8i = 8

Step 3: Find the argument of 8i.

Answer: Argument of 8i = π/2

Step 4: Convert 0 + 8i to polar form.

Answer: 0 + 8i = 8∠(π/2)

## Question:

Convert each of the complex numbers given in the Exercise 3 to 8 in the polar form: 7. √3+i

## Answer:

Answer: 7. √3+i = 2 cis (π/3)

Step 1: Find the modulus (r) of the complex number = 7. √3+i

Step 2: Calculate the argument (θ) of the complex number = 7. √3+i

Step 3: Convert the complex number 7. √3+i to polar form = r cis θ

Step 4: Substitute the value of r and θ in the polar form to get the answer = 2 cis (π/3)

## Question:

Find the modulus and the argument of the complex number z=−√3+i

## Answer:

Answer: Modulus = 2 Argument = 3π/4

## Question:

If z=1+i√3, then ∣arg(z)∣+∣arg(zˉ)∣= A π/3 B 2π/3 C 0 D π/2

## Answer:

Answer: B 2π/3

## Question:

Convert each of the complex numbers given in the Exercise 3 to 8 in the polar form 3 to 8 in the polar form: 3. 1−i

## Answer:

Answer: 3. 1−i = r(cosθ + i sinθ)

r = √2

cosθ = 1/√2

sinθ = -1/√2

Therefore, 1−i = √2 (1/√2 + i(-1/√2)) = √2 (cos(π/4) + i sin(π/4))

## Question:

Convert the given complex number in polar form : −3

## Answer:

Answer: Step 1: Find the modulus (magnitude) of the given complex number, which is equal to 3.

Step 2: Find the argument (angle) of the given complex number, which is equal to 180°.

Step 3: The polar form of the given complex number is 3∠180°.

## Question:

Convert each of the complex numbers given in the Exercise 3 to 8 in the polar form: 4. −1+i

## Answer:

Answer: 4. r = √2, θ = -π/4