Question 1: Find the modulus and argument of the following complex numbers and hence express each one of them in polar form:

i) ii) iii) iv) v)

vi) vii) viii)

Answer:

i)

Let

Since the point lies in the first quadrant, the is given by

Polar form of is given by

ii)

Let

Since the point lies in the first quadrant, the is given by

Polar form of is given by

iii)

Let

Since the point lies in the fourth quadrant, the is given by

Polar form of is given by

iv)

Let

Since the point lies on the negative direction of the imaginary axis, the is given by

Polar form of is given by

v)

Let

Since the point lies in the fourth quadrant, the is given by

Polar form of is given by

vi)

Let

Since the point lies in the second quadrant, the is given by

Polar form of is given by

vii)

Let

Since the point lies in the first quadrant, the is given by

Polar form of is given by

viii)

Let

Since the point lies in the third quadrant, the is given by

Polar form of is given by

Question 2: Write in polar form

Answer:

Let

Since the point lies in the fourth quadrant, the is given by

Polar form of is given by

Question 3: Express the following complex numbers in the form :

i) ii) iii) iv)

Answer:

i) Let

is a periodic function with period .

Hence we take

Let be an acute angle given by

We can see that and , hence lies in first quadrant, .

Thus in the polar form is given by

,

Let be an acute angle given by

We can see that and , hence lies in fourth quadrant, .

Thus in the polar form is given by

ii) Let

is a periodic function with period .

Hence we take

Let be an acute angle given by

We can see that and , hence lies in fourth quadrant, therefore .

Thus in the polar form is given by

,

Let be an acute angle given by

We can see that and , hence lies in third quadrant, therefore .

Thus in the polar form is given by

iii) Let

Both Sine and Cosine functions are periodic function with period

Hence let us take

Let be an acute angle given by

In this case and

and

Clearly, lies in the first quadrant, therefore

Hence the polar form of is

In this case and

and

Clearly, lies in the fourth quadrant, therefore

Hence the polar form of is

In this case and

and

Clearly, lies in the first quadrant, therefore

Hence the polar form of is

iv) Let

Let be an acute angle given by

Clearly, lies in the fourth quadrant, therefore

Hence the polar form of is

Question 4: If and are two complex numbers such that and , then show that

Answer:

Let by and by

Given ; and

Since is a complex number

Hence

Question 5: If and are two pairs of conjugate complex numbers, prove that

Answer:

and are two pairs of conjugate complex numbers.

… … … … … i)

… … … … … ii)

. Hence proved.

Question 6: Express in polar form.

Answer:

Let

Clearly, lies in the first quadrant. Therefore . Hence the polar form of is