# Exercise 13.4

QUESTION 1

Find the modulus and argument of the following complex number and hence express each of them in the polar form :

(i)

Sol :

Let

Since point lies in the first quadrant , the argument of *z *is given by

Polar form

(ii)

Sol :

Let

Since point lies in the quadrant , the argument of *z *is given by

Polar form

(iii)

Sol :

Let

Since point lies in the quadrant , the argument of *z *is given by

Polar form

(iv)

Sol :

Rationalizing the denominator :

= 1

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

Polar form

(v)

Sol :

Rationalizing the denominator

Let

= 1

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

Polar form

(vi)

Sol :

Rationalizing the denominator :

Let

= 1

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

Polar form

(vii)

Sol :

Let

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

Polar form

(viii)

Sol :

Rationalizing the denominator:

= 8

Let

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

Polar form

QUESTION 2

Write in polar form

Sol :

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Let

Then

Let be the argument of *z *and be the acute angle given by

Then ,

Clearly, * z* lies in fourth quadrant. So,

The polar form of *z *is

Thus , the polar form of is

QUESTION 3

Express the following complex in the form

(i)

Sol :

Let

(ii)

Sol :

(iii)

Sol :

(iv)

Sol :