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

Answer:

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

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

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

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

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

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

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

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

Question 2: Write in polar form

Answer:

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

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

Answer:

is a periodic function with period .

Hence we take

We can see that , lies in first quadrant, .

Thus in the polar form is given by

,

We can see that , lies in fourth quadrant, .

Thus in the polar form is given by

is a periodic function with period .

Hence we take

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

Thus in the polar form is given by

,

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

Thus in the polar form is given by

Both Sine and Cosine functions are periodic function with period

Hence let us take

In this case

lies in the first quadrant, therefore

Hence the is

In this case

lies in the fourth quadrant, therefore

Hence the is

In this case

lies in the first quadrant, therefore

Hence the is

lies in the fourth quadrant, therefore

Hence the is

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

Answer:

by by

;

is a complex number

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

Answer:

are two pairs of conjugate complex numbers.

… … … … … i)

… … … … … ii)

. Hence proved.

Question 6: Express in polar form.

Answer: