Question 1: Express the following complex numbers in the form

Answer:

Question 2: Find the real values of , if

Answer:

Therefore comparing, we get

… … … … … i)

… … … … … ii)

Solving i) and ii), multiplying i) by and ii) by and adding we get

Now substituting in ii) we get

Hence and

Therefore comparing, we get

… … … … … i)

… … … … … ii)

Solving i) and ii), multiplying i) by and ii) by and adding we get

Now substituting in i) we get

Hence and

… … … … … i)

… … … … … ii)

Solving i) and ii), multiplying i) by and ii) by and adding we get

Substituting in i) we get

Hence and

Comparing we get

… … … … … i)

… … … … … ii)

Adding i) and ii) we get

Substituting in i) we get

Hence and

Question 3: Find the conjugates of the following complex numbers:

Answer:

Note: , then conjugate of is

Therefore

Therefore

Therefore

Therefore

Therefore

Question 4: Find the multiplicative inverse of the following complex numbers:

Answer:

is the complex number, then multiplication inverse of is or

Answer:

, then

Question 6: , find

Answer:

Answer:

Answer:

Hence proved.

Therefore the general solution is given by

Question 9: Find the least positive integral value of for which is real.

Answer:

For

For

Therefore the least positive integral value of

Answer:

For to be purely Real,

Answer:

For this to be real , the smallest positive value of will be .

Thus , which is real.

Answer:

and

Answer:

Answer:

Answer:

Question 16:

Answer:

Now

Now,

Now

Now,

Now,

Answer:

Question 18: , then show that

Answer:

Hence proved.

Question 19: Solve the system of equations .

Answer:

Let

… … … … … i)

… … … … … ii)

Adding i) and ii) we get

Hence

For positive sign for

For negative sign for

Answer:

Let

If it is purely imaginary,

Question 21: is a complex number other than such that and

Answer:

Let and

… … … … … i)

Since there is no real part, is purely an imaginary numbers

Question 22: , find

Answer:

Let

Comparing,

Question 23: Solve the equation

Answer:

Let

Comparing,

and

Question 24: What is the smallest positive integer for which

Answer:

is a multiple of .

Hence the smallest positive number for which is

Question 25: , are complex numbers such that, .

Answer:

Question 26: Find the number of solutions of

Answer:

Let

Now

Therefore

For

Thus there are infinite many solution of the form