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**Miscellaneous Exercise**

**1. Evaluate: **

**Ans. **Given:

=

=

= = =

=

= =

**2. For any two complex numbers **** and **** prove that **

**Ans. **Let and

Then and

Now,

=

=

=

**3. Reduce **** to the standard form.**

**Ans. **Here:

=

=

= =

=

=

= =

=

**4. If **** prove that **

**Ans. **Given:

Squaring both sides, we get

Squaring both sides, we get

**5. Convert the following in the polar form:**

**(i) **

**(ii) **

**Ans. (i)** Here =

=

and

Squaring both sides and adding both the equations, we get

and

and

[ lies in second quadrant]

Therefore, Polar form of is

**(ii)** Here =

and

Squaring both sides and adding both the equations, we get

and

and

[ lies in second quadrant]

Therefore, Polar form of is

**Solve each of the equations in exercises 6 to 9:**

**6. **

**Ans. **Given:

Comparing with , and

=

= =

Therefore, and

**7. **

**Ans. **Given:

Comparing with , and

=

= =

Therefore, and

**8. **

**Ans. **Given:

Comparing with , and

=

= =

Therefore, and

**9. **

**Ans. **Given:

Comparing with , and

=

= =

Therefore, and

**10. If **** find **

**Ans. **Here and

=

= =

=

**11. If **** prove that **

**Ans. **Here =

Comparing both sides, we have and

= =

= Proved.

**12. Let **** find:**

**(i) **

**(ii) **

**Ans. **Here and

**(i)**

**(ii)**

**13. Find the modulus and argument of the complex number **

**Ans. **Let

=

and

Squaring both sides and adding both the equations, we get

and

and

[ lies in second quadrant]

Therefore, and arg

**14. Find the real numbers **** and **** if **** is the conjugate of **

**Ans. **Here

Now

Comparing both sides, we have and

Solving both equations, we have and

**15. Find the modulus of **

**Ans. **Here =

= =

**16. If **** then show that **

**Ans. **Given:

Comparing both sides, we have and

Now,

=

**17. If **** and **** are different complex numbers with **** then find **

**Ans. **Here =

=

=

=

**18. Find the number of non-zero integral solutions of the equation **

**Ans. **Here

**19. If **** then show that:**

** **

**Ans. **Given:

Taking modulus on both sides,

Squaring both sides, we get

**20. If **** then find the least positive integral value of **

**Ans. **Given: