# NCERT Solutions class-11 Maths Miscellaneous

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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

Therefore, Polar form of is

(ii) Here =

and

Squaring both sides and adding both the equations, we get

and

and

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

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: