NCERT Solutions class-11 Maths Miscellaneous

Miscellaneous Exercise

1. Evaluate:

Ans. Given:

=

=

= = =

=

= =

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2. For any two complex numbers and prove that

Ans. Let and

Then and

Now,

=

=

=

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3. Reduce to the standard form.

Ans. Here:

=

=

= =

=

=

= =

=

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4. If prove that

Ans. Given:

Squaring both sides, we get

Squaring both sides, we get

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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

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Solve each of the equations in exercises 6 to 9:

6.

Ans. Given:

Comparing with , and

=

= =

Therefore, and

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7.

Ans. Given:

Comparing with , and

=

= =

Therefore, and

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8.

Ans. Given:

Comparing with , and

=

= =

Therefore, and

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9.

Ans. Given:

Comparing with , and

=

= =

Therefore, and

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10. If find

Ans. Here and

=

= =

=

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11. If prove that

Ans. Here =

Comparing both sides, we have and

= =

= Proved.

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12. Let find:

(i)

(ii)

Ans. Here and

(i)

(ii)

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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

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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

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15. Find the modulus of

Ans. Here =

= =

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16. If then show that

Ans. Given:

Comparing both sides, we have and

Now,

=

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17. If and are different complex numbers with then find

Ans. Here =

=

=

=

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

Ans. Here

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19. If then show that:

Ans. Given:

Taking modulus on both sides,

Squaring both sides, we get

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20. If then find the least positive integral value of

Ans. Given: