De Moivre's theorem Quiz 1

Question 1

Deepak found his father’s notebook and identified a statement given as, $$ \ $$ Statement: $$\dfrac{cis{{	heta }_{1}}}{cis{{	heta }_{2}}}=cis\left( {{	heta }_{2}}-{{	heta }_{1}} ight)$$ $$ \ $$ He wants to know whether the above statement is correct or wrong and he asked his father. What will be the answer of his father?

Accepted Answer

Given statement is true.

Answer

Given statement is false.

Answer

Given statement is true for only $$	heta >0$$.

Answer

None of these.

Question 2

There is a statement given in a question paper for a student as follows: $$ \ $$ Statement: $${{e}^{i	heta }}+{{e}^{-i	heta }}=2\sin 	heta $$ $$ \ $$ He is asked to find whether the above statement is correct or wrong. What will be his answer?

Accepted Answer

Given statement is false.

Answer

Given statement is true.

Answer

Given statement is true for only $$	heta >0$$.

Answer

None of these.

Question 3

If $$x=cis	heta $$. Then find the value of $$x+\dfrac{1}{x}$$?

Accepted Answer

$$2\cos 	heta $$

Answer

$$2i\sin 	heta $$

Answer

$$\cot 	heta $$

Answer

$${{	an }^{2}}	heta $$

Question 4

Solve $${{\left( a+ib \right)}^{n}}+{{\left( a-ib \right)}^{n}}$$.

Accepted Answer

$$2{{r}^{n}}\cos n\alpha $$

Answer

$$2i\sin n\alpha $$

Answer

$$2\cos n\alpha $$

Answer

$$2{{r}^{n}}\sin n\alpha $$

Question 5

Solve $${{\left( 1+i\sqrt{3} \right)}^{n}}+{{\left( 1-i\sqrt{3} \right)}^{n}}$$?

Accepted Answer

$${{2}^{n}}^{+1}\left( \cos \dfrac{n\pi }{3} \right)$$

Answer

$${{2}^{n}}^{+1}\left( \sec \dfrac{n\pi }{3} \right)$$

Answer

$${{2}^{n}}^{+1}\left( \sin \dfrac{n\pi }{3} \right)$$

Answer

$${{2}^{n}}^{+1}{{\left( \cos \dfrac{n\pi }{3} \right)}^{2}}$$