Euler's Representation

Q.

Real part of $e^{e^{i\theta }}$ is

1. 

2. 

3. 

4. 

Q.

If z=r$e^{i\theta }$, then |$e^{iz}$|=

1. 

2. 

3. 

4. 

Q. If $e^{i\theta }=cos\theta +isin\theta ,$ then in $△ABC$ value of $e^{iA}.e^{iB}.e^{iC}$

1. -i
2. 1
3. -1
4. None of these

Q. The real part of $(1-i{)}^{-i}$ is
[RPET 1999]

1. $e^{-pi/4}cos\left(\frac{1}{2}log2\right)$
2. $e^{-pi/4}sin\left(\frac{1}{2}log2\right)$
3. $e^{pi/4}cos\left(\frac{1}{2}log2\right)$
4. $e^{-pi/4}sin\left(\frac{1}{2}log2\right)$

Q.

Let $f_p\left(\alpha \right)=e^{\frac{i\alpha }{p^2}}.e^{\frac{2i\alpha }{p^2}}.e^{\frac{3i\alpha }{p^2}}.e^{\frac{4i\alpha }{p^2}}\dots e^{\frac{i\alpha }{p}}$, (where $i=\sqrt{-1}$ and $p\in N$) then ${lim}_{n\to \infty }{f}_n\left(\pi \right)$ is

1. 1

2. - 1

3. i

4. - i

Q. The value of ${\sum }_{k=1}^6(sin\frac{2k\pi }{7}-icos\frac{2k\pi }{7})$ is

1. -1
2. -i
3. i
4. 0

Q. Let $z=cos\theta +isin\theta$. The value of ${\sum }_{m=1}^{15}Im\left(z^{2m-1}\right)$ at $\theta =2^{\circ }$

1. $\frac{1}{2sin{2}^{\circ }}$
2. $\frac{1}{sin{2}^{\circ }}$
3. $\frac{1}{3sin{2}^{\circ }}$
4. $\frac{1}{4sin{2}^{\circ }}$

Q.

If $|u|<1,|v|<1$, and $z=\frac{u-v}{1+\overline{u}v}$, then least value of $|z|$ is

1. None of these

2. $\frac{|u|-|v|}{1+|u||v|}$

3. $\frac{|u|+|v|}{1+|u||v|}$

4. $\frac{||u|-|v||}{1-|u||v|}$

Q. If $z_1,z_2$ are complex numbers, then the maximum value of $\frac{z_1\overline{z_2}+\overline{z_1}{z}_2+z_1{z}_2+\overline{z_1}\overline{z_2}}{|z_1{z}_2|}$ is equal to

Q. If $z=i\log (2-\sqrt{3})$, then $\cos z=$

1. $-1$
2. $-\frac{1}{2}$
3. $1$
4. $2$