respectively denote the real and imaginary parts of z, then:

Question:

Let $z=\left(\frac{\sqrt{3}}{2}+\frac{\mathrm{i}}{2}\right)^{5}+\left(\frac{\sqrt{3}}{2}-\frac{\mathrm{i}}{2}\right)^{5}$. If $\mathrm{R}(z)$ and $\mathrm{I}(z)$

respectively denote the real and imaginary parts of $z$, then:

1. (1) $\mathrm{I}(z)=0$

2. (2) $\mathrm{R}(z)>0$ and $\mathrm{I}(z)>0$

3. (3) $\mathrm{R}(z)<0$ and $\mathrm{I}(z)>0$

4. (4) $\mathrm{R}(z)=-(3)$

Correct Option: 1

Solution:

$z=\left(\frac{\sqrt{3}}{2}+\frac{i}{2}\right)^{5}+\left(\frac{\sqrt{3}}{2}-\frac{i}{2}\right)^{5}$

$=\left(\cos \frac{\pi}{6}+i \sin \frac{\pi}{6}\right)^{5}+\left(\cos \frac{\pi}{6}-i \sin \frac{\pi}{6}\right)^{5}$

$=\left(e^{i \frac{\pi}{6}}\right)^{5}+\left(e^{-i \frac{\pi}{6}}\right)^{5}=2 \cos \frac{\pi}{6}=\sqrt{3}$

$\Rightarrow \quad I(z)=0, \operatorname{Re}(z)=\sqrt{3}$