Let $\mathrm{z}=\left(\frac{\sqrt{3}}{2}+\frac{\mathrm{i}}{2}\right)^{5}+\left(\frac{\sqrt{3}}{2}-\frac{\mathrm{i}}{2}\right)^{5}$. If $\mathrm{R}(\mathrm{z})$ and $\mathrm{I}[\mathrm{z}]$
respectively denote the real and imaginary parts of $z$, then :
Correct Option: , 4
$z=\left(\frac{\sqrt{3}+i}{2}\right)^{5}+\left(\frac{\sqrt{3}-i}{2}\right)^{5}$
$\mathrm{z}=\left(\mathrm{e}^{\mathrm{i} \pi / 6}\right)^{5}+\left(\mathrm{e}^{-\mathrm{i} \pi / 6}\right)^{5}$
$=\mathrm{e}^{\mathrm{i} 5 \pi / 6}+\mathrm{e}^{-\mathrm{i} 5 \pi / 6}$
$=\cos \frac{5 \pi}{6}+i \frac{\sin 5 \pi}{6}+\cos \left(\frac{-5 \pi}{6}\right)+i \sin \left(\frac{-5 \pi}{6}\right)$
$=2 \cos \frac{5 \pi}{6}<0$
$\mathrm{I}(\mathrm{z})=0 \quad$ and $\operatorname{Re}(\mathrm{z})<0$
Option (4)
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