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:
Correct Option: 1
$z=\left(\frac{\sqrt{3}}{2}+\frac{i}{2}\right)^{5}+\left(\frac{\sqrt{3}}{2}-\frac{i}{2}\right)^{5}$
$=\left(\cos \frac{\pi}{6}+i \sin \frac{\pi}{6}\right)^{5}+\left(\cos \frac{\pi}{6}-i \sin \frac{\pi}{6}\right)^{5}$
$=\left(e^{i \frac{\pi}{6}}\right)^{5}+\left(e^{-i \frac{\pi}{6}}\right)^{5}=2 \cos \frac{\pi}{6}=\sqrt{3}$
$\Rightarrow \quad I(z)=0, \operatorname{Re}(z)=\sqrt{3}$
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