If the solve the problem
Question:

If $z=\frac{\sqrt{3}}{2}+\frac{i}{2}(i=\sqrt{-1})$,

then $\left(1+i z+z^{5}+i z^{8}\right)^{9}$ is equal to

  1. –1 

  2. 1

  3. 0

  4. $(-1+2 i)^{9}$


Correct Option: 1

Solution:

$z=\frac{\sqrt{3}}{2}+\frac{i}{2}=\cos \frac{\pi}{6}+i \sin \frac{\pi}{6}$

$\Rightarrow z^{5}=\cos \frac{5 \pi}{6}+i \sin \frac{5 \pi}{6}=\frac{-\sqrt{3}+i}{2}$

and $z^{8}=\cos \frac{4 \pi}{3}+i \sin \frac{4 \pi}{3}=-\left(\frac{1+i \sqrt{3}}{2}\right)$

$\Rightarrow\left(1+\mathrm{iz}+\mathrm{z}^{5}+\mathrm{iz}^{8}\right)^{9}=\left(1+\frac{\mathrm{i} \sqrt{3}}{2} \frac{1}{2} \frac{\sqrt{3}}{2}+\frac{\mathrm{i}}{2}-\frac{\mathrm{i}}{2}+\frac{\sqrt{3}}{2}\right)^{9}$

$=\left(\frac{1+i \sqrt{3}}{2}\right)^{9}=\cos 3 \pi+i \sin 3 \pi=-1$

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