Question

Let $z={(\frac{\sqrt{3}}{2}+\frac{i}{2})}^5+{(\frac{\sqrt{3}}{2}-\frac{i}{2})}^5.$
If $R\left(z\right)$ and $I\left(z\right)$ respectively denote the real and imaginary parts of $z,$ then :

• A. $R\left(z\right)>0$ and $I\left(z\right)>0$
• B. $R\left(z\right)<0$ and $I\left(z\right)>0$
• C. $R\left(z\right)=-3$
• D. $I\left(z\right)=0$

Answer 1

The correct option is D $I\left(z\right)=0$

$z={(\frac{\sqrt{3}}{2}+\frac{i}{2})}^5+{(\frac{\sqrt{3}}{2}-\frac{i}{2})}^5\cdots \left(1\right)$
The equation $\left(1\right)$ can be written as
$\Rightarrow z={\left(e^{i\pi /6}\right)}^5+{\left(e^{-i\pi /6}\right)}^5$
$\Rightarrow z=e^{i5\pi /6}+e^{-i5\pi /6}$
$\Rightarrow z=\cos \frac{5\pi }{6}+i\sin \frac{5\pi }{6}+\cos \frac{-5\pi }{6}+i\sin \frac{-5\pi }{6}$
$\Rightarrow z=2\cos \frac{5\pi }{6}$
$\Rightarrow z=2(-\frac{\sqrt{3}}{2})<0$
$Im\left(z\right)=0$ and $Re\left(z\right)<0$