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 $I\left(z\right)=0$

$z={(\frac{\sqrt{3}}{2}+\frac{i}{2})}^5+{(\frac{\sqrt{3}}{2}-\frac{i}{2})}^5\Rightarrow z=(\cos \pi /6+i\sin \pi /6{)}^5+(\cos \pi /6-i\sin \pi /6{)}^5$
By using eulers form,above equation can be written as
$\Rightarrow z={(1\cdot e^{i\pi /6})}^5+{(1\cdot e^{-i\pi /6})}^5$
$\Rightarrow z=e^{i5\pi /6}+e^{-i5\pi /6}$
$\Rightarrow z=(\cos 5\pi /6+i\sin 5\pi /6)+(\cos 5\pi /6-i\sin 5\pi /6)=2\cos 5\pi /6=-\sqrt{3}\therefore R\left(z\right)<0,I\left(z\right)=0$