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

Answer: (D)

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

$z={\left(\frac{\sqrt{3}+i}{2}\right)}^5+{\left(\frac{\sqrt{3}-i}{2}\right)}^5$

$z=(e^{i\pi /6}{)}^5+(e^{-i\pi /6}{)}^5$

$=e^{i5\pi /6}+e^{-i5\pi /6}$

$=\cos \frac{5\pi }{6}+i\frac{\sin 5\pi }{6}+\cos \left(\frac{-5\pi }{6}\right)+i\sin \left(\frac{-5\pi }{6}\right)$

$=2\cos \frac{5\pi }{6}<0$

$I\left(z\right)=0$ and $Re\left(z\right)<0$