Question

If $Z={\{\frac{\sqrt{3}}{2}+\frac{i}{2}\}}^5+{\{\frac{\sqrt{3}}{2}-\frac{i}{2}\}}^{{}^5}$,then

• A. $Re\left(Z\right)=0$
• B. $Im\left(Z\right)=0$
• C. $Re\left(Z\right)>0,Im\left(Z\right)>0$
• D. $Re\left(Z\right)>0,Im\left(Z\right)<0$

Answer 1

The correct option is B $Im\left(Z\right)=0$

Given, $Z=(\frac{\sqrt{3}+i}{2}{)}^5+(\frac{\sqrt{3}-i}{2}{)}^5$

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

$=e^{i(\pi -\frac{\pi }{6})}+e^{-i(\pi -\frac{i\pi }{6})}$

$=2\cos \left(\frac{5\pi }{6}\right)$

$=-\frac{2\sqrt{3}}{2}$

$=-\sqrt{3}$
Hence, $Im\left(Z\right)=0$.