Question

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

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

Answer 1

Answer: (D)

$z=\left(\frac{\sqrt{3}+i}{2}\right)^{5}+\left(\frac{\sqrt{3}-i}{2}\right)^{5} $
$ z=\left(e^{i\pi/6}\right)^{5} +\left(e^{-i \pi/6}\right)^{5} $
$=e^{i5\pi/6} +e^{-i5\pi/6} $
$=\cos \frac{5\pi}{6} +i \frac{\sin5\pi}{6} +\cos\left(\frac{-5\pi}{6}\right) +i\sin\left(\frac{-5\pi}{6}\right) $
$= 2 \cos \frac{5\pi}{6} < 0 $
$I(z) = 0 and Re(z) < 0$