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

 ${{\left( \frac{\sqrt{3}}{2}+\frac{i}{2} \right)}^{5}}={{\left( {{e}^{i\frac{\pi }{b}}} \right)}^{5}}={{e}^{i5\pi /6}}$ $\therefore$      $z=2\cos \frac{5\pi }{6}=2\left( -\frac{\sqrt{3}}{2} \right)=-\sqrt{3}$