KVPY Sample Paper KVPY Stream-SX Model Paper-29

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) \[R(z)=-3\]

Correct Answer: D

Solution :

\[{{\left( \frac{\sqrt{3}}{2}+\frac{i}{2} \right)}^{5}}={{\left( {{e}^{i\frac{\pi }{b}}} \right)}^{5}}={{e}^{i5\pi /6}}\]

\[\therefore \] \[z=2\cos \frac{5\pi }{6}=2\left( -\frac{\sqrt{3}}{2} \right)=-\sqrt{3}\]