Complex Numbers And Quadratic Equations question 208
Question: If $ z={{( \frac{\sqrt{3}}{2}+\frac{i}{2} )}^{5}}+{{( \frac{\sqrt{3}}{2}-\frac{i}{2} )}^{5}} $ , then [MP PET 1997]
Options:
A) $ Re(z)=0 $
B) $ Im(z)=0 $
C) $ Re(z)>0,Im(z)>0 $
D) $ Re(z)>0,Im(z)<0 $
Show Answer
Answer:
Correct Answer: B
Solution:
Given that $ z={{( \frac{\sqrt{3}}{2}+i\frac{1}{2} )}^{5}}+{{( \frac{\sqrt{3}}{2}-i\frac{1}{2} )}^{5}} $ $ ={{[ \cos ( \frac{\pi }{6} )+i\sin ( \frac{\pi }{6} ) ]}^{5}}+{{[ \cos ( \frac{\pi }{6} )-i\sin ( \frac{\pi }{6} ) ]}^{5}} $ $ =\cos \frac{5\pi }{6}+i\sin \frac{5\pi }{6}+\cos \frac{5\pi }{6}-i\sin \frac{5\pi }{6} $ . Hence Im (z) = 0.
BETA
