A) \[\operatorname{Re}(z)=0\]
B) \[\operatorname{Im}(z)=0\]
C) \[\operatorname{Re}(z)>0,\operatorname{Im}(z)>0\]
D) \[\operatorname{Re}(z)>0,\operatorname{Im}(z)<0\]
Correct Answer: B
Solution :
Given that \[z={{\left( \frac{\sqrt{3}}{2}+i\frac{1}{2} \right)}^{5}}+{{\left( \frac{\sqrt{3}}{2}-i\frac{1}{2} \right)}^{5}}\] \[={{\left[ \cos \left( \frac{\pi }{6} \right)+i\sin \left( \frac{\pi }{6} \right) \right]}^{5}}+{{\left[ \cos \left( \frac{\pi }{6} \right)-i\sin \left( \frac{\pi }{6} \right) \right]}^{5}}\] \[=\cos \frac{5\pi }{6}+i\sin \frac{5\pi }{6}+\cos \frac{5\pi }{6}-i\sin \frac{5\pi }{6}\]. Hence Im (z) = 0.
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