A) \[{{2}^{100}}\]
B) \[{{2}^{50}}\]
C) 2/3
D) 3/2
Correct Answer: C
Solution :
[c] : We have, \[z={{(1+i\sqrt{3})}^{100}}={{2}^{100}}{{\left( \cos \frac{\pi }{3}+i\sin \frac{\pi }{3} \right)}^{100}}\] \[={{2}^{100}}{{\left( \cos \frac{100\pi }{3}+i\sin \frac{100\pi }{3} \right)}^{100}}\] \[={{2}^{100}}\left( -\cos \frac{\pi }{3}-i\sin \frac{\pi }{3} \right)={{2}^{100}}\left( -\frac{1}{2}-\frac{i\sqrt{3}}{2} \right)\] Now, \[\frac{2\operatorname{Re}(z)}{\sqrt{3}\operatorname{Im}(z)}=\frac{2\times {{2}^{100}}.(-1/2)}{\sqrt{3}{{.2}^{100}}.(-\sqrt{3}/2)}=\frac{2}{3}\]
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