A) 2100
B) 250
C) \[\frac{1}{\sqrt{3}}\]
D) \[\sqrt{3}\]
Correct Answer: C
Solution :
Let \[z=(1+i\sqrt{3})\] \[r=\sqrt{3+1}=2\] \[\text{and}\,\,r\cos \theta =1,\,\,r\sin \theta =\sqrt{3}\] \[\tan \theta =\sqrt{3}=\tan \frac{\pi }{3}\]\[\Rightarrow \,\theta =\]\[\frac{\pi }{3}.\] \[z=\,2\,\left( \cos \frac{\pi }{3}+i\sin \frac{\pi }{3} \right)\] Þ \[{{z}^{100}}={{\left[ 2\,\left( \cos \frac{\pi }{3}+i\sin \frac{\pi }{3} \right) \right]}^{100}}\] \[={{2}^{100}}\left( \cos \frac{100\,\pi }{3}+i\sin \frac{100\,\pi }{3} \right)\] \[={{2}^{100}}\left( -\cos \frac{\pi }{3}-i\sin \frac{\pi }{3} \right)\]\[={{2}^{100}}\left( -\frac{1}{2}-\frac{i\sqrt{3}}{2} \right)\] \[\therefore \] \[\frac{\operatorname{Re}(z)}{\operatorname{Im}\,(z)}=\frac{-1/2}{-\sqrt{3}/2}=\frac{1}{\sqrt{3}}\].
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