Complex Numbers And Quadratic Equations question 229
Question: Given $ z={{(1+i\sqrt{3})}^{100}}, $ then $ \frac{Re(z)}{Im(z)} $ equals [AMU 2002]
Options:
A) 2100
B) 250
C) $ \frac{1}{\sqrt{3}} $
D) $ \sqrt{3} $
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Answer:
Correct Answer: C
Solution:
Let $ z=(1+i\sqrt{3}) $ $ r=\sqrt{3+1}=2 $ $ andr\cos \theta =1,r\sin \theta =\sqrt{3} $ $ \tan \theta =\sqrt{3}=\tan \frac{\pi }{3} $
$ \Rightarrow \theta = $ $ \frac{\pi }{3}. $ $ z=2( \cos \frac{\pi }{3}+i\sin \frac{\pi }{3} ) $
Þ $ z^{100}={{[ 2( \cos \frac{\pi }{3}+i\sin \frac{\pi }{3} ) ]}^{100}} $ $ =2^{100}( \cos \frac{100\pi }{3}+i\sin \frac{100\pi }{3} ) $ $ =2^{100}( -\cos \frac{\pi }{3}-i\sin \frac{\pi }{3} ) $ $ =2^{100}( -\frac{1}{2}-\frac{i\sqrt{3}}{2} ) $
$ \therefore $ $ \frac{Re(z)}{Im(z)}=\frac{-1/2}{-\sqrt{3}/2}=\frac{1}{\sqrt{3}} $ .
BETA
