Complex Numbers And Quadratic Equations question 834
Question: If $ {{(1+i\sqrt{3})}^{9}}=a+ib, $ then $ b $ is equal to [RPET 1995]
Options:
A) 1
B) 256
C) 0
D) $ 9^{3} $
Show Answer
Answer:
Correct Answer: C
Solution:
$ 1+i\sqrt{3}=2( \frac{1}{2}+i\frac{\sqrt{3}}{2} )=2[ \cos \frac{\pi }{3}+i\sin \frac{\pi }{3} ]=2{e^{i\pi /3}} $ \ $ {{(1+i\sqrt{3})}^{9}}={{(2{e^{i\pi /3}})}^{9}}=2^{9}.{e^{i(3\pi )}} $ $ =2^{9}(\cos 3\pi +i\sin 3\pi )=-2^{9} $ \ $ a+ib={{(1+i\sqrt{3})}^{9}}=-2^{9} $ ; \ $ b=0 $ .
BETA
