Complex Numbers And Quadratic Equations question 246
Question: The product of all the roots of $ {{( \cos \frac{\pi }{3}+i\sin \frac{\pi }{3} )}^{3/4}} $ is [MNR 1984; EAMCET 1985]
Options:
A) $ -1 $
B) 1
C) $ \frac{3}{2} $
D) $ -\frac{1}{2} $
Show Answer
Answer:
Correct Answer: B
Solution:
Given that $ {{[ \cos \frac{\pi }{3}+i\sin \frac{\pi }{3} ]}^{3/4}} $ $ ={{[\cos \pi +i\sin \pi ]}^{1/4}} $ . Since the expression has only 4 different roots, therefore on putting $ n=0,1,2,3 $ in $ \cos [ \frac{2n\pi +\pi }{4} ]+i\sin [ \frac{2n\pi +\pi }{4} ] $ and multiplying them, we get $ =[ \cos \frac{\pi }{4}+i\sin \frac{\pi }{4} ][ \cos \frac{3\pi }{4}+i\sin \frac{3\pi }{4} ] $ $ [ \cos \frac{5\pi }{4}+i\sin \frac{5\pi }{4} ][ \cos \frac{7\pi }{4}+i\sin \frac{7\pi }{4} ] $ $ =[ \frac{1}{\sqrt{2}}+i\frac{1}{\sqrt{2}} ][ -\frac{1}{\sqrt{2}}+i\frac{1}{\sqrt{2}} ][ -\frac{1}{\sqrt{2}}+i\frac{-1}{\sqrt{2}} ][ \frac{1}{\sqrt{2}}-i\frac{1}{\sqrt{2}} ] $ $ =( -\frac{1}{2}-\frac{1}{2} )( -\frac{1}{2}-\frac{1}{2} )=(-1)(-1)=1 $ .
BETA
