Complex Numbers And Quadratic Equations question 275
Question: $ {{( \frac{\sqrt{3}+i}{2} )}^{6}}+{{( \frac{i-\sqrt{3}}{2} )}^{6}} $ is equal to [RPET 1997]
Options:
A) $ -2 $
B) 0
C) 2
D) 1
Show Answer
Answer:
Correct Answer: A
Solution:
$ {{( \frac{\sqrt{3}+i}{2} )}^{6}}+{{( \frac{i-\sqrt{3}}{2} )}^{6}}={{( \frac{-1+\sqrt{3}i}{2i} )}^{6}}+{{( \frac{-1-\sqrt{3}i}{2i} )}^{6}} $ $ =\frac{1}{i^{6}}[{{(\omega )}^{6}}+{{({{\omega }^{2}})}^{6}}]=-[{{({{\omega }^{3}})}^{2}}+{{({{\omega }^{3}})}^{4}}] $ $ ( \because \omega =\frac{-1+\sqrt{3}i}{2},{{\omega }^{2}}=\frac{-1-\sqrt{3}i}{2} ) $ $ =-(1+1)=-2 $ .
BETA
