Complex Numbers And Quadratic Equations question 284
Question: If $ z+{z^{-1}}=1,\text{then }z^{100}+{z^{-100}} $ is equal to [UPSEAT 2001]
Options:
A) i
B) - i
C) 1
D) - 1
Show Answer
Answer:
Correct Answer: D
Solution:
$ z+{z^{-1}} $ $ =1\Rightarrow z^{2}-z+1=0 $
$ \Rightarrow $ $ z=-\omega $ or $ -{{\omega }^{2}} $ For $ z=-\omega , $ $ z^{100}+{z^{-100}}={{(-\omega )}^{100}}+{{(-\omega )}^{-100}} $ = $ \omega +\frac{1}{\omega }=\omega +{{\omega }^{2}}=-1 $ For z = - w2, $ z^{100}+{z^{-100}}={{(-{{\omega }^{2}})}^{100}}+{{(-{{\omega }^{2}})}^{-100}} $ $ ={{\omega }^{200}}+\frac{1}{{{\omega }^{200}}} $ $ ={{\omega }^{2}}+\frac{1}{{{\omega }^{2}}}={{\omega }^{2}}+\omega $ $ =-1. $
BETA
