A) i
B) - i
C) 1
D) - 1
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, \[{{z}^{100}}+{{z}^{-100}}={{(-{{\omega }^{2}})}^{100}}+{{(-{{\omega }^{2}})}^{-100}}\] \[=\,{{\omega }^{200}}+\frac{1}{{{\omega }^{200}}}\]\[={{\omega }^{2}}+\frac{1}{{{\omega }^{2}}}={{\omega }^{2}}+\omega \]\[=-1.\]
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