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\[z=-\omega \], we have \[{{z}^{100}}+{{z}^{-100}}={{(-\omega )}^{100}}\], \[{{(-\omega )}^{-100}}\] \[=\omega +\frac{1}{\omega }=\omega +{{\omega }^{2}}=-1\] For \[z=-{{\omega }^{2}}\] \[{{z}^{100}}+{{z}^{-100}}={{(-{{\omega }^{2}})}^{100}}+{{(-{{\omega }^{2}})}^{-100}}\] \[={{\omega }^{200}}+\frac{1}{{{\omega }^{200}}}={{\omega }^{2}}+\frac{1}{{{\omega }^{2}}}\] \[={{\omega }^{2}}+\omega =-1\]
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