Answer:
Option B
Explanation:
Gtiven equation is x2+x+1=0
$\Rightarrow x=\omega$ and $x=\omega^{2}$
Case I ; when x=$\omega$
Then
$\sum_{n=1}^{6}\left[x^{n}+\frac{1}{x^{n}}\right]^{2}=\sum_{n=1}^{6}\left[\omega^{n}+\omega^{2n}\right]^{2}\left[\because \frac{1}{\omega}=\omega^{2}\right]$
$=(\omega+\omega^{2})^{2}+(\omega^{2}+\omega^{4})^{2}+(\omega^{3}+\omega^{6)^{2}}+(\omega^{4}+\omega^{8)^{2}}+(\omega^{5}+\omega^{10)^{2}}+(\omega^{6}+\omega^{12)^{2}}$
$=(-1)^{2}+(-1)^{2}+(2)^{2}+(-1)^{2}+(-1)^{2}+(2)^{2}=12$
Case II : when $\omega^{2}$
Then
$\sum_{n=1}^{6}\left[x^{n}+\frac{1}{x^{n}}\right]^{2}=\sum_{n=1}^{6}\left[\omega^{2n}+\omega^{n}\right]^{2}\left[\because\frac{1}{\omega^{2}}=\omega\right]$
=12
