Apply $C_1\rightarrow C_1+C_2+C_3$
$\begin{vmatrix}1+\omega+\omega^2&\omega&\omega^2\\1+\omega+\omega^2&\omega^2&1\\1+\omega+\omega^2&1&\omega\end{vmatrix}$
$\Rightarrow \begin{vmatrix}0&\omega&\omega^2\\0&\omega^2&1\\0&1&\omega\end{vmatrix}$
$\Rightarrow 0$
Hence (a) is the correct answer.