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# If $\omega$ is a complex root of unity, one root of the equation $\begin{vmatrix}x+1 &\omega&\omega^2\\\omega&x+\omega^2&1\\\omega^2&1&x+\omega\end{vmatrix}=0$ is

$\begin{array}{1 1}(A)0 \\ (B) 1 \\ (C) \omega \\(D) \omega^2 \end{array}$

$\Delta =\begin{vmatrix}x+1&\omega&\omega^2\\\omega&x+\omega^2&1\\\omega^2&1&x+\omega\end{vmatrix}$
$C_1\rightarrow C_1+C_2+C_3$
$\Rightarrow \Delta=\begin{vmatrix}x+1+\omega+\omega^2&\omega&\omega^2\\x+1+\omega+\omega^2&x+\omega^2&1\\x+1+\omega+\omega^2&1&x+\omega\end{vmatrix}$
$\Rightarrow \Delta=\begin{vmatrix}x&\omega&\omega^2\\x&x+\omega^2&1\\x&1&x+\omega\end{vmatrix}$
$\Rightarrow x=0$ is one root of $\Delta$
Hence (A) is the correct answer.