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$\omega$ is an imaginary cube root of unity and $\left|\begin{array}{ccc}x+\omega^2 & \omega & 1 \\ \omega & \omega^2 & 1+x \\ 1 & x+\omega & \omega^2\end{array}\right|=0$ then one of the values of $x$ is
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$$
\begin{aligned}
\text { Hints: } & \stackrel{C_1^{\prime} \rightarrow C_1+C_2+C_3}{\longrightarrow}\left|\begin{array}{ccc}
x & \omega & 1 \\
x & \omega^2 & 1+x \\
x & x+\omega & \omega^2
\end{array}\right|=x\left|\begin{array}{ccc}
1 & \omega & 1 \\
1 & \omega^2 & 1+x \\
1 & x+\omega & \omega^2
\end{array}\right| \\
\quad & =x\left|\begin{array}{ccc}
1 & \omega & 1 \\
0 & \omega^2-\omega & x \\
0 & x & \omega^2-1
\end{array}\right|=x\left\{\left(\omega^2-\omega\right)\left(\omega^2-1\right)-x^2\right\}=0 \quad \Rightarrow x=0 \quad \text { One value of } x=0
\end{aligned}
$$
\begin{aligned}
\text { Hints: } & \stackrel{C_1^{\prime} \rightarrow C_1+C_2+C_3}{\longrightarrow}\left|\begin{array}{ccc}
x & \omega & 1 \\
x & \omega^2 & 1+x \\
x & x+\omega & \omega^2
\end{array}\right|=x\left|\begin{array}{ccc}
1 & \omega & 1 \\
1 & \omega^2 & 1+x \\
1 & x+\omega & \omega^2
\end{array}\right| \\
\quad & =x\left|\begin{array}{ccc}
1 & \omega & 1 \\
0 & \omega^2-\omega & x \\
0 & x & \omega^2-1
\end{array}\right|=x\left\{\left(\omega^2-\omega\right)\left(\omega^2-1\right)-x^2\right\}=0 \quad \Rightarrow x=0 \quad \text { One value of } x=0
\end{aligned}
$$
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