If $\left| {\,\begin{array}{*{20}{c}}a&b&0\\0&a&b\\b&0&a\end{array}\,} \right| = 0$, then

  • A

    $a $ is one of the cube roots of unity

  • B

    $b$ is one of the cube roots of unity

  • C

    $\left( {\frac{a}{b}} \right)$is one of the cube roots of unity

  • D

    $\left( {\frac{a}{b}} \right)$is one of the cube roots of $ -1$

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The system of equations $(\sin\theta ) x + 2z = 0$ , $(\cos\theta ) x + (\sin\theta )y = 0$ , $(\cos\theta )y + 2z = a$ has

$\Delta = \left| {\,\begin{array}{*{20}{c}}a&{a + b}&{a + b + c}\\{3a}&{4a + 3b}&{5a + 4b + 3c}\\{6a}&{9a + 6b}&{11a + 9b + 6c}\end{array}\,} \right|$where $a = i,b = \omega ,c = {\omega ^2}$, then $\Delta $is equal to