$\left| {\,\begin{array}{*{20}{c}}1&1&1\\1&{{\omega ^2}}&\omega \\1&\omega &{{\omega ^2}}\end{array}\,} \right| = $

Let the system of linear equations $4 x+\lambda y+2 z=0$ ;  $2 x-y+z=0$ ;  $\mu x +2 y +3 z =0, \lambda, \mu \in R$ has a non-trivial solution. Then which of the following is true?

If $a,b,c$ be positive and not all equal, then the value of the determinant $\left| {\,\begin{array}{*{20}{c}}a&b&c\\b&c&a\\c&a&b\end{array}\,} \right|$ is

The number of values of $\alpha$ for which the system of equations:   $x+y+z=\alpha$ ;  $\alpha x+2 \alpha y+3 z=-1$ ;  $x+3 \alpha y+5 z=4$    is inconsistent, is

The maximum value of

$f(x)=\left|\begin{array}{ccc} \sin ^{2} x & 1+\cos ^{2} x & \cos 2 x \\ 1+\sin ^{2} x & \cos ^{2} x & \cos 2 x \\ \sin ^{2} x & \cos ^{2} x & \sin 2 x \end{array}\right|, x \in R \text { is }$

If the system of linear equations $x+ ay+z\,= 3$ ; $x + 2y+ 2z\, = 6$ ; $x+5y+ 3z\, = b$ has no solution, then