If $\omega$ is one of the imaginary cube roots of unity, then the value of the determinant $\left| {\begin{array}{*{20}{c}}1&{{\omega ^3}}&{{\omega ^2}}\\ {{\omega ^3}}&1&\omega \\{{\omega ^2}}&\omega &1\end{array}} \right|$ $=$

Let $k_1$, $k_2$ be the maximum and minimum values of $k$ for which the system of equations given by

$x + ky = 1$ ; $kx + y = 2$;  $x + y = k$  are consistent then $k_1^2 + k_2^2$ is equal to

If $D = \left| {\,\begin{array}{*{20}{c}}1&1&1\\1&{1 + x}&1\\1&1&{1 + y}\end{array}\,} \right|$ for $x \ne 0,y \ne 0$ then $D$ is

Let $S$ be the set of all real values of $k$ for which the system oflinear equations $x +y + z = 2$ ; $2x +y - z = 3$ ; $3x + 2y + kz = 4$ has a unique solution. Then $S$ is

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 $\left| {\,\begin{array}{*{20}{c}}{3x - 8}&3&3\\3&{3x - 8}&3\\3&3&{3x - 8}\end{array}\,} \right| = 0,$ then the values of $x$ are