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

If ${2^{{a_1}}},{2^{{a_2}}},{2^{{a_3}}},{……2^{{a_n}}}$ are in $G.P.$ then $\left| {\begin{array}{*{20}{c}}
  {{a_1}}&{{a_2}}&{{a_3}} \\ 
  {{a_{n + 1}}}&{{a_{n + 2}}}&{{a_{n + 3}}} \\ 
  {{a_{2n + 1}}}&{{a_{2n + 2}}}&{{a_{2n + 3}}} 
\end{array}} \right|$ is equal to

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

Set of equations $a + b – 2c = 0,$ $2a – 3b + c = 0$ and $a – 5b + 4c = \alpha $ is consistent for $\alpha$ equal to

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