If $n \ne 3k$ and 1, $\omega ,{\omega ^2}$ are the cube roots of unity, then $\Delta = \left| {\,\begin{array}{*{20}{c}}1&{{\omega ^n}}&{{\omega ^{2n}}}\\{{\omega ^{2n}}}&1&{{\omega ^n}}\\{{\omega ^n}}&{{\omega ^{2n}}}&1\end{array}\,} \right|$ has the value

If the system of equations $x – ky – z = 0$, $kx – y – z = 0$ and $x + y – z = 0$ has a non zero solution, then the possible value of k are

$\left| {\begin{array}{*{20}{c}}
{4 + {x^2}}&{ – 6}&{ – 2}\\
{ – 6}&{9 + {x^2}}&3\\
{ – 2}&3&{1 + {x^2}}
\end{array}} \right|$ $;(x\neq0)$ is not divisible by

The values of $a$ and $b$, for which the system of equations    $2 x+3 y+6 z=8$   ;  $x+2 y+a z=5$     ;  $3 x+5 y+9 z=b$  has no solution, are:

The value of $\left| {\begin{array}{*{20}{c}}
{\sin \alpha }&{\cos \alpha }&{\sin \left( {\alpha  + \gamma } \right)}\\
{\sin \beta }&{\cos \beta }&{\sin \left( {\beta  + \gamma } \right)}\\
{\sin \delta }&{\cos \delta }&{\sin \left( {\gamma  + \delta } \right)}
\end{array}} \right|$ is