If the system of linear equations $x - 2y + kz = 1$ ; $2x + y + z = 2$ ;  $3x - y - kz = 3$ Has a solution $(x, y, z) \ne 0$, then $(x, y)$ lies on the straight line whose equation is

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

The number of values of $\theta \in (0,\pi)$ for which the system of linear equations
$x + 3y + 7z = 0$
$-x + 4y + 7z = 0$
$(sin\,3\theta )x + (cos\,2\theta )y + 2z = 0$ has a non-trivial solution, is

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 

If $2x + 3y - 5z = 7, \,x + y + z = 6$, $3x - 4y + 2z = 1,$ then  $x =$

$x + ky - z = 0,3x - ky - z = 0$ and $x - 3y + z = 0$ has non-zero solution for $k =$