The system of equations $x + y + z = 6$, $x + 2y + 3z = 10,x + 2y + \lambda z = \mu $, has no solution for

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

Evaluate the determinants : $\left|\begin{array}{ll}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{array}\right|$

If the system of linear equations  $x-2 y+z=-4 $   ;  $2 x+\alpha y+3 z=5 $  ;  $3 x-y+\beta z=3$ has infinitely many solutions, then $12 \alpha+13 \beta$ is equal to

If the system of linear equations $x + y + z = 5$ ; $x = 2y + 2z = 6$ ; $x + 3y + \lambda z = u (\lambda \, \mu \in R)$, has infinitely many solutions then the value of $\lambda  + \mu $ is

The system of equations  $-k x+3 y-14 z=25$  $-15 x+4 y-k z=3$  $-4 x+y+3 z=4$  is consistent for all $k$ in the set