The values of $\mathrm{m}, \mathrm{n}$, for which the system of equations

$ x+y+z=4 $

$ 2 x+5 y+5 z=17 $

$ x+2 y+m z=n$

has infinitely many solutions, satisfy the equation :

Prove that the determinant $\left|\begin{array}{ccc}x & \sin \theta & \cos \theta \\ -\sin \theta & -x & 1 \\ \cos \theta & 1 & x\end{array}\right|$ is independent of $\theta$

A root of the equation $\left| {\,\begin{array}{*{20}{c}}{3 - x}&{ - 6}&3\\{ - 6}&{3 - x}&3\\3&3&{ - 6 - x}\end{array}\,} \right| = 0$ is

The ordered pair $(a, b)$, for which the system of linear equations  $3 x-2 y+z=b$  ;  $5 x-8 y+9 z=3$  ;  $2 x+y+a z=-1$ has no solution, is

$\left| {\,\begin{array}{*{20}{c}}{bc}&{bc' + b'c}&{b'c'}\\{ca}&{ca' + c'a}&{c'a'}\\{ab}&{ab' + a'b}&{a'b'}\end{array}\,} \right|$ is equal to

If $x + y - z = 0,\,3x - \alpha y - 3z = 0,\,\,x - 3y + z = 0$ has non zero solution, then $\alpha = $