The sum of the real roots of the equation $\left| {\begin{array}{*{20}{c}}
x&{ - 6}&{ - 1}\\
2&{ - 3x}&{x - 3}\\
{ - 3}&{2x}&{x = 2}
\end{array}} \right| = 0$ is equal to

The system of equations $\lambda x + y + z = 0,$ $ - x + \lambda y + z = 0,$ $ - x - y + \lambda z = 0$, will have a non zero solution if real values of $\lambda $ are given by

The solutions of the equation $\left| {\,\begin{array}{*{20}{c}}x&2&{ - 1}\\2&5&x\\{ - 1}&2&x\end{array}\,} \right| = 0$ are

If $x, y, z$ are in arithmetic progression with common difference $d , x \neq 3 d ,$ and the
determinant of the matrix $\left[\begin{array}{ccc}3 & 4 \sqrt{2} & x \\ 4 & 5 \sqrt{2} & y \\ 5 & k & z\end{array}\right]$ is zero, then the value of $k ^{2}$ is ..... .

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

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