The greatest value of $c \in R$ for which the system of linear equations

$x - cy - cz = 0 \,\,;\,\, cx - y + cz = 0 \,\,;\,\, cx + cy - z = 0 $ has a non -trivial solution, is

If $A=\left[\begin{array}{lll}1 & 1 & -2 \\ 2 & 1 & -3 \\ 5 & 4 & -9\end{array}\right],$ find $|A|$.

Let $N$ denote the number that turns up when a fair die is rolled. If the probability that the system of equations

$x+y+z=1$  ;  $2 x+N y+2 z=2$  ;  $3 x+3 y+N z=3$

has unique solution is $\frac{k}{6}$, then the sum of value of $k$ and all possible values of $N$ is

The value of $x,$ if $\left| {\,\begin{array}{*{20}{c}}{ - x}&1&0\\1&{ - x}&1\\0&1&{ - x}\end{array}\,} \right| = 0 $ is equal to

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 $a \ne b \ne c,$ the value of $x$ which satisfies the equation $\left| {\,\begin{array}{*{20}{c}}0&{x - a}&{x - b}\\{x + a}&0&{x - c}\\{x + b}&{x + c}&0\end{array}\,} \right| = 0$, is