If $\left| \begin{array}{ccc} 1 & k & 3 \\ 3 & k & -2 \\ 2 & 3 & -1 \end{array} \right| = 0$,then the value of $k$ is

If $A = \begin{bmatrix} x & 1 & 2 \\ 2 & 4 & x \\ -3 & 3 & 2 \end{bmatrix}$ is a singular matrix and the distinct values of $x$ are $x_1$ and $x_2$,then $x_1 + x_2 + x_1 x_2 = $.

If $x^2+y^2+z^2 \neq 0, \quad x=cy+bz, \quad y=az+cx$ and $z=bx+ay$, then $a^2+b^2+c^2+2abc$ is equal to

If $\left|\begin{array}{ccc}x & 4 & 6 \\ 2 & 3 & -9 \\ 5 & 6 & 1\end{array}\right|+\left|\begin{array}{ccc}5 & 6 & 1 \\ 6 & 4 & 5 \\ 2 & 3 & -9\end{array}\right|=\left|\begin{array}{ccc}2 & 3 & -9 \\ 1-2 x & -8 & -11 \\ 5 & 6 & 1\end{array}\right|$,then $x=$ . . . . . .

One of the roots of the given equation $\left| \begin{array}{ccc} x+a & b & c \\ b & x+c & a \\ c & a & x+b \end{array} \right| = 0$ is

Let $A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \alpha & \beta \\ 0 & \beta & \alpha \end{bmatrix}$ and $|2A|^3 = 2^{21}$,where $\alpha, \beta \in \mathbb{Z}$. Then a value of $\alpha$ is: