If $x=\alpha, y=\beta, z=\gamma$ is the solution of the system of equations $2 x+3 y+z=-1,3 x+y+z=4$, $x-3 y-2 z=1$, then the value of $\beta$ is

The positive value of ' $a$ ' for which the system of linear homogeneous equations $x+a y+z=0, a x+2 y-z=0$, $2 x+3 y+z=0$ has non-trivial solution is

If $A=\left[\begin{array}{lll}1 & 2 & 2 \\ 2 & 1 & 1 \\ 1 & 2 & 1\end{array}\right]$ then $|\operatorname{adj}|\left(A^2\right) \mid=$

$K=\left|\begin{array}{cc}3 & 4 \\ 5 & 4\end{array}\right|+\left|\begin{array}{cc}1 & -1 \\ 5 & 4\end{array}\right|+\left|\begin{array}{cc}\frac{1}{3} & \frac{1}{4} \\ 5 & 4\end{array}\right|+\left|\begin{array}{cc}\frac{1}{9} & -\frac{1}{16} \\ 5 & 4\end{array}\right|+\ldots$ to $\infty$, then $K=$