If the point $P(\alpha, \beta, \gamma)$ lies on the plane $2x + y + z = 1$ and $\begin{bmatrix} \alpha & \beta & \gamma \end{bmatrix} \begin{bmatrix} 1 & 9 & 1 \\ 8 & 2 & 1 \\ 7 & 3 & 1 \end{bmatrix} = \begin{bmatrix} 0 & 0 & 0 \end{bmatrix}$,then $\alpha^2 + \beta^2 + \gamma^2 = $

Let $k_1$ and $k_2$ be the maximum and minimum values of $k$ for which the system of equations $x + ky = 1$,$kx + y = 2$,and $x + y = k$ are consistent. Then $k_1^2 + k_2^2$ is equal to:

Let $A = \begin{bmatrix} 12 & 24 & 5 \\ x & 6 & 2 \\ -1 & -2 & 3 \end{bmatrix}$. The value of $x$ for which the matrix $A$ is not invertible is

If $A=\begin{bmatrix} x & y & y \\ y & x & y \\ y & y & x \end{bmatrix}$ is a matrix such that $5 A^{-1}=\begin{bmatrix} -3 & 2 & 2 \\ 2 & -3 & 2 \\ 2 & 2 & -3 \end{bmatrix}$,then $A^2-4 A=$

Consider the simultaneous linear equations $\beta x + \alpha y - z = -1$,$3x - \beta y + \alpha z = 0$,and $\alpha x + \beta y + z = 1$. In the usual notation used in Cramer's rule,given that $\frac{\Delta_1}{\Delta} = -1$,$\frac{\Delta_2}{\Delta} = 1$,and $\frac{\Delta_3}{\Delta} = 2$,then $(\alpha, \beta) = $

If the system of equations $x+y+z=1$,$x+2y+4z=k$ and $x+4y+10z=k^2$ is consistent,then $k$ is equal to