If a point $P(\alpha, \beta, \gamma)$ satisfying $(\alpha \ \beta \ \gamma)\begin{bmatrix} 2 & 10 & 8 \\ 9 & 3 & 8 \\ 8 & 4 & 8 \end{bmatrix} = (0 \ 0 \ 0)$ lies on the plane $2x + 4y + 3z = 5$,then $6\alpha + 9\beta + 7\gamma$ is equal to:

The sum of three numbers is $6$. Thrice the third number when added to the first number gives $7$. On adding three times the first number to the sum of the second and third numbers,we get $12$. The product of these numbers is:

For the equations $x+2y+3z=1$,$2x+y+3z=2$,and $5x+5y+9z=4$,which of the following is true?

Let $\alpha, \beta$ and $\gamma$ be real numbers such that the system of linear equations
$x+2y+3z=\alpha$
$4x+5y+6z=\beta$
$7x+8y+9z=\gamma$
is consistent. Let $|M|$ represent the determinant of the matrix
$M=\begin{bmatrix} \alpha & 2 & \gamma \\ \beta & 1 & 0 \\ -1 & 0 & 1 \end{bmatrix}$
Let $P$ be the plane containing all those $(\alpha, \beta, \gamma)$ for which the above system of linear equations is consistent,and $D$ be the square of the distance of the point $(0,1,0)$ from the plane $P$.
$(1)$ The value of $|M|$ is
$(2)$ The value of $D$ 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 $A^2-4 A=$

The number of solutions of the system of equations $2x + y - z = 7$,$x - 3y + 2z = 1$,and $x + 4y - 3z = 5$ is: