If $\left| {{\kern 1pt} \begin{array}{*{20}{c}}1&2&3\\2&x&3\\3&4&5\end{array}\,} \right| = 0,$ then  $x =$

The values of $x,y,z$ in order of the system of equations $3x + y + 2z = 3,$ $2x - 3y - z = - 3$, $x + 2y + z = 4,$ are

$\left| {\,\begin{array}{*{20}{c}}5&3&{ - 1}\\{ - 7}&x&{ - 3}\\9&6&{ - 2}\end{array}\,} \right| = 0$, then $ x$ is equal to

Let the system of linear equations

$x+y+\alpha z=2$

$3 x+y+z=4$

$x+2 z=1$

have a unique solution $\left(x^{*}, y^{*}, z^{*}\right)$. If $\left(\alpha, x^{*}\right),\left(y^{*}, \alpha\right)$ and $\left(x^{*},-y^{*}\right)$ are collinear points, then the sum of absolute values of all possible values of $\alpha$ is

If a system of the equation ${(\alpha + 1)^3}x + {(\alpha + 2)^3}y - {(\alpha + 3)^3} = 0$ and $(\alpha + 1)x + (\alpha + 2)y - (\alpha + 3) = 0,x + y - 1 = 0$ is constant. what is the value of $\alpha $

Let $\alpha, \beta$ and $\gamma$ be real numbers such that the system of linear equations

$x+2 y+3 z=\alpha$

$4 x+5 y+6 z=\beta$

$7 x+8 y+9 z=\gamma-$

is consistent. Let $| M |$ represent the determinant of the matrix

$M=\left[\begin{array}{ccc}\alpha & 2 & \gamma \\ \beta & 1 & 0 \\ -1 & 0 & 1\end{array}\right]$

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