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Q. Let $\alpha, \beta$ and $\gamma$ be real numbers such that the system of linear equation
$x+2 y+3 z=\alpha$
$4 x+5 y+6 z=\beta $
$7 x+8 y+9 z=\gamma-1$
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 $P$.
The value of $|M|$ is ________.

JEE AdvancedJEE Advanced 2021

Solution:

$x+2 y+3 z=\alpha$
$4 x+5 y+6 z=\beta $
$7 x+8 y+9 z=\gamma-1$
Equation $(1)+(3)-(2)=0$. Equation $(2)$ provides
$\alpha+\gamma-1-2 \beta=0$
$|M|=\begin{vmatrix}\alpha & 2 & \gamma \\ \beta & 1 & 0 \\ -1 & 0 & 1\end{vmatrix}\left(C_{1} \rightarrow C_{1}+C_{3}\right)$
$=\begin{vmatrix}\alpha+\gamma & 2 & \gamma \\ \beta & 1 & 0 \\ 0 & 0 & 1\end{vmatrix}\left(R_{1} \rightarrow R_{1}-2 R_{2}\right)$
$\begin{vmatrix}\alpha+\gamma-2 \beta & 0 & \gamma \\ \beta & 1 & 0 \\ 0 & 0 & 1\end{vmatrix}=\begin{vmatrix}1 & 0 & \gamma \\ \beta & 1 & 0 \\ 0 & 0 & 1\end{vmatrix}=1$