JEE Advanced 2021 — Paper 1 — Question 45

Let $\alpha, \beta$ and $\gamma$ be real numbers such that the system of linear equation $\begin{gathered}x+2 y+3 z=\alpha ,4 x+5 y+6 z=\beta ,7 x+8 y+9 z=\gamma-1\end{gathered}$ is consistent. Let $|\mathrm{M}|$ 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 $P$.

The value of $\left| \text{M} \right|$ is