JEE Advanced 2021 · Paper 1, Mathematics question 8

Question Stem for Question Nos. 7 and 8

Let $\alpha$, $\beta$ and $\gamma$ be real numbers such that the system of linear equations
$$\begin{aligned}x + 2y + 3z &= \alpha\\ 4x + 5y + 6z &= \beta\\ 7x + 8y + 9z &= \gamma - 1\end{aligned}$$
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 $P$.