Question Stem
Let $$\alpha, \beta$$ and $$\gamma$$ be real numbers such that the system of linear equations
$$x + 2y + 3z = \alpha$$
$$4x + 5y + 6z = \beta$$
$$7x + 8y + 9z = \gamma - 1$$
is consistent. Let $$\mid M \mid$$ 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 $$\left(\alpha, \beta, \gamma \right)$$ 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.