1. Tardigrade
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  4. Let α, β and γ be real numbers such that the system of linear equation x+2 y+3 z=α 4 x+5 y+6 z=β 7 x+8 y+9 z=γ-1 is consistent. Let | M | represent the determinant of the matrix M=[α 2 γ β 1 0 -1 0 1] Let P be the plane containing all those ( α, β, γ ) 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 D is .

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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 $D$ 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
$\alpha+\gamma-1-2 \beta=0$
Plane $P$ is $x-2 y+z-1=0$
$D=\left(\frac{|-2-1|}{\sqrt{1+4+1}}\right)^{2}=1.5$