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JEE Advanced 2021
Paper-1
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Let $\alpha, \beta$ and $\gamma$ be real numbers such that the system of linear equations $$ \begin{aligned} & x+2 y+3 z=\alpha \\ & 4 x+5 y+6 z=\beta \\ & 7 x+8 y+9 z=\gamma-1 \end{aligned} $$ 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 $|\mathrm{M}|$ is $\_\_\_\_$ .
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