Let and be real numbers such that the system of linear equations ,

MATHEMATICS · Multi-part · CBSE · Mathematics · Pair of Linear Equations in Two Variables · Pair of Linear Equations in Two Variables

Problem

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

|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? (1) The value of

|M|

is - - - - (2) The value of

D

is - - - -