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 $M=\left[\begin{array}{ccc}\alpha & 2 & \gamma \\ \beta & 1 & 0 \\ -1 & 0 & 1\end{array}\right]$ 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 $|M|$ is ____.

JEE Advanced · 2021 · 03 Oct 2021 · Paper 1 · Determinants

Let

α, β

and

γ

be real numbers such that the system of linear equations

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 β - 1 210 γ 01

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

∣ M ∣

is ____.

Answer: 1

For the system to be consistent, the third equation must be a linear combination of the first two:

7 x + 8 y + 9 z - (γ - 1) = A (4 x + 5 y + 6 z - β) + B (x + 2 y + 3 z - α)

. Comparing coefficients: x: 7=4A+B, y: 8=5A+2B gives A=2, B=-1. From the constant term:

- (γ - 1) = - 2 β - α

, which gives

α - 2 β + γ = 1

. Therefore,

∣ M ∣ = α β - 1 210 γ 01 = α - 2 β + γ = 1

JEE Advanced · 2021 · 03 Oct 2021 · Paper 1 · Numerical answer