JEE 2023 — Mathematics PYQ

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Question

JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

Let the system of linear equations $-x + 2y - 9z = 7$ , $-x + 3y + 7z = 9$ $-2x + y + 5z = 8$ , $-3x + y + 13z = \lambda$ have a unique solution $x = \alpha, y = \beta, z = \gamma$ . Then the distance of the point $(\alpha, \beta, \gamma)$ from the plane $2x - 2y + z = \lambda$ is:

Choose the correct answer:

Explanation

Solution

Augmented Matrix (pehli 3 equations):

$[A:B] = \begin{bmatrix} -1 & 2 & -9 & : & 7 \\ -1 & 3 & 7 & : & 9 \\ -2 & 1 & 5 & : & 8 \end{bmatrix}$

Row operations ke baad: $z = 0, y = 2, x = -3$ .

Point $(\alpha, \beta, \gamma) = (-3, 2, 0)$ .
$\lambda$ nikalna:

Chauthi equation mein value rakhne par: $-3(-3) + 2 + 13(0) = \lambda \implies \lambda = 11$ .
Distance calculation:

Point $(-3, 2, 0)$ ka plane $2x - 2y + z - 11 = 0$ se distance:

$d = \frac{|2(-3) - 2(2) + 0 - 11|}{\sqrt{2^2 + (-2)^2 + 1^2}} = \frac{|-6 - 4 - 11|}{3} = \frac{21}{3} = 7$

Sahi vikalp: (1) 7