JEE 2025 — Mathematics PYQ

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Question

JEE 2025 — Mathematics PYQ

JEE | Mathematics | 2025

If the system of linear equations
$3 x + y + β z = 3$
$2 x + α y - z = - 3$
$x + 2 y + z = 4$
has infinitely many solution, then the value of $22 β - 9 α$ is:

Choose the correct answer:

Explanation

$3 x + y + β z = 3$
$2 x + α y - z = - 3$
$x + 2 y + z = 4$

$D = 321 am p; 1 am p; α am p; 2 am p; β am p; - 1 am p; 1$

$D_{x} = 3 - 3 4 am p; 1 am p; α am p; 2 am p; β am p; - 1 am p; 1$

$D_{y} = 321 am p; 3 am p; - 3 am p; 4 am p; β am p; - 1 am p; 1$

$D_{z} = 321 am p; 1 am p; α am p; 2 am p; 3 am p; - 3 am p; 4$

There are infinite many solution, therefore, $D = 0 = D_{x} = D_{y} = D_{z}$

$D = 0 = 321 am p; 1 am p; α am p; 2 am p; β am p; - 1 am p; 1$

$\Rightarrow 3 α + 4 β - α β + 3 = 0$

$D_{z} = 0 = 321 am p; 1 am p; α am p; 2 am p; 3 am p; - 3 am p; 4$

$\Rightarrow 0 = 9 α + 19$

$α = \frac{- 19}{9}, β = \frac{6}{11}$

$\Rightarrow 22 β - 9 α = 31$