JEE Main 2025 Apr 2 Shift 1, Mathematics Q14: Linear Equations And Cramer’s Rule

JEE Main2025Apr 2, Shift 1Mathematics

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 solutions, then the value of $22 β - 9 α$ is :

Solution

For infinitely many solutions, the coefficient determinant $Δ$ and all the substituted determinants $Δ_{1}, Δ_{2}, Δ_{3}$ must vanish.

Set $Δ = 0$ :

$Δ = 31 β 2 α - 1 121 = 3 (α + 2) - 1 (2 + 1) + β (4 - α)$

$= 3 α + 6 - 3 + β (4 - α) = 3 α + 3 + 4 β - α β = 0 \dots (i)$

Set $Δ_{3} = 0$ (replace the third column with the RHS):

$Δ_{3} = 313 2 α - 3 124 = 3 (4 α + 6) - 1 (8 + 3) + 3 (4 - α)$

$= 12 α + 18 - 11 + 12 - 3 α = 9 α + 19 = 0 \Rightarrow α = - \frac{19}{9}$

Substitute back into (i):

$3 (- \frac{19}{9}) + 3 + 4 β - (- \frac{19}{9}) β = 0$

$- \frac{19}{3} + 3 + 4 β + \frac{19 β}{9} = 0$

Multiply by 9: $- 57 + 27 + 36 β + 19 β = 0 \Rightarrow 55 β = 30 \Rightarrow β = \frac{6}{11}$ .

Compute the required combination:

$22 β - 9 α = 22 \cdot \frac{6}{11} - 9 \cdot (- \frac{19}{9}) = 12 + 19 = 31$

$∴$ Option (2) is correct.

Concept behind this question

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