JEE Main 2025 Jan 23 Shift 1, Mathematics Q19: Linear Equations And Cramer’s Rule

JEE Main2025Jan 23, Shift 1Mathematics

If the system of equations

$(λ - 1) x + (λ - 4) y + λ z = 5$

$λ x + (λ - 1) y + (λ - 4) z = 7$

$(λ + 1) x + (λ + 2) y - (λ + 2) z = 9$

has infinitely many solutions, then $λ^{2} + λ$ is equal to

Solution

For infinitely many solutions, the coefficient determinant $D$ must vanish, and so must all numerator determinants.

$D = λ - 1 λ - 4 λ λλ - 1 λ - 4 λ + 1 λ + 2 - (λ + 2) = 0$ expands to $(λ - 3) (2 λ + 1) = 0$ , so $λ = 3$ or $λ = - \frac{1}{2}$ .

Test $λ = 3$ in $D_{x}$ :

$D_{x} = 5 - 13 72 - 1 95 - 5 = 0$ (verifiable directly), so $λ = 3$ admits infinitely many solutions.

$λ = - 1/2$ fails the consistency check.

$∴ λ = 3$ and $λ^{2} + λ = 9 + 3 = 12$ .

Concept behind this question

Concept-wise practice with instant solutions on Fundamenthol.