JEE Main 2026 Jan 21 Shift 2, Mathematics Q18: Linear Equations And Cramer’s Rule

JEE Main2026Jan 21, Shift 2Mathematics

If the system of equations

$3 x + y + 4 z = 3$

$2 x + α y - z = - 3$

$X + 2 y + z = 4$

has no solution, then the value of $α$ is equal to:

A
19
B
4
C
13
D
23

Solution

For a square linear system to have no solution, the coefficient determinant $Δ = 0$ while at least one of $Δ_{x}, Δ_{y}, Δ_{z} \neq = 0$ .

$Δ = 314 2 α - 1 121 = 3 (α + 2) + 1 (- 1 - (- 2)) \cdot (- 1) + 4 (4 - α)$ .

Carefully: expand along row 1: $3 [α (1) - (- 1) (2)] - 1 [2 (1) - (- 1) (1)] + 4 [2 (2) - α (1)] = 3 (α + 2) - 1 (3) + 4 (4 - α) = 3 α + 6 - 3 + 16 - 4 α = - α + 19$ .

$Δ = 0 \Rightarrow α = 19$ .

Check $Δ_{x} \neq = 0$ at $α = 19$ : replace first column with RHS $(3, - 3, 4)^{T}$ ; computation gives a non-zero value, confirming ‘no solution’.

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JEE Main 2026 Jan 21 Shift 2, Mathematics Q18: Linear Equations And Cramer’s Rule

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