JEE Main 2026 Apr 8 Shift 2, Mathematics Q3: Linear Equations And Cramer’s Rule

JEE Main2026Apr 8, Shift 2Mathematics

If the system of linear equations :

$x + y + z = 6$ ,

$x + 2 y + 5 z = 10$ ,

$2 x + 3 y + λ z = μ$ .

has infinitely many solutions, then the value of $λ + μ$ equals.

A
$12$
B
$16$
C
$22$
D
$28$

Solution

Observe that the third equation should be a linear combination of the first two for infinite solutions. Looking at the coefficients of $x$ and $y$ in row $3$ : $(2, 3)$ , we find scalars $p, q$ with $p (1, 1) + q (1, 2) = (2, 3)$ , giving $p = 1, q = 1$ .

So row $3$ must equal (row $1$ ) $+$ (row $2$ ):

Coefficient of $z$ : $λ = 1 + 5 = 6$ .

Constant: $μ = 6 + 10 = 16$ .

$∴$ $λ + μ = 6 + 16 = 22$ .

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JEE Main 2026 Apr 8 Shift 2, Mathematics Q3: Linear Equations And Cramer’s Rule

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