JEE Main 2026 — Jan 23, Shift 2 — Mathematics solution

JEE Main2026Jan 23, Shift 2Mathematics

The system of linear equations

$x + y + z = 6$

$2 x + 5 y + a z = 36$

$x + 2 y + 3 z = b$

has

A
unique solution for $a = 8$ and $b = 16$
B
infinitely many solutions for $a = 8$ and $b = 14$
C
infinitely many solutions for $a = 8$ and $b = 16$
D
unique solution for $a = 8$ and $b = 14$

Solution

Coefficient determinant: $D = 111 25 a 123$

Expanding: $D = 1 (15 - 2 a) - 1 (6 - a) + 1 (4 - 5) = 15 - 2 a - 6 + a - 1 = 8 - a$ .

For non-unique solutions, $D = 0 \Rightarrow a = 8$ .

With $a = 8$ , check Cramer determinants. Computing $D_{3}$ (replace third column by RHS): $1162536 12 b = 1 (5 b - 72) - 1 (2 b - 36) + 6 (4 - 5) = 5 b - 72 - 2 b + 36 - 6 = 3 b - 42$ .

For consistency we need $D_{3} = 0 \Rightarrow b = 14$ . Similarly $D_{1} = D_{2} = 0$ for the same value.

$∴$ infinitely many solutions when $a = 8$ , $b = 14$ .

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