JEE Main 2025 Jan 22 Shift 2, Mathematics Q14: Linear Equations And Cramer’s Rule

JEE Main2025Jan 22, Shift 2Mathematics

If the system of linear equations :

$x + y + 2 z = 6$ ,

$2 x + 3 y + a z = a + 1$ ,

$- x - 3 y + b z = 2 b$ ,

where $a, b \in R$ , has infinitely many solutions, then $7 a + 3 b$ is equal to :

Solution

$ \Delta= \begin{vmatrix} 1 & 1 & 2\\ 2 & 3 & a\\ -1 & -3 & b \end{vmatrix} =0 $

$ \Rightarrow 2a+b-6=0 \qquad\cdots\cdots(1) $

$ \Delta_1= \begin{vmatrix} 1 & 1 & 6\\ 2 & 3 & a+1\\ -1 & -3 & 2b \end{vmatrix} =0 $

$ \Rightarrow a+b-8=0 \qquad\cdots\cdots(2) $

$ \text{Solving (1) and (2)} $

$ a=-2,\quad b=10 $

$ \Rightarrow 7a+3b=16 $

Concept behind this question

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