If the system of equations x + 5 y + 6 z = 4 , 2 x + 3 y + 4 z = 7 , x + 6 y + az = b has infinitely many solutions, then the point (a, b) lies on the line

Mathematics · Determinants

JEE Main 2026 — 2 April, Evening Shift — Question 25

If the system of equations $\mathrm{x} + 5\mathrm{y} + 6\mathrm{z} = 4$ , $2\mathrm{x} + 3\mathrm{y} + 4\mathrm{z} = 7$ , $\mathrm{x} + 6\mathrm{y} + \mathrm{az} = \mathrm{b}$ has infinitely many solutions, then the point (a, b) lies on the line

Option A:
$\mathrm{y} - \mathrm{x} = 3$
Option B:
$\mathrm{x} - \mathrm{y} = 3$
Correct
Option C:
$\mathrm{x} + \mathrm{y} = 11$
Option D:
$\mathrm{x} + \mathrm{y} = 12$

Answer: B

Step-by-step solution

$D=\left|\begin{array}{lll}1 & 5 & 6\\ 2 & 3 & 4\\ 1 & 6 & a\end{array}\right|=0$ $\Rightarrow \mathrm{a}=\frac{50}{7}$ $\mathrm{D}_{\mathrm{z}}=0 \Rightarrow\left|\begin{array}{ccc}1 & 5 & 4\\ 2 & 3 & 7\\ 1 & 6 & \mathrm{~b}\end{array}\right|=0$ $\Rightarrow \mathrm{b}=\frac{29}{7}$ $\therefore(\mathrm{a}, \mathrm{b})$ lies on $\mathrm{x}-\mathrm{y}=3$

Answer key and solution verified before publishing.

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Exam: JEE Main 2026
Paper: 2 April, Evening Shift
Subject: Mathematics
Chapter: Determinants
Topic: System of Linear Equations using Determinants