Mathematics · Determinants

JEE Main 2026 — 8 April, Evening Shift — Question 23

If the system of linear equations: $x+y+z=6, x+2y+5z=10, 2x+3y+λz=μ$ has infinitely many solutions, then the value of $λ+μ$ equals.

Option A:
$12$
Option B:
$16$
Option C:
$22$
Correct
Option D:
$28$

Answer: C

Step-by-step solution

$D=\left|\begin{array}{lll}1 & 1 & 1\\ 1 & 2 & 5\\ 2 & 3 & \lambda\end{array}\right|=\lambda-6$ $\mathrm{D}_{1}=\left|\begin{array}{ccc}6 & 1 & 1\\ 10 & 2 & 5\\ \mu & 3 & \lambda\end{array}\right|=2 \lambda+3 \mu-60$ $\mathrm{D}_{2}=\left|\begin{array}{ccc}1 & 6 & 1\\ 1 & 10 & 5\\ 2 & \mu & \lambda\end{array}\right|=4(\lambda-\mu+10)$ $D_{3}=\left|\begin{array}{ccc}1 & 1 & 6\\ 1 & 2 & 10\\ 2 & 3 & \mu\end{array}\right|=\mu-16$

For infinite many solution $\mathrm{D}=\mathrm{D}_{1}=\mathrm{D}_{2}=\mathrm{D}_{3}=0$ $\Rightarrow \lambda=6 \& \mu=16$ $\Rightarrow \lambda+\mu=22$

Answer key and solution verified before publishing.

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Exam: JEE Main 2026
Paper: 8 April, Evening Shift
Subject: Mathematics
Chapter: Determinants
Topic: System of Linear Equations using Determinants