Let the system of linear equations $4 x+\lambda y+2 z=0$ $2 x-y+z=0$ $\mu x+2 y+3 z=0, \lambda, \mu \in R$ has a non-trivial solution. Then which of the following is true?

(A) $\mu=6, \lambda \in \mathrm{R}$

(B) $\mu=-6, \lambda \in \mathrm{R}$

(C) $\lambda=2, \mu \in \mathrm{R}$

(D) $\lambda=3, \mu \in \mathrm{R}$

JEE Main · 2021 · 18 Mar 2021 · Shift 2 · Matrices

Let the system of linear equations

4 x + λ y + 2 z = 0

2 x - y + z = 0

μx + 2 y + 3 z = 0, λ, μ \in R

has a non-trivial solution. Then which of the following is true?

Answer: (A)

A homogeneous system has a non-trivial solution exactly when the determinant of its coefficient matrix vanishes:

42 μ λ - 1 2 213 = 4 (- 3 - 2) - λ (6 - μ) + 2 (4 + μ) = λ μ + 2 μ - 6 λ - 12

This factorises:

λ μ + 2 μ - 6 λ - 12 = μ (λ + 2) - 6 (λ + 2) = (λ + 2) (μ - 6)

The determinant vanishes for every real

λ

precisely when

μ = 6

, so the system has a non-trivial solution with

μ = 6

and

λ

unrestricted in

R

JEE Main · 2021 · 18 Mar 2021 · Shift 2 · Single correct option