If the system of equations \(\mathrm{x}+4 \mathrm{y}-\mathrm{z}=\lambda\), \(7 \mathrm{x}+9 \mathrm{y}+\mu \mathrm{z}=-3\), \(5 \mathrm{x}+\mathrm{y}+2 \mathrm{z}=-1\) has infinitely many solutions, then \((2 \mu +3 \lambda)\) is equal to :

JEE Main · 2024 · 08 Apr 2024 · Shift 2 · Matrices

If the system of equations

x + 4 y - z = λ

7 x + 9 y + μ z = - 3

5 x + y + 2 z = - 1

has infinitely many solutions, then

(2 μ + 3 λ)

is equal to :

Answer: (B)

Infinitely many solutions require the coefficient determinant to vanish:

175491 - 1 μ 2 = (18 - μ) - 4 (14 - 5 μ) - (7 - 45) = (18 - 56 + 38) + 19 μ = 19 μ = 0 ⟹ μ = 0

With

μ = 0

, consistency also requires the determinant formed by replacing the

x

-column with the constants

(λ, - 3, - 1)

to vanish:

λ - 3 - 1 491 - 1 02 = 18 λ - 4 (- 6) - (- 3 + 9) = 18 λ + 24 - 6 = 18 λ + 18 = 0 ⟹ λ = - 1

2 μ + 3 λ = 2 (0) + 3 (- 1) = - 3

JEE Main · 2024 · 08 Apr 2024 · Shift 2 · Single correct option