The system of equations $\mathrm{x}+\mathrm{y}+\mathrm{z}=6$, $x+2 y+5 z=9$, $\mathrm{x}+5 \mathrm{y}+\lambda \mathrm{z}=\mu$, has no solution if

(B) $\lambda \neq 17, \mu \neq 18$

JEE Main · 2025 · 23 Jan 2025 · Shift 2 · Matrices

The system of equations

x + y + z = 6

x + 2 y + 5 z = 9

x + 5 y + λ z = μ

,
has no solution if

Answer: (A)

For the system

x + y + z = 6, x + 2 y + 5 z = 9, x + 5 y + λ z = μ

the coefficient determinant is

D = 111125 15 λ = (2 λ - 25) - (λ - 5) + (5 - 2) = λ - 17

For no solution,

D = 0

, so

λ = 17

.

With

λ = 17

, replace the

z

-column with the constants to get

D_{z}

D_{z} = 111125 69 μ = (2 μ - 45) - (μ - 9) + 6 (3) = μ - 18

The system has no solution (rather than infinitely many) only when

D_{z} \neq = 0

, i.e.

μ \neq = 18

.

Hence the system has no solution precisely when

λ = 17, μ \neq = 18

JEE Main · 2025 · 23 Jan 2025 · Shift 2 · Single correct option