Question

The value of λ and µ for which the equation x +y + z = 3,

x + 3y + 2z = 6 and x + λy + 3z = µ have –

• A. A unique solution ; if λ = 5 , µ ∈ R
• B. No. solution; if λ≠ 5, µ = 9
• C. Infinitely many solution ; if λ = 5, µ = 9
• D. None of these

Answer 1

Answer: (C)

Infinitely many solution ; if λ = 5, µ = 9

Answer 2

∆ = $\left| \begin{matrix} 1 & 1 & 1 \\ 1 & 3 & 2 \\ 1 & \lambda & 3 \end{matrix} \right|$ = 5 – λ

For unique solution , λ ≠ 5,

Now , ∆₁ = – µ + 9, ∆₂ = – µ + 9, ∆₃ = 2 (µ –9)

∴ If ∆ = ∆₁ = ∆₂ = ∆₃ = 0 ⇒ infinite solution

i.e. λ = 5, µ = 9