Question

The values of λ for which the system of equations x + y – 3 = 0 (1 + λ)x + (2 + λ)y – 8 = 0 x – (1 + λ)y + (2 + λ) = 0 is consistent are –

• A. –5/3, 1
• B. 2/3, –3
• C. –1/3, –3
• D. 0, 0 1

Answer 1

Answer: (A)

–5/3, 1

Answer 2

The given system of equations will be consistent if

∆= $\left| \begin{matrix} 1 & 1 & - 3 \\ 1 + \lambda & 2 + \lambda & - 8 \\ 1 & - (1 + \lambda) & 2 + \lambda \end{matrix} \right|$= 0

Applying C₂ → C₂ – C₁ and C₃ → C₃ + 3 C₁ ,

we get ∆ =$\left| \begin{matrix} 1 & 0 & 0 \\ 1 + \lambda & 1 & - 5 + 3\lambda \\ 1 & - 2–\lambda & 5 + \lambda \end{matrix} \right|$= 0

⇒ (5 + λ) + (2 + λ) (3λ – 5) = 0

⇒ 5 + λ + 6λ – 10 + 3λ² – 5λ = 0

⇒ 3λ² + 2λ – 5 = 0 ⇒ (3λ + 5) (λ – 1) = 0

⇒ λ = –5/3 or λ = 1.