Question

If the system of equations 2x – 3y + 5z =12, 3x + y+λz

=μ, x–7y + 8z = 17 has infinitely many real solutions, then

λ + μ =

• A. 5
• B. 9
• C. 3
• D. None

Answer 1

Answer: (B)

9

Answer 2

For infinitely many solution

∆ = 0, ∆₁ = 0, ∆₂ = 0, ∆₃ = 0

∆ =$\left| \mspace{6mu}\begin{matrix} 2 & –3 & 5 \\ 3 & 1 & \lambda \\ 1 & –7 & 8 \end{matrix}\mspace{6mu} \right|$ = 0

⇒ 2(8 + 7λ) + 3(24 – λ) + 5(–21 – 1) = 0

⇒ 16 + 14λ + 72 – 3λ – 110 = 0

⇒ 11λ = 22 ⇒ λ = 2

∆₁ =$\left| \begin{matrix} 12 & –3 & 5 \\ \mu & 1 & \lambda = 2 \\ 17 & –7 & 8 \end{matrix} \right|$ = 0

⇒ 12(8 + 14) + 3(8μ – 34) + 5(–7μ –17) = 0

⇒ 264 + 24μ – 102 – 35 μ – 85 = 0

⇒ 11 μ = 77 ⇒ μ = 7

∴ λ + μ = 2 + 7 = 9