Consider the system of linear equation x + y + z = 4 μ , x + 2 y + 2 λ z = 10 μ , x + 3 y + 4 λ 2 z = μ 2 + 15 , where λ , μ ∈ R . Which one of the following statements is NOT correct?

Step-by-step solution

Given: System of linear equations, x + y + z = 4 μ , x + 2 y + 2 λ z = 10 μ , x + 3 y + 4 λ z 2 = μ 2 + 15 . ⇒ Δ = 1 1 1 1 2 2 λ 1 3 4 λ 2 ⇒ Δ = 8 λ 2 - 6 λ - 4 λ 2 - 2 λ + 3 - 2 ⇒ Δ = 8 λ 2 - 6 λ - 4 λ 2 + 2 λ + 1 ⇒ Δ = 4 λ 2 - 4 λ + 1 ⇒ Δ = ( 2 λ - 1 ) 2 For unique solution, Δ ≠ 0 . ⇒ 2 λ - 1 ≠ 0 ⇒ λ ≠ 1 2 Let, Δ = 0 , λ = 1 2 We know that, for infinite solutions, Δ x = Δ y = Δ z = 0 ⇒ Δ x = 4 μ 1 1 10 μ 2 1 μ 2 + 15 3 1 ⇒ Δ x = 4 μ 2 - 3 - 10 μ - μ 2 - 15 + 30 μ - 2 μ 2 - 30 ⇒ Δ x = - 4 μ - 10 μ + μ 2 + 15 + 30