Consider the system of linear equations x + y + z = 5 , x + 2 y + λ 2 z = 9 and x + 3 y + λ z = μ , where λ , μ ∈ R . Then, which of the following statement is NOT correct ?

Step-by-step solution

Given: x + y + z = 5 x + 2 y + λ 2 z = 9 x + 3 y + λ z = μ ⇒ △ = 1 1 1 1 2 λ 2 1 3 λ ⇒ △ = 2 λ - 3 λ 2 - λ + λ 2 + 1 ⇒ △ = - 2 λ 2 + λ + 1 We know that for infinite solutions, △ = 0 & △ 1 = △ 2 = △ 3 = 0 ⇒ - 2 λ 2 + λ + 1 = 0 ⇒ 2 λ 2 - λ - 1 = 0 ⇒ λ - 1 2 λ + 1 = 0 ⇒ λ = 1 , - 1 2 Now, finding △ 1 for λ = 1 we get, Δ 1 = 5 1 1 9 2 1 μ 3 1 ⇒ Δ 1 = μ - 13 And for λ = - 1 2 we get, Δ 1 = 5 1 1 9 2 1 4 μ 3 - 1 2 = 1 4 5 1 4 9 2 1 μ 3 - 2 = 7 4 13 - μ So, for λ ≠ 1 & μ ≠ 13 system has no solution, Hence, option C is the