For α , β ∈ ℝ , suppose the system of linear equations x - y + z = 5 2 x + 2 y + α z = 8 3 x - y + 4 z = β has infinitely many solutions. Then α and β are the roots of

Step-by-step solution

Given, α , β ∈ ℝ , suppose the system of linear equations x - y + z = 5 2 x + 2 y + α z = 8 3 x - y + 4 z = β has infinitely many solutions, So, finding △ = 0 we get, 1 - 1 1 2 2 α 3 - 1 4 = 0 ⇒ 8 + α - 2 ( - 4 + 1 ) + 3 ( - α - 2 ) = 0 ⇒ 8 + α + 6 - 3 α - 6 = 0 ⇒ α = 4 Now finding △ z = 0 we get, 1 - 1 5 2 2 8 3 - 1 β = 0 ⇒ 3 - 18 + 1 - 2 + β 4 = 0 ⇒ β = 14 Hence, quadratic equation having roots 4