The number of real values of λ , such that the system of linear equations 2 x - 3 y + 5 z = 9 x + 3 y - z = - 18 3 x - y + λ 2 - λ z = 16 has no solutions, is

Step-by-step solution

For the system of linear equations 2 x - 3 y + 5 z = 9 x + 3 y - z = - 18 3 x - y + λ 2 - λ z = 16 to have no solutions, ∆ should be equal to 0 . i.e. Δ = 2 - 3 5 1 3 - 1 3 - 1 λ 2 - λ = 2 3 λ 2 - 3 λ - 1 + 3 λ 2 - λ + 3 + 5 - 1 - 9 = 9 λ 2 - 9 λ - 43 = 9 λ 2 - 9 λ - 43 Now ∆ = 0 for 2 values of λ as for the above quadratic the value of determinant is greater than zero. So, 2 values of λ satisfy the system of equations to obtain