Two fair dice are thrown. The numbers on them are taken as λ and μ , and a system of linear equations x + y + z = 5 x + 2 y + 3 z = μ x + 3 y + λ z = 1 is constructed. If p is the probability that the system has a unique solution and q is the probability that the system has no solution, then:

Step-by-step solution

1 1 1 1 2 3 1 3 λ = λ - 5 Δ 1 = 1 1 5 2 3 μ 3 λ 1 = 10 λ + 3 μ - λμ - 44 Δ 2 = 1 1 5 1 3 μ 1 λ 1 = 5 λ + μ - λμ - 13 Δ 3 = 1 1 5 1 2 μ 1 3 1 = - 2 μ + 6 For unique solution λ ≠ 5 , μ ∈ 1 , 2 , 3 , 4 , 5 , 6 ⇒ p = 5 6 × 1 = 5 6 For No Solution λ = 5 ,   μ ≠ 3 ⇒    q = 1 6 × 5 6 = 5 36