Let N denote the number that turns up when a fair die is rolled. If the probability that the system of equations x + y + z = 1 2 x + N y + 2 z = 2 3 x + 3 y + N z = 3 has unique solution is k 6 , then the sum of value of k and all possible values of N is

Step-by-step solution

Given equations are x + y + z = 1 2 x + N y + 2 z = 2 3 x + 3 y + N z = 3 So, ∆ = 1 1 1 2 N 2 3 3 N Applying C 2 ↔ C 2 - C 1 and C 3 ↔ C 3 - C 1 ⇒ ∆ = 1 0 0 2 N - 2 0 3 0 N - 3 ⇒ ∆ = N - 2 N - 3 For unique solution ∆ ≠ 0 , so N ≠ 2 , 3 Also, N ∈ 1 , 2 , 3 , 4 , 5 , 6 and for unique solution N can take values 1 , 4 , 5 , 6 . ⇒ P (system has unique solution ) = 4 6 = k 6 So, k = 4 Therefore, required sum is = 4 + 1 + 4 + 5 + 6 = 20