Let N denote the number that turns up when a fair die is rolled. If the probability that the system of equations
has unique solution is , then the sum of value of k and all possible values of N is
has unique solution is , then the sum of value of k and all possible values of N is
- A.21
- B.19
- C.18
- D.20
Show answer and solution
Correct option: D
Solution
For unique solution, the determinant of the coefficient matrix must be non-zero.
D = |1, 1, 1; 2, N, 2; 3, 3, N|
Expanding: 1(N²- 6) - 1(2N - 6) + 1(6 - 3N) = N² - 6 - 2N + 6 + 6 - 3N = N² - 5N + 6 = (N-2)(N-3)
So D = 0 when N = 2 or N = 3. For all other values of N ∈ {1,2,3,4,5,6}, the system has a unique solution.
Values giving unique solution: N ∈ {1, 4, 5, 6} → 4 out of 6 outcomes.
So probability = 4/6 = k/6, giving k = 4.
Possible values of N (where unique solution exists): 1, 4, 5, 6 → sum = 16.
Total = k + sum of all possible N = 4 + 16 = 20.
D = |1, 1, 1; 2, N, 2; 3, 3, N|
Expanding: 1(N²- 6) - 1(2N - 6) + 1(6 - 3N) = N² - 6 - 2N + 6 + 6 - 3N = N² - 5N + 6 = (N-2)(N-3)
So D = 0 when N = 2 or N = 3. For all other values of N ∈ {1,2,3,4,5,6}, the system has a unique solution.
Values giving unique solution: N ∈ {1, 4, 5, 6} → 4 out of 6 outcomes.
So probability = 4/6 = k/6, giving k = 4.
Possible values of N (where unique solution exists): 1, 4, 5, 6 → sum = 16.
Total = k + sum of all possible N = 4 + 16 = 20.
[JEE Main 2023 · 24 Jan · Shift 1 · Paper-1(B.E./B. Tech.)]