JEE Main 2023 · 24 January · morning · Single correct · +4 / −1
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
2x + Ny + 2z = 2
3x + 3y + Nz = 3
has unique solution is k/6, then the sum of value of k and all possible values of N is :
- (a) 18
- (b) 21
- (c) 20
- (d) 19
Answer and solution
Answer (c) 20
Solution
Maths sir · 3 steps
Unique solution means the coefficient determinant is non-zero, so find which die faces make it zero.
- Condition for uniqueness. A system of three linear equations in three unknowns has a unique solution exactly when the determinant of its coefficient matrix is not zero.
|1, 1, 1; 2, N, 2; 3, 3, N| = (N² - 6) - (2N - 6) + (6 - 3N) = N² - 5N + 6
- Factorise. N² - 5N + 6 = (N-2)(N-3). It becomes zero only for N = 2 or N = 3.
N (N-2)(N-3) Unique solution? 1 2 yes 2 0 no 3 0 no 4 2 yes 5 6 yes 6 12 yes - So. The favourable faces are 1, 4, 5, 6, so the probability is 4/6 and k = 4. The required sum is 4 + (1 + 4 + 5 + 6) = 4 + 16 = 20.
The idea to keep. For a square linear system, unique solution means Δ ≠ 0. Find where Δ = 0 and remove those cases. It is often easier to count the bad cases and subtract.
Answer from the NTA final answer key. Solution by PrepWiser.