Question
Let n be the number obtained on rolling a fair die. If the probability that the system x- n y+z=6 x+( n -2) y+( n +1) z=8 ( n -1) y+z=1 has a unique solution is k 6 , then the sum of k and all possible values of n is :
JEE Main
Mathematics
Determinants
2026
JEE Main 2026 (22 January Shift 2)
JEE Main Mathematics Question (2026) — Solution
Options
- A. 20
- B. 24
- C. 21
- D. 22
Answer
D. 22
Step-by-step solution
The coefficient matrix has determinant (A) = -(n-1)(n-2). The system has a unique solution when (A) 0, which requires n 1 and n 2. For a fair die, n \ 1,2,3,4,5,6\ , so the unique solution occurs for n \ 3,4,5,6\ . The probability is 4 6 = 2 3 , giving k = 4. The sum of k and all possible values of n where unique solution exists is 4 + (3+4+5+6) = 22.
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