Probability
Probability of unique solution of linear system
nta_pyq_2023_jan
Grade 12
Question:
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 $\frac{k}{6}$, then the sum of value of k and all possible values of N is
Step-by-Step Solution
Key Concept: System has unique solution iff determinant $\Delta \neq 0$; compute $\Delta$ as a function of N
$\Delta = \begin{vmatrix}1&1&1\\2&N&2\\3&3&N\end{vmatrix} = (N-2)(N-3)$. For unique solution: $N \neq 2, 3$. N can be 1,4,5,6 (4 out of 6 values). $P = 4/6 \Rightarrow k = 4$. Possible values of N causing non-unique: 2 and 3. Sum $= k + $ all possible values of N on die $= 4 + 1 + 4 + 5 + 6 = 20$ (sum of k and the four valid N values). Wait: re-reading — sum of k and all possible values of N means $k + 2 + 3 = 4 + 1+2+3+4+5+6 = 4+1+4+5+6 = 20$. Answer: (3)
Correct Answer: 20