Question

\[ x - ny + z = 6
x - (n - 2)y + (n + 1)z = 8
(n - 1)y + z = 1 \] Let \( n \) be the number on the die when rolled randomly. Then \( P \) (that system equation has a unique solution) = \( \frac{k}{6} \). Then sum of value of \( k \) and all possible values of \( n \) is:

• A. 22
• B. 21
• C. 20
• D. 24

Hint:

To find the condition for a unique solution, calculate the determinant of the coefficient matrix and ensure it is non-zero.

Answer 1

The Correct Option is D

Step 1: Analyze the system of equations.
The system of equations can be written in matrix form as: \[ \begin{pmatrix} 1 & -n & 1
1 & -(n-2) & (n+1)
(n-1) & 1 & 1 \end{pmatrix} \begin{pmatrix} x
y
z \end{pmatrix} = \begin{pmatrix} 6
8
1 \end{pmatrix} \] To ensure that the system has a unique solution, the determinant of the coefficient matrix must be non-zero. Step 2: Calculate the determinant.
After calculating the determinant and solving, we get the possible values for \( k \) and \( n \). Step 3: Conclusion.
The sum of the values of \( k \) and \( n \) gives 24. Final Answer: \[ \boxed{24}. \]