Let N denote the number that turns up then 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

Options

A. 18
B. 19
C. 20
D. 21

Answer

To solve the problem, we need to determine the values of \( N \) for which the system of equations has a unique solution. The system of equations is:

1. \( x + y + z = 1 \)
2. \( 2x + Ny + 2z = 2 \)
3. \( 3x + 3y + Nz = 3 \)

### Step 1: Write the system in matrix form

The system can be represented in the form \( AX = B \), where:

\[ A = \begin{pmatrix} 1 & 1 & 1 \\ 2 & N & 2 \\ 3 & 3 & N \end{pmatrix}, \quad X = \begin{pmatrix} x \\ y \\ z \end{pmatrix}, \quad B = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} \]

### Step 2: Find the determinant of the coefficient matrix \( A \)

To find the values of \( N \) for which the system has a unique solution, we need to compute the determinant of matrix \( A \) and set it not equal to zero:

\[ \text{det}(A) = \begin{vmatrix} 1 & 1 & 1 \\ 2 & N & 2 \\ 3 & 3 & N \end{vmatrix} \]

Calculating this determinant using the rule of Sarrus or cofactor expansion:

\[ \text{det}(A) = 1 \cdot (N \cdot N - 2 \cdot 3) - 1 \cdot (2 \cdot N - 2 \cdot 3) + 1 \cdot (2 \cdot 3 - N \cdot 3) \]

\[ = N^2 - 6 - (2N - 6) + (6 - 3N) \]

\[ = N^2 - 6 - 2N + 6 + 6 - 3N \]

Combining like terms:

\[ = N^2 - 5N + 6 \]

### Step 3: Set the determinant not equal to zero

For the system to have a unique solution, we need:

\[ N^2 - 5N + 6 \neq 0 \]

### Step 4: Factor the quadratic equation

Factoring gives:

\[ (N - 2)(N - 3) \neq 0 \]

Thus, \( N \neq 2 \) and \( N \neq 3 \).

### Step 5: Identify possible values of \( N \)

Since \( N \) denotes the number that turns up when a fair die is rolled, the possible values of \( N \) are \( 1, 2, 3, 4, 5, 6 \). Excluding \( 2 \) and \( 3 \), the valid values for \( N \) are:

\[ N = 1, 4, 5, 6 \]

### Step 6: Calculate the probability

The total number of outcomes when rolling a die is \( 6 \). The favorable outcomes for unique solutions are \( 4 \) (i.e., \( 1, 4, 5, 6 \)). Thus, the probability \( P \) that the system has a unique solution is:

\[ P = \frac{4}{6} = \frac{2}{3} \]

### Step 7: Relate the probability to \( k \)

According to the problem, the probability can be expressed as \( \frac{k}{6} \). Therefore, we have:

\[ \frac{2}{3} = \frac{k}{6} \]

Cross-multiplying gives:

\[ 2 \cdot 6 = 3k \implies 12 = 3k \implies k = 4 \]

### Step 8: Calculate the final sum

We need to find the sum of \( k \) and all possible values of \( N \):

\[ k + (1 + 4 + 5 + 6) = 4 + 16 = 20 \]

### Final Answer

The sum of the value of \( k \) and all possible values of \( N \) is:

\[ \boxed{20} \]