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(C) The given system of equations is:$x+y+z=1$$2x+Ny+2z=2$$3x+3y+Nz=3$The system has a unique solution if the determinant of the coefficient matrix $\

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$20$

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(C) The given system of equations is:$x+y+z=1$$2x+Ny+2z=2$$3x+3y+Nz=3$The system has a unique solution if the determinant of the coefficient matrix $\Delta 
eq 0$.$\Delta = \begin{vmatrix} 1 & 1 & 1 \ 2 & N & 2 \ 3 & 3 & N \end{vmatrix}$Expanding along the first row:$\Delta = 1(N^2 - 6) - 1(2N - 6) + 1(6 - 3N)$$\Delta = N^2 - 6 - 2N + 6 + 6 - 3N$$\Delta = N^2 - 5N + 6 = (N-2)(N-3)$For a unique solution,$\Delta 
eq 0$,which implies $N 
eq 2$ and $N 
eq 3$.Since $N$ is the outcome of a fair die,$N \in \{1, 2, 3, 4, 5, 6\}$.The values of $N$ for which the system has a unique solution are $\{1, 4, 5, 6\}$.There are $4$ such values,so the probability is $\frac{4}{6}$,which gives $k = 4$.The sum of $k$ and all possible values of $N$ for which the system has a unique solution is $4 + (1 + 4 + 5 + 6) = 4 + 16 = 20$.

Let $N$ denote the number that turns up when a fair die is rolled. If the probability that the system of equations $x+y+z=1$,$2x+Ny+2z=2$,and $3x+3y+Nz=3$ has a unique solution is $\frac{k}{6}$,then the sum of the value of $k$ and all possible values of $N$ is:

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