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

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answer is C.

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Detailed Solution

$|\begin{array}{ccc} 1 & 1 & 1 \\ 2 & N & 2 \\ 3 & 3 & N \end{array}| \neq 0 \Rightarrow (N - 2) (N - 3) \neq 0 N = 1, 4, 5, 6$

$∴$ Probability = 4/6
$∴$ Sum = (1+ 4 + 5 + 6) + 4= 20

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