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 + N y + 2 z = 2 3 x + 3 y + N z = 3 has a unique solution is k 6 , then the sum of the value of k and all possible

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Solution

Q.
Let $N$ denote the number that turns up when a fair die is rolled. If the probability that the system of equations

$x + y + z = 1$

$2 x + N y + 2 z = 2$

$3 x + 3 y + N z = 3$

has a unique solution is $\frac{k}{6}$ , then the sum of the value of $k$ and all possible values of $N$ is [2023]

1 18

2 19

3 20

4 21

Ans.
(3)

For unique solution, determinant $\neq 0$

$| \begin{matrix} 1 & 1 & 1 \\ 2 & N & 2 \\ 3 & 3 & N \end{matrix} |$ $\neq 0$

$\Rightarrow N^{2} - 5 N + 6 \neq 0$

$\Rightarrow (N - 2) (N - 3) \neq 0 \Rightarrow N \neq 2, 3$

$∴ Possible values of N are 1, 4, 5, 6 and \frac{k}{6} = \frac{4}{6} \Rightarrow k = 4$

$Required sum = 1 + 4 + 5 + 6 + 4 = 20$

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