Two fair dice are thrown. The numbers on them are taken as $λ a n d μ$ and a system of linear equations. $x + y + z = 5, x + 2 y + 3 z = μ, x + 3 y + λ z = 1,$ is constructed. If P is the probability that the system has a unique solution and q is the probability that the system has no solution, then

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$p = \frac{1}{6} a n d q = \frac{5}{36}$

$p = \frac{5}{6} a n d q = \frac{1}{36}$

$p = \frac{1}{6} a n d q = \frac{1}{36}$

$p = \frac{5}{6} a n d q = \frac{5}{36}$

answer is D.

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Detailed Solution

$n (S) = 36$

given equations can be written Augment matrix

$[\begin{matrix} 1 & 1 & 1 & 1 \\ 1 & 2 & 3 & μ \\ 1 & 3 & λ & 1 \end{matrix}] \begin{matrix} R_{2} \to R_{2} - R_{1} \\ R_{3} \to R_{3} - R_{1} \end{matrix} [\begin{matrix} 1 & 1 & 1 & 5 \\ 0 & 1 & 2 & μ - 5 \\ 0 & 2 & λ - 1 & - 4 \end{matrix}] R_{3} \to R_{3} - 2 R_{2} [\begin{matrix} 1 & 1 & 1 & 5 \\ 0 & 1 & 2 & μ - 5 \\ 0 & 0 & λ - 5 & 6 - 2 μ \end{matrix}]$

$P = Unique sol \Rightarrow λ \neq 5, μ \in R P = \frac{5 \times 6}{36} = \frac{5}{6} q = no Solution \Rightarrow λ = 5, μ \neq 3 q = \frac{1 \times 5}{36} = \frac{5}{36}$