Two fair dice are thrown the number on them are taken as $λ$ and $μ$ and the 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, q is the probability that system has no solution and ‘r’ is the probability that system has infinitely many solution then true statement of the following

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$p = \frac{5}{6}; q = \frac{1}{36}; r = \frac{5}{36}$

$p = \frac{1}{36}; q = \frac{5}{36}; r = \frac{5}{6}$

$p = \frac{5}{36}; q = \frac{1}{36}; r = \frac{1}{6}$

$p = \frac{5}{6}; q = \frac{5}{36}; r = \frac{1}{36}$

answer is A.

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Detailed Solution

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

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

$\begin{array}{l} λ = 5; μ = 3 \Rightarrow i n f i n i t e \Rightarrow r = \frac{1}{6} . \frac{1}{6} \Rightarrow r = \frac{1}{36} \\ λ \neq 5; μ \in R \Rightarrow u n i q u e s \Rightarrow P = 1 - \frac{1}{6} \Rightarrow P = \frac{5}{6} \\ λ = 5; μ \neq 3 \Rightarrow n o . s o l u t i o n \Rightarrow q = \frac{1}{6} (1 - \frac{1}{6}) = q = \frac{5}{36} \end{array}$