The values of λ and μ for which the system of linear equations x + y + z = 2 , x + 2y + 3z = 5 , x + 3y + = μ has infinitely many solutions, are respectively

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Correct answer: C. 5 and 8

For infinite number of solutions, all determinants $D, {D}_{1}, {D}_{2}\text{and}{D}_{3}$ must be zero and planes represented by these equations should not be parallel. So, we have,

$D = \left|$ \begin{cases}1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & \lambda\end{cases} $\right| = 0 ⇒ \lambda = 5$

${D}_{1} = \left|$ \begin{cases}2 & 1 & 1 \\ 5 & 2 & 3 \\ \mu & 3 & \lambda\end{cases} $\right| = 0$

${D}_{2} = \left|$ \begin{cases}1 & 2 & 1 \\ 1 & 5 & 3 \\ 1 & \mu & \lambda\end{cases} $\right| = 0$

And ${D}_{3} = \left|$ \begin{cases}1 & 1 & 2 \\ 1 & 2 & 5 \\ 1 & 3 & \mu\end{cases} $\right| = 0 ⇒ \mu = 8$

The values of λ and μ for which the system of linear equations x +...

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