If the system of equations 2x + y - z = 5 2x - 5y + = μ x + 2y - 5z...

Given system of equations.

$2x + y - z = 5$

$2x - 5y + \lambdaz = \mu$

$x + 2y - 5z = 7$

For infinite solution $\Delta = 0 = {\Delta}_{1} = {\Delta}_{2} = {\Delta}_{3}$

⇒ \Delta = \left|\begin{cases}2 & 1 & - 1 \\ 2 & - 5 & \lambda \\ 1 & 2 & - 5\end{cases}\right| = 0

$⇒ 51 - 3\lambda = 0$

$⇒ \lambda = 17$

⇒ {\Delta}_{3} = \left|\begin{cases}5 & 2 & 1 \\ \mu & 2 & - 5 \\ 7 & 1 & 2\end{cases}\right| = 0

$⇒ - 3 ( \mu + 13 ) = 0$

$⇒ \mu = - 13$

Now $( \lambda + \mu{)}^{2} + ( \lambda - \mu{)}^{2}$

$= ( 17 + 13{)}^{2} + ( 17 - 13{)}^{2}$

$= 900 + 16$

$= 916$

Hence this is the correct option.