If the system of equations 2 x + y - z = 5 2 x - 5 y + λ z = μ x + 2 y - 5 z = 7 has infinitely…

Given system of equations.

$2x+y-z=5$

$2x-5y+\lambda z=\mu$

$x+2y-5z=7$

For infinite solution $\Delta =0={\Delta }_1={\Delta }_2={\Delta }_3$

$\Rightarrow \Delta =\left|\begin{array}{ccc}2& 1& -1\\ 2& -5& \lambda \\ 1& 2& -5\end{array}\right|=0$

$\Rightarrow 51-3\lambda =0$

$\Rightarrow \lambda =17$

$\Rightarrow {\Delta }_3=\left|\begin{array}{ccc}5& 2& 1\\ \mu & 2& -5\\ 7& 1& 2\end{array}\right|=0$

$\Rightarrow -3(\mu +13)=0$

$\Rightarrow \mu =-13$

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

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

$=900+16$

$=916$

Hence this is the correct option.