If the system of equations $2x+y-z=5;2x-5y+\lambda z=\mu ;x+2y-5z=7$  has infinitely many solutions, then ${(\lambda +\mu )}^2+{(\lambda -\mu )}^2$ is equal to 

Given equations are $2x+y-z=5;2x-5y+\lambda z=\mu ;x+2y-5z=7$

It has infinitely many solutions then

$$\begin{gathered} ∆=0\Rightarrow \left|\begin{array}{ccc}2& 1& -1\\ 2& -5& \lambda \\ 1& 2& -5\end{array}\right|=0 \\ \Rightarrow 2(25-2\lambda )-1(-10-\lambda )-1(4+5)=0 \\ \Rightarrow \lambda =17 \\ Now {∆}_3=0 \\ \Rightarrow \left|\begin{array}{ccc}2& 1& 5\\ 2& -5& \mu \\ 1& 2& 7\end{array}\right|=0 \\ \Rightarrow \mu =-13 \\ Now {(\lambda +\mu )}^2+{(\lambda -\mu )}^2=2({\lambda }^2+{\mu }^2)=916 \end{gathered}$$