If A and B are the two real values of k for which the system of equations x + 2y + z = 1, x + 3y + 4z = k , x + 5y +10z = k² is consistent, then A + B =

Augmented matrix 
$$\begin{gathered} \left[\mathrm{AB}\right]=\left[\begin{array}{cccc}1& 2& 1& 1\\ 1& 3& 4& \mathrm{k}\\ \mathrm{k}& 1& 5& {\mathrm{k}}^2\end{array}\right]\begin{array}{c}{\mathrm{R}}_2-{\mathrm{R}}_1\\ {\mathrm{R}}_3-{\mathrm{R}}_1\\ ~\end{array}\left|\begin{array}{cccc}1& 2& 1& 1\\ 0& 1& 3& \mathrm{k}-1\\ 0& 3& 9& {\mathrm{k}}^2-1\end{array}\right|{\mathrm{R}}_3-3{\mathrm{R}}_2 \\ \left|\begin{array}{cccc}1& 2& 1& 1\\ 0& 1& 3& \mathrm{k}-1\\ 0& 0& 0& {\mathrm{k}}^2-3\mathrm{k}+2\end{array}\right| \end{gathered}$$
Given system of equations is consistent
$\begin{array}{l}\Rightarrow \mathrm{rank}\mathrm{A}=\mathrm{rank}\left[\mathrm{AB}\right]\Rightarrow {\mathrm{k}}^2-3\mathrm{k}+2=0\Rightarrow \mathrm{k}=1,2\\ \Rightarrow \mathrm{A}+\mathrm{B}=1+2=3.\end{array}$