The sum of the values of p for which the equation  $x+y+z=1, x+2y+4z=p$ and  $x+4y+10z=p^2$ have a solution is ___

$$\begin{gathered} x+y+z=1, x+2y+4z=p and x+4y+10z=p^2 \\ \Rightarrow \Delta =\left|\begin{array}{ccc}1& 1& 1\\ 1& 2& 4\\ 1& 4& 10\end{array}\right|\Rightarrow R_1\to R_1-R_2,R_2\to R_2-R_2 \\ =\left|\begin{array}{ccc}0& -1& -3\\ 0& -2& -6\\ 1& 4& 10\end{array}\right|=0 \end{gathered}$$

Since  $\Delta =0$, solution is not unique solution
The system will have infinite solutions is  ${\Delta }_1=0,{\Delta }_2=0,{\Delta }_3=0$
$$\begin{gathered} \Rightarrow {\Delta }_1=\left|\begin{array}{ccc}1& 1& 1\\ p& 2& 4\\ p^2& 4& 10\end{array}\right|=0\Rightarrow C_3\to C_3-C_2 \\ \Rightarrow {\Delta }_1=\left|\begin{array}{ccc}1& 1& 0\\ p& 2& 2\\ p^2& 4& 6\end{array}\right|=0.;p^2-3p+2=0 \end{gathered}$$

$\Rightarrow p=1$  or  2 also for these values of p,  ${\Delta }_2,{\Delta }_3=0$