An ordered pair $(\alpha ,\beta )$ for which the system of linear equations $(1+\alpha )x+\beta y+z=2,\alpha x+(1+\beta )y+z=3$, $\alpha x+\beta y+2z=2$ has a unique solutions, is

For unique solution

 $\begin{array}{l}\left|\begin{array}{clc}1+\alpha & \beta & 1\\ \alpha & 1+\beta & 1\\ \alpha & \beta & 2\end{array}\right|\ne 0\\ \Rightarrow (1+\alpha )(2+2\beta -\beta )-\beta (2\alpha -\alpha )+1(\alpha \beta -\alpha -\alpha \beta )\ne 0\\ \Rightarrow 2+2\beta -\beta +2\alpha +2\alpha \beta -\alpha \beta -2\alpha \beta +\alpha \beta -\alpha \ne 0\\ \Rightarrow 2+\alpha +\beta \ne 0\end{array}$

$\therefore (2,4)$ satisfies the above condition.