An ordered pair  $\left(\alpha ,\beta \right)$ for which the system of linear equations 

$\left(1+\alpha \right)x+\beta y+z=2$

$\alpha x+\left(1+\beta \right)y+z=3$

$ax+\beta y+2z=2$

Has a unique solution , is 

Given system of linear equations , 

$\begin{array}{r}(1+\alpha )x+\beta y+z=2\\ \alpha x+(1+\beta )y+z=3\\ \alpha x+\beta y+2z=2\end{array}$

Has a unique solution , if 

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

$\text{ Apply }{R}_1\to R_1-R_3\text{ and }{R}_2\to R_2-R_3$

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

Note that, only (2,4) satisfy the Eq . (i)