Mathematics - Determinant Question with Solution | TestHub

For unique solution $\left|\begin{array}{ccc}1+\alpha & \beta & 1\\ \alpha & 1+\beta & 1\\ \alpha & \beta & 2\end{array}\right|\ne 0$

Applying $C_1\to C_1+C_2+C_3$, we get

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

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

$\Rightarrow \alpha +\beta +2\ne 0$

Clearly, point $\left(2, 4\right)$ satisfying the given condition.