Question:
An ordered $\operatorname{pair}(\alpha, \beta)$ for which the system of linear equations
$(1+\alpha) x+\beta y+z=2$
$\alpha x+(1+\beta) y+z=3$
$\alpha x+\beta y+2 z=2$ has a unique solution is
Correct Option: , 3
Solution:
For unique solution
$\Delta \neq 0 \Rightarrow\left|\begin{array}{ccc}1+\alpha & \beta & 1 \\ \alpha & 1+\beta & 1 \\ \alpha & \beta & 2\end{array}\right| \neq 0$
$\left|\begin{array}{ccc}1 & -1 & 0 \\ 0 & 1 & -1 \\ \alpha & \beta & 2\end{array}\right| \neq 0 \Rightarrow \alpha+\beta \neq-2$
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