A) \[(-3,\text{ }1)\]
B) \[(2,\text{ 4})\]
C) \[(1,\text{ }-3)\]
D) \[(-4,\text{ 2})\]
Correct Answer: B
Solution :
For unique solution\[\left| \begin{matrix} 1+\alpha & \beta & 1 \\ \alpha & 1+\beta & 1 \\ \alpha & \beta & 2 \\ \end{matrix} \right|\ne 0\] \[\Rightarrow \]\[(1+\alpha )(2+2\beta -\beta )-\beta (2\alpha -\alpha )+1(\alpha \beta -\alpha -\alpha \beta )\ne 0\] \[\Rightarrow \]\[2+2\beta -\beta +2\alpha +2\alpha \beta -\alpha \beta -2\alpha \beta +\alpha \beta -\alpha \ne 0\] \[\Rightarrow \]\[2+\alpha +\beta \ne 0\] \[\therefore \](2, 4) satisfies the above condition.
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