JEE Main 12 Jan 2019 Morning Question 11
Question: 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 solution, is [JEE Main Online Paper Held On 12-Jan-2019 Morning]
Options:
A) $ (-3,1) $
B) $ (2,\text{ 4}) $
C) $ (1,-3) $
D) $ (-4,\text{ 2}) $
Show Answer
Answer:
Correct Answer: B
Solution:
For unique solution $ \begin{vmatrix} 1+\alpha & \beta & 1 \\ \alpha & 1+\beta & 1 \\ \alpha & \beta & 2 \\ \end{vmatrix} \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.
