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  4. An ordered pair (α , β) for which the system of linear equations (1 + α )x + β y + z = 2 α x+(1+ β )y+z = 3 α x+ β y+2z = 2 has a unique solution is

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Q. 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 MainJEE Main 2019Determinants

Solution:

For unique solution
$\Delta \ne0 \Rightarrow \begin{vmatrix}1+\alpha&\beta&1\\ \alpha&1+\beta&1\\ \alpha&\beta&2\end{vmatrix} \ne0 $
$ \begin{vmatrix}1&-1&0\\ 0&1&-1\\ \alpha&\beta&2\end{vmatrix} \ne0 \Rightarrow \alpha+\beta \ne- 2$