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

• A. $(1,-3)$
• B. $(-3,1)$
• C. $(2,4)$
• D. $(-4,2)$

Answer 1

The correct option is C $(2,4)$

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

For Option C

$\alpha +\beta =2+4=6$ which is $\ne -2$

An ordered pair (2,4) for which the system of linear equations has a unique solution.