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. $(-4,2)$
• D. $(2,4)$

Answer 1

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

The system of linear equation are
$(1+\alpha )x+\beta y+z=2\alpha x+(1+\beta )y+z=3\alpha x+\beta y+2z=2$
for unique solution, $\Delta \ne 0$
$\Rightarrow \left|\begin{array}{ccc}1+\alpha & \beta & 1\\ \alpha & 1+\beta & 1\\ \alpha & \beta & 2\end{array}\right|\ne 0$
Apply $R_1\to R_1-R_2$
$\Rightarrow \left|\begin{array}{ccc}1& -1& 0\\ \alpha & 1+\beta & 1\\ \alpha & \beta & 2\end{array}\right|\ne 0$
$\Rightarrow 1(2+2\beta -\beta )+1(2\alpha -\alpha )+0\ne 0\Rightarrow \alpha +\beta +2\ne 0\Rightarrow \alpha +\beta \ne -2$
Hence, $(2,4)$ will satisfy this condition.