Question

For $\alpha, \beta \in R$, suppose the system of linear equations
$ x-y+z=5 $
$2 x+2 y+\alpha z=8 $
$3 x-y+4 z=\beta$
has infinitely many solutions. Then $\alpha$ and $\beta$ are the roots of

• A. $x^2-18 x+56=0$
• B. $x^2+14 x+24=0$
• C. $x^2-10 x+16=0$
• D. $x^2+18 x+56=0$

Answer 1

Answer: (A)

$\begin{vmatrix}1 & -1 & 1 \\ 2 & 2 & \alpha \\ 3 & -1 & 4\end{vmatrix}=0 ; 8+\alpha-2(-4+1)+3(-\alpha-2)=0$
$ 8+\alpha+6-3 \alpha-6=0 $
$ \alpha=4$