Mathematics - Determinant Question with Solution | TestHub

Given,

For the system of linear equations

$2x-y+3z=5$

$3x+2y-z=7$

$4x+5y+\alpha z=\beta$,

Now finding,

$D=\left|\begin{array}{ccc}2& -1& 3\\ 3& 2& -1\\ 4& 5& \alpha \end{array}\right|$

$\Rightarrow D=2\left(2\alpha +5\right)+\left(3\alpha +4\right)+3\left(7\right)$

$\Rightarrow D=7\alpha +35$

Now we know that, for unique solution$D\ne 0\Rightarrow \alpha \ne –5$

So, in option $\left(B\right)$ $\alpha =–6$ corresponds to infinitely many solution, which is completely wrong hence, this is answer.