Mathematics - Determinant Question with Solution | TestHub

$\begin{array}{rl}& \Delta =\left|\begin{array}{ccc}1& 2& -3\\ 2& \lambda & 5\\ 14& 3& \mu \end{array}\right|=0\Rightarrow \lambda \mu +42\lambda -4\mu +107=0\\ & {\Delta }_1=\left|\begin{array}{ccc}2& 2& -3\\ 5& \lambda & 5\\ 33& 3& \mu \end{array}\right|=0\Rightarrow 2\lambda \mu +99\lambda -10\mu +255=0\\ & {\Delta }_2=\left|\begin{array}{ccc}1& 2& -3\\ 2& 5& 5\\ 14& 33& \mu \end{array}\right|=0\Rightarrow \mu =13\end{array}$
Also, $\lambda =-1$
Hence, $\lambda +\mu =13-1=12$