Mathematics - Determinant Question with Solution | TestHub

$\Delta =0\Rightarrow \left|\begin{array}{ccc}2& \lambda & 3\\ 3& 2& -1\\ 4& 5& \mu \end{array}\right|=0$
$\Rightarrow 2(2\mu +5)+\lambda (-4-3\mu )+3(7)=0$
$\Rightarrow 4\mu -3\lambda \mu -4\lambda +31=0$ ...(1)
$\begin{array}{rl}& {\Delta }_3=0\Rightarrow \left|\begin{array}{lll}2& \lambda & 5\\ 3& 2& 7\\ 4& 5& 9\end{array}\right|=0\\ & \Rightarrow 2(-17)+\lambda (1)+5(7)=0\\ & \Rightarrow \lambda =-1\end{array}$
from equation (1)
$4\mu +3\mu +4+31=0\Rightarrow \mu =-5$
$\therefore {\lambda }^2+{\mu }^2=26$