Mathematics - Determinant Question with Solution | TestHub

$\begin{array}{rl}& 11x+y+\lambda z=-5\\ & 2x+3y+5z=3\\ & 8x-19y-39z=\mu \end{array}$ for infinite sol. $\begin{array}{rl}& \mathrm{D}=\left|\begin{array}{ccc}11& 1& \lambda \\ 2& 3& 5\\ 8& -19& -39\end{array}\right|=0\\ & \Rightarrow 11(-117+95)-1(-78-40)+\lambda (-38-24)\\ & \Rightarrow 11(-22)+118-\lambda (62)=0\\ & \Rightarrow 62\lambda =118-242\\ & \Rightarrow \lambda =\frac{-124}{62}=-2\end{array}$ $\begin{array}{rl}& {\mathrm{D}}_1=\left|\begin{array}{ccc}-5& 1& -2\\ 3& 3& 5\\ \mu & -19& -39\end{array}\right|=0\\ & \Rightarrow -5(-117+95)-1(-117-5\mu )-2(-57-3\mu )=0\\ & \Rightarrow -5(-22)+117+5\mu +114+6\mu =0\\ & \Rightarrow 11\mu =-110-231=-341\\ & \Rightarrow \mu =-31\\ & {\lambda }^4-\mu =(-2{)}^4-(-31)=16+31=47\end{array}$