Mathematics - Determinant Question with Solution | TestHub

For infinitely many solution
$\begin{array}{rl}& \Delta =0\\ & \left|\begin{array}{ccc}1& 5& -1\\ 4& 3& -3\\ 24& 1& \lambda \end{array}\right|=0\\ & \Rightarrow 1(3\lambda +3)-5(4\lambda +72)-1(4-72)=0\\ & \Rightarrow -17\lambda +3-4\times 72-4=0\\ & \Rightarrow 17\lambda =-289\end{array}$

$\begin{array}{rl}& \Delta 1=0\\ & \Rightarrow \left|\begin{array}{ccc}1& 5& -1\\ 7& 3& -3\\ \mu & 1& -17\end{array}\right|=0\\ & \Rightarrow 1(-51+3)-5(-119+3\mu )-1(7-3\mu )=0\\ & \Rightarrow -48+595-15\mu -7+3\mu =0\\ & \Rightarrow 12\mu =540\end{array}$
$\begin{array}{rl}& x+5y-z=1\\ & 4x+3y-3z=7\\ & 24x+y-17z=45\\ & \text{ Let }z=1\\ & x+5y=1+\lambda ]\times 4\\ & 4x+3y=7+3\lambda \\ & 4x+20y=4+4\lambda \\ & \frac{-17y=3-\lambda }{-17}\end{array}$
$\begin{array}{rl}& \begin{array}{rl}\mathrm{y}& =\frac{\lambda -3}{17},\mathrm{x}=1+\lambda -\frac{5\lambda -15}{17}\\ & =\frac{32-12\lambda }{17}\\ 7& \le \frac{\lambda -3}{17}+\frac{32+12\lambda }{17}+\lambda \le 77\\ 7& \le \frac{30\lambda +29}{17}\le 77\\ 3& \le \lambda \le 42\\ \lambda & =3,20,37\end{array}\end{array}$