JEE Main · 2020 · 07 Jan 2020 · Shift 2 · Matrices

More than two solutions means infinitely many, so the coefficient determinant must vanish. Expanding $\left|\begin{array}{ccc}1& 1& 1\\ 1& 2& 3\\ 3& 2& \lambda \end{array}\right|=\lambda -1=0$ gives $\lambda =1$. Consistency then forces the replaced determinant to vanish too; using ${\Delta }_x$, $6(2-6)-(10-3\mu )+(20-2\mu )=\mu -14=0$, so $\mu =14$. Hence $\mu -{\lambda }^2=14-1=13$.