JEE Main 2017 · 8 Apr · Shift 1, Mathematics question 64

A homogeneous system has infinitely many solutions when its determinant is zero:
$$\begin{vmatrix}2&4&-\lambda\\4&\lambda&2\\\lambda&2&2\end{vmatrix}=\lambda^3+4\lambda-40=0.$$
$\lambda^3+4\lambda-40$ is strictly increasing, so it has exactly one real root.