JEE Main · 2021 · 24 Feb 2021 · Shift 1 · Matrices

The determinant of the coefficient matrix is $24-8k$, which vanishes only when $k=3$; for any other $k$ the system has a unique solution and is consistent, so inconsistency can arise only at $k=3$. Putting $k=3$, the first two equations give $9+4y=10$, so $y=\frac{1}{4}$, while the second and third give $4y=5m-3$, so $y=\frac{5m-3}{4}$. These agree only when $5m-3=1$, i.e. $m=\frac{4}{5}$, for which the equations are consistent with infinitely many solutions; for every other $m$ they contradict one another. Hence the system is inconsistent precisely when $k=3$ and $m\ne \frac{4}{5}$.