Step 1: Check molecular symmetry and dipole moment.
- BF$_3$: Trigonal planar ($D_{3h}$ symmetry), no permanent dipole moment → microwave inactive.
However, its asymmetric B–F stretching vibrations change dipole moment → infrared active.
- CH$_4$: Tetrahedral ($T_d$ symmetry), no permanent dipole moment → microwave inactive.
Some vibrational modes (asymmetric stretches) change dipole moment → infrared active.
Step 2: Conclusion.
Both BF$_3$ and CH$_4$ are microwave inactive but infrared active.
Hence, the most appropriate choice is **(B)**.