How to show mathematically (using mathematical equations) that a Band Pass Filter (BPF) passes input voltage of a particular frequency and eliminates the other frequency voltages.
I think you should find the frequency response of the Band Pass Filter (BPF) i.e. nothing but the Transfer function (TF), which it self represents the range of frequency it is going to pass and eliminate. The nature of the BPF can be find out by drawing its frequency response plot (Bode diagram).