〜ものか・〜もんか

ものか is a strong, emotional refusal or denial: あんな店、二度と行くものか, "as if I'd ever go to that shop again!" It looks like a question, but it means "never".

It's used for the speaker's own determination (負けるもんか, "there's no way I'm losing!") or disbelief (信じるものか, "as if I'd believe that!").

In casual speech, it's もんか; a softer, more feminine version is ものですか.

Compare ことか, which is an exclamation of degree ("how very!"). ものか is always a flat rejection. The meaning is close to わけがない, but much more emotional.