〜ものか／〜もんか

Verb dictionary form + ものか (行くものか). い-adjective + ものか (おいしいものか). な-adjective + な + ものか (簡単なものか). Noun + な + ものか (本物なものか). Casual: もんか. Polite (a little softer, often used by women): ものですか / もんですか. The verb or adjective before it is in the affirmative, but the whole sentence means 'not'.

• 負けるもんか。最後までがんばるぞ。

  まけるもんか。さいごまでがんばるぞ。

  There's no way I'll lose. I'll keep going to the end!

• 「この仕事は簡単でしょう？」「簡単なものか。毎日大変だよ。」

  「このしごとはかんたんでしょう？」「かんたんなものか。まいにちたいへんだよ。」

  "This job is easy, right?" "Easy? Not at all. It's tough every day."

• 「あの映画、おもしろかった？」「おもしろいものか。途中で寝ちゃったよ。」

  「あのえいが、おもしろかった？」「おもしろいものか。とちゅうでねちゃったよ。」

  "Was that movie interesting?" "Interesting? No way. I fell asleep halfway through."

• 彼の言うことなんか、信じるものか。

  かれのいうことなんか、しんじるものか。

  As if I'd believe anything he says!

• 誰がこんな高い服を買うもんか。

  だれがこんなたかいふくをかうもんか。

  Who would buy such expensive clothes? Nobody!

• こんなに安い指輪が、本物なものか。

  こんなにやすいゆびわが、ほんものなものか。

  A ring this cheap can't possibly be real!

• 母は「あなたに負けるものですか」と笑った。

  ははは「あなたにまけるものですか」とわらった。

  My mother laughed and said, "I won't lose to you!"

• Putting the verb in the negative to mean 'I won't': ×二度と行かないものか → ○二度と行くものか. ものか already makes the sentence negative, so the verb stays affirmative. (〜ないものか is a different, more advanced pattern meaning 'I wish there were some way to ...'.)
• Forgetting な after a な-adjective or a noun: ×簡単ものか → ○簡単なものか, ×本物ものか → ○本物なものか.
• Using it with a boss, a customer or in writing. ものか / もんか is emotional spoken language. To refuse politely, just say 行きません or 申し訳ありませんが、…

〜わけがない (there's no way that ...)

わけがない is a judgment from reasons or facts: 毎日練習している彼が負けるわけがない (there's no reason he'd lose, since he practices every day). ものか is emotional: it shows the speaker's strong feeling or will, often about himself or herself: 絶対に負けるもんか (I'm definitely not going to lose!). わけがない can be used in writing; ものか is for emotional speech.

〜もん at the end (because ...)

Both are casual and end the sentence, but the meaning is different. もん gives a reason or an excuse: だって、眠いんだもん (Well, it's because I'm sleepy). もんか, with か, is a strong 'no': 寝るもんか (I won't sleep!). Listen for the か at the end.

In N3 grammar-form questions, ものか is often the answer at the end of a sentence that shows anger or strong will (もう〜, 二度と〜, 絶対に〜), with ものだ, はずだ or ことか as wrong choices. Some questions check the form before it: the affirmative dictionary form, and な after な-adjectives and nouns. In listening, a falling ものか / もんか means a strong 'no', not a real question.