The Miracle Quiz Finals will be available once you have earned the "Breaking Through the Fog" trophy (secret trophy). When you start the game, go to TV listings and look for it and tap the screen on that spot in the listings.
You'll have to do three rounds of the quiz - prelims, playoffs and finals to get the trophy.
Questions are random, so can't really give a listing of what the questions and answers will be, but questions will be repeated at times. Also you can do each round an unlimited number of times if you mess up and the correct answer is given if answered wrong.
The best strategy is to only answer questions you are for sure about being right on. You lose a point if you are wrong, and chances are if another character tries to answer they will be wrong and lose points instead. They only tend to be right if the question is about them or their persona specifically and usually will buzz in before you get a chance in that case. So it's truly better to not guess and risk going into the negative and let the others go negative and shoot for just getting a few points.
One thing to keep in mind though, if you exit the game you lose all progress. You can suspend or put your Vita in sleep mode. But overall it's pretty easy to get this, the harder part is getting it to actually be available.