Rook vs. Queen and Rook Checkmate.

I want to understand three things:

1-How to win a rook vs lonely king endgame.

2-How to draw a rook vs queen endgame.

3-How to win two rooks vs queen endgame.

Any help from you fellow chess players and analyzers would be appreciated. Thanks

(Note: Black is the side trying to win or draw.)

1) lone Rook checkmate is an easy one to teach and there are many resources to learn this checkmate.

2) Actually, it is a loss, not a draw! Queen vs Rook is better for the Queen side, but the winning technique isn't so easy for most human players. If you want to learn how to defend in this endgame, then I recommend learning how to win from the Queen side. You can then see the most stubborn defenses and recall them for when you have the Rook side.

3) This one is probably a draw in most cases too. Usually the leftover pieces and pawns help the 2 Rooks side. This is because the Queen usually gives many annoying checks and force a draw via repetition or 50 move rule. With all other pieces and pawns gone, the Queen is able to harass the enemy King and the ending is most likely a draw. 

If you would like some insight into the easier King and Rook vs King checkmate, then feel free to message me. As for the other two endgames you named, they are far more complex - but both are probably not the result you were expecting. 

Thanks KeSetoKaiba. Where do I find those resources that teach these endgames?

I'll message you

3rd example mate in 1 (easy) puzzle

Qh6#  Yes, this is one of those not "most cases" example. Usually, this endgame should be a draw I think.

In all examples, pretend it is Black to move.

 I know what you mean but this doesn't change anything significant. Position #1 is a win for Black either way (as I correctly noted before). Position #2 has White in check, so it can't legally be Black to move. The funniest one is position #3 though; with Black to move, the draw is super easy - just keep playing Rook checks for repetition/perpetual because the Queen is "on the wrong side" to interpose or do anything defensive.

I know what you mean though: analyze those endings without getting into specifics of an exact piece configuration.

It actually only takes 31 moves to mate or capture the rook, after which at most 10 moves are needed to mate with the queen, so you're off by nearly a factor of 2.

yes