You can play the queen vs rook endgame for a win.
In most situations the queen should be winning.
The winning process can be divided into three steps.
First: force the rook into a third-rank defence and break that down. The defender then has to opt for a second-rank defence.
Second: play against the second-rank defence until you reach the Philidor position.
Third: win the rook from the Philidor Position.
Because of the difficult nature of this endgame, you'll have to know all these steps by heart. If you don't know them by heart, you may not be able to win the queen vs rook endgame.
If you fail to win the rook quickly, you may end up playing this endgame for more than fifty moves. And if you don't know some essential parts of this endgame, you may not be able to find the solution during the game.
The good news is, you can discover the way to play this queen vs rook endgame here. Let's start with the third stage: winning the rook from the Philidor Position.
Philidor's position of the queen vs rook endgame has been known for quite a while. When you know how to play this position, you'll be well on your way towards mastering this endgame.
If Black is to play in this Philidor Position, he can't prevent the loss of his rook because of all the possible double attacks. Let's examine the black possibilities together.
You can see that this loses to Qd8#.
You can see this loses to1.Qxc7+, Ka8; 2.Qb7#.
You'll find this loses the rook to 1.Kxd7, without stalemate. The only task left for White now is mating the black king. This shouldn't be too difficult.
This loses the rook because of the double attack Qd8+ (or Qb4+).
1. Qb4+, Kc8 (or Ka8; 2.Qa3+, Kb8; 3.Qb3+ winning the rook); 2. Qd6,
Now the black king can't move and when the rook moves there are two options.
If the rook moves somewhere on the seventh rank black will be mated starting with Qc7, Qd7 or Qf8.
If the rook moves away from the seventh rank, black will be mated with Qc7#.
Here White wins the rook through 1.Qd8+, Ka7; 2.Kd4+ attacking king and rook with a double attack.
Here 1.Qe5+, Ka8; 2.Qa1+, Kb8; 3.Qb1+, does the job and wins the rook.
By now, you might be able to find the solution without my help.
You could play 1. Qe5+, Ka7; 2. Qg7+,
2....Ka8 (Ka6 allows Qa1+ and mate next move); 3. Qg8+ (double attack), Rb8 (not allowing the rook to be captured); 4.Qb2#
Here the simple 1.Qe5+ wins the rook
This final rook move loses almost the same as Rb3.
1.Qe5+, Ka7; 2.Qd4+, Ka8;
3.Qh8+, Ka7; 4.Qh7+ the double attack and winning the rook.
All the rook moves lose. So you might want to try moving the king.
But after 1....Kc8
White wins with 2.Qa6. The rook is pinned, so it is lost next move.
Now we go back to the Philidor Position.
You've seen that Black to move loses. But what if it's White's turn now?
If you're playing White and it's your turn, you'll have to transfer the turn to Black somehow.
This can be done starting 1. Qe5+
Black will have no useful defending moves.
1....Kc8; allows 2.Qe8#
1.. .Rc7; allows 2.Qxc7+, and mate next move.
1....Ka8, is followed by 2. Qa1+, Kb8 (Rb8 allows Qh8#); 3. Qa5.
White has succesfully transferred the move to Black.
Now it's Black's move in the Philidor Position and as you know by now, he will lose the rook.
This was the first part of the queen vs rook endgame.
You'll know by now how to win the Philidor Position. If you can bring this position on the board, you'll have no problem mating your opponent.
So how do you manage to end up with the Philidor Position?
Therefore you'll have to break down your opponent's second rank defence.