1. Keep an eye on the 50 move rule.
2. The rook could be won in fork, pin or x-ray attack and may be sometimes to avoid mate threat.
3. Push the opponent's king to the edge.
4. Keep your king diagonally closer to the rook, attacking the rook and limiting opponent king's movements.
5. Use some quiet moves to force the seperation of the rook.
6. Use the queens ability to move diagonally as often as possible.
7.Avoid checks to your own king..
1.Ke2
Rf4
2.Qd8
Rf6
[If 2...Re4+
3.Kd3
anothr check...rook is gone! 3...Rf4
4.Qe7+
Kf5
5.Ke3
again...no check..from the rook.. 5...Rg4
6.Qf7+
Kg5
7.Kf3
Rh4
8.Qe7+
Kh5
9.Qd8
Rh3+
10.Kf4
Kg6
(10...Rh4+
11.Kf5
Black faces checkmate or loss of rook next move...) 11.Qd6+
Kh5
12.Qe5+
Kg6
13.Qe6+
fork 13...Kh5
14.Qxh3+
winning]
3.Ke3
Rd6
4.Qg5+
Ke6
5.Ke4
pushing the king to the edge of the board
5...Kd7
6.Qb5+
Kc7
7.Ke5
attacking the rook
7...Rb6
8.Qc5+
Rc6
9.Qe7+
Kb6
10.Kd5
attacking the rook
10...Rc7
11.Qb4+
Ka6
nearly there...but the king is still close to the rook..and can offer protection
12.Kd6
attacking the rook
12...Rb7
13.Qa4+
Kb6
14.Qa8
such a quiet move! The Black king can not move..
14...Rh7
[If 14...Rg7
15.Qc6+
Ka7
16.Qe4
quiet move (16.Qa4+
Kb8
17.Qb4+
Rb7
18.Qe4
Rb6+
19.Kd7
Rb7+
20.Kc6
Ka8
21.Qd5
Ka7
22.Qd8
Ka6
23.Qc8
pinning and winning) 16...Rg3
(16...Kb8
17.Kc6
Rb7
18.Qe8+
Ka7
19.Qd8
Rh7
20.Qa5+
Kb8
21.Qe5+
Ka7
22.Qa1+
Kb8
23.Qb1+
fork..) 17.Kd5
Kb7
18.Qb4+
Ka8
19.Qa4+
Kb7
20.Qc6+
Ka7
21.Qc7+
forking and winning the rook; 14...Rf7
15.Qc6+
Ka7
16.Qa4+
Kb6
17.Qb3+
Ka5
18.Qxf7
forking and winning the rook; If 14...Ra7
15.Qc6+
Ka5
16.Kc5
threatening mate with the queen 16...Rc7
distraction 17.Qxc7+
Ka4
18.Qg3
Ka5
19.Qa3#
]
15.Qc6+
Ka5
[15...Ka7
16.Qe4
attacking the rook 16...Rb7
17.Kc6
attacking the rook 17...Kb8
18.Qe8+
Ka7
19.Qd8
Rh7
20.Qa5+
Kb8
21.Qe5+
Ka7
22.Qa1+
Kb8
23.Qb1+
fork]
16.Qd5+
Kb6
17.Qb3+
Ka5
18.Qa2+
Does not matter where the Black King goes now on the 'b' file(..It could be black square as well!! ...) the rook will be gone!
18...Kb4
19.Qb1+
fork!!..While acting as a rook, the queen gives a check to the Black king and at the same time, acting as a Bishop, she attacks the Black Rook sitting on the other corner.
19...Kc3
20.Qxh7
forking and winning the rook.
What is the moral of the story?
If you are down with material finish off opponent's pawns even if it has to be a rook vs queen ending. When put under pressure with the clock ticking over and exhausting the 50 moves, your human opponent is most likely to falter. You do not have to resign and may get away with a draw!
Be prepared that your opponent might do this to you! Studying like this will make it look easy and hence it is a good idea to test your skills, playing similar positions, with stronger opponents with clock. *