How to mate with the king and queen
The idea is to place your queen so that your opponent's king is cut off from the main part of the board. In time he will only have 2 squares to move about in. Then bring up your king to help checkmate.  White plays Qb4 The white queen has moved to b4 and the king cannot move outside the red line. He cannot cross the red line because if he did cross he would be in check from the queen. There are now only 23 squares he can move about in. It is his move. It does not matter where he moves his king, you must move your queen to keep him in an even smaller part of the board.
Black plays Ke6. The king has moved to e6. You now need to move your queen to shut him into a smaller area and make him move nearer the edge of the board Qc5 The white queen has moved to c5 and the king has now a smaller part of the board to move about in. There are only 14 squares inside the red line.
Kd7 The king has moved to d7. You can now move your queen to keep him on the last 2 ranks of the board. Qb6 The white queen has moved to b6 and the black king has now only 11 squares to move about in.
Now you need to keep doing the same thing. Every time the black king moves, you should move your queen so that he has fewer squares to move about in. In time you will get to a position like this. .Black has only the square in the corner to move into.
The only move he can play is Kh8
Do not play Qf6

It is important not to bring the queen in and trap the king in the corner. He has no moves left so this is stalemate and the game is a draw.

Leave your queen alone and bring your king nearer. You might start by playing Kg3
This will take a few moves but while you are doing this, all the black king can  do is move from h8 to g8 and then from g8 to h8, time after time.
Black plays Kg8 After a few moves with your king you will reach a position like this and you can checkmate
Like this.
Now look at the video showing how to checkmate with the queen. Here»
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