Lesson
White to move. Main line: 1. Qb7 Kg8 2. Qg7#
The quiet 1.Qb7! takes away the seventh rank and leaves the black king only g8. Then 2.Qg7# is mate: the queen sits right next to the king, protected by its own king on f6, and every escape square is covered.
This is the mate you must reach with king and queen against a lone king. The safe method is to keep the queen a knight’s-move away to shepherd the king to the edge, then bring your own king up to guard the mating square. (Take care not to stalemate — the queen must never leave the king only illegal moves.)