The typical clue to the back rank mate is that your opponent has castled and has not moved the pawns in front of his king. But that is not the only way a king can get trapped on the back rank, as we have seen; you want to be alert as well to other ways its flight squares may be sealed off, especially by attacks your pieces may launch against empty squares on the seventh rank. This brings us to the current batch of studies.
A common way to fend off the threat of a back rank mate is to advance the h-pawn one square as Black has in this first frame, giving the king a flight square. Here Black’s advance also serves to threaten the knight on g5. But it's too late for such threats to be effective; the White knight seals off h7 just as a Black pawn would, so White can disregard the danger to his knight and work with checks that keep Black busy elsewhere. He has two resources usable against the back rank—his rook and queen—and can start with either of them. It might go 1. Qb8+, QxQ; 2. RxQ+, Re8 (a useless interposition); 3. RxR#.