Hello:

If each player has one rook and his king at the end, is it possible to mate or is that going to be a draw?

I imagine it depends how each player plays, but I don't really know.  :(

Unless one of the players makes a blunder it will be a draw

It will be a draw unless and until one blunders his rook or trapped due to not playing positionally.

The same is true of each player having a king and a bishop, or each player having a king and a queen. Assuming the start position is balanced (that is, not leading to an iimediate mate) then it should be a draw.

Yes, this is a draw. Unless the position is something like one of these:

It is a draw, unless a blunder or bad position.

speaking from experience it will be a draw unless one of the players fall asleep

Thank you, guys.  I especially found  oinquarki's illustrations to be helpful.

good question innkeeper, i was just about to ask it.