The second problem looks easier than the first one to me. Just play c7 and black cant stop you from queening. Black can give some checks, Rd6 check, Kb5, Rd5 check, Kb4, Rd4 check, Kb3, Rd3 check Kc2 and queen next move.
The hard part was seeing Rd6 check, Kc5 is a mistake because of Rd1 and c8=queen loses to Rc1 check. White can still make a draw with Kc6 or Kb6 instead of c8 = queen
The second problem looks easier to you because you think you solved it! You have not completed the problem.
"Kc2 and queen next move" is not correct with best play by black. Try to figure out why. After you play Kc2, black will play Rd4. Spend more time thinking about what will happen next.
The first problem touches on one of the most important ideas in king and pawn endgames (along with opposition) the square of the pawn.
Are you saying that I should be solving all the way to mate? I know that King and Queen against King and Rook is supposed to be enough to win. Eventually the person with the Queen can either win the rook or force checkmate.
Read the clue I gave you again. If I say more I will give it away. Look at the position after Kc2 and Rd4.
Ok. I added a few more moves to the end to complete my answer.
Just play c7 and black cant stop you from queening. Black can give some checks, Rd6 check, Kb5, Rd5 check, Kb4, Rd4 check, Kb3, Rd3 check Kc2 and queen next move.
Based on your comment of Kc2 Rd4 from my solution above I add a few more moves: c8 = queen, Ra4 (to stop Qa8 checkmate) then Kb3 threatening Kxa4 and Qc1 checkmate. Black can't save both the rook and checkmate.
