For which database operation would you need the database to be in the MOUNT state?

Renaming control files is performed at the OS level while the database is fully shut down, followed by updating the CONTROL_FILES parameter - it does not require MOUNT state to execute.

Re-creating control files with CREATE CONTROLFILE requires the database to be in NOMOUNT state, not MOUNT, because the control file itself is being constructed from scratch.

Dropping a user is a data dictionary operation that requires the database to be fully OPEN so that the data dictionary and system tablespace are accessible.

Dropping a tablespace requires the database to be in OPEN state so Oracle can update the data dictionary and confirm the tablespace is offline and empty.

The command ALTER DATABASE ARCHIVELOG or ALTER DATABASE NOARCHIVELOG requires the database to be mounted but not open. In MOUNT state, Oracle has parsed the control file and can persist the archiving mode change there, but the data files are not yet online, which is the prerequisite state for this structural database configuration change.