Step 1: Think of the OS as a team of specialised managers.
A modern operating system splits its responsibilities among different subsystems, one manages memory, another manages files, another manages devices and I/O, and one manages the CPU itself.
Step 2: Focus on the CPU manager's job.
That CPU manager is the scheduler. Since there is usually only one or a few CPUs but many processes wanting to run, the scheduler's job is to decide which ready process gets the CPU next and for how long, cycling through long term, short term, and sometimes medium term scheduling decisions to keep the system responsive and efficient.
Step 3: Match the other options to their real owners.
Managing memory allocation is the memory manager's job, managing file systems belongs to the file manager, and handling I/O operations is done by the I/O manager and device drivers, the scheduler is not responsible for any of these.
Step 4: Conclude.
The scheduler's defining responsibility is
\[ \boxed{\text{To manage CPU time for processes}} \]