Whether same output is obtained for the two queries
where Eid field is empty.


What do you mean by "empty"? Perhaps you mean 'null'?

What words exactly does your "solution manual" use? "empty" or "null"?

If the Eid field is CHAR there could easily be a row with Eid "empty" -- that is: zero length, or containing only space characters. count(*) and count(Eid) return the same value in that case.

If the Eid field is NULLABLE there could easily be a row with Eid null. count(*) will include that row, whereas count(Eid) will not.

Usually fields in keys are expected to be NOT NULL. Some vendors' implementations of the SQL standard insist on NOT NULL. But some don't. It is good practice to make key fields NOT NULL. Presumably your manual is getting you to think about why.