This section is dependent on your Academics attribute; you need it to be at least level 3 (Above Average) to have a chance at scoring at the top of your class. Today is the last day of exams, so you'll go back to your normal schedule and be able to do what you want.
During the night, you'll see a cutscene and you'll now have Akihiko available for battle. Akihiko mentions that they have found another Persona user and mentions her name, Fuuka Yamagishi. Also, the police station will get new items in stock (gloves) since Akihiko is now a party member.
