To start this trophy requires having your knowledge stat at rank 5 so you'll more than likely not get this until the December exam unless you're playing on NG+.
The answers for each exam are as follows:
May Exam
May 11th
Minamoto no Yoshitsune
Minamoto no Toritomo, Yoritomo wonn, the weak
May 12th
Cognition
Both
May 13th
The Devil's Dictionary
Femma Fatale
July Exam
July 13th
64 Degrees.
Zhuge Liange.
Head.
Head Replacement.
July 14th
King Crab/Hermit Crab
The economy was confused.
July 15th
Raining Dogs and Cats.
Demon Entrails.
October Exam
October 17th
32 Sides.
Originally one colour.
Monochrome.
October 18th
Charles-Henri Sanson.
Because of its history.
October 19th
Forced Labour.
European Honey Bee.
December Exam
December 20th
D.
10 billion or more.
Prison gate.
It was like a performance.
December 21st
Hearts.
Assurance.
December 22nd
Japan.
Dreadnought of “Do”.