What can syntax demonstrate about a character? a. it can demonstrate that the character is interested in writing. b. it can demonstrate that the character does not like the other speaker. c. it can demonstrate that the character is from a different place or time. d. it can demonstrate that the character has learned about mathematics.

C. It can demonstrate that the character is from a different place of time.

Syntax basically means how a well-formed sentence is put together and like most parts of language, is something that changes based on place and time period.  Therefore the choices A, B and D doesn't make sense as answers. The arrangement of words has changed a lot since the 15th century after all, and so might the syntax change as well if you went to another country. So if a character uses a certain syntax, it might show that the character is from a different place or time.