is it possible to know the composer?

Okay, so I'm joining the ABRSM examination this year and for some
reason, my teacher asked me to listen to different composers from
different eras. I think she's expecting me to guess the composer of a
music..

so, is it possible to do so?
If yes, how will I do this?

by the way, do any of you know the significance of composers? like
J.S. Bach, Handel, Beethoven, Mozart, Haydn, Chopin, Liszt, Debussy?
(this extra info will surely help me identify the composers)
=)