A) False because - L = a*b*, L' = a^nb^n Here L/L' = a*. L/L' is regular, but L' is not.
B) True. If L is regular, L/L' is prefix of language. Regular languages are closed under Qoutient/Prefix. So this is correct.
C) False L' = Empty set. Then L/L' is Empty set whatever L is. Here L can be say a^nb^n. See defination of L/L' to see why L/L' should be empty set.
D) False because L/L' can accept prefixes of string of Language L, which may or may not be accepted by L itself. So L/L' is not subset. ( It is not Superset either , because L' can be empty set )
E) False. Same explanation as C.
Answer :- B.