I may be wrong but none of the answers seem to be correct to me.

First of all, it is not clear what each state represent (it isn’t mentioned) but if I assume each state represent the value of the vector from the inputs as it is defined in the question mod 5, then if for example I am in state q4 and the input is a 0 I should go to state q3 (8 mod 5 =3) and if the input is a 1 then I should go to state q4 (9 mod 5 =4). None of the answers have both of these transitions.

Handout 12 Q4 also contradict these options (in there is also a slight mistake regarding the output but the transitions are correct unlike in here).