@Dlveen Muhammad In the phrase "Neither of the answers," "neither" is a singular term, referring to one out of two items. Therefore, it takes a singular verb, which is why "is" is correct. The sentence should read: "Neither of the answers is correct."
@Dlveen Muhammad "Do" is correct in the sentence "He is one of those people who do not like to travel." Since "people" is plural, "do" is the appropriate verb form.