Subject-verb agreement and correct pronoun usage are fundamental in English grammar. Let's examine each provided sentence:
Neither of the answers are correct.
The pronoun "neither" is singular, requiring a singular verb. The correct construction is "Neither of the answers is correct," as "neither" signifies not one.
Each of the boys have their own books.
"Each" is singular, thus demanding a singular verb ("has" instead of "have"). The pronoun also needs to align with "each," so "their" should be "his" or "his or her." The corrected sentence is: "Each of the boys has his own book."
The jury were divided in their opinions.
Collective nouns such as "jury" can be treated as singular or plural depending on context. When the group functions as a single entity, the singular form is used: "The jury was divided in its opinion." If the context implies individual actions, "were divided" might be acceptable, but the pronoun should then reflect individual opinions if that is the intended meaning.
It is I who am responsible for the mistake.
This sentence is grammatically sound. "I" functions as the predicate nominative after "is," and the relative clause "who am responsible" correctly uses "am" to agree with "I."
Therefore, the grammatically accurate sentence is: It is I who am responsible for the mistake.