Let's review some of the ins and outs of subject-verb agreement and pronoun-antecedent agreement so you can select the right answers for each of these sentences.
We'll start with subject-verb agreement. Subjects can be either singular or plural, and the verb you choose must agree with the proper number. You would always say, for instance, “The cat is black” and not “The cat are black.” In the second sentence, you would say instead, “The cats are black.”
So far, so good. But what happens when you have a compound subject in which two or more nouns or pronouns form the full subject? Here's an example: “The cats and the dog are running around the yard.” Here there are at least three individuals in the subject, at least two cats and one dog. You are describing all of them together, so you would use the plural verb “are” to agree.
That works great if the conjunction in the subject is “and,” but what if it's “or”? Then how do you make the verb agree? Let's look at an example: “My brother or my sister is coming to the concert.” Notice the singular verb “is.” We use that because, in the end, only one person is going to come to the concert, either the brother or the sister, not both. Here's another one: “The cats or the dog is making a lot of noise.” We need to use “is” again because the part of the subject closest to the verb is singular, one dog. If we were to write the subject the other way around, we would need this: “The dog or the cats are making a lot of noise.” Remember that when there is an “or” in the subject to make the verb agree with the part of the subject that is closest to the verb.
Now let's look at another situation you might run into. Many sentences begin with “There are...” or “There is...” How do you determine which verb to use? First, remember that “there” is not the subject of the sentence and that the verb must agree with the true subject. Here's an example: “There are six children playing outside.” What is the true subject? It's “children.” The verb must agree with the plural “children,” so we choose “are.”
How about if a phrase or clause sneaks between the subject and the verb? If we say, “The person that the children saw running away is the thief,” the subject is “person,” which is singular, so we choose the singular verb “is.” Be careful not to mistake one of the words in the intervening phrase or clause as the subject. Let's try one more example: “The dogs that the cat was chasing were running in terror.” The subject is the plural “dogs,” so the verb must be plural, too: “were.”
Now let's turn our attention to pronoun-antecedent agreement. Pronouns must agree with their antecedents in number. Let's look at an example: “The woman took her dog to the vet.” The pronoun “her” is singular just like its antecedent, “woman.” If we say, “The children are enjoying their day off,” we also have made the plural pronoun “their” agree with the plural subject “children.”
But what if we have something a bit trickier like this: “Anyone could have taken his or her dog for a walk.” “Anyone” is the antecedent here, and it is a singular indefinite pronoun, just like “everyone,” “everybody,” “someone,” “something,” “no one,” “nobody,” and “anything.” Therefore, the pronoun that replaces “anyone” later in the sentence must also be singular. That's why we choose “his or her” (since we don't know whether “anyone” refers to a male or a female). Even though it is becoming common, it is not correct to say, “Anyone could have taken their dog for a walk” because “their” is a plural pronoun, and it does not agree with the singular “anyone.”
Now take a look at your sentences, remember what we've just reviewed, and see if you can figure out which are correct and which have errors in subject-verb or pronoun-antecedent agreement.