The main idea tested is pronoun case: objective pronouns are used when a pronoun functions as the object of a verb or a preposition. Objective pronouns include me, you, him, her, it, us, and them. Among the options, the one in objective form is me. The others are subjective forms used when the pronoun is the subject of a sentence or clause: I, we, they. For example, in "The coach praised me," me is the object of the verb praised. After a preposition, we also use the objective form, as in "Between you and me." Using the subject form in these positions—"I" or "we" or "they"—would be incorrect. So the word that is an objective pronoun is me.

The key idea here is choosing the correct case for who/whom in a question that uses did-support, and keeping the verb in its base form after did. In the embedded clause, the pronoun is the object of the verb invite, so the proper object form is used: whom. The auxiliary did requires the main verb to be in base form, so invite, not invited, follows did. So the sentence Whom did you invite to the party? uses whom as the object of invite and keeps the verb in its base form after did, which is exactly what formal grammar calls for in questions. The other options run into two kinds of issues: using who instead of whom when the pronoun is the object, or using invited after did, or punctuation that makes the sentence read as a statement rather than a question.

An introductory participial phrase should clearly attach to the same subject that performs the action in the main clause, and it’s usually followed by a comma to separate the introductory element from the main statement. In the sentence that works, the introductory phrase After finishing the project clearly describes the subject who then does the next action. The subject right after the comma is she, who goes on to publish the report. This keeps the action of finishing the project tied to the same person who publishes, avoiding any confusion about who did what. The other options run into a classic problem: the introductory phrase ends up dangling or attaching to the wrong noun. In the first, After finishing the project, the report was published, the action of finishing the project isn’t done by the report, so the modifier misleads you about who finished it. In the third, Finishing the project, the report was published, the phrase is attached to the report itself, which would imply the report finished the project—nonsensical. The last option, The report was published finishing the project, places the participial idea after the main clause and without a separating comma, making the sentence awkward and unclear about who performed the finishing action. Tip: when you start with an introductory participial phrase, choose a subject in the following main clause that can plausibly be the one doing the action in the phrase, and set it off with a comma.

When multiple adjectives describe the same noun, English follows a natural order, and size usually comes before color. Here, the ball is described first by its size (big) and then by its color (red), with the noun (ball) at the end. This arrangement—big red before the noun—reads smoothly and matches the common pattern for describing objects. Swapping the adjectives to red before big sounds off because it goes against that usual sequence. Placing an adjective after the noun isn’t typical for a straightforward noun phrase, and moving both adjectives to after the noun doesn’t fit standard grammar for describing the object. If you wanted to treat the adjectives as equally independent, you’d typically use a comma (A big, red ball), but the simplest, most natural order here is big red before ball.