"Paltry" means having little meaning, importance, or worth or very small or meager. "Insignificant" is the only choice that has a definition similar to that of "paltry."

"Trivial" means unimportant, so "unimportant" is the synonym.

The word "crucial" refers to something that is necessary to achieve a certain end. For example, if one more loss will eliminate your team from the baseball championship, then winning the next game is crucial. Basically, we are looking for a word that implies great importance. "Vital," which means essential, is the best choice here. "Unimportant" is the opposite of the answer that we want, and "extra" implies that something is present in addition to a certain quantity and unnecessary, not that it is necessary. Finally, "diminutive" means short or unusually small.

If something is pivotal, it means that it is important.  For example, the last regular football game of a season is pivotal in determining whether the team will play in the semifinals.

Something that is “eminent” is well-known or important. This is opposed to something that is "imminent," which means about to happen. To provide further guidance, "futile" means pointless; "worthless" means not worth anything; "morose" means miserable and glum; "novelty" means innovation, something original.

"Tangential" is an adjective that means either "diverging from a previous course or line; erratic," or "hardly touching a matter; peripheral." So, we need to pick out an answer choice that means something like "erratic" or "peripheral." "Essential" cannot be the correct answer, then, because "essential" means "absolutely necessary; extremely important," making it an antonym of "tangential," not a synonym. "Nonessential," however, is an adjective that means "not absolutely necessary," and because "nonessential" is the answer choice closest in meaning to "tangential," "nonessential" is the correct answer.

"Petty" is and adjective that means relatively unimportant, so the answer choice closest in meaning to "petty" is "insignificant," an adjective meaning of no significance, unimportant, or irrelevant. None of the other answer are close in meaning to "petty": "lucrative" is an adjective meaning profitable; "important" is an antonym of "petty" meaning of great significance; and "dark" is an adjective meaning having little light by which to see or closer to black than to white in shade.

"Substantial" means of great weight or significance. Therefore, "significant" is the correct synonym. 

Something that is "imperative" is crucial or essential, so "essential" is the correct answer choice. As for the other answer choices, "somber" means sad and serious; "abrupt" means sudden; and "optional" means not necessary.

“Pertinent” means relevant or able to be applied and is therefore most similar in meaning to “applicable.” To provide further help, “irrelevant” means not related to the subject; “extraneous” means not needed, additional, and wasted; “intravenous” means injected into the veins; and “immaterial” means not related to the subject, irrelevant, or not pertinent.