The word "precipitate" is most often associated with the phenomenon of meteorological precipitation—as when it rains or snows. This is a specific case of the more general process of something falling out of solution or suspension. We can say that a person makes a "precipitate decision" when he or she makes a quick and unmeasured decision—proceeding as quickly as the dry skies begin to fill with rain. Thus, such a hasty decision is well contrasted one that is thought out and measured—i.e. to a prudent decision.

Something that is protracted is drawn out or lasting for too long of a time. Do not associate this word with the mathematical/geometrical options given in the question—all of these are trying to get you to make a false association. In contrast to being "protracted," "brief" is an excellent antonym. A "protracted affair" is one that is too long, but one that is brief is quite short in contrast.

Several of the wrong answers attempt to have you associate "brilliant" with the sense of being intelligent—a standard and acceptable usage. However, none of these wrong options are well enough matched as antonyms—they would have to be explicitly opposed to such intellectual ability. Instead, "brilliant" is being used here in a more literal and physical sense—describing something that is bright in the light. Something can be physically "obscure" when it is shadowed or darkened. (From this, we talk about an "obscure idea," which is perhaps the more standard use of the word "obscure" in our day-to-day talk.)

The word "copious" comes from the Latin for store or plenty. When something is "copious," it exists in great quantities. The best antonym is "meager," meaning little in quantity or quality.

We tend to think of "vulgar" in the sense of "vulgarity" or the use of swear words; however, this notion comes from the general notion or meaning of "vulgar," which comes from the Latin meaning pertaining to the ordinary people. Something vulgar is lacking in refinement or culture. Therefore, the best antonym among the options provided is "refined," meaning elegant or cultured.

When an action is "blatant," it is done without any attempt to hide it. Perhaps you have heard the expression used, "That was a blatant lie," (or some other close expression like that). Such a sentence means "That was a total lie—without any attempt to hide the fact." In contrast to this, a concealed act is one that is hidden—or at least an attempt is made to hide the act. This is a good contrast to something done blatantly.

The verb "inoculate" is likely more familiar than the adjective "innocuous." When we get inoculated for a virus, we receive a vaccine (or other kind of treatment) to remove the danger of getting the illness in question. Something that is "innocuous" is therefore something that is not dangerous or harmful. When something is "detrimental," it causes harm. Often, we use this word in expressions like "It was much to his detriment that he did not study a language at a younger age." The idea in such expressions is that it harms his present state of life that he did not study a language when he was younger. Something detrimental is not innocuous.

The word "brisk" can refer to cool weather, but it can also be used as an adjective describing fast movement. The adverb "briskly" means quickly or with energy. A brisk motion is one that is fast, and walking briskly means walking quickly with haste. In contrast to this, a leisurely pace is one that is slow—taking its time. Leisure is the free time used for enjoyable activities. Such time is not usually stressful. Therefore, "leisurely" activities are quite slow in their pace.

We are looking for something that simply implies irregularity as opposed to regularity. When something is done inconsistently, it is done without a regular manner or mode. A consistent tempo is one that is regular and repeated. Therefore, "inconsistently" is a good antonym for "regularly."

To be "coarse" means to lack refinement or to be "rough." Indeed, the word first has the physical sense of being rough—like a coarse, rough piece of sandpaper. In contrast to this, something that is delicate is fine in quality. Thus, to do something "delicately" is to do that thing in a way that is opposite to being "coarsely" done.