For the first blank, we are looking for a word which means to rid of impurities, so we should focus on "rarefying," which means just that, and "homogenizing," which means to make uniform. For the second blank, we need a word that means to combine, since the metals are being poured into one mold. Since "amalgamates" means combines different things into one, the correct answer is "rarefying . . . amalgamates." 

The key points to heed are that the topics are "difficult" and that Paschal did not fear directly addressing issues. Really, the easiest way to begin with this is to see that "circumlocute" particularly fits the bill for the first blank and to note that "disconcerting" likewise works well. Circumlocute comes from the Latin for to talk ("locute") and around ("circum"). The other options are tempting, but none of them catch the sense in such a manner that so directly matches Paschal's desire to directly address issues.

Because medieval treatments were risky, we know that they were as likely to hurt patients as they were to help them. So, because the missing words both refer to symptoms, we’re looking for a synonym of cause or increase for the first blank, and a synonym for lessen for the second blank. We can eliminate the answer choices that include “hamper,” “nullify,” and “abrogate” as options for the first blank, leaving us with “exacerbate . . . allay” and “engender . . . worsen.” Since we’re looking for a word for the second blank that means lessen, we can eliminate “engender . . . worsen,” since “worsen” means the opposite of “lessen.” So, the correct answer is “exacerbate . . . allay.” 

For the first blank, we can infer that we need to pick out a verb that means something like "ruin the reputation of," because we know that the reporter exposed the newspaper's corruption. Either "demonize" ("portray as wicked and threatening") or "vilify" ("speak or write about in an abusively disparaging manner") could be potentially correct. For the second blank, we need to pick out a word that means something like "write things intended to damage one's reputation," so either "slandering" ("making false and damaging statements about (someone)") or "disparaging" ("regarding or representing as being of little worth") could be correct. Of the possible words that we've identified as potentially correct for each blank, only "vilify" and "slandering" appear in the same answer choice, so the correct answer is "vilify . . . slandering."

If this restaurant's popularity has increased, we would expect it to be difficult to reserve a table for the same night; conversely, if its popularity has declined, it would be relatively easy to get a same-day reservation. The end of the sentence ("we were able to") tells us that the latter scenario (diminished popularity/ better odds of getting a reservation) is the case here. For the first blank, either "waned" or "ebbed" would fit, as these are synonyms meaning decreased. "Secure" and "procure" in the second blank would both mean that you were able to obtain a reservation. "Proffer" (offer) would not make sense. Thus, the pairing of waned/secure is the correct answer.

The best clue in this sentence is the word "upon," which indicates that the monks apparently came to the city in protest. Likewise, it is important to notice that they came as a "group." Thus, we should say that they gathered together, not that they fled together. The general idea is that they gathered and then all "stormed" the city in protest. The word "congregate" means come together in a crowd.  A church "congregation" is called such because it is a gathering of people.  Based upon its Latin roots, it is literally a "flocking together." The monks gathered and then all together "swarmed"—i.e. moved as one large group—upon the capital city in protest as a swarming group. Swarming insects are so called because they act as a group made up of many individuals.

"Mitigate" means to lessen damage, while "impute" means to attribute or ascribe. Only "abstain" means to stop from doing something bad. In the second blank, the student is trying to undo the damage, so "horrify" does not make sense, and "recuperate" means to recover. Only "placate" means to lessen damage.

The best clue for starting this question is "only the tiniest quantity of vapor." It seemed that the inlet for the pipe could allow in only a small quantity of gas. However, it seems that more had happened—for Heather was surprised at something. The general idea is that gas had spread throughout the house. Hence, "diffused" is an excellent first choice, and "emitting" well describes the general act of allowing a substance to "go forth" (as would happen with the gas coming out of the pipe). ("Emit" comes from Latin roots literally meaning to send forth. To "trans-mit" means to send ("-mit") across ("trans-").)

Clearly, the first blank has to be something describing the act of stating an opinion and (likely) defending it. Many of the choices provided would fit that role; therefore, the best thing to look at in a case like this is the second blank and the options provided for it. Almost all of the wrong answers have something to do with pleasing or displeasing. This is not the point of disagreement between the two brothers. They disagree as to whether or not the meal will be filling. Therefore, the best option for the second blank is "satiate," meaning to satisfy or fill.

Sentence completion questions sometimes count on students studying roots, then throw in answers that seem like they could be the right answer based on the roots, but do not actually have the appropriate definition, as a way of separating the wheat from the chaff. This is one such question.

The best strategy for a compound sentence such as this, which doesn't have clear indications of meaning from conjunctions, is to use the text's logic to reach your answers. This means when you read along, it can be helpful to "fill in the blank," i.e. guess what kind of word would be in the blank before looking at the answer choices, so you can choose the best answer upon seeing it.  

For the first blank, you need to look at the first half of this sentence. It relates da Vinci to the helicopter, which is a twentieth-century invention; thus, it would make sense for da Vinci to have "foreseen" the existence of modern inventions. "Forecasted" (predicted) makes sense in this context, but "precasted" (cast something in a different location than where it will be installed) does not.

For the second blank, the same trick works. Reading this half of the sentence again, the art, which is described by the word that will go in the second blank, is seen by thousands of people every year. Thus, it must be well-known or well-liked, which "esteemed" (admired) accurately conveys, while "estimated" (roughly judged the value or total of something) does not.