A tried-and-true approach to text completion questions is the process of elimination approach: get rid of answer choices first due to their definitions, then due to their connotations and intensities. On many questions, just the first part of the process will suffice, though on some, like this one, you will have to consider connotations as well.

First, we can get rid of "wandering" (traveling aimlessly) because the word doesn't make much sense in context. Then, we're left with "flabbergasted," "bewildered," "bemused," and "disbelieving," which all express some level of shock and/or confusion, an emotion that makes sense if a book's sequel is announced fifty years after it was published.  

However, it is here that you must pay special attention to context clues. The sentence explicitly states that "no one expected" the sequel, but says nothing about anyone being confused, caught off guard, or the like. There is simply surprise. Thus, the words that imply confusion—"bewildered" and "bemused"—can be thrown out. The sentence also doesn't explicitly state anything about skepticism, which "disbelieving" implies, at the very least. Thus, we are left with "flabbergasted," which is the correct answer.

This sentence indicates a kind of intensification. Many people seemed to think that Harry would merely be happy. The word "merely" provides the best indication that we want to describe him has being something more than "merely happy." A great intensifier for "happy" is "elated," which describes someone who is extremely happy.

Acumen—keen insight; shrewness.

Founder—to sink; to fail utterly.

"Spate" is the correct answer, as it means a group of events occurring one after another in a short amount of time. The construction site is implied to have had enough accidents to justify increased caution.

For this blank, we're looking for a noun that refers to pieces of broken pottery. "Shards," which means pieces of broken pottery, is the best answer.

"Requisite" means that something is necessary and most accurately completes the sentence. The sentence indicates that self confidence is not merely helpful or sufficient for achieving success, but is essential. For that reason, "requisite" is a better answer than "condition" or "quality," the only other answers that could potentially complete the sentence.

None of the answers to this question are ideal; however, the sense of the second sentence indicates that it is a question of "learning." Perhaps the most tempting words are "stupor" or "indolence." The first, however, deals with a state of being dazed, not necessarily lack of intellectual acumen. The latter indicates laziness. The closest word on the list would be "stultification," which can mean the process of making of one become stupid.

The key word in the second sentence is "both." It is particularly tempting to choose "premises," "conclusions," or "implications;" However, these do not deal two cases—a "both" fulfilling the "whence" and "whither." The only word that really fits that is "termini" —as strange as it might seem in this context.

The fact that the professor was ostracized means his calculations were at the very least controversial and against the prevailing thought of the scientific community. Breakthrough and achievement do not have the kind of disagreement indicated in the sentence, while restoration and reformation do not convey the appropriate level of hatred for the professor's ideas.