We're looking for an adjective that conveys how easily the duke is bribed. Since "venal" means "susceptible to bribery," it is the correct answer.

We're looking for an adverb that describes how the explorers didn't receive supplies on a regular schedule, but rather, received them at random. Because "sporadic" means "occurring at irregular intervals or only in a few places; scattered or isolated," "sporadically" is the best answer choice.

The professor's solution to the problematic conversations is to encourage "a more open and unstructured approach." This means the correct answer will stand in opposition to "open and unstructured." "Pedantic," meaning overly focused on minor details of book learning, is the best choice among the answers.

The subject is in "seeming great health," but still has to be worried about heart attacks. The missing word should mean vulnerable to or likely to have, the definition of "prone."

The sentence contrasts the missing word with "good will." This means the correct answer will in some way mean the opposite of "good will." "Malevolent," meaning intrinsically and irreversibly evil, is the best choice among the answers.

The boy's has "always bothered his parents," which the sentence's structure indicates is a near definition for the missing word.  The correct answer should mean persistent and annoying, which is the definition of "importunate." 

The jury "believed the police's account more," which means they felt the defendant was less trustworthy. The correct answer must indicate that the jury was not entirely certain about the defendant's story. "Disingenuous," meaning insincere or deceitful, is the best choice among the answers.

The "good moods" are explicitly contrasted with her "sour temperament" being what "she usually walked around with" in her life. The correct answer will indicate the "good moods" are not permanent or customary.  "Evanescent," meaning short lived and fleeting, is the best choice among the answers.

While there are some clues in the sentence, particularly that the recruiter thought "resumes did not always help," the answer choices provide an opportunity to rule out answers.  "Throwaway" and "insincere" are synonyms, as are "concrete" and "material," which means they would give the same meaning to the sentence and cannot be correct answers.  "Intangible," meaning abstract or impossible to define, is the best choice.

Since the communities described by the missing word have "strongly objected," then "indifferent" and "callous" can be ruled out. Similarly, a "powerful" or "distant" community would have no need to object over a waste problem. "Proximate," meaning near and immediate, is the best choice.