In order to arrive at the correct answer, one must effectively combine the two independent clauses. Because they contrast in meaning, using the conjunctive adverb however after a semi-colon is appropriate. The other answers incorrectly implement conjunctions or punctuation to resolve the issue. Using a semi-colon to join two independent clauses that contrast will work when using a conjunctive adverb like "however" to convey the appropriate relationship between the clauses. Note that comma usage also comes into play here.

In order to arrive at the correct answer, one must effectively combine the two independent clauses. Because they contrast in meaning, using the conjunctive adverb however after a semi-colon is appropriate. The other answers incorrectly implement conjunctions or punctuation to resolve the issue. Using a semi-colon to join two independent clauses that contrast will work when using a conjunctive adverb like "however" to convey the appropriate relationship between the clauses. Note that comma usage also comes into play here.

The word “which_”_ does not have an antecedent noun, and so the construction is incorrect in the two answer choices in which it is present. Similarly, in the sentence implementing “this,” there is no noun to which it refers. The final incorrect answer leaves the modifier “passing through the manifold” dangling too far from the noun to which the modifier refers.

The correct correlative conjunction pair here is "neither/nor." No comma is needed. The other pairings listed are either mismatched or inappropriate. The correct correlative conjunction for "either" is "or."

Although the constituents in the list are a bit convoluted, this can be boiled down to a simple either A or B. A and B each have an "and," which can make the sentence seem more confusing, but, when simplified, it can be easier to tell that the sentence is describing a choice BETWEEN two options (both of which include two items and an "and"), so the "and" between "quizzes" and "one" needs to be replaced with "or."

Although the constituents in the list are a bit convoluted, this can be boiled down to a simple either A or B. A and B each have an "and," which can make the sentence seem more confusing, but, when simplified, it can be easier to tell that the sentence is describing a choice BETWEEN two options (both of which include two items and an "and"), so the "and" between "quizzes" and "one" needs to be replaced with "or."

The word “which_”_ does not have an antecedent noun, and so the construction is incorrect in the two answer choices in which it is present. Similarly, in the sentence implementing “this,” there is no noun to which it refers. The final incorrect answer leaves the modifier “passing through the manifold” dangling too far from the noun to which the modifier refers.

The correct correlative conjunction pair here is "neither/nor." No comma is needed. The other pairings listed are either mismatched or inappropriate. The correct correlative conjunction for "either" is "or."

"While I was busy working on the grant proposal" is a subordinate clause, so the sentence requires the second clause—an independent clause—in order to be a complete grammatical sentence. Because "While I was busy working on the grant proposal" is a dependent clause, it should thus be followed with a comma. Semicolons and periods can only be used to punctuate independent clauses. The correct version of the sentence reads, "While I was busy working on the grant proposal, Kevin was taking care of the day-to-day operations of the organization."

The central issue here is the proper implementation of the correlative conjunction "neither...nor." The use of the preposition "of" and the conjunctions "or," "and," etc., appropriately resolve the issue. If anything, these other modifications result in unidiomatic phrases or awkward syntax.