First part of sentence conveys a positive idea, therefore revered (choice A) would be best fit. The opening word "though" says, it is presenting something contrary to the rest of sentence. The other two words have a negative connotation. Vilified and slighted both mean to insult.

Second part of the sentence conveys the meaning that the officials want to discredit Socrates work, undermine (choice E) works best in the context.
One may challenge the work but that doesn't discredit the work. Acknowledge is the exact opposite of what the sentences are trying to convey

The first sentence presents a contrast. Since, the second half mentions threat, we need to have a positive word for the first blank - revered.

Since, the intent was to not completely destroy, but still have a negative impact, it was to undermine.

I picked undermine but is there a clear way to reject challenge for the 2nd blank ? It seemed close.

Detailed Analysis

1. Analyze Blank (i): Students' Reaction (Positive)
- Context: Socrates was _______ (i) by his students who found truth in his teachings. This requires a word showing respect, admiration, or high regard.
- A. revered: To regard or treat with deep respect. (Excellent Fit. This captures the high esteem students would hold for a teacher whose philosophy offered truth.)
- B. slighted: Treated with discourtesy; ignored. (Opposite meaning.)
- C. vilified: Spoken or written about in an abusively disparaging manner. (Opposite meaning.)
2. Analyze Blank (ii): Officials' Action (Negative/Mitigated)
- Context: His philosophy constituted a threat to the government. Consequently, officials sought to ______ (ii) the value of his premises, but they did so "without the extreme position of completely destroying Socrates' position."
- This requires a word meaning to reduce the importance of, attack, or weaken the philosophy's value, but not eliminate it entirely.
- D. challenge: To dispute the truth or validity of something. (Plausible, but perhaps too weak, as "challenge" is less consequential than "destroying.")
- E. undermine: To lessen the effectiveness, power, or ability of, especially gradually or insidiously. (Excellent Fit. This fits the goal of officials trying to weaken the value of the philosophy without totally destroying it.)
- F. acknowledge: Accept or admit the existence or truth of. (Opposite meaning; officials saw his work as a threat.)