In the following question, out of the four alternatives, select the word that is opposite in meaning to the given word Nugatory, which means of no value or importance.

Introduction / Context:
Antonym questions test a student's ability to recognise not only the meaning of a word, but also which word expresses the opposite idea. The term Nugatory is somewhat formal and is often found in written English, particularly in legal or academic contexts, to describe something that is of no value, ineffective, or insignificant. To answer correctly, one needs to choose an option that represents usefulness or effectiveness, rather than any negative or empty quality.

Given Data / Assumptions:

• Target word: Nugatory.
• Options: Delusive, Futile, Unreal, Productive.
• We need the word that expresses the opposite meaning.
• Standard dictionary sense of Nugatory is assumed: trivial, worthless, or of no practical effect.

Concept / Approach:
Nugatory is used to describe something that fails to produce a useful result, is of no importance, or is invalid. A clear antonym will therefore express the idea that something is useful, effective, or capable of producing good results. Among the given options, Productive clearly indicates generating useful or beneficial outcomes. The other choices are more similar to Nugatory in implying illusion, uselessness, or lack of reality, and hence cannot be antonyms.

Step-by-Step Solution:
Step 1: Recall that Nugatory means of no real value, trivial, or ineffective. Step 2: Consider Delusive, which refers to something that deceives or misleads and does not directly express usefulness or value. Step 3: Examine Futile, which explicitly means useless or serving no practical purpose, and is therefore similar in meaning to Nugatory, not opposite. Step 4: Examine Unreal, which suggests something that does not exist in reality, again not conveying usefulness or value. Step 5: Examine Productive, which describes something that yields good results, generates output, or creates value. Step 6: Recognise that Productive is directly opposed to the idea of worthless or ineffective. Step 7: Conclude that Productive is the correct antonym for Nugatory.

Verification / Alternative check:
To verify, imagine two sentences: The discussion proved nugatory and The discussion proved productive. The first suggests it achieved nothing, while the second suggests it led to useful outcomes. These two meanings clearly oppose each other. If we substitute Futile in place of Nugatory, the meaning remains quite close, which proves Futile cannot be an antonym. Delusive and Unreal describe different types of negative qualities but not the direct opposite of having value and effect. Therefore, Productive stands as the only correct opposite.

Why Other Options Are Wrong:
Delusive is wrong because it focuses on deception or illusion, which does not directly counter the idea of lack of value. Futile is wrong because it is almost a synonym of Nugatory, meaning pointless or useless. Unreal is wrong because it describes something that is not real rather than something that is effective or valuable. None of these options express the positive idea of creating worthwhile results.

Common Pitfalls:
Students often confuse synonyms with antonyms when the choices are all somewhat negative sounding. Words like Futile and Nugatory can be mistakenly treated as opposites of more neutral words, but here they both point in the same direction of uselessness. To avoid such errors, examine whether a choice truly reverses the sense of the original word in a sentence, rather than merely sounding different. Training with pairs like useless and productive helps solidify the correct association.

Final Answer:
The word that is opposite in meaning to Nugatory is Productive.