In relation to the fundamental duties enshrined in the Constitution of India, which one of the following is NOT correct?

The correct answer is They may not be used for determining constitutionality of laws, because this statement is false. The Supreme Court has held that when a court tests whether a law is constitutional, and finds that the law gives effect to a Fundamental Duty, it may treat the law as "reasonable" under Article 14 or Article 19 and uphold it. So the duties can indeed help determine the constitutionality of laws, even though they cannot themselves be enforced.

The other three statements are correct, which is what makes the question hard. The Fundamental Duties were not part of the original Constitution of 1950; they were added by the 42nd Amendment in 1976. Safeguarding public property and abjuring violence is a Fundamental Duty, clause (i) of Article 51A. And the duties are not enforceable through writs, because they are non-justiciable: no court can order a citizen to perform them, though Parliament can enforce them through ordinary laws. In a "NOT correct" question, test each statement separately before choosing.