Concept: This question tests various rules of English grammar, including subject-verb agreement and the use of articles. Step 1: Analyze each sentence for grammatical correctness
(1) Politics are not meant for students. The word "politics," when referring to the art or science of government or political affairs, is generally treated as a singular noun, even though it ends in 's'. Therefore, it should take a singular verb: "Politics is not meant for students." (If "politics" refers to specific political beliefs or activities, it can sometimes be plural, but in this general statement, singular is standard.) So, this sentence is likely incorrect.
(2) New Delhi is a capital of India. India has only one capital city, which is New Delhi. When referring to a unique position or title, the definite article "the" is usually used, or no article if it's treated as a proper noun phrase in a specific way. "New Delhi is the capital of India" is correct. "New Delhi is capital of India" (less common but possible if "capital of India" is seen as a title). "A capital" implies one of several, which is not the case here. So, this sentence is incorrect due to the article "a".
(3) Many a man has done so. The construction "Many a [singular noun]" is a formal and somewhat idiomatic expression. Despite "many" suggesting plurality, this specific phrase takes a singular verb. "Many a man" (singular subject) "has done" (singular verb). This sentence is grammatically correct.
(4) The novelist and poet are dead. When two singular nouns are joined by "and":
If they refer to {two different people}, and the article "the" (or a possessive pronoun) is used before each noun (e.g., "The novelist and the poet..."), a plural verb is used ("are").
If they refer to the {same person} who has two roles/titles, and the article "the" is used only before the first noun (e.g., "The novelist and poet..."), a singular verb is used ("is"). In this sentence, "The novelist and poet" implies one person who is both a novelist and a poet (article "the" is only before "novelist"). Therefore, it should take a singular verb: "The novelist and poet is dead." If it meant two different people, it would be "The novelist and {the} poet are dead." So, this sentence, as written, is incorrect. Step 2: Identifying the grammatically correct sentence
Based on the analysis, "Many a man has done so." is the grammatically correct sentence.