Choose the options which demonstrates an appropriate usage of article…

The correct answer is Option (3) → She would make a better statesman than a philosopher

Articles with singular countable nouns:

• A singular countable noun (statesman, philosopher) cannot stand alone; it needs an article or another determiner.
• The article comes before the adjective that describes the noun: a + better + statesman.
• This sentence: "make a better statesman" has the article in its normal place, and "than a philosopher" gives the second singular noun its own article.
  Example: He would make a better teacher than a lawyer.

Why the Other Options are Incorrect?

Option 1: She would make better a statesman than philosopher — Incorrect. The article is placed after the adjective: "better a statesman". In a plain noun phrase the order is article + adjective + noun (a better statesman).

Option 2: She would make better statesman than philosopher — Incorrect. The singular countable noun "statesman" has no article; "make better statesman" is ungrammatical.

Option 4: She would makes better statesman than a philosopher — Incorrect. A modal (would) is followed by the base form of the verb, so "would makes" is wrong; "statesman" also lacks its article.