The correct option is A Relative pronoun - whose; Antecedent - The artist
A relative pronoun is a pronoun (who, whose, that, what, which, etc.) that performs as a pronoun as well as a conjunction. When two clauses are combined and there is a common noun between them, repetition of the noun must be avoided. Hence the relative pronoun is used to refer to that noun, as well as join the clauses as one. For example, 'They caught the girl. The girl stole some bread.' Here the girl is repeated in both sentences, when combined, 'the girl' is replaced by 'who' in the second clause: They caught the girl who stole some bread. Here, 'who' refers to the girl. Hence 'who' is a relative pronoun and 'the girl' is called the antecedent. The antecedent is a word, a phrase or a clause that the relative pronoun replaces.
In the given sentence, 'whose' is a relative pronoun, it refers to the artist and connects two clauses together (The artist's design is approved by all. The artist will be given the contract.) 'whose' refers to 'the artist', hence the latter is called the antecedent and option A is correct. ('whose design is approved by all' is a relative clause that defines the subject 'the artist)
Option B is incorrect because the roles of the pronoun and antecedent have been reversed. 'The artist' cannot be a relative pronoun, it refers to a specific person whereas the relative pronoun replaces that noun to avoid repetition. The same goes for 'whose', which cannot be the antecedent as it is the relative pronoun itself.
The given sentence has an antecedent, 'the artist' is the subject of the sentence which is referred to, using 'whose', as it adds essential information 'whose design is approved by all' which defines it. Hence option C is incorrect.
Option D is incorrect because the sentence has both a relative pronoun: whose, and an antecedent: the artist.