Question

The sentence "John wants to really do well on this exam" contains which of the following errors?

Answer 1

The answer to your question would be that the sentence "John wants to really do well on this exam" contains the following error: Split infinitive.

A split infinitive, also known as as cleft infinitive, is a grammatical construction in which a word or phrase comes between the to and the bare infinitive form of the to form of the infinitive verb. Usually, an adverb or adverbial phrase comes between them. In this case, the word between the infinitive particle and the bare form is the adverb "really".  

Answer 2

This sentence contains split infinitive. John wants to really do well - you see that the infinitive is split here by introducing really between to and do. If you can, you should avoid doing that, it would be better to say John really wants to do well...