How should you finish the sentence: To be admitted to the university, he __________ to make application by May 15?

In this sentence, "To be admitted to the university, he __________ to make application by May 15," the correct choice is to use a form that indicates obligation or necessity. The word "has" establishes that he is required to take action before a specific deadline; it aligns with the context of making an application by a certain date.

The construction "he has to make application" indicates a strong obligation or requirement, suggesting that it is essential for him to complete this action in order to be considered for admission. In contrast, other choices do not convey the same level of necessity or clarity. For instance, "have" would be grammatically incorrect in this context because "he" requires a singular verb form. "Might" implies uncertainty or possibility, which does not fit the context of a requirement for admission. "Should" indicates a recommendation rather than an obligation, making it less suitable for this specific implication of needing to make the application by a set date.

Thus, using "has" effectively communicates the necessity of making the application by the specified deadline for the purpose of university admission.