The correct option is D that occur as a result of coal mining is the increasing incidence
There are two verbs in the given sentence- ‘occur’ and ‘are’. Let us look at each of these in turn to identify the subject associated with the verb.
‘One of the problems’ is singular, as it refers to one of many problems. So, the right verb to be used with this subject is ‘is’, not ‘are’. The correct usage is “One of the problems……..is….”
Now, consider the verb ‘occur’. The subject that ‘occur’ is associated with is ‘problems’, not ‘one of the problems’. So, the subordinate clause - the clause beginning with ‘that’- takes a plural verb. Why is this so? The verb must agree with the subject of the clause, which is the relative pronoun ‘that’. The pronoun has the same number as its antecedent, which is ‘problems’. Since ‘problems’ is plural, ‘that’ is also plural, and so the verb associated with it must agree with ‘problems’. Problems occur. So, the correct usage is “one of the problems that occur”.
Another way to test this is to reword the sentence: Of the problems that occur, one is…
Also note that ‘incidence’ is the frequency of occurrence of something. As the sentence refers to the frequency of occurrence of a single metric- lung cancer- ‘incidence’ has to be used in singular.