The correct option is
C believe
In order to fill in the blank with the most suitable verb, we need to understand the usage of 'make'. 'Make' means to force someone to do something. If 'make' is used in a passive (voice) sentence, we use the word 'to' along with the verb. In an active sentence, we do not use 'to' with the verb. The verbs without the marker 'to', are called bare infinitives.As the given sentence is in the active voice, we will use the bare infinitive here.
NOTE: Here, 'made' is used as the main verb, which describes the sentence in past tense. Though the sentence is in past tense, the bare infinitive will always be in its original form.
Option B - As explained above, we need the verb in its original form. 'Believes' represents the third person form of the verb 'believe'. Hence, option B is incorrect.
Option C - As the sentence is in the active voice, we can't use the marker 'to' with the verb. Hence, we can't use 'to believe' here. Thus option C is incorrect.
Option D - 'Believed' is the past form of 'Believe'. As we need the verb in its original form, we can't use 'believed'. Hence, option D is incorrect.
Option A - 'Believe' is a verb in its original form and it is a bare infinitive. Hence, option A is the correct option.
'He has made me believe that I can pass the exam.'