The correct option is
A to study
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 passive voice, we will not use the bare infinitive here; we will use the verb with the marker 'to'.
NOTE: Here, 'made' is used as the main verb, which describes the sentence in past tense. Though the sentence is in past tense, the verb with the marker 'to' will always be in its original form.
Option B - 'Study' is a verb in its regular form, which is used in sentences with active voice. As the given sentence is in the passive voice, we must use the infinitive with the marker 'to'. Hence, option B is incorrect.
Option C - 'Studies' is the third person form of the verb 'study'. As explained above, we need the verb to in its original form and to be accompanied by the marker 'to'. Hence, option C is incorrect.
Option D - 'To be studying' can't be used here as the verb with 'to' is not in its original form. 'Be' is often used to represent one's state of being and it is not required here. 'Studying' is the participle form of the verb 'study'. Hence, option D is incorrect.
Option A - 'To study' can be used, as the infinitive 'study' is in its original form and is accompanied by the marker 'to'. Hence, option A is the correct answer.
'The children were made to study even during their vacations.'