Either believe or believes should be used. As far as I know majority goes with both singular and plural verbs, but the above question was asked in my exam and I had to choose only one option, and no further context was provided. There is an example sentence in OALD 'The majority was/were in favor of banning smoking'
The majority is singular, however, it can be used as a collective noun and be plural. Was your exam from India? They tend to follow British patterns.
Commented Sep 20, 2020 at 17:08The collective noun "majority" typically has plural override, especially in examples like "The majority of her friends are *is Irish" / "The fact that the overwhelming majority of Americans don't *doesn't want the President impeached does not mean that that would be the right decision".
Commented Sep 20, 2020 at 17:18The following is the only example of "majority" in M-W's Learners Dictionary where "majority" is the subject and there is a present tense verb. It shows a plural verb form:
A clear majority of the voters support the policy. (M-W Learners Dictionary)
In the Cambridge Advanced Learner's Dictionary, the clear majority of examples show plural agreement:
The majority of the employees have university degrees. (Cambridge)
A large majority of people approve of the death sentence.
The majority of young professionals in the capital have moved there from provincial towns.
In the past, the majority of women were consigned to a lifetime of servitude and poverty.
But the following example is also given:
A large majority of the population is demanding the restoration of the former government.
Google NGrams shows that "the majority of voters are" and "the majority of people are" are much more common than "the majority of voters is" and "the majority of people is".
As you point out, the Oxford Advanced Learner's Dictionary explicitly allows both and gives an example with both (separated by a slash):
The majority was/were in favour of banning smoking. (OALD)
Overall, the plural seems the safer bet.