NOTE: The word 'cost' is a singular subject and hence we need a singular form of the auxiliary verb. 'Increased' indicates that the sentence can be in simple past, past perfect or present perfect tense.
Option B - 'Is' is the singular form of the auxiliary verb used with a singular subject. As the sentence requires the verb to be simple past, past perfect or present perfect tense, we can't use 'is' here. Hence, option B is incorrect.
Option C - 'Am' is the plural form of the auxiliary verb which is used with the subject 'I'. As the subject is 'The cost', we cant use 'am' here. Hence, option C is incorrect.
Option D - 'Were' is the plural form of the auxiliary verb which can be used with a plural subject only. As the subject of the sentence is singular, we can't use 'were' here. Hence, option D is incorrect.
Option A - 'Has' is the singular form of the auxiliary verb used with a singular subject. It indicates the subject's state of being in present perfect or present tense. Hence, we can use 'has' here. Thus option A is the correct answer.
'The cost of living has increased a lot.'