In order to fill in the blank with the correct auxiliary verb, let's understand what auxiliary verbs are. Auxiliary verbs are used to form tenses, moods and forms of other verbs. For example - I am a woman. Here, 'am' is an auxiliary verb as it indicates that the sentence is in simple present tense.
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.'