Solution:
'content writer' and 'graphic designer' are two different person. Both words are singular noun and hence indefinite article 'A/an' shall be used. Both words begin with consonant sound and therefore 'A' will be used.

Solution:
Here, 'test' is a general term and therefore indefinite article shall be used. 'A' article is correct because the word after the blank begins with a consonant sound.

Solution:
Use 'an' before words such as "hour" which sound like they start with a vowel even if the first letter is a consonant. Also use 'an' before letters and numbers which sound like they begin with a vowel, such as "F" or "8". 'An' will be used before 'S.D.O.'.

Solution:
'Aeroplane' is a singular countable noun starting with a vowel sound and therefore indefinite article 'an' will be used.

Solution:
Here, a specific 'black cap' that was in the shop is being talked about. Definite article 'the' will be used.

Solution:
Both 'country' and 'town' are specific in the sentence. Hence, 'the' article will be used.

Solution:
'The' will be used here because the sentence talks about a particular case.