Past perfect tense is used to explain an action that happened before an action in the past. For example: 'Ram had finished studying before you walked in.' This explains: The action of studying was completed before someone came in. Hence, we use 'had' to explain past perfect tense and the V2 form of the verb to explain the simple past tense.
Here, the action of doing homework was completed before the mother came home in the past. Hence, we will use the past perfect tense.
Option A - 'Already did' is used in the simple past tense. Thus option A is incorrect.
Option B - 'Have already done' is used in the present perfect tense. Thus option B is incorrect.
Option D - 'Has already done' is used in the present perfect tense. Also, 'has' can't be used with the subject 'I' as it is a plural subject. Thus option D is incorrect.
Option E - 'Already do' is used in the present tense. Thus option E is incorrect.
Option C - 'Had already done' is used in the past perfect tense. Thus option C is the correct answer.
'I had already done my homework when my mother came home.'