The correct answer is (A) Your mother had been waiting for you when you went to your friend's house. This sentence correctly uses the past perfect continuous tense ("had been waiting") to describe an action that started in the past and continued up to another point in the past (when you went to your friend's house), which is correctly expressed with the simple past ("went"). This sequence of tenses accurately conveys that the waiting was ongoing before and possibly at the time of the second past action.
Distractor (B) "Your mother had being waited for you..." is grammatically incorrect. The auxiliary verb "being" is not used in this construction for the past perfect continuous; it should be "been." Additionally, "waited" is the past participle, but the continuous form requires the present participle ("waiting").
Distractor (C) "Your mother have been waited for you..." is also incorrect. "Have" is used for the present perfect tense, not the past perfect. The correct auxiliary for past perfect continuous is "had." Furthermore, "waited" here implies a passive voice, which doesn't fit the active meaning of "waiting."