The correct answer is C. Where does your mother live?
The other options are not appropriate because they are not about Liz’s mother’s current residence.
A. Where has your mother lived? is not appropriate because it asks about Liz’s mother’s past residences, not her current residence.
B. Where will your mother live? is not appropriate because it asks about Liz’s mother’s future residence, not her current residence.
D. Where was your mother living? is not appropriate because it asks about Liz’s mother’s past residence, not her current residence.
The best way to ask Liz about her mother’s current residence is to ask her directly, “Where does your mother live?” This question is clear and concise, and it will give Liz the opportunity to provide you with the information you are looking for.