The sentence is:
"My mother __________ in this house for ten years."
The phrase "for ten years" indicates an action that started in the past and is still continuing now.
For this, we use the present perfect continuous tense.
Structure:
has/have + been + verb(-ing)
So:
π My mother has been living in this house for ten years.
Why others are wrong:
was living β past continuous (doesnβt show duration up to now)
had been living β past perfect continuous (used when referring to another past event)
had lived β past perfect (completed action in the past)