Yes, you have to determine your cost basis.
You didn't buy the house for $10 ($10 isn't your basis). You got a gift (gift of equity).
But, it might be an inheritance, for tax purposes. The usual rule, for a gift, is that the recipient's basis is the giver's basis (what you mother paid for it*). But there is an exception for the gift of her home, where she retained the right to live there ("life estate"). "If you give away an asset and keep a life estate in that asset..... the cost basis of the house is "stepped-up" to the value of the house on date of death [IRC 2036]")
More info: http://www.law.cornell.edu/cfr/text/26/20.2036-1
Any time you spend in a care facility may count as still being in her primary residence if it’s a licensed facility.
You have a complicated situation. You (and your siblings) may want to seek professional tac assistance for filing this year
*If your father was half owner and died before your mother, his half of the basis stepped up on his date of death (in community property states, the whole basis steps up).
