Mode A is used and if received could result in TA but not RA.
In the original scenario there isn't enough info to make a call. If the transponder has failed completely then aircraft A doesn't exist to aircraft B. If the transponder can still receive then it could provide TA for aircraft A.
RA is a contract and both sides need to be able to negotiate it. That means listen and talk. TA is one sided and that side only needs to be aware of the other.