(a) will leave the UK at the end of their visit; and
(c) is genuinely seeking entry for a purpose that is permitted by the visitor routes (these are listed in Appendices 3, 4 and 5); and
I think you should provide evidence that she is your mother, something mention that explicitly.
You should provide a return ticket to comply with point "a"
If you say that she has no body back home, so they might be thinking the same, if she has no body back home then she will remain in the UK. So, you have to provide some supporting documents to explain why she will be back home and will not remain in the UK to handle point "c"