The first thing you need to do is to send for your mothers death certificate. It seems to me that you lost contact with your mother because one would know or any of your living relatives would know when your mother died. If for example your mother died when you were very young a relative would be the best person to approach. If your mother and father separated and you have lost contact with your father you need to start searching for him as he would have the answers for you. If you can't find him then procede to the next step.
It may take time, but a search through the entire 1980s would find an exact date of death through the Tasmanian , Birth, Deaths and Marriages Register. You will find the address on the internet. You send off the appropriate forms to them and they will do the searching and this wouldn't take long.
Once that is done then send a copy of the death certificate and your verification that you were her daughter ( your birth certificate )to the Probates Court of Tasmania as they will then send you a copy of your mothers Will.
This all costs time and money but that is what you need to do. Hoping everything works out.
Should all this fail then contact your Solicitor.