False. Australia became a Federation with its own constitution in 1901. No. Not false.
While Australia did federate in 1901 and the Australian Constitution came into effect on that date, the British Parliament retained the right to pass laws concerning. Australia under the Colonial Laws Validity Act of 1865. This Statute of Westminster of 1931 amended that situation by stating that the British government could no longer make laws for the Dominions (at that time, Australia, New Zealand, Canada, Newfoundland, South Africa, and the Irish Free State) but also contained provision that the Statute of Westminster would not take effect in Australia until the Australian government ratified it. The Australian government finally ratified the Statute of Westminster on the 9th October 1942.
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That was just a formality -- The Brits passed their own version in 1931. These statues did not affect autonomy and independence of Australia, which was granted in 1901. In 1901, Australia accepted as default legislation, the centuries of British law as legal precedent. In a way today, that still hasn't changed. In the 1970s, there was a movement to remove/repeal some of the very old - unused legislation that had been accepted in Medieval Britain. For example --it had been illegal for a woman to gossip -- there was a piece of legislation that dated back to the Plantagenet Dynasty and one of the many pieces of default legislation that was accepted with Federation. The gossiping legislation was never enforced and therefore was easy to repeal.
Other pieces of legislation date back from significant moments in British history -- the Great Fire of London in 1666, for example. After that event, the Parliament of Charles II made it illegal for any new structure to be built of wood on its first floor -- it must be brick. I'm not sure if that law is repealed today in Australia - but the habit of building brick buildings remained a large custom, even when we have sprinkler technology, fire hydrants etc.
Australian law is a very different beast. Although it does have the basis of British legal precedent, there are elements of Australian law that were only forged in Australia and that was well before 1901 -- the incident at the Eureka Stockade has left an indelible mark. No Australian to this day is required to have any papers on him/her at any time (including driver's license). But when asked for the license, has several days to produce it. In Australia, you can leave the house without your driver's license, run a red light, be pulled over and have 3 days to produce your legal right to drive at the local police station.