Yes  buyers have statutory rights,  but they are different to statutory warranties.

yes fit for purpose,  under statutory conditions, but there is no set period for such and is very dependent on price and condition of the item purchased.  Statutory conditions are different to statutory warranties, which virtually only covers free title.

The statutory warranties require that:
1) The consumer will enjoy quiet possession of
the goods.
2) The goods are free from any charge or
encumbrance not disclosed or known to the
consumer.

There is no legal  requirement  for a seller of new or used goods, to provide what is known as a voluntary warranty, Commonly known  as manufacturers warranties, store warranties, express warranties etc.

Voluntary warranties
‘Voluntary warranties’ are the written and verbal
promises sellers and manufacturers make about
their goods and services. However, they are also
known by other names, including:
• express warranties
• store warranties
• manufacturers’ warranties
• money-back guarantees
• store refund policies
• store exchange policies.
The law does not require any business to provide
a voluntary warranty.

As sourced from  ACCC