My iMac shuts down randomly.

My 27-inch iMac (EMC 2639 with a 3.2GHz quad-core Intel Core i5 processor) will shut down randomly. When I go to restart there is nothing. If I switch it off at the mains and leave it for 30 minutes and go to restart then still nothing. If I change the socket it is plugged into then it will boot back up.

The length of time between shut downs seems to relate to how long it has been switched off.

I have had advice to reboot PRAM or change motherboard battery...

I would appreciate any ideas as this is a mystery!