Why does machine power off after partial start up.

The problem started with the device screen fuzzing out from the left and then freezing. After powering off and restarting nothing happens apart from repeating three beeps continuously. When left for approximately 1 hour machine started normally and lasted about 30 min before the problem re-occurred. This became progressively worse and the machine was taken to the supplier. After being returned twice, declared fixed, the problem persisted. A different Apple repair shop did a test and diagnosed a faulty graphics card ( integrated with the mother board).

They don't carry stock and referred me to an agent who wants $1 560-00 to supply and replace mother board. A new machine is available for $1 065-00. Since the machine was returned to me, all I get is the welcome tune on start up before the machine shuts itself off. If stared again, a beeping. To me this points to an over heating problem, with the processor thermal protection shutting it down when over heating. The beeping being the alarm for the thermal protection before it re sets and allows switch on.