iMac sometimes fails to start

I have an early 2008 iMac running OS X 10.10.5 which fails to start up about half the time following a shut down. Instead of the Apple symbol I get a rectangular shape containing a question mark. It starts up mostly if I remove the power cord and leave it to cool down for about 10 minutes, although it doesn't seem unusually hot.

I have also had a difficult time with software updates. The display first appears to freeze, but the mouse cursor can still be moved, then clicks stop having any effect and the spinning beach ball appears. Nothing then works except powering off.

Any suggestions?