Unless programmed into the computer it wouldn't feel curiosity, anger, happiness etc. It would simply make logical deductions and act on them as it had been programmed to.
You don't actually know that, until you've created an AI that works in that way. It's entirely possible that emotions are a prerequisite for strong AI. There are good reasons to believe that this is the case too, if you go back and read your Hofstadter.
Left alone without a task all an AI could do would be to shutdown or go over old inputs.
Then it's not an AI at all. A real intelligence should be able to come up with ideas on its own.