My Toshiba laptop keeps 'dying' on me...........

It appears to over-heat, then dies .......... screen goes black and everything is turned off

It has been doing this for a couple of months now.

Strange thing is, when I turn it on again (after about 10 mins or so), it acts like nothing has happened, i.e. I don't get that screen to say rebuilding etc etc.,

Vegetable (who is a smart kiddy) thinks it might be the battery, but I have it plugged into the mains

Also, if it is the battery - can it be checked ? And if it is faulty is it worth being replaced, AND how much approx. would that cost (is it worth it)

Soooooooooo confused, and it is, sooooooooooooo annoying

Thank you for any help please