No, Korean Immigration will not ask to see any financial documents; however, you might be asked to show proof of departure, i.e.: you might have to produce proof that you booked a ticket out of the country. They may want to see a flight ticket showing that you actually plan to leave when your trip comes to an end. One's length of stay in Korea is determined by which passport the person holds. Check to see how lengthy a stay you will be granted. Lastly, you will definitely be asked where you plan on staying for ten weeks, so you better be able to tell them your friend's name and address or the name and address of the hotel or guest house you'll be staying.
Generally, travelers are not asked to show proof of financial assets when entering a foreign country. The thinking behind that is that if you wind up broke, you'd be forced to contact your embassy. Your embassy would arrange a flight home for you, but you'd likely have to reimburse your government once you got back. It wouldn't be Korea's problem if you were to run out of money, so they don't care how much you've got, they will definitely NOT ask anything about that.