Ahh we're back to some very very useful expressions. My significant other used this one on me just this morning to my great chagrin.
If you know the only kanji present in the expression in the title line, then you've got some idea what the word might mean.
For those that don't know, 知 (し、shi) is the kanji for "know" or "understand" as in you know when Columbus landed on the Eastern shores of the US. A fact or some piece of information considered knowledge (not how to make 300 different kinds of shots - that is another skill set entirely).
知るか literally means "how in the @#%^ should I know?!" and should be said/yelled as such in moments of extreme stress. My partner was taking the piss this morning, but she nailed me with one of my own words from the past. She doesn't really like "colorful" language. It seems like no matter where you go - you can't find girls that allow you to curse like a sailor - even in their native tongue.
Bust this one out to your Japanese friends. Brilliant reactions await.