Similar to Sakura, Naruto really had no techniques at the time (except for Kage Bunshin, which isn't exactly subtle).
Of course, he could make 50 Shadow Clones and attack all the examiners. While they're occupied fighting his shadow clones, he could look off someone's paper.

I could see the #1 most unpredictable ninja doing that.
Some people just don't have good information gathering techniques. Would Naruto even have something to pull that off right after the timeskip? I can't think of a way off the top of my head.