Recently I had a problem with PDF’s in IE10. The only thing I saw when opening a PDF was the no symbol, a circle with a line through it. After looking through many posts on adobe.com and other sites, after uninstalling and reinstalling Adobe Acrobat Reader 11 several times, I finally found this message board post that fixed that fixed the problem: http://forums.adobe.com/message/5262172. That foum post referenced this page: http://superuser.com/questions/561036/unable-to-open-ie-10-in-32-bit-mode.
The answer that worked for me, and found in the links above, was to set the registry key value HKCU\Software\Microsoft\Internet Explorer\Main\TabProcGrowth to 1. After restarting Internet Explorer, PDF’s opened just fine.
You will also want to make sure the Adobe Reader ActiveX add-on is enabled. If the above solution does not work, then disable IE10’s protected mode under the security settings. Worst case scenario is to revert to IE9 or, ideally, use Mozilla Firefox or Google Chrome. It is a good idea to use Firefox or Chrome anyway.