The short answer is yes they can. But it is more complicate then it looks. If you are familiar with ad-block extention (which prevent banner and advertisment popups), you can see there are some web sites that can detect if your browser using this extention and they not allow you to see their content. But add-block is very well known extention and I guess the developers who wrote the code that detect if it is install did some reaserch how to do it.
In your case, if your extention will stay only for your private use, it is much harder to fight aganist somethine you (the developer of the web site) doesn't know and can test.
There are some known scenarios that they can test to see if the answers came from script. Like to calculate the times that you answered the question and to see if it is not too fast or to check if there are a lot of attempts of tring to solve question in very short period.