There are past exam questions often freely available on exam sites, like SAT sites, BMAT sample papers, AP exam papers. Can I use these questions for YouTube videos to show how one would solve them?
You may have issues if you take their content wholesale. Even if they freely distribute them, they still retain copyright. As such, they absolutely can claim copyright. Whether they will or not is another question.
Your best bet around this is Fair Use doctrine. You can take a part of their work (e.g: a single question) and do your video based on how you work out your answer, with your video mainly focusing on the 'working out' part (thus satisfying the 'educational purposes' part)