My main character is going to a highly advanced school but I have no idea how to make up hard questions. I'm terrible at math, so I was wondering do I need permission to copy my older sister's college questions. (Not sure if this question was already posted.)
You say reading is a hobby. Do some more of it. Read a book and instead of getting lost in it and enjoying it, read it with critical eyes. You should pay attention to whether most settings are actually described, or just referred to. When someone quickly runs into the school bathroom, does the book tell you exactly what it looks like and how it works? When a character has a test, does the book include the test? When a character hums a song, does the book include all the lyrics? Notice and observe how books work. They work by drawing on your knowledge. People take the bus, the subway, or a plane flight. People sing and take tests and pee. Detailed descriptions, diagrams, and sample questions are not needed. They get in the way. If they're wrong, they throw off your enjoyment of the story. What you need is just enough to connect to what your reader already knows, so your reader fills in the details. It doesn't matter what colour the carpet is in the airport, how many urinals a typical high school boy's bathroom has, or the exact wording of a really hard question your protagonist faces on a school exam.
So for the case your very hard questions, I searched for "AP calculus question" and clicked the first link. There, I found this:
The limit of a function may be found by using algebraic manipulation, alternate forms of trigonometric functions, or the squeeze theorem.
The concept of a limit can be extended to include one-sided limits, limits at infinity, and infinite limits.
Finding these sentences is easy. Putting them as-is into your book may not be ok. But that's fine. You don't want them in your book. Instead, try something like this:
She picked up the practice test with a smile, which quickly faded. "Limits, sure. Algebraic what? Squeeze theorem? Limits at infinity?" Turning to her mother, she whispered "I'm not ready."
She picked up the practice test. Her smile widened. "Squeeze theorem! I can do that!"
You don't need the questions. You just need a few words. Enough that people who have taken those classes will recognize them, and people who haven't will think "wow, that sure sounds hard."