If you create a quiz, then 'add a new question' rather than add 'from question bank' (in the Quiz administration > Edit quiz screen), you may stumble into this problem. If you leave the Category as 'Default for [topic name]' or choose another category that you have already created, there won't be a problem. But if you choose the category ‘Default for [quiz name]’ from the pull-down menu, the questions will belong to the quiz and you won't be able to see or use them for any another quiz in the topic (eg in a revision quiz, or an end-of-topic exam that draws from questions used in formative quizzes throughout the topic).
Solution to prevent this problem
Do one of the following:
Go to Question bank from your topic homepage (Administration > Question bank). Create a category and questions first, before you create the quiz, then add questions into your quiz 'from question bank'
Create a category, then create the quiz and 'add a new question', making sure the new question goes under the category you have created
We strongly advise solution 1 as the best approach to creating quizzes.
Solution if you have this problem
If you already have this problem and need to use your questions in other quizzes:
From your topic home page, go to Question bank > Categories
Create a new category (probably best to make the parent category 'top')
Go to your existing quiz, then Administration > Question bank
Select all questions which are in the quiz's default category, then using the 'With selected' menu at the bottom of the page, move to your new category
Note: Questions can be moved to different categories even if they are already used in quizzes, though be careful if they are used as random questions in a quiz.