Some HSK mock exams (those written by an author unaffiliated with Hanban) are mediocre quality. I remember encountering this fill-in-the-blanks question on a HSK6 mock exam:
There are 5 blanks, and 4 words you can fill in the blanks with. It's impossible! Just learn what you can learn from the mock exams, and move on.
likely means what you think it means:
Mother is warm-hearted, and always helps people. So when anyone has a problem, they come find her.
So is 妈妈现在有问题 true or false?
False, the snippet does not directly imply this. (The given answer is incorrect.)
True, mother always has people asking her for help.
Both answers are possible depending on your point of view. I (like the OP) think the most natural answer is "false" (it's 大家有问题 not 妈妈有问题), but who knows what the author was thinking.