It seems to me the first one is fine, as we always say "I have some work to do". Do we say "I have some work to be done"?
If they are both correct, in which scenarios they should be used respectively?
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"Questions need to answer" is wrong (Sorry for the harshness).
Questions is the subject, need is the verb, so the following clause relates to "the questions". That is because you use the active (active means that the subject performes the action). So you could ask: "what do the questions need to do?". In this case it would mean that the questions themselves give the answer. But even if the questions could answer themselves, it is still a grammatically incomplete sentence, because you lack an object. "The questions need to answer...what?".
On the other hand, "questions need to be answered" is a complete sentence, because you use the passive (passive means that some other entity performs the action on the subject).
In the case of "I have some work to do", you are the one doing the work, so active tense is the right choice.
I would try to avoid "I have some work to be done", because i think you can regard it as incorrect, as you could say that "to be done" is an incomplete clause. "I have some work" is already a complete clause, you yould have to add a subordinate clause, so the right phrasing youd be: "I have some work that has to be done".
"Questions need to answer" doesn't sound right to me. It seems to be stating that questions are the thing that need to answer something.
"Questions need to be answered" is stating in general that there are questions that need to be answered.