Mrs. Woods ....... New York very well. She has lived there all her life.
a. must have known
b. must know
c. should have known
d. should know
The test answer key says b is the right option. But I think both b and d can be correct.
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Must X means X is required to or has to happen, without a doubt.
Should X means X is expected to happen, but there could be a small chance it doesn't - or the information is second hand so you don't actually 100% know for sure.
Mrs. Woods must know New York very well. She has lived there all her life.
We are saying that Mrs. Woods has to know New York very well.
Mrs. Woods should know New York very well. She has lived there all her life.
We are saying that we expect Mrs. Woods to know New York very well. Perhaps we haven't met Mrs. Woods personally or for some other reason there's a bit (even if just a tiny, tiny bit) of doubt. Other conversation would have to establish this doubt.
This is a tricky test question, but in test questions, you are supposed to go by the information in the question. So this is why the test says must know because there's nothing casting doubt in the provided information of the question.