Q- How the following sentences are different? 
a) They will not have passed the examination.
 b) They Shall not have passed the examination.

Please explain the use of shall and will.  Thank you.
 

Dear Student

'Will' is used for making requests or when we want to make an offer or suggestion whereas 'shall' is used to describe obligations. 
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The sentences above are not grammatically correct. They would not have passed the examination means that there wasn't a possibility of passing the examination. It is a suggestion.

On the other hand, they shall not have passed the examination isn't correct because shall is used to describe obligations.

​​​​​​​Regards.