For this question we need to figure out which word best fits the blank space in the sentence.

When a policeman is sent somewhere, it's usually to look into or gather more information about something.

Option a, b, c and d all express this idea of looking into or investigating an issue - "investigate", "examine", "probe" and "enquire" work well. 

Option e "observe" is too passive and doesn't capture the meaning of actively investigating or inquiring.

So based on closely matching the context, i'd say the best choice here is choice a "investigate". the full sentence with the blank filled in would be:

"the policeman was sent to investigate the allegation made by the man." 

investigate expresses most directly that the policeman was sent there specifically to actively look into and gather facts about the claim or allegation that was made.