With regard to the present perfect tense, it seems correct to me.We can check by saying the statement version of the question
The police have tried to solve the case...
I think the textbook may be trying to get at the need for FOR
The police have tried to solve the case for ... 5 years, 6 years, 10 minutes, etc.
So the proper construction of the question should be
For how long have the police tried to solve the case?
Edit: Just thinking about this some more
We can also say "the police have been trying to solve the case"
and so, as a question say "For how long have the police been trying to solve the case?"
And use the present perfect progressive