Will math always give you the correct answers?

Answer 1

Generally speaking, yes, but ...

Kurt Godel proved the incompleteness of mathematics. According to him in any axiomatic system one can make statements that cannot be proven to be true or untrue within the system. In such a case there is no correct answer.

The axiomatic system must be appropriate. For example, non-parallel lines must meet in plane geometry (2-d) but in 3-d non-parallel line need not meet. In projective geometry, all lines must meet - even parallel ones.