Returning to university I graduated from an undergraduate maths course from a prestigious university. However, I was rejected from attending it's master's program due to poor showing in the final exams. I'd like to study at said university again sometime within the next few years, applying for the same course. In applying though, I will almost definitely have to use the same final exam results. What can I do to show that I have improved, if I indeed have? I am not currently enrolled in any graduate level program.
At many schools, particularly those rated highly, the most important parts of your PhD application are your letters of recommendation.  What many applicants don't grasp is that a PhD program is so different from undergrad that past performance in the latter is only a poor indicator of future performance in the former.  For most students, the hard part of a PhD is doing original research.  So, as an applicant, you must convince the committee that you will excel at doing research.  

One way to make your case is to have already successfully done research (or some "research-like" project).  In the absence of this, you must somehow convince your letter writers that you will succeed in research (and then they convince the admissions committee).  Meet with your letter writers and ask them what they suggest.  You might even ask them if you could work on some research with them (prior to being admitted to the PhD program).