If I were a Spanish Major as an undergraduate and decide to pursue a PhD in a completely unrelated field (like Theoretical Physics), it makes sense to give a qualifying exam to check that I had the necessary skills to begin the program. But if I'm coming from a B.S in math to a PhD program also in math, it doesn't seem to make sense to give a qualifying exam, as if the knowledge I gained in my undergraduate was insufficient. I presume that one is accepted into a PhD program because he/she has already demonstrated the "qualifying" skills. Thus, I'm baffled by the notion of the qualifying / prelim exam. I'm curious about the ultimate goals of these exams, and how they relate to the professional development of a graduate student.