Should I ask the instructor for a make-up exam for attending a PhD interview at another university (and risk my standing)? I am a first year PhD student at an American university and since I was not satisfied with my present department, I had applied to a few universities last semester (it was my first semester here). My recommenders were all from my previous university and no one in this university knows I applied. I recently received an interview call from one of the universities I applied to. It is a 2 day program, the first day being the interviews and the second day for the faculty presentations and interaction with current students. Unfortunately, the second day clashes with a major exam for a core PhD course I am taking here. 

I asked the official of the interviewing university as to what I should do and she said that I could ask for a make up exam in order to attend the second day as well. My question is, should I? My fear is that the instructor could tell about this to other people in my department (or she may actually be bound to do so by some school law?) and that could seriously jeopardize my situation here. I did like the PI for my first rotation here, and might want to join his lab for my PhD in the event that I am not selected by the interviewing university OR I do not like the interviewing university after visiting it.

I thought that I could go to the instructor of this course and ask her if she discuss a matter with me confidentially (keeping the matter confidential at least till the end of this semester), and then ask her about the make-up exam. Is this a good idea? 
Another option is to not tell her anything and miss the 2nd day, since the interviews happen on the first day. That way no one would probably know, although I will miss the crucial part of the faculty interviews and student interaction which is so important for me to make my decision.

Is there something else I should consider that I am missing? Any help could be life-saving!
You don't have to tell your instructor why you want to reschedule your exam.  Just tell her you have a conflict with a personal matter and ask if it is possible to take the exam on a different date.

She might say no, but if she does, then she would probably have said no even if you had told her the real reason.  Usually exam makeup policies are either "flexible" or "no makeups except for medical reasons or conflict with another course".  Attending an admissions event at another university isn't any more compelling as an excuse than "generic personal business".

If she does say no, then you're going to have to choose between the two.  It probably makes the most sense to attend the interview and then come back for your exam, skipping the second day of the event.  It's probably not wise to skip the exam - you don't want to get bad grades if you are looking to apply elsewhere.