"Be" is a main verb here. || Modal verbs are the following: can, may, will, should, must, would, could, shall, might || "Be" would be an auxiliary only in these cases: if its used in present progressive (I am reading a book) or in past progressive (I was reading a book) or in passive (The house was built).

Norbi is correct. "Be" is never a modal verb. Your example sentence uses "be" in a typical manner. However, your use of the word, "take" in the infinitive form is incorrect. You need to use the gerund here (don't capitalize 'institute') : He is in the institute taking an examination now.