ANSWER: You can do this just to pass the exam while believing that all of this goes against the teachings of Islam.  https://youtu.be/ryu9r8TNTpA

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ANSWER: If this is applied externally and it is medically safe, this is permissible. But if it is consumed and it is intoxicating if taken in large quantities, this is prohibited. 

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ANSWER: If this hair is coming in the eye, causing nuisance or blocking your view then no problem in cutting it short due to necessity. However, you must not touch any other hair at all. 

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ANSWER: Yes, lower your gaze and listen to the lecture without seeing her.  https://www.assimalhakeem.net/are-boys-allowed-to-study-at-mixed-universities-knowing-that-in-our-country-all-universities-are-mixed-2/

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