She is going to fail her exam because of her hijab

The Muslim woman is obliged to wear hijab in front of non-mahram men, because of a great deal of well-known evidence, such as the verses (interpretation of the meaning): 

“O Prophet! Tell your wives and your daughters and the women of the believers to draw their cloaks (veils) all over their bodies (i.e. screen themselves completely except the eyes or one eye to see the way). That will be better, that they should be known (as free respectable women) so as not to be annoyed. And Allaah is Ever Oft-Forgiving, Most Merciful”

[al-Ahzaab 33:59] 

“And tell the believing women to lower their gaze (from looking at forbidden things), and protect their private parts (from illegal sexual acts) and not to show off their adornment except only that which is apparent (like both eyes for necessity to see the way, or outer palms of hands or one eye or dress like veil, gloves, headcover, apron), and to draw their veils all over Juyoobihinna (i.e. their bodies, faces, necks and bosoms) and not to reveal their adornment except to their husbands, or their fathers, or their husband’s fathers, or their sons, or their husband’s sons, or their brothers or their brother’s sons, or their sister’s sons, or their (Muslim) women (i.e. their sisters in Islam), or the (female) slaves whom their right hands possess, or old male servants who lack vigour, or small children who have no sense of feminine sex. And let them not stamp their feet so as to reveal what they hide of their adornment. And all of you beg Allaah to forgive you all, O believers, that you may be successful”

[al-Noor 24:31]

It is not permissible to take off hijab because of studying, because that is not a necessity that permits doing something that Allaah has forbidden. So this girl should adhere to her religion and keep her hijab, even if that means giving up her studies. 

It should be noted that whoever puts his trust in Allaah, Allaah will suffice him and protect him, and that whoever fears Allaah, Allaah will make things easy for him, as Allaah says (interpretation of the meaning): 

“And whosoever fears Allaah and keeps his duty to Him, He will make a way for him to get out (from every difficulty).

And He will provide him from (sources) he never could imagine. And whosoever puts his trust in Allaah, then He will suffice him. Verily, Allaah will accomplish his purpose. Indeed Allaah has set a measure for all things”
[al-Talaaq 65:2-3]

She should bear patiently the annoyance and mockery that she may face, remembering the reward that Allaah has prepared for those who adhere to His religion, as the Prophet (peace and blessings of Allaah be upon him) said: “There will come a time of patience when the one who adheres (to Islam) will have the reward of fifty martyrs from amongst you.” Narrated by al-Tabaraani from the hadeeth of Ibn Mas’ood; classed as saheeh by al-Albaani in Saheeh al-Jaami’, no. 2234. 

And Allaah knows best.