She studied hard so as not to fail the entrance exam.
Literal
Entrance-exam [at-に] fail-[negative purpose-ないように], she [topic-は] desperately studied.
入学試験に落ちないように 'so as not to fail the entrance exam' uses the negative purpose pattern. 一生懸命に 'with all one's might, desperately.' A straightforward example of purpose + effort — common in language textbooks.