1) _____for the chance to go to university, the young man devoted himself to study.
A. Desperate B. Being desperate
2) Being ill for a few days, she doesn't feel like eating anything.
1. Either A or B is marginally grammatical, but not recommended.
2. Needs changes. One possibility is this: Having been ill for the last few days, she doesn't feel like eating anything.
Another possibility: Being ill for a few days has made her reluctant to eat anything.
Student or Learner