a) in spite (of) — despite
Both the items have got the same meaning — not prevented by, but they function differently in contexts. In spite is always followed by the preposition of. In spite of his illness he came to our meeting. Despite his illness he came to our meeting.
Dictionaries stress that in spite о/is often followed by the gerund. In spite of being hungry he refused to go to the restaurant.
The usual context for in spite is in the construction in spite of the fact. Jane nodded in spite of the fact that she couldn't agree.
b) require - demand
Demand means to ask firmly for sth. Cf.: I demand an explanation (Я требую объяснений). Require means:
1) to need. Cf.: Is there anything you require? (Тебе что-нибудь нужно?) It is also possible (but less frequently) to use demand to express the same idea. Cf.: This requires/demands careful investigation.
2) to ask or order someone to do something (often in the passive voice). Cf.: All passengers are required to show their tickets.
c) seek (for) — look for
Both the verbs have the same meaning — try to find sth — but they differ in style. Seek is a formal word, look (for) belongs to the basic vocabulary. "We are earnestly seeking for the truth, " said the lawyer. — I can't find my glasses. Will you look for them ?
d) feast — holiday
e) feast — meal
f) pupil — disciple