Students in British (and British Commonwealth) schools and universities can be said to 'offer' the subjects in which they take examinations. The usage is confined to academic institutions, and can replaced by 'take' or 'sit'. School pupils used to sit O-Level (Ordinary Level) examinations at age 16, and A-Level (Advanced Level, university entrance) exams at age 18 or 19. Colin Dexter was a teacher from 1959 to 1966.
A Comparison Of The Orders Of Merit Of H.S.C. Candidates Offering Two
British Journal of Educational Psychology (1945 article)
O-levels and A-Levels arrived in the 1950s; before that there were the School Certificate and Higher School Certificate ("H.S.C").
The Scholarship examination will be in two sessions, one for those
offering the Humanities, and the other for those hoping to be made an
award on the basis of their skills in a Mathematical or Science based
Uppingham School Academic Scholarships (2019)