When did gentleman scholars end in China and what kinds of things were they required to answer on their exams?

Answer 1

While the answer given by the second person is correct about the four natures of the gentleman scholar, he/she is incorrect in maintaining that these four areas were subject to formal testing in government service. Only Confucius and Calligraphy were tested.

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The formal exams ended in 1909, This was the last year.
There were three levels of examinations: Local, Provincial, and in the Forbidden City. All examinations were based on a study of the writings of Confucius, and how priciples of the Analects were carried out in the conduct of government. In addition, calligraphy was especially important.

Hope this answer helps.

Answer 2

they had to know more than just confucius!

http://en.wikipedia.org/wiki/Four_Arts_of_the_Chinese_Scholar