Answer: 1. Became 2. will study.
Explanation:
Simple past tense is used for event that started and ended at a definite time in the part. "During the 19th century" which is already in the past. The 19th century has ended therefore the tense to use is the simple past tense "became".
A future tense expresses an event occurring in the future or a future state of being. In the above question, the teacher "tells" which is a present tense that if they do something i.e something to be done in the future, she won't give them extra assignment. Therefore, "will study" is the correct answer.
Simple past tense and future tense are tenses used in English.