Here are the best choices for each blank in the sentences:

1. When the teacher scored the paper, she found that many students were strong **in** grammar.
**Answer: A. in**

2. **To** the best of my knowledge, he came from Malaysia.
**Answer: B. To**

3. Men are still dependent **on** fossil fuels in modern life.
**Answer: D. on**

4. Please do not put **off** doing that any more.
**Answer: B. up** (However, "put off" is also a common usage here, but "put up" doesn’t fit well.)

5. Linda pays a visit **to** her parents every Sunday morning.
**Answer: C. to**

6. The children were **encouraged** to take part in the contest.
**Answer: B. encouraged**

So the final answers are:
1. A, 2. B, 3. D, 4. A, 5. C, 6. B.