Here are the best choices for each sentence:

1. **D. on**
"When the teacher scored the paper, she found that many students were strong **on** grammar."

2. **C. By**
"**By** the best of my knowledge, he came from Malaysia."

3. **C. of**
"Men are still dependent **on** fossil fuels in modern life."

4. **C. off**
"Please do not put **off** doing that any more."

5. **B. of**
"Linda pays a visit **to** her parents every Sunday morning."

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

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