Let's go through each question one by one to identify the correct answers:

1. **Correct sentence**: B. The crime is a problem in many big cities.
Explanation: This sentence is correct, as "the" is appropriately used to specify the type of crime being discussed.

2. **Correct verb form**: D. are taking
Explanation: Since "neither...nor" refers to more than one subject (father and grandparents), the plural verb "are" is used.

3. **Correct preposition**: C. to
Explanation: The correct expression is "he prefers soccer to tennis."

4. **Correct conditional**: C. wouldn’t make
Explanation: The phrase "If they had worked more carefully" indicates a past unreal conditional meaning they did not work carefully.

5. **Correct verb form**: B. was going
Explanation: The past progressive tense "was going" is appropriate here when indicating an action that was in progress when another action occurred.

6. **Correct phrase**: B. we are being followed
Explanation: This is the correct passive voice structure for the sentence.

7. **Strongest advice**: D. You really must get a hair-cut.
Explanation: "Must" conveys the strongest obligation or suggestion compared to "should," "ought to," and "if I were you."

8. **Correct question form**: B. Do
Explanation: The correct form to ask if someone minds is "Do you mind...?"

9. **Correct relative pronoun**: C. who
Explanation: "Who" is used to introduce a clause that gives more information about Mr. Vo Van Kiet.

10. **Correct phrase**: A. get rid of
Explanation: The expression "get rid of" correctly indicates the action of disposing of rubbish.

To summarize, here are the correct options for each question:
1. B
2. D
3. C
4. C
5. B
6. B
7. D
8. B
9. C
10. A