Sure! Let's go through each question one by one, providing explanations for the correct answers.

### 6. The teacher said that I would be able to speak English fluently ________ six months.
**Answer: B. by**
**Explanation:**
- "By" indicates a deadline or endpoint by which something will happen. In this context, it means that by the end of six months, the speaker will be able to speak English fluently.
- "In" usually indicates the duration or time taken to achieve something (e.g., "in six months"). However, "by" is the correct choice here as it refers to reaching the ability in that timeframe.

### 7. The police began an ________ into the disappearance of the painting five days ago.
**Answer: A. investigation**
**Explanation:**
- "Investigation" is a noun that describes the act or process of investigating something. It is the correct form that fits in this context.
- The other options (B, C, D) are not proper noun forms to fit this sentence structure.

### 8. They are having their house ________ by a construction company.
**Answer: D. painted**
**Explanation:**
- "Painted" is the correct form in the passive voice. The sentence implies that the construction company is the one performing the action (painting) on their house, so the passive construction is appropriate here.
- The other options do not fit grammatically or contextually for this sentence.

### 9. Only after doing his chores ________ to go to meet his friends.
**Answer: D. was Jim allowed**
**Explanation:**
- The structure "was Jim allowed" is the correct passive voice form in this context.
- It emphasizes that Jim did not have the liberty to go until he completed his chores. The other options either change the meaning or do not maintain proper grammatical structure.

### 10. I hope you will ________ notice of what I am going to tell you.
**Answer: D. take**
**Explanation:**
- "Take notice" is a standard phrase meaning to pay attention or be aware of something. "Take" in this context is the right verb to collocate with "notice."
- The other options (gain, keep, get) do not form common English phrases that would fit appropriately here.

### Summary of Correct Answers:
6. B. by
7. A. investigation
8. D. painted
9. D. was Jim allowed
10. D. take

If you have any further questions or need more assistance, feel free to ask!