Sure! Let's analyze each question one by one and explain why each correct option is the best choice.

### Question 6
**The study showed that one in twelve women is_______ to develop breast cancer.**

**Options:**
A. likely
B. alike
C. unlike
D. like

**Correct Answer: A. likely**

**Explanation:**
- The phrase "is likely to develop breast cancer" is a common expression used to convey the probability or likelihood of an event occurring.
- "Likely" functions as an adjective indicating that there is a good chance or probability that something (in this case, developing breast cancer) will happen.

### Question 8
**Art critics do not all agree on what _______ a painting great.**

**Options:**
A. qualities to make
B. are the qualities for making
C. qualities make
D. do the qualities that make

**Correct Answer: C. qualities make**

**Explanation:**
- The sentence is asking what characteristics or traits contribute to making a painting great.
- "Qualities make a painting great" is a correct, concise structure indicating that the qualities themselves are the ones that create greatness in a painting. Options A, B, and D are grammatically incorrect or awkward.

### Question 11
**Had it not been for the intolerable heat in the hall, they _____ much longer.**

**Options:**
A. will stay
B. would stay
C. would be staying
D. would have stayed

**Correct Answer: D. would have stayed**

**Explanation:**
- This sentence is structured using the third conditional, which describes a situation that didn't happen in the past. The phrase "Had it not been for" indicates a hypothetical situation.
- "Would have stayed" correctly indicates that they would have remained longer if it weren't for the heat. The other options do not fit the conditional structure appropriately.

### Question 13
**Fast internet service in rural areas provided a new ________ for prosperity.**

**Options:**
A. convenience
B. opportunity
C. possibility
D. chance

**Correct Answer: B. opportunity**

**Explanation:**
- "Opportunity" is the most suitable word here, as it implies a favorable circumstance for progress or development (like economic prosperity).
- While "possibility," "chance," and "convenience" might seem relevant, they do not capture the idea of creating favorable conditions for growth as well as "opportunity" does.

In summary, the correct options for the questions are:
6. A. likely
8. C. qualities make
11. D. would have stayed
13. B. opportunity