Given the set of data 1,1,2,2,2,3,3,x,y, where x and y represent two different integers. If the mode is 2, which of the following statements must be true?

Options

A. If x=1 or 3, then y must=2
B. Both x and y must be gt 3
C. Either x or Y must =2
D. It does not matter what values x and y have

Explanation

To solve the problem, we need to determine the conditions under which the mode of the given data set is 2. The data set is: 1, 1, 2, 2, 2, 3, 3, x, y, where x and y are two different integers.

### Step-by-Step Solution:

1. **Understanding Mode**: The mode of a data set is the number that appears most frequently. For the mode to be 2, the number 2 must appear more times than any other number in the data set.

2. **Counting Current Frequencies**:
- The current frequencies of the numbers in the data set (without x and y) are:
- 1 appears 2 times
- 2 appears 3 times
- 3 appears 2 times
- So, currently, the frequency of 2 is the highest (3 times).

3. **Adding x and y**: We need to consider how the values of x and y can affect the frequency of the numbers:
- If x and y are both numbers greater than 2 (for example, 4, 5, etc.), they will not affect the frequency of 2.
- If either x or y is 2, the frequency of 2 will increase, making it even more likely to remain the mode.

4. **Conditions for x and y**:
- If x and y are both greater than 3, the frequency of 2 will still be 3, and it will remain the mode.
- If either x or y is 1 or 3, the frequency of 2 can still remain the highest, but we need to ensure that they do not exceed the frequency of 2.

5. **Conclusion**:
- For the mode to remain 2, we can conclude that if x is 1 or 3, then y must be 2. This ensures that the frequency of 2 remains the highest (4 times if y is 2).
- Therefore, the statement that must be true is: "If x is 1 or 3, then y must be 2."

### Final Answer:
The correct statement that must be true is: **A. If x is 1 or 3, then y must be 2.**