To solve the problem, we need to analyze the given statements based on the conditions provided: the range of a set of integers is 2, and the mean is 50.
### Step-by-Step Solution:
1. **Understanding the Range**:
- The range is defined as the difference between the largest and smallest values in the dataset.
- Given that the range is 2, if we let the smallest value be \( x \), then the largest value must be \( x + 2 \).
2. **Identifying Possible Values**:
- The integers in the dataset must include \( x \) (smallest), \( x + 1 \), and \( x + 2 \) (largest).
- Therefore, the possible values in the dataset could be \( x, x + 1, x + 2 \).
3. **Calculating the Mean**:
- The mean is given as 50. The mean of a set of numbers is calculated as the sum of the numbers divided by the count of the numbers.
- For the integers \( x \), \( x + 1 \), and \( x + 2 \), the mean can be calculated as:
\[
\text{Mean} = \frac{x + (x + 1) + (x + 2)}{3} = \frac{3x + 3}{3} = x + 1
\]
- Setting this equal to the given mean:
\[
x + 1 = 50 \implies x = 49
\]
- Therefore, the values in the dataset must be \( 49, 50, 51 \).
4. **Evaluating the Statements**:
- **Statement I**: The mode is 50.
- The mode is the most frequently occurring number in the dataset. In our case, we have \( 49, 50, 51 \) each occurring once, so there is no mode. This statement is **not necessarily true**.
- **Statement II**: The median is 50.
- The median is the middle value when the numbers are arranged in order. For the dataset \( 49, 50, 51 \), the median is indeed 50. This statement is **true**.
- **Statement III**: There are exactly three data values.
- While we have identified three values \( 49, 50, 51 \), the conditions do not restrict the dataset to exactly three values. We could have more occurrences of 49 or 51, so this statement is **not necessarily true**.
5. **Conclusion**:
- Based on the evaluations:
- Statement I is false.
- Statement II is true.
- Statement III is false.
- Therefore, the only statement that must be true is **Statement II**.
### Final Answer:
The correct answer is **only statement II is true**.
