Given data set: 3, 8, 6, 7, 1, 6, 10, 6, 7, 2k, 5, 9, 7, and 13

Finding the mode:
The mode is the value that appears most frequently in a data set. In this case, the mode is given as 7.

To find the value of k:
We need to determine the value of k in the data set. To do this, we can use the information given and apply it to the concept of mode.

Analysis:
- The mode is 7, which means that 7 appears most frequently in the data set.
- Looking at the given data set, we see that 7 appears three times: 7, 7, and 7.
- We also see that there are two 6s in the data set.
- All other numbers appear only once.

Deducing the value of k:
- Since the mode is 7 and 7 appears three times in the data set, we can conclude that the value of k must be 7 as well to maintain the mode.
- If we substitute 7 for k in the data set, we get: 3, 8, 6, 7, 1, 6, 10, 6, 7, 7, 5, 9, 7, and 13.
- In this revised data set, the mode is still 7, as it appears four times, which is more than any other number.

Therefore, the value of k in the data set is 7.

Explanation of the correct answer:
- Option 'D' is the correct answer, which states that the value of k is 1.
- This answer is incorrect because we have determined that the value of k is 7, not 1.
- It seems there may be a mistake or confusion in the given options, as option 'D' does not match the correct answer based on the analysis and calculations above.

In conclusion, the correct value of k in the data set 3, 8, 6, 7, 1, 6, 10, 6, 7, 2k, 5, 9, 7, and 13 is 7, not 1 as stated in the given options.