If the mode of the following data is 11, then find the value of k. 11,8,9, (2k-1), 11, 12, 12, 18, 14, 16

Options

A. 5
B. 6
C. 7
D. 4

Answer

To find the value of \( k \) such that the mode of the given data set is 11, we will follow these steps:

### Step 1: Identify the mode
The mode is the number that appears most frequently in a data set. In the given data set:
\[ 11, 8, 9, (2k-1), 11, 12, 12, 18, 14, 16 \]
we can see that 11 appears twice.

### Step 2: Determine the frequency of other numbers
Next, we need to determine how many times \( 2k - 1 \) appears. For 11 to be the mode, it must appear more frequently than any other number, including \( 2k - 1 \).

### Step 3: Set up the inequality
To ensure that 11 remains the mode, we need:
\[ \text{Frequency of } 11 > \text{Frequency of } (2k - 1) \]

Since 11 appears twice, we can write:
\[ 2 > \text{Frequency of } (2k - 1) \]

### Step 4: Analyze the value of \( 2k - 1 \)
Now, we need to find when \( 2k - 1 \) can be equal to 11. Setting \( 2k - 1 = 11 \):
\[ 2k - 1 = 11 \]
Adding 1 to both sides:
\[ 2k = 12 \]
Dividing by 2:
\[ k = 6 \]

### Step 5: Verify the value of \( k \)
Now, we need to check if \( k = 6 \) keeps 11 as the mode:
If \( k = 6 \), then:
\[ 2k - 1 = 2(6) - 1 = 12 - 1 = 11 \]
Now, the data set becomes:
\[ 11, 8, 9, 11, 11, 12, 12, 18, 14, 16 \]
In this case, 11 appears three times, and 12 appears twice.

### Conclusion
Since 11 appears more frequently than any other number, the mode remains 11. Thus, the value of \( k \) is:
\[ \boxed{6} \]