Question

If the mean of 3, 4, 9, 2k, 10, 8, 6 and $(k + 6)$ is 8, and the mode of 2, 2, 3, $2p$, $(2p + 1)$, 4, 4, 5 and 6 (where $p$ is a natural number) is 4, then the value of $k - 2p$ is:

• A. 2
• B. 1
• C. 0
• D. 3

Answer 1

The Correct Option is A

To find the value of \(k - 2p\), we must first solve for \(k\) and \(p\). 

Step 1: Find \(k\)

The mean of the numbers 3, 4, 9, 2k, 10, 8, 6, and \(k + 6\) is 8. The formula for the mean is:

\(\text{Mean} = \frac{\text{Sum of all elements}}{\text{Number of elements}}\)

For these numbers:

\(\frac{3+4+9+2k+10+8+6+(k+6)}{8} = 8\)

Simplify and solve for \(k\):

\(46 + 3k = 64\)

\(3k = 64 - 46\)

\(3k = 18\)

\(k = \frac{18}{3} = 6\)

Step 2: Find \(p\)

The mode of the numbers 2, 2, 3, \(2p\), \(2p + 1\), 4, 4, 5, and 6 is given as 4. Mode is the number that appears most frequently.

For 4 to be the mode, it must appear more frequently than any other number. Consider permutations of \(2p\):

If \(2p = 4\)	then \(p = 2\), and sequence becomes 2, 2, 3, 4, 5, 4, 4, 5, 6
If \(2p = 3\)	or any value other than 4, mode would differ

Since 4 is the mode, \(2p = 4\) means \(p = 2\).

Step 3: Calculate \(k - 2p\)

With \(k = 6\) and \(p = 2\), calculate:

\(k - 2p = 6 - 2 \times 2 = 6 - 4 = 2\)

Thus, the value of \(k - 2p\) is 2.

Answer 2

The mean of 3, 4, 9, 2k, 10, 8, 6 and (k + 6) is 8. So the sum of these numbers divided by 8 is 8.

\(\frac{3 + 4 + 9 + 2k + 10 + 8 + 6 + (k+6)}{8} = 8\)

\(\frac{46 + 3k}{8} = 8\)

\(46 + 3k = 64\)

\(3k = 18\)

\(k = 6\)

The mode of 2, 2, 3, 2p, (2p + 1), 4, 4, 5 and 6 is 4, meaning 4 appears most often. 

Since 2,3,5,6 only appear once, either 2p or 2p+1 needs to be 4 in order for 4 to be the mode.

If 2p=4, then p=2. If 2p+1=4, this will lead to p=3/2 which is not a natural number. 

Thus, the mode equals 4 when p=2 Now we calculate k - 2p = 6 - 2 \(\times\) 2 = 6 - 4 = 2