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 $$ 6 $$ ($$ p $$ is a natural number) is 4, then the value of $$ k - 2p $$ is:

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Accepted Answer

1. **Mean Condition:**  
   The mean of $$ 3, 4, 9, 2k, 10, 8, 6 $$, and $$ (k + 6) $$ is given as $$ 8 $$. Using the formula for mean:  
   $$
   	ext{Mean} = \frac{	ext{Sum of all terms}}{	ext{Number of terms}}
   $$  
   Substituting the given values:  
   $$
   \frac{3 + 4 + 9 + 2k + 10 + 8 + 6 + (k + 6)}{8} = 8
   $$  
   Simplify:  
   $$
   \frac{46 + 3k}{8} = 8
   $$  
   Multiply through by $$8$$:  
   $$
   46 + 3k = 64
   $$  
   Solve for $$k$$:  
   $$
   k = \frac{64 - 46}{3} = \frac{18}{3} = 6
   $$

2. **Mode Condition:**  
   The mode of $$2, 2, 3, 2p, (2p + 1), 4, 4, 5$$, and $$6$$ is given as $$4$$. For the mode to be $$4$$, it must occur more frequently than any other number. Thus:  
   - The frequency of $$4$$ must be at least $$3$$. This implies that one of the terms $$2p$$ or $$2p +1$$ must equal $$4$$.

If $$2p = 4$$:  
   $$
   p = \frac{4}{2} = 2
   $$

3. **Calculate $$k - 2p$$:**  
   Substitute the values of $$k = 6$$ and $$p = 2$$:  
   $$
   k - 2p = 6 - (2 	imes 2) = 6 - 4 = 2
   $$

Thus, the value of $$k - 2p$$ is **2**, making option **(3)** correct.

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