If the mean of 3, 4, 9, 2k, 10, 8, 6 and (k + 6) is 8, and 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 :

Correct option is C

For 4 to be the mode, it must occur more frequently than any other number. We already have two 4's, so there must be fewer occurrences of any other value including 2p and 2p + 1.
If 2p = 4 then p = 2
If p = 2 then 2p + 1 = 5
The list then becomes: 2 , 2 , 3 , 4 , 5 , 4 , 4 , 5 , 6. This includes three 4's, satisfying the mode condition.
Then, k-2p=6-2×2=6-4=2