'P' is the smallest positive integer such that every positive integer N greater than 'P' can be written as a sum of two composite numbers. Then:

The correct option is D $P=11$

Let us examine each option.
Option A $⟶p=3$
$\therefore p+1=4=2+2=1+3\Rightarrow$ They are not composite numbers.
$\therefore$ Option A is incorrect.
Option B $⟶p=6$

$\therefore p+1=7=1+6=2+5=3+4$
$\Rightarrow$ none of the additive pairs is sum of composite numbers.
$\therefore$ Option B is incorrect.
Option C $⟶p=6$
$\therefore p+1=11=1+10=2+9=3+8=4+7=5+6$
$\Rightarrow$ none of the additive pairs is sum of composite numbers.
$\therefore$ Option C is incorrect.
Option D $⟶p=11$
$\therefore p+1=12=1+11=2+10=3+9=4+8=5+7=6+6$
Here, Both the additive pair $(4,8)$ and $(6,6)$ are composite numbers.
$\therefore$ Option D is Correct.