If $$\sqrt 2 $$ : $$\left( {1 + \sqrt 3 } \right)$$  :: $$\sqrt 6 $$ : $$x$$, then $$x$$ is equal to -

$$\eqalign{ & = \sqrt 2 :\left( {1 + \sqrt 3 } \right)::\sqrt 6 :x \cr & \Rightarrow \sqrt {2}x = \sqrt 6 \left( {1 + \sqrt 3 } \right) \cr & \Rightarrow x = \frac{{\sqrt 6 \left( {1 + \sqrt 3 } \right)}}{{\sqrt 2 }} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\, = \sqrt 3 \left( {1 + \sqrt 3 } \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\, = \sqrt 3 + 3 \cr} $$