If 3 + 1 3 , then the value of ( x - 126 42 ) ( x - 1x - 2 3 3 ) is?

Answer: Option C

$$\begin{array}{rl}& \because x=\sqrt{3}+\frac{1}{\sqrt{3}}\\ & \Rightarrow x=\frac{3+1}{\sqrt{3}}\\ & \Rightarrow x=\frac{4}{\sqrt{3}}\\ & \text{So, }\left(x-\frac{\sqrt{126}}{\sqrt{42}}\right)\left(x-\frac{1}{x-\frac{2\sqrt{3}}{3}}\right)\\ & =\left(\frac{4}{\sqrt{3}}-\frac{\sqrt{126}}{\sqrt{42}}\right)\left(\frac{4}{\sqrt{3}}-\frac{1}{\frac{4}{\sqrt{3}}-\frac{2}{\sqrt{3}}}\right)\\ & =\left(\frac{4\sqrt{42}-\sqrt{126}\times \sqrt{3}}{\sqrt{3\times }\sqrt{42}}\right)\left(\frac{4}{\sqrt{3}}-\frac{1}{\frac{2}{\sqrt{3}}}\right)\\ & =\frac{4\sqrt{42}-3\sqrt{42}}{\sqrt{3\times }\sqrt{42}}\left(\frac{4}{\sqrt{3}}-\frac{\sqrt{3}}{2}\right)\\ & =\left(\frac{\sqrt{42}}{\sqrt{3\times }\sqrt{42}}\right)\left(\frac{8-3}{2\sqrt{3}}\right)\\ & =\frac{1}{\sqrt{3}}\times \frac{5}{2\sqrt{3}}\\ & =\frac{5}{6}\end{array}$$