If x = 5 + 2 6 , then the value of ( x + 1 x )\, is?

Answer: Option C

$$\begin{array}{rl}& x=5+2\sqrt{6}\\ & \iff x=3+2+2\sqrt{3}\times \sqrt{2}\\ & \iff x={\left(\sqrt{3}\right)}^2+{\left(\sqrt{2}\right)}^2+2\sqrt{3}\times \sqrt{2}\\ & \iff x={\left(\sqrt{3}+\sqrt{2}\right)}^2\\ & \iff \sqrt{x}=\sqrt{3}+\sqrt{2}\\ & \text{Similarly,}\\ & \iff \frac{1}{\sqrt{x}}=\sqrt{3}-\sqrt{2}\\ & \therefore \left(\sqrt{x}+\frac{1}{\sqrt{x}}\right)\\ & =\sqrt{3}+\sqrt{2}+\sqrt{3}-\sqrt{2}\\ & =2\sqrt{3}\end{array}$$