The mean proportion between ( 3 + 2 ) and ( 12 - 32 ) is = ?

Answer: Option B

$$\begin{array}{rl}& \left(3+\sqrt{2}\right):x:\left(12-\sqrt{32}\right)\\ & a:b:c\\ & \text{mean proportion}\\ & \Rightarrow b^2=a\times c\\ & \Rightarrow x^2=\left(3+\sqrt{2}\right)\times \left(12-\sqrt{32}\right)\\ & \Rightarrow x^2=\left(3+\sqrt{2}\right)\times \left(12-4\sqrt{2}\right)\\ & \Rightarrow x^2=\left(3+\sqrt{2}\right)\times 4\left(3-\sqrt{2}\right)\\ & \Rightarrow x^2=4\times \left\{{\left(3\right)}^2-{\left(\sqrt{2}\right)}^2\right\}\\ & \Rightarrow x^2=28\\ & \Rightarrow x=2\sqrt{7}\end{array}$$