The mean proportion between $$\left( {3 + \sqrt 2 } \right)$$   and $$\left( {12 – \sqrt {32} } \right)$$   is = ?
a) $$\sqrt 7 $$
b) $$2\sqrt 7 $$
c) 6
d) $$\frac{{15 – 3\sqrt 2 }}{2}$$

Answer: b
Explanation:
$$\eqalign{ & \left( {3 + \sqrt 2 } \right):x:\left( {12 – \sqrt {32} } \right) \cr & a:b:c \cr & {\text{mean proportion}} \cr & {b^2} = a \times c \cr & {x^2} = \left( {3 + \sqrt 2 } \right) \times \left( {12 – \sqrt {32} } \right) \cr & {x^2} = \left( {3 + \sqrt 2 } \right) \times \left( {12 – 4\sqrt 2 } \right) \cr & {x^2} = \left( {3 + \sqrt 2 } \right) \times 4\left( {3 – \sqrt 2 } \right) \cr & {x^2} = 4 \times \left\{ {{{\left( 3 \right)}^2} – {{\left( {\sqrt 2 } \right)}^2}} \right\} \cr & {x^2} = 28 \cr & x = 2\sqrt 7 \cr} $$