If x = 2 + √3, y = 2 - √3, z = 1 then what is the value of $$\frac{x}{{yz}} + \frac{y}{{xz}} + \frac{z}{{xy}} + 2\left[ {\frac{1}{x} + \frac{1}{y} + \frac{1}{z}} \right]?$$

Solution (By Examveda Team)

$$\eqalign{ & x = 2 + \sqrt 3 \cr & y = 2 - \sqrt 3 \cr & z = 1 \cr & \frac{x}{{yz}} + \frac{y}{{xz}} + \frac{z}{{xy}} + 2\left[ {\frac{1}{x} + \frac{1}{y} + \frac{1}{z}} \right] \cr & = \frac{{{x^2} + {y^2} + {z^2}}}{{xyz}} + 2\left[ {\frac{1}{x} + \frac{1}{y} + \frac{1}{z}} \right] \cr & = \frac{{{{\left( {2 + \sqrt 3 } \right)}^2} + {{\left( {2 - \sqrt 3 } \right)}^2} + {{\left( 1 \right)}^2}}}{{\left( {2 + \sqrt 3 } \right)\left( {2 - \sqrt 3 } \right)\left( 1 \right)}} + 2\left[ {\frac{1}{{\left( {2 + \sqrt 3 } \right)}} + \frac{1}{{\left( {2 - \sqrt 3 } \right)}} + 1} \right] \cr & = \frac{{4 + 3 + 4\sqrt 3 + 4 + 3 - 4\sqrt 3 + 1}}{1} + 2\left[ {2 - \sqrt 3 + 2 + \sqrt 3 + 1} \right] \cr & = 15 + 10 \cr & = 25 \cr} $$