b
(b) \(y = {1 \over x}\) \( \Rightarrow \) \(xy = 1\)

\(\therefore 3{x^2} + 4xy - 3{y^2} = 3\,(x - y)\,(x + y + 4)\)

\( = 3\,.\,\left( {{{\sqrt 5 + \sqrt 2 } \over {\sqrt 5 - \sqrt 2 }} - {{\sqrt 5 - \sqrt 2 } \over {\sqrt 5 + \sqrt 2 }}} \right)\,\,\left( {{{\sqrt 5 + \sqrt 2 } \over {\sqrt 5 - \sqrt 2 }} + {{\sqrt 5 - \sqrt 2 } \over {\sqrt 5 + \sqrt 2 }}} \right)\, + 4\)

\( = {{3\,[{{(\sqrt 5 + \sqrt 2 )}^2} - {{(\sqrt 5 - \sqrt 2 )}^2}]} \over {(5 - 2)\,(5 - 2)}}\,[{(\sqrt 5 + \sqrt 2 )^2} + {(\sqrt 5 - \sqrt 2 )^2}] + 4\)

\( = {1 \over 3}.4\sqrt {10} \,.\,2\,(5 + 2) + 4 = {{56} \over 3}\sqrt {10} + 4 = {1 \over 3}(56\sqrt {10} + 12)\).