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જો $x = {{\sqrt 5 + \sqrt 2 } \over {\sqrt 5 - \sqrt 2 }},y = {{\sqrt 5 - \sqrt 2 } \over {\sqrt 5 + \sqrt 2 }},$ તો $3{x^2} + 4xy - 3{y^2} = $
${1 \over 3}[56\sqrt {10} - 12]$
${1 \over 3}[56\sqrt {10} + 12]$
${1 \over 3}[56 + 12\sqrt {10} ]$
એકપણ નહીં
Solution
(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)$.
