Question : Let $x=\frac{\sqrt{13}+\sqrt{11}}{\sqrt{13}-\sqrt{11}}$ and $y=\frac{1}{x}$, then the value of $3x^{2}-5xy+3y^{2}$ is:
Option 1: 1717
Option 2: 1177
Option 3: 1771
Option 4: 1171
Correct Answer: 1717
Solution :
The value of $x=\frac{\sqrt{13}+\sqrt{11}}{\sqrt{13}-\sqrt{11}} $
After rationalizing this, we get,
$ =\frac{(\sqrt{13}+\sqrt{11})(\sqrt{13}+\sqrt{11})}{(\sqrt{13}-\sqrt{11})(\sqrt{13}+\sqrt{11})} $ $=\frac{(13+2\sqrt{143}+11)}{(13-11)} =\frac{24+2\sqrt{143}}{2} =12+\sqrt{143}$
The value of $y=\frac{1}{x} =\frac{1}{12+\sqrt{143}} $
After rationalizing this, we get,
$ =\frac{(12-\sqrt{143})}{(12+\sqrt{143})(12-\sqrt{143})} =\frac{(12-\sqrt{143})}{1} =
12-\sqrt{143}$
$⇒3x^{2}-5xy+3y^{2}$
$=3(12+\sqrt{143})^{2}-5(12+\sqrt{143})(12-\sqrt{143})+3(12-\sqrt{143})^{2}$
$=3(144+24\sqrt{143}+143)-5(144-143)+3(144-24\sqrt{143}+143)$
$=3(287+24\sqrt{143})-5+3(287-24\sqrt{143})$
$=861+72\sqrt{143}-5+861-72\sqrt{143}$
$=1717$
Therefore, the value of $3x^{2}-5xy+3y^{2}$ is 1717.
Hence, the correct answer is 1717.
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