| Let \[x=\frac{\sqrt{13}+\sqrt{11}}{\sqrt{13}-\sqrt{11}}\] and \[y=\frac{1}{x},\] then the value of \[3{{x}^{2}}-5xy+3{{y}^{2}}\] is |
A) 1771
B) 1177
C) 1717
D) 1171
Correct Answer: C
Solution :
| Given, \[x=\frac{\sqrt{3}+\sqrt{11}}{\sqrt{13}-\sqrt{11}}\] |
| and \[y=\frac{\sqrt{13}-\sqrt{11}}{\sqrt{13}+\sqrt{11}}\] \[\left[ \because y=\frac{1}{x} \right]\] |
| \[\therefore \] \[x+y=\frac{\sqrt{13}+\sqrt{11}}{\sqrt{13}-\sqrt{11}}+\frac{\sqrt{13}-\sqrt{11}}{\sqrt{13}+\sqrt{11}}\] |
| \[=\,\,\frac{{{(\sqrt{13}+\sqrt{11})}^{2}}+{{(\sqrt{13}-\sqrt{11})}^{2}}}{{{(\sqrt{13})}^{2}}+{{(\sqrt{11})}^{2}}}\] |
| \[=\,\,\frac{2\,\,[\,{{(\sqrt{13})}^{2}}+{{(\sqrt{11})}^{2}}]}{13-11}=13+11=24\] |
| \[\therefore \] \[3{{x}^{2}}-5xy+3{{y}^{2}}=3\,{{(x+y)}^{2}}-11xy\] |
| \[=3\,{{(24)}^{2}}-11=1717\] |
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