A) 64
B) 62
C) 66
D) 68
Correct Answer: B
Solution :
Given expression \[={{\left( \frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}} \right)}^{2}}+{{\left( \frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}} \right)}^{2}}\] Now, \[{{\left( \frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}} \right)}^{2}}=\frac{{{(\sqrt{5}+\sqrt{3})}^{2}}}{{{(\sqrt{5}-\sqrt{3})}^{2}}}\] \[=\frac{{{(\sqrt{5})}^{2}}+{{(\sqrt{3})}^{2}}+2\sqrt{5}\times \sqrt{3}}{{{(\sqrt{5})}^{2}}+{{(\sqrt{3})}^{2}}-2\sqrt{5}\times \sqrt{3}}\] \[=\frac{5+3+2\sqrt{15}}{5+3-2\sqrt{15}}\] \[=\frac{8+2\sqrt{15}}{8-2\sqrt{15}}=\frac{4+\sqrt{15}}{4-\sqrt{15}}\] Similarly, \[{{\left( \frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}} \right)}^{2}}=\frac{4-\sqrt{15}}{4+\sqrt{15}}\] Therefore, given expression\[=\frac{4+\sqrt{15}}{4-\sqrt{15}}+\frac{4-\sqrt{15}}{4+\sqrt{15}}\] \[=\frac{16+15+8\sqrt{15}+16+15-8\sqrt{15}}{16-15}=62\]
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