A) 1
B) \[\sqrt{2}\]
C) \[\frac{1}{\sqrt{2}}\]
D) 0
Correct Answer: C
Solution :
\[\frac{1}{\sqrt{2}+\sqrt{3}-\sqrt{5}}=\frac{\sqrt{2}+\sqrt{3}+\sqrt{5}}{[\sqrt{2}+\sqrt{3}+\sqrt{5}]}=\frac{\sqrt{2}+\sqrt{3}+\sqrt{5}}{2+3+2\sqrt{6}-5}\] |
\[=\frac{\sqrt{2}+\sqrt{3}+\sqrt{5}}{2\sqrt{6}}\] |
Similarly, \[=\frac{1}{\sqrt{2}-\sqrt{3}-\sqrt{5}}\] |
\[=\frac{\sqrt{2}-\sqrt{3}+\sqrt{5}}{[(\sqrt{2}-\sqrt{3})-\sqrt{5}](\sqrt{2}-\sqrt{3})+\sqrt{5}}\] |
\[=\frac{\sqrt{2}-\sqrt{3}+\sqrt{5}}{-\,\,2\sqrt{6}}\] |
Expression\[=\frac{\sqrt{2}+\sqrt{3}+\sqrt{5}}{2\sqrt{6}}-\frac{\sqrt{2}-\sqrt{3}+\sqrt{5}}{2\sqrt{6}}\] |
\[=\frac{\sqrt{2}+\sqrt{3}+\sqrt{5}-\sqrt{2}+\sqrt{3}-\sqrt{5}}{2\sqrt{6}}\] |
\[=\frac{\sqrt{3}}{\sqrt{6}}=\frac{1}{\sqrt{2}}\] |
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