A) 0
B) 1
C) 2
D) 3
Correct Answer: A
Solution :
Expression \[=\frac{1}{\sqrt{12-\sqrt{140}}}-\frac{1}{\sqrt{8-\sqrt{60}}}-\frac{1}{\sqrt{10+\sqrt{84}}}\] \[=\frac{1}{\sqrt{12-\sqrt{4\times 35}}}-\frac{1}{\sqrt{8-\sqrt{4\times 15}}}-\frac{2}{\sqrt{10+\sqrt{4\times 21}}}\] \[x=\frac{{{m}_{1}}+{{x}_{2}}+{{m}_{2}}{{x}_{1}}}{{{m}_{1}}+{{m}_{2}}}=\frac{2\,\,(4)+3(-1)}{2+3}=\frac{8-3}{5}=\frac{5}{3}=1\] \[-\frac{2}{\sqrt{10+2\times \sqrt{7}\times \sqrt{3}}}\] \[=\frac{1}{\sqrt{{{(\sqrt{7})}^{2}}+{{(\sqrt{5})}^{2}}-2\times \sqrt{7}\times \sqrt{5}}}\]\[-\frac{1}{\sqrt{{{(\sqrt{5})}^{2}}+{{(\sqrt{3})}^{2}}-2\times \sqrt{5}\times \sqrt{3}}}\] \[-\frac{1}{\sqrt{{{(\sqrt{7})}^{2}}+{{(\sqrt{3})}^{2}}-2\times \sqrt{7}\times \sqrt{3}}}\] \[=\frac{1}{\sqrt{{{(\sqrt{7}-\sqrt{5})}^{2}}}}-\frac{1}{\sqrt{{{(\sqrt{5}-\sqrt{3})}^{2}}}}-\frac{2}{\sqrt{{{(\sqrt{7}+\sqrt{3})}^{2}}}}\] \[=\frac{1}{\sqrt{7}-\sqrt{5}}-\frac{1}{\sqrt{5}-\sqrt{3}}-\frac{2}{\sqrt{7}+\sqrt{3}}\] \[=\frac{\sqrt{7}+\sqrt{5}}{{{(\sqrt{7})}^{2}}+{{(\sqrt{5})}^{2}}}-\frac{\sqrt{5}+\sqrt{3}}{{{(\sqrt{5})}^{2}}-{{(\sqrt{3})}^{2}}}-\frac{2(\sqrt{7}-\sqrt{3})}{{{(\sqrt{7})}^{2}}-{{(\sqrt{3})}^{2}}}\] [Rationalizing each term by respective conjugates] \[=\frac{\sqrt{7}+\sqrt{5}}{2}-\frac{\sqrt{5}+\sqrt{3}}{2}-\frac{2(\sqrt{7}-\sqrt{3})}{4}\] \[=\frac{\sqrt{7}+\sqrt{5}-\sqrt{5}-\sqrt{3}-\sqrt{7}+\sqrt{3}}{2}=0\]
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