| If \[\frac{\frac{1}{\sqrt{9}}-\frac{1}{\sqrt{11}}}{\frac{1}{\sqrt{9}}+\frac{1}{\sqrt{11}}}\times \frac{10+\sqrt{99}}{x}=\frac{1}{2},\] then the value of \[x\]is |
A) 2
B) 3
C) 4
D) 5
Correct Answer: A
Solution :
| \[\frac{\frac{1}{\sqrt{9}}-\frac{1}{\sqrt{11}}}{\frac{1}{\sqrt{9}}+\frac{1}{\sqrt{11}}}=\frac{\sqrt{11}-\sqrt{9}}{\sqrt{11}+\sqrt{9}}=\frac{\sqrt{11}-\sqrt{9}}{\sqrt{11}+\sqrt{9}}\times \frac{\sqrt{11}-\sqrt{9}}{\sqrt{11}-\sqrt{9}}\] |
| \[=\frac{11+9-2\sqrt{99}}{11-9}=\frac{2\,\,(10-\sqrt{99})}{2}=10-\sqrt{99}\] |
| \[\therefore \]\[\frac{(10-\sqrt{99})\times (10+\sqrt{99})}{x}=\frac{1}{2}\]\[\Rightarrow \]\[\frac{100-99}{x}=\frac{1}{2}\] |
| \[\Rightarrow \] \[\frac{1}{x}=\frac{1}{2}\]\[\Rightarrow \]\[x=2\] |
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