A) \[\frac{4\sqrt{3}-9}{33}\]
B) \[\frac{4\sqrt{3}+9}{33}\]
C) \[\frac{9\sqrt{3}-4}{33}\]
D) \[\frac{9\sqrt{3}+4}{33}\]
Correct Answer: A
Solution :
| [a] We have, \[\frac{\sin 60{}^\circ +\cot 45{}^\circ -\cos ec\,\,30{}^\circ }{\sec 60{}^\circ -\cos 30{}^\circ +\tan 45{}^\circ }\] |
| \[=\frac{\frac{\sqrt{3}}{2}+1-2}{2-\frac{\sqrt{3}}{2}+1}=\frac{\frac{\sqrt{3}+2-4}{2}}{\frac{4-\sqrt{3}+2}{2}}=\frac{\sqrt{3}-2}{6-\sqrt{3}}\] |
| \[=\frac{\sqrt{3}-2}{6-\sqrt{3}}\times \frac{6+\sqrt{3}}{6+\sqrt{3}}=\frac{(\sqrt{3}-2)\,\,(6+\sqrt{3})}{{{6}^{2}}-{{(\sqrt{3})}^{2}}}\] |
| \[=\frac{6\sqrt{3}+3-12-2\sqrt{3}}{36-3}=\frac{4\sqrt{3}-9}{33}\] |
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