A) \[1+2\sqrt{3}\]
B) \[2+\sqrt{3}\]
C) \[3+\sqrt{2}\]
D) \[2\sqrt{3}-1\]
Correct Answer: A
Solution :
\[x=1+\sqrt{2}+\sqrt{3}\] (Given) \[\therefore \] \[x+\frac{1}{x}=1+\sqrt{2}+\sqrt{3}+\frac{1}{\sqrt{2}+\sqrt{3}}\] \[=1+\sqrt{2}+\sqrt{3}+\frac{\sqrt{3}-\sqrt{2}}{(\sqrt{3}+\sqrt{2})(\sqrt{3}-\sqrt{2})}\] \[=1+\sqrt{2}+\sqrt{3}+\frac{\sqrt{3}-\sqrt{2}}{(3-2)}\] \[=1+\sqrt{2}+\sqrt{3}+\sqrt{3}-\sqrt{2}\] \[=1+2\sqrt{3}\]
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