A) \[8\sqrt{5}\]
B) \[2\sqrt{5}\]
C) \[5\sqrt{5}\]
D) \[7\sqrt{5}\]
Correct Answer: B
Solution :
(b): \[x+\frac{1}{x}=\sqrt{5}\] ....(1) \[{{x}^{3}}+\frac{1}{{{x}^{3}}}=x+\frac{1}{x}\left\{ {{x}^{2}}-x.\frac{1}{x}+\frac{1}{{{x}^{2}}} \right\}\] \[=\sqrt{5}\left\{ {{x}^{2}}-1+\frac{1}{{{x}^{2}}} \right\}\] Also, squaring (1). \[{{x}^{2}}+2.x.\frac{1}{x}+\frac{1}{{{x}^{2}}}=5\] \[\Rightarrow {{x}^{2}}+\frac{1}{{{x}^{2}}}+2=5\] \[\Rightarrow {{x}^{2}}+\frac{1}{{{x}^{2}}}=3\] \[\Rightarrow \left\{ {{x}^{2}}-1+\frac{1}{{{x}^{2}}} \right\}=2\] \[\therefore {{x}^{3}}+\frac{1}{{{x}^{3}}}=\sqrt{5}\times 2=2\sqrt{5}\].
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