Question : If $x+\frac{1}{x}=\sqrt{3}$, the value of $\left (x^{3}+\frac{1}{x^{3}} \right )$ is:
Option 1: $\sqrt{3}$
Option 2: $\frac{1}{\sqrt{3}}$
Option 3: $0$
Option 4: $1$
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Correct Answer: $0$
Solution :
Given: $x+\frac{1}{x}=\sqrt{3}$
Cubing both sides and simplifying the equation,
⇒ $(x+\frac{1}{x})^3=(\sqrt{3})^3$
⇒ $x^3+\frac{1}{x^3}+3(x×\frac{1}{x})(x+\frac{1}{x})=(\sqrt{3})^3$
⇒ $x^3+\frac{1}{x^3}+3(x+\frac{1}{x})=3\sqrt{3}$
Putting the value of $x+\frac{1}{x}=\sqrt{3}$
⇒ $x^3+\frac{1}{x^3}+3\sqrt{3}=3\sqrt{3}$
⇒ $x^3+\frac{1}{x^3}=3\sqrt{3}-3\sqrt{3}$
⇒ $x^3+\frac{1}{x^3}=0$
Hence, the correct answer is $0$.
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