Question : If $2 x+\frac{2}{x}=5$, then the value of $\left(x^3+\frac{1}{x^3}+2\right)$ will be:
Option 1: $\frac{81}{11}$
Option 2: $\frac{81}{7}$
Option 3: $\frac{71}{8}$
Option 4: $\frac{81}{8}$
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Correct Answer: $\frac{81}{8}$
Solution : $2 x+\frac{2}{x}=5$ $⇒2(x+\frac{1}{x})=5$ $⇒(x+\frac{1}{x})=\frac{5}{2}$ Cubing both sides we get, $⇒x^3+\frac{1}{x^3}+3\times x \times \frac{1}{x}(x+\frac{1}{x})=\frac{125}{8}$ $⇒x^3+\frac{1}{x^3}+3(\frac{5}{2})=\frac{125}{8}$ $⇒x^3+\frac{1}{x^3} = \frac{125}{8} - \frac{15}{2}$ $⇒x^3+\frac{1}{x^3} = \frac{125-60}{8}$ $⇒x^3+\frac{1}{x^3} = \frac{65}{8}$ Adding 2 on both sides, we get $⇒x^3+\frac{1}{x^3}+2=\frac{65}{8}+2$ $⇒x^3+\frac{1}{x^3}+2=\frac{65+16}{8}$ $⇒x^3+\frac{1}{x^3}+2=\frac{81}{8}$ Hence, the correct answer is $\frac{81}{8}$.
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