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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