Question : If $x\left(5-\frac{2}{x}\right)=\frac{5}{x}$, then the value of $x^2+\frac{1}{x^2}$ is:
Option 1: $\frac{54}{25}$
Option 2: $\frac{53}{28}$
Option 3: $\frac{53}{27}$
Option 4: $\frac{54}{23}$
Correct Answer: $\frac{54}{25}$
Solution :
Given: $x\left(5-\frac{2}{x}\right)=\frac{5}{x}$
⇒ $5x-2 = \frac{5}{x}$
⇒ $5x-\frac{5}{x} =2$
⇒ $5(x-\frac{1}{x})=2$
⇒ $x- \frac{1}{x} = \frac{2}{5}$
Squaring both sides.
⇒ $x^2+\frac{1}{x^2}-2 = \frac{4}{25}$
⇒ $x^2+\frac{1}{x^2} = \frac{4}{25}+2$
⇒ $x^2+\frac{1}{x^2} = \frac{54}{25}$
Hence, the correct answer is $\frac{54}{25}$.
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