Question : If $\left(x-\frac{1}{x-2}\right)=7$, then the value of $(x-2)^3-\frac{1}{(x-2)^3}$ is:
Option 1: 130
Option 2: 115
Option 3: 140
Option 4: 125
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Correct Answer: 140
Solution :
Given:
$\left(x-\frac{1}{x-2}\right)=7$
⇒ $\left(x-2-\frac{1}{x-2}\right)=5$
⇒ $\left(x-2-\frac{1}{x-2}\right)^3=5^3$
⇒ $(x-2)^3 - (\frac{1}{x-2})^3-3×(x-2)×(\frac{1}{x-2})\left(x-2-\frac{1}{x-2}\right)=125$
⇒ $(x-2)^3 - (\frac{1}{x-2})^3=125+(3×5)$
$\therefore (x-2)^3 - (\frac{1}{x-2})^3=140$
Hence, the correct answer is 140.
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