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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


Team Careers360 23rd Jan, 2024
Answer (1)
Team Careers360 25th Jan, 2024

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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