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Question : If $a-\frac{1}{a-5}=10$, then the value of $(a-5)^3-\frac{1}{(a-5)^3}$ is:

Option 1: 140

Option 2: 70

Option 3: 100

Option 4: 120


Team Careers360 3rd Jan, 2024
Answer (1)
Team Careers360 10th Jan, 2024

Correct Answer: 140


Solution : $a-\frac{1}{a-5}=10$ .............(1)
Subtract 5 in equation (1) on both sides
$(a-5)-\frac{1}{a-5}=10-5$
⇒ $(a-5)-\frac{1}{a-5}=5$
Now, take $(a - 5) = x$
So, the equation becomes
⇒ $x-\frac{1}{x}=5$
Cubing both sides
⇒ $x^3-\frac{1}{x^3}-3(x-\frac{1}{x}) = 125$
⇒ $x^3-\frac{1}{x^3} = 125+3(5)$
⇒ $x^3-\frac{1}{x^3} = 140$
Hence, the correct answer is 140.

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