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
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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