Question : If $m+\frac{1}{m-2}=4$, then find the value of $(m-2)^2+\frac{1}{(m-2)^2}$.
Option 1: – 2
Option 2: 4
Option 3: 0
Option 4: 2
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Correct Answer: 2
Solution :
Given, $m+\frac{1}{m-2}=4$
⇒ $(m-2)+\frac{1}{m-2}=2$
Squaring both sides, we get:
$(m-2)^2+\frac{1}{(m-2)^2}+2(m-2)(\frac{1}{m-2})=4$
⇒ $(m-2)^2+\frac{1}{(m-2)^2}+2=4$
$\therefore(m-2)^2+\frac{1}{(m-2)^2}=2$
Hence, the correct answer is 2.
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