Question : If $2x+\frac{1}{2x}=2,$ what is the value of $\sqrt{2\left (\frac{1}{x}\right)^{4}+\left (\frac{1}{x}\right)^{5}}\; ?$
Option 1: 1
Option 2: 2
Option 3: 4
Option 4: 8
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Correct Answer: 8
Solution :
Given: $2x+\frac{1}{2x}=2$
$⇒4x^{2}-4x+1=0$
$⇒\left (2x-1 \right)^{2}=0$
$⇒x=\frac{1}{2}$
So, $\sqrt{2\left (\frac{1}{x}\right)^{4}+\left (\frac{1}{x}\right)^{5}}\;=\sqrt{2\left (2\right)^{4}+\left (2\right )^{5}}\;=\sqrt{32+32}\;=8$
Hence, the correct answer is 8.
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