Question : If $x>1$ and $x+\frac{1}{x}=2\frac{1}{12}$, then the value of $x^{4}-\frac{1}{x^{4}}$ is:
Option 1: $\frac{58975}{20736}$
Option 2: $\frac{59825}{20736}$
Option 3: $\frac{57985}{20736}$
Option 4: $\frac{57895}{20736}$
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Correct Answer: $\frac{58975}{20736}$
Solution :
Given that: $x+\frac{1}{x}=\frac{25}{12}$
⇒ $12(x^2+1)=25x$
⇒ $12x^2-25x+12=0$
⇒ $12x^2-25x+12=0$
⇒ $12x^2-16x-9x+12=0$
⇒ $(4x-3)(3x-4)=0$
⇒ $x=\frac{4}3$ since $x>1$
$\therefore x^4-\left(\frac{1}{x}\right)^4$
Substituting the values,
$x^4-\left(\frac{1}{x}\right)^4= (\frac{4}3)^4-(\frac{3}4)^4$
$= \frac{256}{81}-\frac{81}{256}$
$= \frac{58975}{20736}$
Hence, the correct answer is $\frac{58975}{20736}$.
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