Question : If $x+\frac{1}{x}=6$, then find the value of $\frac{3 x}{2 x^2-5 x+2}$.
Option 1: $1$
Option 2: $\frac{3}{7}$
Option 3: $\frac{2}{3}$
Option 4: $0$
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Correct Answer: $\frac{3}{7}$
Solution :
Given, $x+\frac{1}{x}=6$
$\frac{3 x}{2 x^2-5 x+2} $
Taking $x$ common from the denominator,
$= \frac{3x}{x(2x-5+\frac{2}{x})}$
$=\frac{3}{2x+\frac{2}{x} - 5}$ ------------(1)
Here, $2x + \frac{2}{x} = 2(x+\frac{1}{x}) = 2\times 6$
⇒ $2(x+\frac{1}{x}) = 12$
Putting this value in (1), we get,
$\frac{3}{12-5} = \frac{3}{7}$
Hence, the correct answer is $\frac{3}{7}$.
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