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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