Question : If $x+\frac{1}{x}=5$, then the value of $\frac{x}{1+x+x^2}$ is:
Option 1: $\frac{1}{5}$
Option 2: $\frac{1}{6}$
Option 3: $5$
Option 4: $6$
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Correct Answer: $\frac{1}{6}$
Solution : Given: $x+\frac{1}{x}=5$ Now, $\frac{x}{1+x+x^2}$ Divide the above expression by $x$ in numerator and denominator, ⇒ $\frac{1}{\frac{1}{x}+1+x}$ $=\frac{1}{x+\frac{1}{x}+1}$ $=\frac{1}{5+1}=\frac{1}{6}$ Hence, the correct answer is $\frac{1}{6}$.
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Question : If $x=2+\sqrt3$, then the value of $\frac{x^{2}-x+1}{x^{2}+x+1}$ is:
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Question : If $x+\frac{1}{x}=6$, then find the value of $\frac{3 x}{2 x^2-5 x+2}$.
Question : If $x-\frac{1}{x}=5, x \neq 0$, then what is the value of $\frac{x^6+3 x^3-1}{x^6-8 x^3-1} ?$
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