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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