Question : If $\left (4x+\frac{1}{x} \right)=5,x\neq 0,$ then the value of $\frac{5x}{4x^{2}+10x+1}$ is:
Option 1: $\frac{1}{2}$
Option 2: $\frac{1}{3}$
Option 3: $\frac{2}{3}$
Option 4: $3$
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Correct Answer: $\frac{1}{3}$
Solution : Given: $(4x+\frac{1}{x})=5$ ⇒ $\frac{4x^{2}+1}{x}=5$ ⇒ $4x^{2}+1=5x$ ⇒ $4x^{2}=5x-1$ Now, putting the value of $4x^{2}$ in $\frac{5x}{4x^{2}+10x+1}$ $=\frac{5x}{5x–1+10x+1}=\frac{5x}{15x}=\frac{1}{3}$ Hence, the correct answer is $\frac{1}{3}$.
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