If 3x + 12x = 5 , then the value of 8x^3 + 127x^3\, is?

Answer: Option B

$$\begin{array}{rl}& 3x+\frac{1}{2x}=5\\ & \Rightarrow \text{Multiply both sides by }\frac{2}{3}\\ & \therefore 3x\times \frac{2}{3}+\frac{1}{2}x\times \frac{2}{3}=5\times \frac{2}{3}\\ & \Rightarrow 2x+\frac{1}{3x}=\frac{10}{3}\\ & \therefore \text{Taking cube on both sides}\\ & \Rightarrow \text{8}{x}^3\text{ + }\frac{1}{27x^3}+3.2x.\frac{1}{3x}\left(\text{ 2}x\text{ + }\frac{1}{3x}\right)={\left(\frac{10}{3}\right)}^3\\ & \Rightarrow \text{8}{x}^3\text{ + }\frac{1}{27x^3}+2\left(\frac{10}{3}\right)=\left(\frac{1000}{27}\right)\\ & \Rightarrow \text{8}{x}^3\text{ + }\frac{1}{27x^3}=\frac{1000}{27}-\frac{20}{3}\\ & \Rightarrow \text{8}{x}^3\text{ + }\frac{1}{27x^3}=\frac{1000-180}{27}\\ & \Rightarrow \text{8}{x}^3\text{ + }\frac{1}{27x^3}=\frac{820}{27}\\ & \Rightarrow \text{8}{x}^3\text{ + }\frac{1}{27x^3}=30\frac{10}{27}\end{array}$$