If a - 1a - 3 = 5 , then the value of ( a - 3 )^3 - 1 ( a - 3 )^3 = ?

Answer: Option B

$$\begin{array}{rl}& a-\frac{1}{a-3}=5\\ & \Rightarrow a-3-\frac{1}{a-3}=5-3\\ & \Rightarrow \left(a-3\right)-\frac{1}{\left(a-3\right)}=2\\ & \text{Cubing both sides}\\ & \Rightarrow {\left[\left(a-3\right)-\frac{1}{\left(a-3\right)}\right]}^3={\left(2\right)}^3\end{array}$$
  $$\Rightarrow {\left(a-3\right)}^3-\frac{1}{{\left(a-3\right)}^3}-3\times \left(a-3\right)\times$$       $$\frac{1}{\left(a-3\right)}$$ $$\left[\left(a-3\right)-\frac{1}{\left(a-3\right)}\right]$$    $$=8$$
$$\begin{array}{rl}& \Rightarrow {\left(a-3\right)}^3-\frac{1}{{\left(a-3\right)}^3}-3\left(2\right)=8\\ & \Rightarrow {\left(a-3\right)}^3-\frac{1}{{\left(a-3\right)}^3}=8+6\\ & \Rightarrow {\left(a-3\right)}^3-\frac{1}{{\left(a-3\right)}^3}=14\end{array}$$