If a - 1a - 5 = 10 , then the value of ( a - 5 )^3 - 1 ( a - 5 )^3 is:

Answer: Option A

$$a-\frac{1}{a-5}=10$$
We can write it is as (subtracting 5 from both sides)
$$\begin{array}{rl}& a-5-\frac{1}{a-5}=10-5\\ & \text{Now, }\left(a-5\right)-\frac{1}{\left(a-5\right)}=5\end{array}$$
So, take the cube of this equation,
$$\begin{array}{rl}& \left[{\left(a-5\right)}^3-\frac{1}{{\left(a-5\right)}^3}\right]=125+3\times 5\\ & \left[{\left(a-5\right)}^3-\frac{1}{{\left(a-5\right)}^3}\right]=140\end{array}$$