Step 1: The problem.
A railway track is banked, meaning the outer rail is lifted a little higher than the inner rail. The banking angle is $\theta = \tan^{-1}(1/20)$. The track is a metre gauge, so the two rails are $1$ m apart. We want how much the outer rail is raised.
Step 2: The geometry.
The two rails and the height between them make a right triangle. The rail spacing is the base (width $w = 1$ m) and the lift is the height $h$.
Step 3: Relate angle to height.
For the banking angle,
\[ \tan\theta = \frac{h}{w} \]
The angle is small, so this simple relation works well.
Step 4: List the values.
Width $w = 1$ m $= 100$ cm, and $\tan\theta = \dfrac{1}{20}$.
Step 5: Form the equation.
\[ \frac{1}{20} = \frac{h}{100\ \text{cm}} \]
Step 6: Solve for the lift.
Cross multiply:
\[ h = \frac{100}{20} = 5\ \text{cm} \]
So the outer rail is raised $5$ cm, which is option (4).
\[ \boxed{h = 5\ \text{cm}} \]