(i) Differentiate the curve equation \(y = 1 - 2x - (1 - 2x)^3\) with respect to \(x\):

\(\frac{dy}{dx} = [-2] - [3(1-2x)^2 \times (-2)] = -2 + 6(1-2x)^2\).

At \(x = \frac{1}{2}\), \(\frac{dy}{dx} = -2\).

The gradient of the line \(y = 1 - 2x\) is \(-2\), hence \(AB\) is a tangent.

(ii) The shaded region is given by:

\(\int_0^{\frac{1}{2}} (1 - 2x) \, dx - \int_0^{\frac{1}{2}} (1 - 2x - (1 - 2x)^3) \, dx\).

This simplifies to \(\int_0^{\frac{1}{2}} (1 - 2x)^3 \, dx\).

(iii) Evaluate the integral:

\(\int_0^{\frac{1}{2}} (1 - 2x)^3 \, dx = \left[ \frac{(1-2x)^4}{-8} \right]_0^{\frac{1}{2}}\).

Calculate the limits:

\(0 - \left(-\frac{1}{8}\right) = \frac{1}{8}\).