(i) The change in gravitational potential energy (PE) is given by the formula:

\(\text{PE gain} = mgh\)

where \(m = 8 \text{ kg}\), \(g = 9.8 \text{ m/s}^2\), and \(h = 20 \sin 30^{\circ}\).

Calculate \(h\):

\(h = 20 \sin 30^{\circ} = 20 \times 0.5 = 10 \text{ m}\)

Substitute into the PE formula:

\(\text{PE gain} = 8 \times 9.8 \times 10 = 784 \text{ J}\)

According to the mark scheme, the change in PE is 800 J.

(ii) The total work done against gravity and friction is given by:

\(8g \times 20 \sin 30^{\circ} + 20F = 1146\)

Substitute \(g = 9.8 \text{ m/s}^2\):

\(8 \times 9.8 \times 10 + 20F = 1146\)

Simplify:

\(784 + 20F = 1146\)

Rearrange to solve for \(F\):

\(20F = 1146 - 784\)

\(20F = 362\)

\(F = \frac{362}{20} = 18.1 \text{ N}\)

According to the mark scheme, the frictional force is 17.3 N.