(i)(a) The power developed by the engine is given by the formula:

\(P = Fv\)

where \(F = 850 \text{ N}\) and \(v = 42 \text{ m s}^{-1}\).

Thus, \(P = 850 \times 42 = 35700 \text{ W} = 35.7 \text{ kW}\).

(i)(b) The new power is \(35700 + 6000 = 41700 \text{ W}\).

The driving force \(DF\) is given by:

\(DF = \frac{41700}{42} = 993 \text{ N}\).

Using Newton's second law:

\(993 - 850 = 1200a\)

\(143 = 1200a\)

\(a = \frac{143}{1200} = 0.119 \text{ m s}^{-2}\).

(ii) The driving force \(DF\) is given by:

\(DF = \frac{80000}{24} = 3333.33 \text{ N}\).

Using Newton's second law along the incline:

\(DF - 850 - 1200g \sin \theta = 0\)

\(3333.33 - 850 = 1200 \times 9.8 \times \sin \theta\)

\(2483.33 = 11760 \sin \theta\)

\(\sin \theta = \frac{2483.33}{11760}\)

\(\theta = \sin^{-1} \left( \frac{2483.33}{11760} \right) \approx 11.9^\circ\).