(i) To find the car's gain in kinetic energy, we use the formula for kinetic energy gain:

\(\text{KE gain} = \frac{1}{2} m (v_{\text{top}}^2 - v_{\text{bottom}}^2)\)

First, find the velocities at the bottom and top using the power formula \(P = Fv\):

\(30000/v - 1000 - 1250g \times 30/500 = 1250a\)

At the bottom, \(a = 4 \text{ m/s}^2\):

\(v_{\text{bottom}} = \frac{30000}{1250 \times 4 + 1000 + 750}\)

At the top, \(a = 0.2 \text{ m/s}^2\):

\(v_{\text{top}} = \frac{30000}{1250 \times 0.2 + 1000 + 750}\)

Substitute these velocities into the kinetic energy formula to find the increase in KE:

\(\text{Increase in KE} = 128000 \text{ J}\)

(ii) To find the work done by the car's engine, consider the total work done:

\(\text{WD}_{\text{car}} = \text{KE gain} + \text{PE gain} + \text{WD against resistance}\)

Potential energy gain:

\(\text{PE gain} = 1250g \times 30\)

Work done against resistance:

\(\text{WD against resistance} = 1000 \times 500\)

Substitute these values:

\(\text{WD}_{\text{car}} = 128000 + 375000 + 500000 = 1000000 \text{ J}\)