(i) When the force is parallel to the plane, resolve the forces parallel to the plane. The component of the weight acting down the plane is given by:

\(W \sin 40^{\circ} = 12 \sin 40^{\circ}\)

Equating this to the force P, we have:

\(P = 12 \sin 40^{\circ}\)

Calculating gives:

\(P = 7.71 \text{ N}\)

(ii) When the force is horizontal, resolve the forces horizontally and vertically. The horizontal component of P is \(P \cos 40^{\circ}\) and the vertical component of the weight is \(W \sin 40^{\circ}\). Equating the horizontal component of P to the vertical component of the weight gives:

\(P \cos 40^{\circ} = 12 \sin 40^{\circ}\)

Solving for P gives:

\(P = \frac{12 \sin 40^{\circ}}{\cos 40^{\circ}}\)

Calculating gives:

\(P = 10.1 \text{ N}\)