(a) The forces acting on the particle include:

• The gravitational force \(12g\) acting vertically downward.
• The normal reaction \(R\) perpendicular to the plane.
• The frictional force \(F\) acting up the plane.
• The force \(P\) acting parallel to the plane.

(b) To find the least possible value of \(P\), resolve the forces parallel and perpendicular to the plane:

Parallel to the plane:

\(P + F = 12g \sin 25^{\circ}\)

Perpendicular to the plane:

\(R = 12g \cos 25^{\circ}\)

The frictional force \(F\) is given by:

\(F = \mu R = 0.35 \times 12g \cos 25^{\circ}\)

Substitute \(F\) into the equation for forces parallel to the plane:

\(P + 0.35 \times 12g \cos 25^{\circ} = 12g \sin 25^{\circ}\)

Calculate \(F\):

\(F = 0.35 \times 12 \times 9.8 \times \cos 25^{\circ} \approx 38.1 \text{ N}\)

Substitute \(F\) back into the equation:

\(P + 38.1 = 12 \times 9.8 \times \sin 25^{\circ}\)

\(P + 38.1 = 50.7\)

\(P = 50.7 - 38.1 = 12.6 \text{ N}\)

Thus, the least possible value of \(P\) is \(12.6 \text{ N}\).