To solve for \(P\) and \(\theta\), we resolve the forces horizontally and vertically.

For horizontal equilibrium:

\(12\cos75^\circ + P\cos\theta^\circ = 18\cos65^\circ\)

For vertical equilibrium:

\(18\sin65^\circ + 12\sin75^\circ = 15 + P\sin\theta^\circ\)

We can solve these simultaneous equations by eliminating either \(\theta\) or \(P\).

Using the given mark scheme:

\(P^2 = (18\sin65^\circ + 12\sin75^\circ - 15)^2 + (18\cos65^\circ - 12\cos75^\circ)^2\)

or

\(\theta = \arctan\left(\frac{18\sin65^\circ + 12\sin75^\circ - 15}{18\cos65^\circ - 12\cos75^\circ}\right)\)

Solving these gives:

\(P = 13.7\) or \(\theta = 70.8\)