(a) To find the speed of P when it has travelled halfway, we first calculate the potential energy (PE) lost. The initial height of P is given by the vertical component of the string length:

\(h = 4 \cos 45^{\circ}\)

When P is halfway, the height is:

\(h = 4 \cos 22.5^{\circ}\)

The PE lost is:

\(\text{PE lost} = 35g(4 \cos 22.5^{\circ} - 4 \cos 45^{\circ})\)

This PE lost is converted into kinetic energy (KE):

\(\frac{1}{2} \times 35 \times v^2 = 35g(4 \cos 22.5^{\circ} - 4 \cos 45^{\circ})\)

Solving for \(v\), we find:

\(v = 4.16 \text{ m/s}\)

(b) Using the work-energy principle, the equation is:

\(\frac{1}{2} \times 35 \times 4^2 = 35g(4 - 4 \cos 45^{\circ}) - X\)

Given the speed at the lowest point is 4 m/s, the KE at the lowest point is:

\(\frac{1}{2} \times 35 \times 4^2\)

Solving for \(X\), we find:

\(X = 130 \text{ J}\)