(a)
Rest condition: \( 3 \sin(a)(1 + \cos(a)) = 0 \)
Solve: \( \sin(a) = 0 \implies a = 0, \pi, 2\pi, \ldots \)
Or: \( 1 + \cos(a) = 0 \implies a = \pi, 3\pi, \ldots \)
Next rest after \( t = 0 \): \( a = \pi \)
Verify: \( \sin(\pi) = 0 \implies v = 0 \)
Marks: (M1) for equation, (A1) for solutions, (A1) for \( a = \pi \)
Result: \( a = \pi \approx 3.14 \) [3]

(b)
Velocity: \( v = 3 \sin(t)(1 + \cos(t)) \), interval \( 0 \leq t \leq \pi \)
Derivative: \( \frac{dv}{dt} = 3 [\cos(t)(1 + \cos(t)) – \sin(t) \sin(t)] \)
Simplify: \( \frac{dv}{dt} = 3 [\cos(t) + \cos(2t)] \)
Set \( \frac{dv}{dt} = 0 \): \( \cos(t) + \cos(2t) = 0 \)
Solve: \( 2 \cos^2(t) + \cos(t) – 1 = 0 \)
Roots: \( \cos(t) = \frac{1}{2} \implies t = \frac{\pi}{3} \), or \( \cos(t) = -1 \implies t = \pi \)
Evaluate: At \( t = \frac{\pi}{3} \), \( v = \frac{9 \sqrt{3}}{4} \approx 3.897 \)
At endpoints: \( v(0) = 0 \), \( v(\pi) = 0 \)
Marks: (M1) for derivative, (A1) for critical points, (A1) for maximum
Result: Maximum velocity = \( \frac{9 \sqrt{3}}{4} \approx 3.90 \, \text{ms}^{-1} \) [3]

(c)
Distance: \( \int_0^\pi 3 \sin(t)(1 + \cos(t)) \, dt \)
Substitute: \( u = 1 + \cos(t) \), \( du = -\sin(t) \, dt \)
Limits: \( t = 0 \), \( u = 2 \); \( t = \pi \), \( u = 0 \)
Integral: \( \int_0^2 3u \, du = 3 \cdot 2 = 6 \)
Average speed: \( \frac{6}{\pi} \approx 1.91 \)
Marks: (M1) for integral, (A1) for substitution, (A1) for average speed
Result: Average speed = \( \frac{6}{\pi} \approx 1.91 \, \text{ms}^{-1} \) [3]