2025 IGCSE Additional Mathematics May/June 0606/13 Q11

A particle \(P\) moves in a straight line and passes through a fixed point \(O\).

At time \(t\) seconds, its displacement from \(O\), \(s\) metres, is given by

\(s=t+6t^2-t^3\quad\text{for}\hspace{0.5em}0\le t\le3\)

\(s=12t-\dfrac13t^2-3\quad\text{for}\hspace{0.5em}3\le t\le k\), where \(k\) is a constant.

It is given that, for \(3\le t\le k\), the velocity of \(P\) is positive and its acceleration is negative.

1. The maximum velocity of \(P\) occurs when \(t=2\).
   On the axes below, sketch a velocity–time graph for the first \(k\) seconds of the motion of \(P\).[4]
2. The total distance travelled by \(P\) for \(0\le t\le k\) is \(57\) metres.
   Given that when \(t=3\) the distance and displacement of \(P\) from \(O\) are equal, find the value of \(k\).[6]