2025 IGCSE Additional Mathematics May/June 0606/21 Q10

A particle \(P\) moves in a straight line.

\(t\) seconds after passing a fixed point, \(O\), the acceleration of \(P\), \(a\text{ m s}^{-2}\), is given by

\(a=t^2-2\quad\text{for}\hspace{0.5em}0\le t\le4\)

\(a=19-5\mathrm e^{8-2t}\quad\text{for}\hspace{0.5em}4\le t\le10\).

When \(t=3\), the velocity of \(P\) is \(-\dfrac13\text{ m s}^{-1}\) and its displacement from \(O\) is \(-\dfrac14\text{ m}\).

1. 1. Find the velocity of \(P\) when \(t=4\).[3]
   2. Find the displacement of \(P\) from \(O\) when \(t=4\).[3]
2. Find the displacement of \(P\) from \(O\) when \(t=10\).[5]