2025 IGCSE Additional Mathematics October/November 0606/21 Q8

In this question, the units are metres and seconds.

A particle \(P\) is travelling in a straight line through a fixed point \(O\).

At time \(t\) its acceleration, \(a\), is given by \(a=(2t-3)^2\), where \(t\ge0\).

When \(t=3\), \(P\) has a velocity of 6.

1. 1. Find an expression for the velocity, \(v\), of \(P\) at time \(t\).[3]
   2. Find the time when \(P\) is at rest.[3]
      When \(t=\dfrac52\), the displacement of \(P\) from \(O\) is 4.
   3. Find the displacement of \(P\) from \(O\) when \(t=3\).[4]
2. Use calculus to find the approximate change in \(v\) when \(t\) increases from \(\dfrac52\) by the small amount 0.02.[3]