2025 May TZ1 P1 Q7

An object moves in a straight line.
Its velocity \(\mathrm{v}\) m s\(^{-1}\), at time \(t\) seconds, is given by \(\mathrm{v}(t) = 30 + 20t - 10t^2\) for \(0 \leq t \leq 5\).
The graph of \(\mathrm{v}\) is shown in the following diagram.

The graph of \(\mathrm{v}\) has a local maximum point where \(t = 1\) and intersects the \(t\)-axis at \(t = 3\).

1. Determine the object's

   [4]

   1. maximum velocity;

   2. maximum speed.

At \(t = T\), the object changes direction.

2. [5]

   1. Write down the value of \(T\).

   2. Find the distance travelled by the object in the first \(T\) seconds.

3. Determine whether the object returns to its initial position during the time period \(0 \leq t \leq 5\), justifying your answer.

   [4]