Mathematics

The velocity \(\mathrm{v}\) ms\(^{-1}\) of a particle at time \(t\) seconds, is given by \(\mathrm{v} = 2t + \cos{2t}\), for \(0 \leq t \leq 2\).

1. Write down the velocity of the particle when \(t=0\).[1]

When \(t = k\), the acceleration is zero.

2. 1. Show that \(k = \frac{\pi}{4}\).
   2. Find the exact velocity when \(t = \frac{\pi}{4}\).[8]
3. When \(t < \frac{\pi}{4}\), \(\frac{\mathrm{dv}}{\mathrm{d}t}>0\) and when \(t > \frac{\pi}{4}, \frac{\mathrm{dv}}{\mathrm{d}t}>0\).
   Sketch a graph of \(\mathrm{v}\) against \(t\).[4]
4. Let \(d\) be the distance travelled by the particle for \(0 \leq t \leq 1\).
   1. Write down an expression for \(d\).
   2. Represent \(d\) on your sketch.[3]