- A particle \(P\) is moving along the \(x\)-axis.
At time \(t\) seconds, \(t \geqslant 0 , P\) has acceleration \(a \mathrm {~ms} ^ { - 2 }\) and velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction of \(x\) increasing, where
$$v = \mathrm { e } ^ { 2 t } + 6 \mathrm { e } ^ { t } - k t$$
and \(k\) is a positive constant.
When \(t = \ln 2\), \(a = 0\)
- Find the value of \(k\).
When \(t = 0\), the particle passes through the fixed point \(A\).
When \(t = \ln 2\), the particle is \(d\) metres from \(A\). - Showing all stages of your working, find the value of \(d\) correct to 2 significant figures.
[0pt]
[Solutions relying entirely on calculator technology are not acceptable.]