Challenging +1.2 This is a first-order linear differential equation in vector form requiring integration factor method (e^{-2t}) and application of initial conditions. While it involves multiple steps (recognizing the DE type, finding integrating factor, integrating, applying boundary conditions, solving for t), the techniques are standard M5 material with no novel insight required. The 7 marks reflect computational length rather than conceptual difficulty.
At time \(t = 0\), the position vector of a particle \(P\) is \(-3j\) m. At time \(t\) seconds, the position vector of \(P\) is \(\mathbf{r}\) metres and the velocity of \(P\) is \(\mathbf{v}\) m s\(^{-1}\). Given that
$$\mathbf{v} - 2\mathbf{r} = 4e^t \mathbf{j},$$
find the time when \(P\) passes through the origin.
[7]
At time $t = 0$, the position vector of a particle $P$ is $-3j$ m. At time $t$ seconds, the position vector of $P$ is $\mathbf{r}$ metres and the velocity of $P$ is $\mathbf{v}$ m s$^{-1}$. Given that
$$\mathbf{v} - 2\mathbf{r} = 4e^t \mathbf{j},$$
find the time when $P$ passes through the origin.
[7]
\hfill \mbox{\textit{Edexcel M5 Q1 [7]}}