| Exam Board | Edexcel |
|---|---|
| Module | M1 (Mechanics 1) |
| Year | 2013 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors Introduction & 2D |
| Type | When is one object due north/east/west/south of another |
| Difficulty | Standard +0.3 This is a standard M1 vectors question requiring velocity calculation from displacement, forming a position vector equation, and using bearing information to find a position. All steps are routine applications of basic vector concepts with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.10d Vector operations: addition and scalar multiplication1.10e Position vectors: and displacement3.02a Kinematics language: position, displacement, velocity, acceleration |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{(i - 4j) - (4i - 8j)}{0.5}, (\pm 6i \pm 8j)\) | M1 A1 | |
| \(\sqrt{(\pm 6)^2 + (\pm 8)^2} = 10\) | M1 A1 | (4) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\mathbf{r} = (4i - 8j) + t(-6i + 8j)\) | M1 | |
| \(= (4i - 8j) - 6ti + 8tj\) | ||
| \(= (4 - 6t)i + (8t - 8)j\) \* | A1 | (2) |
| Answer | Marks | Guidance |
|---|---|---|
| At 10 am, \(\mathbf{r} = -2i\) | M1 A1 | |
| At 10.30 am, \(\mathbf{r} = -5i + 4j\) | A1 | |
| \(l = ki\), \(k < -2\) | DM1 | |
| \(k = -5 - 4 = -9\) | A1 | (5) |
### Part (a)
$\frac{(i - 4j) - (4i - 8j)}{0.5}, (\pm 6i \pm 8j)$ | M1 A1 |
$\sqrt{(\pm 6)^2 + (\pm 8)^2} = 10$ | M1 A1 | (4)
### Part (b)
$\mathbf{r} = (4i - 8j) + t(-6i + 8j)$ | M1 |
$= (4i - 8j) - 6ti + 8tj$ | |
$= (4 - 6t)i + (8t - 8)j$ **\*** | A1 | (2)
### Part (c)
At 10 am, $\mathbf{r} = -2i$ | M1 A1 |
At 10.30 am, $\mathbf{r} = -5i + 4j$ | A1 |
$l = ki$, $k < -2$ | DM1 |
$k = -5 - 4 = -9$ | A1 | (5)
**Total for Question 6: 11**
---[In this question, $\mathbf{i}$ and $\mathbf{j}$ are horizontal unit vectors due east and due north respectively and position vectors are given with respect to a fixed origin.]
A ship sets sail at 9 am from a port $P$ and moves with constant velocity. The position vector of $P$ is $(4\mathbf{i} - 8\mathbf{j})$ km. At 9.30 am the ship is at the point with position vector $(\mathbf{i} - 4\mathbf{j})$ km.
\begin{enumerate}[label=(\alph*)]
\item Find the speed of the ship in km h$^{-1}$. [4]
\item Show that the position vector $\mathbf{r}$ km of the ship, $t$ hours after 9 am, is given by $\mathbf{r} = (4 - 6t)\mathbf{i} + (8t - 8)\mathbf{j}$. [2]
\end{enumerate}
At 10 am, a passenger on the ship observes that a lighthouse $L$ is due west of the ship. At 10.30 am, the passenger observes that $L$ is now south-west of the ship.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Find the position vector of $L$. [5]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M1 2013 Q6 [11]}}