| Exam Board | Edexcel |
|---|---|
| Module | M1 (Mechanics 1) |
| Year | 2004 |
| Session | January |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors Introduction & 2D |
| Type | When is one object due north/east/west/south of another |
| Difficulty | Moderate -0.3 This is a standard M1 mechanics question on relative velocity and position vectors. Part (a) requires basic trigonometry (arctan), parts (b-c) involve routine vector arithmetic and setting up position equations, part (d) is algebraic manipulation to reach a given result, and part (e) solves a quadratic equation. All techniques are textbook exercises with clear scaffolding and no novel insight required, making it slightly easier than average. |
| Spec | 1.10a Vectors in 2D: i,j notation and column vectors1.10b Vectors in 3D: i,j,k notation1.10c Magnitude and direction: of vectors1.10d Vector operations: addition and scalar multiplication1.10e Position vectors: and displacement1.10f Distance between points: using position vectors1.10g Problem solving with vectors: in geometry |
| Answer | Marks | Guidance |
|---|---|---|
| - \(d^2 = | \mathbf{b}-\mathbf{a} | ^2 = (3t-10)^2 + 16t^2 = 25t^2 - 60t + 100\) (*) |
## Part (a)
**Answer/Working:** $\tan \theta = \frac{3}{5} \Rightarrow \theta = 031°$
**Marks:** M1, A1 (2 marks)
---
## Part (b)
**Answer/Working:** $\mathbf{a} = 9t \mathbf{j}$
**Marks:** B1 (1 mark)
---
## Part (c)
**Answer/Working:**
- $B$ south of $A \Rightarrow -10 + 3t = 0$
- $t = 3\frac{1}{3} \Rightarrow 1520 \text{ hours}$
**Marks:** M1, A1 (2 marks)
---
## Part (d)
**Answer/Working:**
- $\mathbf{AB} = \mathbf{b} - \mathbf{a} = (3t-10)\mathbf{i} + 5t \mathbf{j}$
- $d^2 = |\mathbf{b}-\mathbf{a}|^2 = (3t-10)^2 + 16t^2 = 25t^2 - 60t + 100$ (*)
**Marks:** M1, A1, M1, A1 (4 marks)
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## Part (e)
**Answer/Working:**
- $d = 10 \Rightarrow d^2 = 100 \Rightarrow 25t^2 - 60t = 0$
- $\Rightarrow t = (0 \text{ or } 2.4)$
- $\Rightarrow \text{time } 1424 \text{ hours}$
**Marks:** M1, A1, A1 (3 marks)[In this question the vectors $\mathbf{i}$ and $\mathbf{j}$ are horizontal unit vectors in the direction due east and due north respectively.]
Two boats $A$ and $B$ are moving with constant velocities. Boat $A$ moves with velocity $9\mathbf{j}$ km h$^{-1}$. Boat $B$ moves with velocity $(3\mathbf{i} + 5\mathbf{j})$ km h$^{-1}$.
\begin{enumerate}[label=(\alph*)]
\item Find the bearing on which $B$ is moving. [2]
\end{enumerate}
At noon, $A$ is at point $O$, and $B$ is 10 km due west of $O$. At time $t$ hours after noon, the position vectors of $A$ and $B$ relative to $O$ are $\mathbf{a}$ km and $\mathbf{b}$ km respectively.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find expressions for $\mathbf{a}$ and $\mathbf{b}$ in terms of $t$, giving your answer in the form $p\mathbf{i} + q\mathbf{j}$. [3]
\item Find the time when $B$ is due south of $A$. [2]
\end{enumerate}
At time $t$ hours after noon, the distance between $A$ and $B$ is $d$ km. By finding an expression for $\overrightarrow{AB}$,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item show that $d^2 = 25t^2 - 60t + 100$. [4]
\end{enumerate}
At noon, the boats are 10 km apart.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{4}
\item Find the time after noon at which the boats are again 10 km apart. [3]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M1 2004 Q7 [14]}}