| Exam Board | Edexcel |
|---|---|
| Module | Paper 2 (Paper 2) |
| Year | 2023 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors Introduction & 2D |
| Type | Geometric properties using vectors |
| Difficulty | Moderate -0.8 This is a straightforward vectors question requiring basic vector subtraction to show parallelism (part a) and simple distance/speed calculations (part b). Both parts use routine techniques with no problem-solving insight needed, making it easier than average but not trivial due to the multi-step calculation in part (b). |
| Spec | 1.10a Vectors in 2D: i,j notation and column vectors1.10b Vectors in 3D: i,j,k notation1.10e Position vectors: and displacement1.10f Distance between points: using position vectors |
| Answer | Marks | Guidance |
|---|---|---|
| \(\overrightarrow{AD} = 10\mathbf{i} + 24\mathbf{j}\) and \(\overrightarrow{BC} = 50\mathbf{i} + 120\mathbf{j}\) | M1 | Attempts to find both \(\overrightarrow{AD} = \pm(10\mathbf{i}+24\mathbf{j})\) and \(\overrightarrow{BC} = \pm(50\mathbf{i}+120\mathbf{j})\). At least one correct component per vector if no method shown |
| \(\overrightarrow{AD} = \frac{1}{5}\overrightarrow{BC}\) therefore \(AD\) is parallel to \(BC\ ^*\) | A1\*cso | Correct work showing parallel e.g. \(\overrightarrow{AD} = \pm\frac{1}{5}\overrightarrow{BC}\) or equal ratios/gradients, plus a minimal conclusion |
| Answer | Marks | Guidance |
|---|---|---|
| Attempt to find at least two lengths from \(AB\), \(BC\), \(CD\), \(AD\): \(\lvert\overrightarrow{BC}\rvert = \sqrt{50^2+120^2} = 130\), \(\lvert\overrightarrow{DA}\rvert = \sqrt{10^2+24^2} = 26\) | M1 | Attempts Pythagoras for at least two lengths of the quadrilateral |
| \(\lvert\overrightarrow{AB}\rvert = \sqrt{12^2+16^2} = 20\), \(\lvert\overrightarrow{CD}\rvert = \sqrt{28^2+112^2} = 28\sqrt{17}\) (awrt 115 m) | A1 | At least 2 correct lengths |
| Average speed \(= \dfrac{2(26+130+20+28\sqrt{17})}{1000} \div \dfrac{5}{60}\) | dM1 | Attempt at average speed ignoring units; must have attempted all 4 lengths; indication of dividing by 5; depends on M1 |
| awrt \(= 6.99\) (km/h) | A1 | awrt 6.99 (km/h). Units not required but if given must be correct. Exact answer: \(4.224 + 0.672\sqrt{17}\) |
## Question 6:
### Part (a):
| $\overrightarrow{AD} = 10\mathbf{i} + 24\mathbf{j}$ and $\overrightarrow{BC} = 50\mathbf{i} + 120\mathbf{j}$ | **M1** | Attempts to find both $\overrightarrow{AD} = \pm(10\mathbf{i}+24\mathbf{j})$ and $\overrightarrow{BC} = \pm(50\mathbf{i}+120\mathbf{j})$. At least one correct component per vector if no method shown |
| $\overrightarrow{AD} = \frac{1}{5}\overrightarrow{BC}$ therefore $AD$ is parallel to $BC\ ^*$ | **A1\*cso** | Correct work showing parallel e.g. $\overrightarrow{AD} = \pm\frac{1}{5}\overrightarrow{BC}$ or equal ratios/gradients, plus a minimal conclusion |
### Part (b):
| Attempt to find at least two lengths from $AB$, $BC$, $CD$, $AD$: $\lvert\overrightarrow{BC}\rvert = \sqrt{50^2+120^2} = 130$, $\lvert\overrightarrow{DA}\rvert = \sqrt{10^2+24^2} = 26$ | **M1** | Attempts Pythagoras for at least two lengths of the quadrilateral |
| $\lvert\overrightarrow{AB}\rvert = \sqrt{12^2+16^2} = 20$, $\lvert\overrightarrow{CD}\rvert = \sqrt{28^2+112^2} = 28\sqrt{17}$ (awrt 115 m) | **A1** | At least 2 correct lengths |
| Average speed $= \dfrac{2(26+130+20+28\sqrt{17})}{1000} \div \dfrac{5}{60}$ | **dM1** | Attempt at average speed ignoring units; must have attempted all 4 lengths; indication of dividing by 5; depends on M1 |
| awrt $= 6.99$ (km/h) | **A1** | awrt 6.99 (km/h). Units not required but if given must be correct. Exact answer: $4.224 + 0.672\sqrt{17}$ |\begin{enumerate}
\item Relative to a fixed origin $O$,
\end{enumerate}
\begin{itemize}
\item $A$ is the point with position vector $12 \mathbf { i }$
\item $B$ is the point with position vector $16 \mathbf { j }$
\item $C$ is the point with position vector $( 50 \mathbf { i } + 136 \mathbf { j } )$
\item $D$ is the point with position vector $( 22 \mathbf { i } + 24 \mathbf { j } )$\\
(a) Show that $A D$ is parallel to $B C$.
\end{itemize}
Points $A , B , C$ and $D$ are used to model the vertices of a running track in the shape of a quadrilateral.
Runners complete one lap by running along all four sides of the track.\\
The lengths of the sides are measured in metres.
Given that a particular runner takes exactly 5 minutes to complete 2 laps,\\
(b) calculate the average speed of this runner, giving the answer in kilometres per hour.
\hfill \mbox{\textit{Edexcel Paper 2 2023 Q6 [6]}}