| Exam Board | Edexcel |
|---|---|
| Module | Paper 1 (Paper 1) |
| Year | 2022 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vector Product and Surfaces |
| Type | Area of parallelogram using vector product |
| Difficulty | Standard +0.3 This is a straightforward Further Maths vector product question requiring (a) showing two adjacent sides have equal magnitude (simple calculation) and (b) computing area via |a × b|. Both parts are routine applications of standard techniques with no conceptual challenges, making it slightly easier than average even for Further Maths. |
| Spec | 1.10c Magnitude and direction: of vectors1.10g Problem solving with vectors: in geometry |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Attempts both \(\ | \overrightarrow{PQ}\ | =\sqrt{2^2+3^2+(-4)^2}\) and \(\ |
| States \(\ | \overrightarrow{PQ}\ | =\ |
| Total: (2) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Attempts BOTH \(\overrightarrow{PR}=\overrightarrow{PQ}+\overrightarrow{QR}=7\mathbf{i}+3\mathbf{j}-6\mathbf{k}\) AND \(\overrightarrow{QS}=-\overrightarrow{PQ}+\overrightarrow{PS}=3\mathbf{i}-3\mathbf{j}+2\mathbf{k}\) | M1 | For attempting to find both key vectors |
| Correct \(\overrightarrow{PR}=7\mathbf{i}+3\mathbf{j}-6\mathbf{k}\) and \(\overrightarrow{QS}=3\mathbf{i}-3\mathbf{j}+2\mathbf{k}\) | A1 | Allow either way around |
| Correct method for area \(PQRS\), e.g. \(\frac{1}{2}\times\ | \overrightarrow{PR}\ | \times\ |
| \(=\sqrt{517}\) | A1 | |
| Total: (4) |
## Question 9:
### Part (a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Attempts both $\|\overrightarrow{PQ}\|=\sqrt{2^2+3^2+(-4)^2}$ and $\|\overrightarrow{QR}\|=\sqrt{5^2+(-2)^2}$ | M1 | Also accept attempts at $\overrightarrow{PR}\bullet\overrightarrow{QS}$ or $PM^2, MQ^2$ and $PQ^2$ where $M$ is midpoint of $PR$ |
| States $\|\overrightarrow{PQ}\|=\|\overrightarrow{QR}\|=\sqrt{29}$ so $PQRS$ is a rhombus | A1 | Must show calculations and state $PQRS$ is a rhombus; requires reason and conclusion |
| **Total: (2)** | | |
### Part (b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Attempts BOTH $\overrightarrow{PR}=\overrightarrow{PQ}+\overrightarrow{QR}=7\mathbf{i}+3\mathbf{j}-6\mathbf{k}$ AND $\overrightarrow{QS}=-\overrightarrow{PQ}+\overrightarrow{PS}=3\mathbf{i}-3\mathbf{j}+2\mathbf{k}$ | M1 | For attempting to find both key vectors |
| Correct $\overrightarrow{PR}=7\mathbf{i}+3\mathbf{j}-6\mathbf{k}$ and $\overrightarrow{QS}=3\mathbf{i}-3\mathbf{j}+2\mathbf{k}$ | A1 | Allow either way around |
| Correct method for area $PQRS$, e.g. $\frac{1}{2}\times\|\overrightarrow{PR}\|\times\|\overrightarrow{QS}\|$ | dM1 | Dependent on previous M; constructs rigorous method for area |
| $=\sqrt{517}$ | A1 | |
| **Total: (4)** | | |
---9.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{129adfbb-98fa-4e88-b636-7b4d111f3349-20_406_515_246_776}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{center}
\end{figure}
Figure 3 shows a sketch of a parallelogram $P Q R S$.\\
Given that
\begin{itemize}
\item $\overrightarrow { P Q } = 2 \mathbf { i } + 3 \mathbf { j } - 4 \mathbf { k }$
\item $\overrightarrow { Q R } = 5 \mathbf { i } - 2 \mathbf { k }$
\begin{enumerate}[label=(\alph*)]
\item show that parallelogram $P Q R S$ is a rhombus.
\item Find the exact area of the rhombus $P Q R S$.
\end{itemize}
\end{enumerate}
\hfill \mbox{\textit{Edexcel Paper 1 2022 Q9 [6]}}