| Exam Board | Edexcel |
|---|---|
| Module | P4 (Pure Mathematics 4) |
| Year | 2021 |
| Session | January |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Lines & Planes |
| Type | Triangle and parallelogram areas |
| Difficulty | Standard +0.3 This is a straightforward application of standard vector techniques: part (a) uses the scalar product formula to find an angle, and part (b) uses the cross product magnitude for parallelogram area. Both are routine procedures covered in any Further Maths vectors course with no problem-solving insight required, making it slightly easier than average. |
| Spec | 1.10b Vectors in 3D: i,j,k notation1.10c Magnitude and direction: of vectors1.10g Problem solving with vectors: in geometry |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\overrightarrow{BA}\cdot\overrightarrow{BC}=-6\times2+2\times5-3\times8=(-26)\) | M1 | Attempts scalar product of \(\pm\overrightarrow{AB}\cdot\pm\overrightarrow{BC}\); condone slips as long as intention is clear |
| Uses \(\overrightarrow{BA}\cdot\overrightarrow{BC}=\ | \overrightarrow{BA}\ | \ |
| \(\theta=112.65°\) | A1 | Versions finishing with \(\theta=\) awrt \(67.35°\) score M1 dM1 A0. Angles in radians score A0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Attempts \(\ | \overrightarrow{AB}\ | \ |
| Area \(=\) awrt \(62.3\) | A1 | If achieved from \(\theta=\) awrt \(67.35°\) full marks can be scored |
## Question 2:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\overrightarrow{BA}\cdot\overrightarrow{BC}=-6\times2+2\times5-3\times8=(-26)$ | M1 | Attempts scalar product of $\pm\overrightarrow{AB}\cdot\pm\overrightarrow{BC}$; condone slips as long as intention is clear |
| Uses $\overrightarrow{BA}\cdot\overrightarrow{BC}=\|\overrightarrow{BA}\|\|\overrightarrow{BC}\|\cos\theta \Rightarrow -26=\sqrt{49}\times\sqrt{93}\cos\theta\Rightarrow\theta=\ldots$ | dM1 | Must attempt to find correct angle. Expect at least one correct modulus calculation e.g. $\sqrt{2^2+5^2+8^2}=\sqrt{93}$ or $\sqrt{6^2+2^2+3^2}=7$ |
| $\theta=112.65°$ | A1 | Versions finishing with $\theta=$ awrt $67.35°$ score M1 dM1 A0. Angles in radians score A0 |
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### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Attempts $\|\overrightarrow{AB}\|\|\overrightarrow{BC}\|\sin\theta$ with their $\theta$ | M1 | Or attempts magnitude of vector product e.g. $\sqrt{3881}$ |
| Area $=$ awrt $62.3$ | A1 | If achieved from $\theta=$ awrt $67.35°$ full marks can be scored |
---2.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{216f5735-a7ad-4d70-9da9-ae1f098a97d9-04_511_506_264_721}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
Figure 1 shows a sketch of parallelogram $A B C D$.\\
Given that $\overrightarrow { A B } = 6 \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k }$ and $\overrightarrow { B C } = 2 \mathbf { i } + 5 \mathbf { j } + 8 \mathbf { k }$
\begin{enumerate}[label=(\alph*)]
\item find the size of angle $A B C$, giving your answer in degrees, to 2 decimal places.
\item Find the area of parallelogram $A B C D$, giving your answer to one decimal place.
\end{enumerate}
\hfill \mbox{\textit{Edexcel P4 2021 Q2 [5]}}