| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2024 |
| Session | March |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Lines & Planes |
| Type | Triangle and parallelogram areas |
| Difficulty | Standard +0.3 This is a straightforward vector geometry question requiring standard techniques: computing vectors AB, BC, OC to verify perpendicularity and equal opposite sides for the rectangle, then finding diagonals OB and AC and using the scalar product formula for the angle between them. All steps are routine applications of A-level vector methods with no novel 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 | Marks | Guidance |
| Find the scalar product of a pair of adjacent sides | M1 | \(\overrightarrow{OA} = (5,-2,1)\), \(\overrightarrow{OB} = (8,2,-6)\), \(\overrightarrow{OC} = (3,4,-7)\), \(\overrightarrow{CB} = (5,-2,1)\), \(\overrightarrow{AB} = (3,4,-7)\) |
| Show that the sides are perpendicular | A1 | e.g. \(\overrightarrow{OA}.\overrightarrow{OC} = 15 - 8 - 7 = 0\). Need to see working of numerator, ignore denominator. |
| Compare a pair of opposite sides | M1 | \(\overrightarrow{OA}\) and \(\overrightarrow{CB}\) or \(\overrightarrow{OC}\) and \(\overrightarrow{AB}\) |
| Show that they are parallel and equal in length and hence \(OABC\) is a rectangle | A1 | e.g. \(\overrightarrow{AB} = \overrightarrow{AO} + \overrightarrow{OB} = 3\mathbf{i} + 4\mathbf{j} - 7\mathbf{k} = \overrightarrow{OC}\). If show \(\overrightarrow{AB} = 3\mathbf{i} + 4\mathbf{j} - 7\mathbf{k} = \overrightarrow{OC}\), then M1 A1 since this implies parallel and of equal length. If only show lengths equal M1. If repeat for other pair of opposite sides then A1. |
| Total | 4 | Without calculation of scalar product max is M1 A1. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Show the diagonals \(\overrightarrow{OB}\) and \(\overrightarrow{AC}\) are equal in length \((\sqrt{104})\) | \(\overrightarrow{AC} = (-2, 6, -8)\) | |
| Show the diagonals bisect each other at \((4, 1, -3)\) | \(\frac{\overrightarrow{OB}}{2} = \overrightarrow{OC} + \frac{1}{2}(\overrightarrow{OA} - \overrightarrow{OC}) = (4,1,-3)\) | |
| Show the quadrilateral is a parallelogram | e.g. \(\overrightarrow{OB} = \overrightarrow{OA} + \overrightarrow{OC}\) | |
| Show both pairs of opposite sides are equal in length and a pair of adjacent sides are perpendicular |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\overrightarrow{AC} = \pm(-2\mathbf{i} + 6\mathbf{j} - 8\mathbf{k})\) or \(\frac{\overrightarrow{AC}}{2} = \pm(-1\mathbf{i} + 3\mathbf{j} - 4\mathbf{k})\) | B1 | Seen or implied using diagonals. |
| Scalar product of a pair of relevant vectors | M1 | e.g. \(\overrightarrow{AC}.\overrightarrow{OB} = -16 + 12 + 48\) |
| Using the correct process for the moduli, divide the scalar product by the product of the moduli and obtain the inverse cosine of the result | M1 | \(\pm\cos^{-1}\left(\frac{44}{104}\right)\). For any two vectors. |
| Obtain answer \(65.(0)°\) | A1 | Accept \(1.13\) radians. |
| Total | 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Scalar product of a pair of relevant vectors | M1 | e.g. \(\overrightarrow{OA}.\overrightarrow{OB} = 40 - 4 - 6\) using one side and a diagonal, or \(\overrightarrow{OC}.\overrightarrow{OB} = 24 + 8 + 42\). Must use scalar product. |
| Using the correct process for the moduli, divide the scalar product by the product of the moduli and obtain the inverse cosine of the result. Any two vectors. | M1 | \(\pm\cos^{-1}\left(\frac{\sqrt{30}}{\sqrt{104}}\right)\) or \(\cos^{-1}\left(\frac{\sqrt{74}}{\sqrt{104}}\right)\) |
| Required angle \(= 180° - 2 \times 57.5°\) or \(180° - 2 \times 32.5° = 115°\) and \(180° - 115°\) or \(2 \times 32.5°\) | B1 | OE SOI. Complete method to find the acute angle. |
| Obtain answer \(65.0°\) | A1 | Accept \(1.13\) radians. |
| Total | 4 |
## Question 9:
### Part 9(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Find the scalar product of a pair of adjacent sides | M1 | $\overrightarrow{OA} = (5,-2,1)$, $\overrightarrow{OB} = (8,2,-6)$, $\overrightarrow{OC} = (3,4,-7)$, $\overrightarrow{CB} = (5,-2,1)$, $\overrightarrow{AB} = (3,4,-7)$ |
| Show that the sides are perpendicular | A1 | e.g. $\overrightarrow{OA}.\overrightarrow{OC} = 15 - 8 - 7 = 0$. Need to see working of numerator, ignore denominator. |
| Compare a pair of opposite sides | M1 | $\overrightarrow{OA}$ and $\overrightarrow{CB}$ or $\overrightarrow{OC}$ and $\overrightarrow{AB}$ |
| Show that they are parallel and equal in length and hence $OABC$ is a rectangle | A1 | e.g. $\overrightarrow{AB} = \overrightarrow{AO} + \overrightarrow{OB} = 3\mathbf{i} + 4\mathbf{j} - 7\mathbf{k} = \overrightarrow{OC}$. If show $\overrightarrow{AB} = 3\mathbf{i} + 4\mathbf{j} - 7\mathbf{k} = \overrightarrow{OC}$, then M1 A1 since this implies parallel and of equal length. If only show lengths equal M1. If repeat for other pair of opposite sides then A1. |
| **Total** | **4** | Without calculation of scalar product max is M1 A1. |
**Alternative solution for 9(a):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Show the diagonals $\overrightarrow{OB}$ and $\overrightarrow{AC}$ are equal in length $(\sqrt{104})$ | | $\overrightarrow{AC} = (-2, 6, -8)$ |
| Show the diagonals bisect each other at $(4, 1, -3)$ | | $\frac{\overrightarrow{OB}}{2} = \overrightarrow{OC} + \frac{1}{2}(\overrightarrow{OA} - \overrightarrow{OC}) = (4,1,-3)$ |
| Show the quadrilateral is a parallelogram | | e.g. $\overrightarrow{OB} = \overrightarrow{OA} + \overrightarrow{OC}$ |
| Show both pairs of opposite sides are equal in length and a pair of adjacent sides are perpendicular | | |
### Part 9(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\overrightarrow{AC} = \pm(-2\mathbf{i} + 6\mathbf{j} - 8\mathbf{k})$ or $\frac{\overrightarrow{AC}}{2} = \pm(-1\mathbf{i} + 3\mathbf{j} - 4\mathbf{k})$ | B1 | Seen or implied using diagonals. |
| Scalar product of a pair of relevant vectors | M1 | e.g. $\overrightarrow{AC}.\overrightarrow{OB} = -16 + 12 + 48$ |
| Using the correct process for the moduli, divide the scalar product by the product of the moduli and obtain the inverse cosine of the result | M1 | $\pm\cos^{-1}\left(\frac{44}{104}\right)$. For any two vectors. |
| Obtain answer $65.(0)°$ | A1 | Accept $1.13$ radians. |
| **Total** | **4** | |
**Alternative solution for 9(b):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Scalar product of a pair of relevant vectors | M1 | e.g. $\overrightarrow{OA}.\overrightarrow{OB} = 40 - 4 - 6$ using one side and a diagonal, or $\overrightarrow{OC}.\overrightarrow{OB} = 24 + 8 + 42$. Must use scalar product. |
| Using the correct process for the moduli, divide the scalar product by the product of the moduli and obtain the inverse cosine of the result. Any two vectors. | M1 | $\pm\cos^{-1}\left(\frac{\sqrt{30}}{\sqrt{104}}\right)$ or $\cos^{-1}\left(\frac{\sqrt{74}}{\sqrt{104}}\right)$ |
| Required angle $= 180° - 2 \times 57.5°$ or $180° - 2 \times 32.5° = 115°$ and $180° - 115°$ or $2 \times 32.5°$ | B1 | OE SOI. Complete method to find the acute angle. |
| Obtain answer $65.0°$ | A1 | Accept $1.13$ radians. |
| **Total** | **4** | |
---9 Relative to the origin $O$, the position vectors of the points $A , B$ and $C$ are given by
$$\overrightarrow { \mathrm { OA } } = 5 \mathbf { i } - 2 \mathbf { j } + \mathbf { k } , \quad \overrightarrow { \mathrm { OB } } = 8 \mathbf { i } + 2 \mathbf { j } - 6 \mathbf { k } \quad \text { and } \quad \overrightarrow { \mathrm { OC } } = 3 \mathbf { i } + 4 \mathbf { j } - 7 \mathbf { k }$$
\begin{enumerate}[label=(\alph*)]
\item Show that $O A B C$ is a rectangle.\\
\includegraphics[max width=\textwidth, alt={}, center]{446573d3-73b1-482a-a3f6-1abddfdd90d0-14_67_1573_557_324}\\
\includegraphics[max width=\textwidth, alt={}, center]{446573d3-73b1-482a-a3f6-1abddfdd90d0-14_68_1575_648_322}\\
\includegraphics[max width=\textwidth, alt={}]{446573d3-73b1-482a-a3f6-1abddfdd90d0-14_70_1573_737_324} ....................................................................................................................................... .........................................................................................................................................
\item Use a scalar product to find the acute angle between the diagonals of $O A B C$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2024 Q9 [8]}}