| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2019 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors 3D & Lines |
| Type | Vector geometry in 3D shapes |
| Difficulty | Standard +0.3 This is a straightforward 3D vector question requiring standard techniques: finding position vectors by addition/subtraction, calculating magnitudes for distance comparison, and using scalar product formula for angles. While it involves multiple steps and 3D visualization, all methods are routine A-level procedures with no novel problem-solving 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 multiplication1.10f Distance between points: using position vectors1.10g Problem solving with vectors: in geometry |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\overrightarrow{PB} = 5\mathbf{i} + 8\mathbf{j} - 5\mathbf{k}\) | B2,1,0 | B2 all correct, B1 for two correct components |
| \(\overrightarrow{PQ} = 4\mathbf{i} + 8\mathbf{j} + 5\mathbf{k}\) | B2,1,0 | B2 all correct, B1 for two correct components. Accept column vectors. SC B1 for each vector if all components multiplied by \(-1\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Length of \(PB = \sqrt{5^2 + 8^2 + 5^2} = (\sqrt{114} \approx 10.7)\); Length of \(PQ = \sqrt{4^2 + 8^2 + 5^2} = (\sqrt{105} \approx 10.2)\) | M1 | Evaluation of both lengths. Other valid complete comparisons can be accepted |
| \(P\) is nearer to \(Q\) | A1 | WWW |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \((\overrightarrow{PB} \cdot \overrightarrow{PQ}) = 20 + 64 - 25\) | M1 | Use of \(x_1x_2 + y_1y_2 + z_1z_2\) on their \(\overrightarrow{PB}\) and \(\overrightarrow{PQ}\) |
| \(({\rm their}\ \sqrt{114})({\rm their}\ \sqrt{105})\cos BPQ = ({\rm their}\ 59)\) | M1 | All elements present and in correct places |
| \(BPQ = 57.4(°)\) or \(1.00\) (rad) | A1 | AWRT. Calculating the obtuse angle and then subtracting gets A0 |
## Question 7(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\overrightarrow{PB} = 5\mathbf{i} + 8\mathbf{j} - 5\mathbf{k}$ | B2,1,0 | B2 all correct, B1 for two correct components |
| $\overrightarrow{PQ} = 4\mathbf{i} + 8\mathbf{j} + 5\mathbf{k}$ | B2,1,0 | B2 all correct, B1 for two correct components. Accept column vectors. **SC** B1 for each vector if all components multiplied by $-1$ |
---
## Question 7(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Length of $PB = \sqrt{5^2 + 8^2 + 5^2} = (\sqrt{114} \approx 10.7)$; Length of $PQ = \sqrt{4^2 + 8^2 + 5^2} = (\sqrt{105} \approx 10.2)$ | M1 | Evaluation of both lengths. Other valid complete comparisons can be accepted |
| $P$ is nearer to $Q$ | A1 | WWW |
---
## Question 7(iii):
| Answer | Mark | Guidance |
|--------|------|----------|
| $(\overrightarrow{PB} \cdot \overrightarrow{PQ}) = 20 + 64 - 25$ | M1 | Use of $x_1x_2 + y_1y_2 + z_1z_2$ on their $\overrightarrow{PB}$ and $\overrightarrow{PQ}$ |
| $({\rm their}\ \sqrt{114})({\rm their}\ \sqrt{105})\cos BPQ = ({\rm their}\ 59)$ | M1 | All elements present and in correct places |
| $BPQ = 57.4(°)$ or $1.00$ (rad) | A1 | AWRT. Calculating the obtuse angle and then subtracting gets A0 |
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\includegraphics[max width=\textwidth, alt={}, center]{567c3d72-c633-4ae0-8605-f63f93d718c4-12_784_677_260_735}
The diagram shows a three-dimensional shape $O A B C D E F G$. The base $O A B C$ and the upper surface $D E F G$ are identical horizontal rectangles. The parallelograms $O A E D$ and $C B F G$ both lie in vertical planes. Points $P$ and $Q$ are the mid-points of $O D$ and $G F$ respectively. Unit vectors $\mathbf { i }$ and $\mathbf { j }$ are parallel to $\overrightarrow { O A }$ and $\overrightarrow { O C }$ respectively and the unit vector $\mathbf { k }$ is vertically upwards. The position vectors of $A , C$ and $D$ are given by $\overrightarrow { O A } = 6 \mathbf { i } , \overrightarrow { O C } = 8 \mathbf { j }$ and $\overrightarrow { O D } = 2 \mathbf { i } + 10 \mathbf { k }$.\\
(i) Express each of the vectors $\overrightarrow { P B }$ and $\overrightarrow { P Q }$ in terms of $\mathbf { i } , \mathbf { j }$ and $\mathbf { k }$.\\
(ii) Determine whether $P$ is nearer to $Q$ or to $B$.\\
(iii) Use a scalar product to find angle $B P Q$.\\
\hfill \mbox{\textit{CAIE P1 2019 Q7 [9]}}