| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2017 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors 3D & Lines |
| Type | Position vectors and magnitudes |
| Difficulty | Standard +0.3 This question involves routine vector operations (finding displacement vectors and magnitudes) followed by applying geometric progression properties and parametric line equations. While it has multiple parts and requires careful calculation, all techniques are standard P1 material with no novel problem-solving insight required. The GP aspect adds mild complexity but is straightforward once |AB| and |BC| are calculated. |
| Spec | 1.04i Geometric sequences: nth term and finite series sum1.04j Sum to infinity: convergent geometric series |r|<11.10a 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 vectors |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\overrightarrow{AB} = +/-\begin{pmatrix}-18\\9\\-18\end{pmatrix}\), \(\overrightarrow{BC} = +/-\begin{pmatrix}12\\-6\\12\end{pmatrix}\) | B1 B1 | Allow i, j, k form throughout |
| \( | \overrightarrow{AB} | = 27\), \( |
| \( | \overrightarrow{CD} | = \left(\frac{18}{27}\right) \times 18\) OR \(\left(\frac{18}{27}\right)^2 \times 27 = 12\) |
| Total | 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\overrightarrow{CD} = (\pm)\text{their}\frac{18}{27} \times \text{their}\ \overrightarrow{BC}\) SOI | M1 | Expect \((\pm)\begin{pmatrix}8\\-4\\8\end{pmatrix}\) |
| \(\overrightarrow{OD} = \begin{pmatrix}2\\-3\\-1\end{pmatrix} (\pm)\text{ their }\frac{18}{27}\begin{pmatrix}12\\-6\\12\end{pmatrix} = \begin{pmatrix}10\\-7\\7\end{pmatrix}, \begin{pmatrix}-6\\1\\-9\end{pmatrix}\) | M1 A1 A1 | Other methods possible for \(\overrightarrow{OD}\), e.g. \(\overrightarrow{OB} + \frac{5}{2}\overrightarrow{CD}\), \(\overrightarrow{OB} + \frac{1}{2}\overrightarrow{CD}\) (One soln M2A1, 2nd soln A1) OR \(\overrightarrow{OB} + \frac{5}{3}\overrightarrow{BC}\), \(\overrightarrow{OB} + \frac{1}{3}\overrightarrow{BC}\) (One soln M2A1, 2nd soln A1) |
| Total | 4 |
## Question 9(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\overrightarrow{AB} = +/-\begin{pmatrix}-18\\9\\-18\end{pmatrix}$, $\overrightarrow{BC} = +/-\begin{pmatrix}12\\-6\\12\end{pmatrix}$ | B1 B1 | Allow **i**, **j**, **k** form throughout |
| $|\overrightarrow{AB}| = 27$, $|\overrightarrow{BC}| = 18$ | B1 FT B1 FT | FT on *their* $\overrightarrow{AB}$, *their* $\overrightarrow{OD}$ |
| $|\overrightarrow{CD}| = \left(\frac{18}{27}\right) \times 18$ OR $\left(\frac{18}{27}\right)^2 \times 27 = 12$ | B1 | |
| **Total** | **5** | |
---
## Question 9(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\overrightarrow{CD} = (\pm)\text{their}\frac{18}{27} \times \text{their}\ \overrightarrow{BC}$ SOI | M1 | Expect $(\pm)\begin{pmatrix}8\\-4\\8\end{pmatrix}$ |
| $\overrightarrow{OD} = \begin{pmatrix}2\\-3\\-1\end{pmatrix} (\pm)\text{ their }\frac{18}{27}\begin{pmatrix}12\\-6\\12\end{pmatrix} = \begin{pmatrix}10\\-7\\7\end{pmatrix}, \begin{pmatrix}-6\\1\\-9\end{pmatrix}$ | M1 A1 A1 | Other methods possible for $\overrightarrow{OD}$, e.g. $\overrightarrow{OB} + \frac{5}{2}\overrightarrow{CD}$, $\overrightarrow{OB} + \frac{1}{2}\overrightarrow{CD}$ (One soln M2A1, 2nd soln A1) OR $\overrightarrow{OB} + \frac{5}{3}\overrightarrow{BC}$, $\overrightarrow{OB} + \frac{1}{3}\overrightarrow{BC}$ (One soln M2A1, 2nd soln A1) |
| **Total** | **4** | |9 Relative to an origin $O$, the position vectors of the points $A , B$ and $C$ are given by
$$\overrightarrow { O A } = \left( \begin{array} { r }
8 \\
- 6 \\
5
\end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r }
- 10 \\
3 \\
- 13
\end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { r }
2 \\
- 3 \\
- 1
\end{array} \right)$$
A fourth point, $D$, is such that the magnitudes $| \overrightarrow { A B } | , | \overrightarrow { B C } |$ and $| \overrightarrow { C D } |$ are the first, second and third terms respectively of a geometric progression.\\
(i) Find the magnitudes $| \overrightarrow { A B } | , | \overrightarrow { B C } |$ and $| \overrightarrow { C D } |$.\\
(ii) Given that $D$ is a point lying on the line through $B$ and $C$, find the two possible position vectors of the point $D$.\\
\hfill \mbox{\textit{CAIE P1 2017 Q9 [9]}}