| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2012 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors 3D & Lines |
| Type | Equal length conditions |
| Difficulty | Moderate -0.3 This is a straightforward two-part question testing standard vector operations: (i) requires computing a dot product and using the cosine formula for angles, which is routine; (ii) involves finding the magnitude of a vector and solving a quadratic equation. Both parts are direct applications of basic vector formulas 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 vectors4.04c Scalar product: calculate and use for angles |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\begin{pmatrix}1\\0\\2\end{pmatrix} \cdot \begin{pmatrix}2\\-2\\4\end{pmatrix} = 10\) | M1 | Use of \(x_1x_2 + y_1y_2 + z_1z_2\) |
| \(= \sqrt{5} \times \sqrt{24} \cos\theta\) | M1 | Product of 2 moduli |
| \(\rightarrow \theta = 24.1°\) | M1 A1 | All connected correctly, co |
| [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\overrightarrow{AB} = \begin{pmatrix}k-1\\-k\\2k-2\end{pmatrix}\) allow each component \(\pm\) | M1 | Correct for either \(\mathbf{AB}\) or \(\mathbf{BA}\) |
| \((k-1)^2 + k^2 + (2k-2)^2\) | M1 | Sum of 3 squares (doesn't need \(=1\)) |
| \(\rightarrow 6k^2 - 10k + 4 = 0\) | A1 | Correct quadratic |
| \(\rightarrow k = 1\) or \(\frac{2}{3}\) | A1 | co |
| [4] |
## Question 7:
### Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\begin{pmatrix}1\\0\\2\end{pmatrix} \cdot \begin{pmatrix}2\\-2\\4\end{pmatrix} = 10$ | M1 | Use of $x_1x_2 + y_1y_2 + z_1z_2$ |
| $= \sqrt{5} \times \sqrt{24} \cos\theta$ | M1 | Product of 2 moduli |
| $\rightarrow \theta = 24.1°$ | M1 A1 | All connected correctly, co |
| **[4]** | | |
### Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\overrightarrow{AB} = \begin{pmatrix}k-1\\-k\\2k-2\end{pmatrix}$ allow each component $\pm$ | M1 | Correct for either $\mathbf{AB}$ or $\mathbf{BA}$ |
| $(k-1)^2 + k^2 + (2k-2)^2$ | M1 | Sum of 3 squares (doesn't need $=1$) |
| $\rightarrow 6k^2 - 10k + 4 = 0$ | A1 | Correct quadratic |
| $\rightarrow k = 1$ or $\frac{2}{3}$ | A1 | co |
| **[4]** | | |
---7 The position vectors of the points $A$ and $B$, relative to an origin $O$, are given by
$$\overrightarrow { O A } = \left( \begin{array} { l }
1 \\
0 \\
2
\end{array} \right) \quad \text { and } \quad \overrightarrow { O B } = \left( \begin{array} { r }
k \\
- k \\
2 k
\end{array} \right)$$
where $k$ is a constant.\\
(i) In the case where $k = 2$, calculate angle $A O B$.\\
(ii) Find the values of $k$ for which $\overrightarrow { A B }$ is a unit vector.
\hfill \mbox{\textit{CAIE P1 2012 Q7 [8]}}