| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2010 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Lines & Planes |
| Type | Angles between vectors |
| Difficulty | Moderate -0.3 This is a straightforward application of standard vector techniques: finding vectors BA and BC, computing their scalar product to find an angle, then calculating magnitudes for the perimeter. All steps are routine A-level procedures with no conceptual challenges, making it slightly easier than average. |
| Spec | 1.10c Magnitude and direction: of vectors1.10d Vector operations: addition and scalar multiplication1.10f Distance between points: using position vectors |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \((\pm) 2\mathbf{i} + 4\mathbf{j} + 4\mathbf{k}\) | B1 | co |
| \((\pm) 4\mathbf{i} + 4\mathbf{j} - 2\mathbf{k}\) | B1 | co |
| \(\vec{AB}\cdot\vec{CB} = 16\) | M1 | Needs to be scalar. |
| \(\vec{AB}\cdot\vec{CB} = \sqrt{36}\sqrt{36}\cos\theta\) | M1 | For product of 2 moduli and cosine |
| \(\theta = 63.6°\) | M1 A1 | All correct. |
| [6] | ||
| (ii) Perimeter \(= 6 + 6 + \sqrt{40}\) or \(6 + 6 + 6\sin 31.8° × 2\) | M1 | Correct overall method for perimeter. |
| \(\to 18.32\) | A1 | co |
| [2] |
$\vec{OA} = \mathbf{i} - 2\mathbf{j} + 4\mathbf{k}, \vec{OB} = 3\mathbf{i} + 2\mathbf{j} + 8\mathbf{k}, \vec{OC} = -\mathbf{i} - 2\mathbf{j} + 10\mathbf{k}$
(i) $(\pm) 2\mathbf{i} + 4\mathbf{j} + 4\mathbf{k}$ | B1 | co
$(\pm) 4\mathbf{i} + 4\mathbf{j} - 2\mathbf{k}$ | B1 | co
$\vec{AB}\cdot\vec{CB} = 16$ | M1 | Needs to be scalar.
$\vec{AB}\cdot\vec{CB} = \sqrt{36}\sqrt{36}\cos\theta$ | M1 | For product of 2 moduli and cosine
$\theta = 63.6°$ | M1 A1 | All correct.
| [6] |
(ii) Perimeter $= 6 + 6 + \sqrt{40}$ or $6 + 6 + 6\sin 31.8° × 2$ | M1 | Correct overall method for perimeter.
$\to 18.32$ | A1 | co
| [2] |
---6 Relative to an origin $O$, the position vectors of the points $A , B$ and $C$ are given by
$$\overrightarrow { O A } = \mathbf { i } - 2 \mathbf { j } + 4 \mathbf { k } , \quad \overrightarrow { O B } = 3 \mathbf { i } + 2 \mathbf { j } + 8 \mathbf { k } , \quad \overrightarrow { O C } = - \mathbf { i } - 2 \mathbf { j } + 10 \mathbf { k }$$
(i) Use a scalar product to find angle $A B C$.\\
(ii) Find the perimeter of triangle $A B C$, giving your answer correct to 2 decimal places.
\hfill \mbox{\textit{CAIE P1 2010 Q6 [8]}}