| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2013 |
| Session | November |
| Marks | 6 |
| 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 vectors question requiring coordinate setup from a diagram, vector subtraction, and a standard scalar product calculation for an angle. While it involves multiple steps, each step uses routine A-level techniques with no novel insight required. The rectangular base and clear coordinate system make it slightly easier than average. |
| Spec | 1.10b Vectors in 3D: i,j,k notation1.10d Vector operations: addition and scalar multiplication1.10h Vectors in kinematics: uniform acceleration in vector form |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(DB = 6i + 4j - 3k\) cao | B1 | |
| \(DE = 3i + 2j - 3k\) cao | B1 | [2] |
| (ii) \(DB \cdot DE = 18 + 8 + 9 = 35\) | M1 | Use of \(x_1x_2 + y_1y_2 + z_1z_2\) |
| \( | DB | = \sqrt{61}\), \( |
| \(35 = \sqrt{61} \times \sqrt{22} \times \cos\theta\) oe | M1 | All connected correctly |
| \(\theta = 17.2°\) (0.300 rad) cao | A1 | Use of e.g. \(BD \cdot DE\) can score M marks (leads to obtuse angle) |
(i) $DB = 6i + 4j - 3k$ cao | B1 |
$DE = 3i + 2j - 3k$ cao | B1 | [2]
(ii) $DB \cdot DE = 18 + 8 + 9 = 35$ | M1 | Use of $x_1x_2 + y_1y_2 + z_1z_2$
$|DB| = \sqrt{61}$, $|DE| = 22$ | M1 | Correct method for moduli
$35 = \sqrt{61} \times \sqrt{22} \times \cos\theta$ oe | M1 | All connected correctly
$\theta = 17.2°$ (0.300 rad) cao | A1 | Use of e.g. $BD \cdot DE$ can score M marks (leads to obtuse angle) | [4]3\\
\includegraphics[max width=\textwidth, alt={}, center]{02da6b6a-6db1-4bc3-ad4e-537e4f61dcac-2_397_949_657_596}
The diagram shows a pyramid $O A B C D$ in which the vertical edge $O D$ is 3 units in length. The point $E$ is the centre of the horizontal rectangular base $O A B C$. The sides $O A$ and $A B$ have lengths of 6 units and 4 units respectively. The unit vectors $\mathbf { i } , \mathbf { j }$ and $\mathbf { k }$ are parallel to $\overrightarrow { O A } , \overrightarrow { O C }$ and $\overrightarrow { O D }$ respectively.\\
(i) Express each of the vectors $\overrightarrow { D B }$ and $\overrightarrow { D E }$ in terms of $\mathbf { i } , \mathbf { j }$ and $\mathbf { k }$.\\
(ii) Use a scalar product to find angle $B D E$.
\hfill \mbox{\textit{CAIE P1 2013 Q3 [6]}}