| 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 basic position vector manipulation and scalar product application. Students must find midpoint D, express vectors in component form, then use the standard scalar product formula to find an angle. All steps are routine techniques with no novel insight required, making it slightly easier than average. |
| Spec | 1.10b Vectors in 3D: i,j,k notation1.10c Magnitude and direction: of vectors1.10d Vector operations: addition and scalar multiplication |
| Answer | Marks | Guidance |
|---|---|---|
| (i) OD \(= 4i + 3j\) | B1 | |
| CD \(= 4i + 3j - 10k\) | B1♦ | ♦ for OD \(- 10k\) |
| [2] | ||
| (ii) OD.CD \(= 9 + 16 = 25\) | M1 | Use of \(x_1x_2 + y_1y_2 + z_1z_2\) |
| \(\mid \text{OD} \mid = \sqrt{25}\) or \(\mid \text{CD} \mid = \sqrt{125}\) | M1 | Correct method for moduli |
| \(25 = \sqrt{25} \times \sqrt{125} \times \cos \theta\) oe | M1 | All connected correctly |
| ODC \(= 63.4°\) (or 1.11 rads) | A1 | cao |
| [4] |
**(i)** OD $= 4i + 3j$ | B1 |
CD $= 4i + 3j - 10k$ | B1♦ | ♦ for OD $- 10k$
| | [2]
**(ii)** OD.CD $= 9 + 16 = 25$ | M1 | Use of $x_1x_2 + y_1y_2 + z_1z_2$
$\mid \text{OD} \mid = \sqrt{25}$ or $\mid \text{CD} \mid = \sqrt{125}$ | M1 | Correct method for moduli
$25 = \sqrt{25} \times \sqrt{125} \times \cos \theta$ oe | M1 | All connected correctly
ODC $= 63.4°$ (or 1.11 rads) | A1 | cao
| | [4]4\\
\includegraphics[max width=\textwidth, alt={}, center]{16a5835e-002f-4c49-aacf-cda41c37f214-2_711_643_900_753}
The diagram shows a pyramid $O A B C$ in which the edge $O C$ is vertical. The horizontal base $O A B$ is a triangle, right-angled at $O$, and $D$ is the mid-point of $A B$. The edges $O A , O B$ and $O C$ have lengths of 8 units, 6 units and 10 units respectively. The unit vectors $\mathbf { i } , \mathbf { j }$ and $\mathbf { k }$ are parallel to $\overrightarrow { O A } , \overrightarrow { O B }$ and $\overrightarrow { O C }$ respectively.\\
(i) Express each of the vectors $\overrightarrow { O D }$ and $\overrightarrow { C D }$ in terms of $\mathbf { i } , \mathbf { j }$ and $\mathbf { k }$.\\
(ii) Use a scalar product to find angle ODC.
\hfill \mbox{\textit{CAIE P1 2013 Q4 [6]}}