| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2006 |
| Session | June |
| Marks | 8 |
| 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 vector problem requiring coordinate setup from a symmetric diagram, basic vector arithmetic, magnitude calculation, and a standard scalar product application for angles. While it involves multiple steps, each is routine for A-level: identifying coordinates from symmetry (D and E are centered above the rectangle), computing a vector and its magnitude, then applying the dot product formula. No novel insight or complex problem-solving required—just systematic application of standard techniques. |
| Spec | 1.10a Vectors in 2D: i,j notation and column vectors1.10b Vectors in 3D: i,j,k notation1.10c Magnitude and direction: of vectors1.10f Distance between points: using position vectors |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Vector \(OD = 4i + 4j + 5k\) | B2,1 | One off for each error. Column vectors ok. |
| Magnitude \(= \sqrt{(4^2+4^2+5^2)} = \sqrt{57}\) | M1 | Correct use of Pythagoras. Accept \(\sqrt{57}\). |
| → Magnitude \(= 7.55m\) | A1 | co. |
| (ii) Vector \(OB = 14i + 8j\) | B1 | co |
| \(OD.OB = 4×14 + 4×8 = 88\) | M1 | Use of \(x_1x_2 + y_1y_2 + z_1z_2\) for his vectors |
| \(OD.OB = \sqrt{57}.√{260}\cos\theta\) | M1 | Used correctly |
| → Angle \(DOB = 43.7°\) | A1 | co |
**(i)** Vector $OD = 4i + 4j + 5k$ | B2,1 | One off for each error. Column vectors ok.
Magnitude $= \sqrt{(4^2+4^2+5^2)} = \sqrt{57}$ | M1 | Correct use of Pythagoras. Accept $\sqrt{57}$.
→ Magnitude $= 7.55m$ | A1 | co.
**(ii)** Vector $OB = 14i + 8j$ | B1 | co
$OD.OB = 4×14 + 4×8 = 88$ | M1 | Use of $x_1x_2 + y_1y_2 + z_1z_2$ for his vectors
$OD.OB = \sqrt{57}.√{260}\cos\theta$ | M1 | Used correctly
→ Angle $DOB = 43.7°$ | A1 | co
**Total: [4] [4]**
---8\\
\includegraphics[max width=\textwidth, alt={}, center]{cbcb15b4-1870-4dfd-b6e9-839aa4601511-3_517_1117_1362_514}
The diagram shows the roof of a house. The base of the roof, $O A B C$, is rectangular and horizontal with $O A = C B = 14 \mathrm {~m}$ and $O C = A B = 8 \mathrm {~m}$. The top of the roof $D E$ is 5 m above the base and $D E = 6 \mathrm {~m}$. The sloping edges $O D , C D , A E$ and $B E$ are all equal in length.
Unit vectors $\mathbf { i }$ and $\mathbf { j }$ are parallel to $O A$ and $O C$ respectively and the unit vector $\mathbf { k }$ is vertically upwards.\\
(i) Express the vector $\overrightarrow { O D }$ in terms of $\mathbf { i } , \mathbf { j }$ and $\mathbf { k }$, and find its magnitude.\\
(ii) Use a scalar product to find angle $D O B$.
\hfill \mbox{\textit{CAIE P1 2006 Q8 [8]}}