| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2011 |
| Session | June |
| 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 vector question requiring coordinate setup from a diagram, vector arithmetic, and a standard scalar product calculation for an angle. While it involves multiple steps and 3D visualization, the techniques are routine for A-level: finding position vectors by adding displacements, then applying the cosine formula with dot products. No novel insight or complex problem-solving required. |
| 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) \(\overrightarrow{CP} = -6\mathbf{i} + 6\mathbf{j} - 2\mathbf{k}\) | B1 | |
| \(\overrightarrow{CQ} = -6\mathbf{i} + 6\mathbf{j} + 3\mathbf{k}\) | B1 [2] | |
| (ii) Scalar product = 36 + 36 − 6 | M1 | Use of \(x_1x_2 + y_1y_2 + z_1z_2\) |
| 66 = \( | \overrightarrow{CP} | |
| \( | \overrightarrow{CP} | = \sqrt{76}\), \( |
| Angle PCQ = 32.7° (or 0.571 rad) | A1 [4] |
**(i)** $\overrightarrow{CP} = -6\mathbf{i} + 6\mathbf{j} - 2\mathbf{k}$ | B1 |
$\overrightarrow{CQ} = -6\mathbf{i} + 6\mathbf{j} + 3\mathbf{k}$ | B1 [2] |
**(ii)** Scalar product = 36 + 36 − 6 | M1 | Use of $x_1x_2 + y_1y_2 + z_1z_2$
66 = $|\overrightarrow{CP}| |\overrightarrow{CQ}| \cos \theta$ | M1 | Linking everything correctly
$|\overrightarrow{CP}| = \sqrt{76}$, $|\overrightarrow{CQ}| = \sqrt{81}$ | M1 | Correct magnitude for either cao 147.3° converted to 32.7° gets A0
Angle PCQ = 32.7° (or 0.571 rad) | A1 [4] |
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\includegraphics[max width=\textwidth, alt={}, center]{53839c8c-07ea-4545-9c00-a6884aa2afc3-2_750_855_902_646}
The diagram shows a prism $A B C D P Q R S$ with a horizontal square base $A P S D$ with sides of length 6 cm . The cross-section $A B C D$ is a trapezium and is such that the vertical edges $A B$ and $D C$ are of lengths 5 cm and 2 cm respectively. Unit vectors $\mathbf { i } , \mathbf { j }$ and $\mathbf { k }$ are parallel to $A D , A P$ and $A B$ respectively.\\
(i) Express each of the vectors $\overrightarrow { C P }$ and $\overrightarrow { C Q }$ in terms of $\mathbf { i } , \mathbf { j }$ and $\mathbf { k }$.\\
(ii) Use a scalar product to calculate angle $P C Q$.
\hfill \mbox{\textit{CAIE P1 2011 Q4 [6]}}