| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2009 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors 3D & Lines |
| Type | Vector geometry in 3D shapes |
| Difficulty | Moderate -0.3 This is a straightforward 3D vector question in a cube with clear geometric setup. Part (i) requires basic position vector calculations using given ratios, and part (ii) involves standard dot product formula for angles. The cube structure simplifies all calculations, making this slightly easier than average A-level vector questions which often involve more complex geometric configurations or require additional insight. |
| Spec | 1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta1.07n Stationary points: find maxima, minima using derivatives |
| Answer | Marks | Guidance |
|---|---|---|
| \(\overrightarrow{PQ} = -3\mathbf{i} + \mathbf{j} + 6k\) | B1, B2, 1 [3] | co. Loses one for each error. |
| Answer | Marks | Guidance |
|---|---|---|
| \(\theta = 53.0°\) | M1, M1, M1A1 [4] | Use of \(\mathbf{x_1 x_2} + y_1y_2 + z_1z_2\) co. Correct method for modulus (once) and all correctly linked. co. nb \(\overrightarrow{OQ} \cdot \overrightarrow{OP}\) can gain 4/4. but \(\overrightarrow{OQ} \cdot \overrightarrow{PQ}\) can only gain 3/4. Use of other vectors (e.g. \(\overrightarrow{OP} \cdot \overrightarrow{OQ}\)) M3 ok. |
**(i)** $\overrightarrow{OQ} = 3\mathbf{i} + 3\mathbf{j} + 6k$
$\overrightarrow{PQ} = -3\mathbf{i} + \mathbf{j} + 6k$ | B1, B2, 1 [3] | co. Loses one for each error.
**(ii)** $(3\mathbf{i} + 3\mathbf{j} + 6k) \cdot (-3\mathbf{i} + \mathbf{j} + 6k) = -9 + 3 + 36 = 30$
$30 = \sqrt{54}\sqrt{46}\cos\theta$
$\theta = 53.0°$ | M1, M1, M1A1 [4] | Use of $\mathbf{x_1 x_2} + y_1y_2 + z_1z_2$ co. Correct method for modulus (once) and all correctly linked. co. nb $\overrightarrow{OQ} \cdot \overrightarrow{OP}$ can gain 4/4. but $\overrightarrow{OQ} \cdot \overrightarrow{PQ}$ can only gain 3/4. Use of other vectors (e.g. $\overrightarrow{OP} \cdot \overrightarrow{OQ}$) M3 ok.
**Total: [7]**
---6\\
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In the diagram, $O A B C D E F G$ is a cube in which each side has length 6 . Unit vectors $\mathbf { i } , \mathbf { j }$ and $\mathbf { k }$ are parallel to $\overrightarrow { O A } , \overrightarrow { O C }$ and $\overrightarrow { O D }$ respectively. The point $P$ is such that $\overrightarrow { A P } = \frac { 1 } { 3 } \overrightarrow { A B }$ and the point $Q$ is the mid-point of $D F$.\\
(i) Express each of the vectors $\overrightarrow { O Q }$ and $\overrightarrow { P Q }$ in terms of $\mathbf { i } , \mathbf { j }$ and $\mathbf { k }$.\\
(ii) Find the angle $O Q P$.
\hfill \mbox{\textit{CAIE P1 2009 Q6 [7]}}