| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2011 |
| Session | June |
| Marks | 7 |
| 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 position vector identification from a diagram, basic vector arithmetic (finding PQ and RQ), and a standard scalar product calculation to find an angle. While it involves multiple steps and 3D visualization, all techniques are routine A-level procedures with no novel problem-solving required, making it slightly easier than average. |
| 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 vectors |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(\overrightarrow{PQ} = 3i + 6j - 3k\) | B2,1 B1 | Allow B2,1 for either one, B1 for the other. |
| \(\overrightarrow{RQ} = -3i + 8j + 3k\) | ||
| [3] | ||
| (ii) \(\overrightarrow{PQ} \cdot \overrightarrow{RQ} = -9 + 48 - 9 = 30\) | M1 | Use of \(x_1x_2 + y_1y_2 + z_1z_2\) |
| \(= \sqrt{54} \sqrt{82} \cos RQP\) | M1 | Correct use of modulus |
| All linked correctly | M1 | |
| \(\rightarrow RQP = 63.2°\) | A1 | co |
| [4] |
(i) $\overrightarrow{PQ} = 3i + 6j - 3k$ | B2,1 B1 | Allow B2,1 for either one, B1 for the other.
$\overrightarrow{RQ} = -3i + 8j + 3k$ | |
| [3] |
(ii) $\overrightarrow{PQ} \cdot \overrightarrow{RQ} = -9 + 48 - 9 = 30$ | M1 | Use of $x_1x_2 + y_1y_2 + z_1z_2$
$= \sqrt{54} \sqrt{82} \cos RQP$ | M1 | Correct use of modulus
All linked correctly | M1 |
$\rightarrow RQP = 63.2°$ | A1 | co
| [4] |5\\
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In the diagram, $O A B C D E F G$ is a rectangular block in which $O A = O D = 6 \mathrm {~cm}$ and $A B = 12 \mathrm {~cm}$. The 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 the mid-point of $D G , Q$ is the centre of the square face $C B F G$ and $R$ lies on $A B$ such that $A R = 4 \mathrm {~cm}$.\\
(i) Express each of the vectors $\overrightarrow { P Q }$ and $\overrightarrow { R Q }$ in terms of $\mathbf { i } , \mathbf { j }$ and $\mathbf { k }$.\\
(ii) Use a scalar product to find angle $R Q P$.
\hfill \mbox{\textit{CAIE P1 2011 Q5 [7]}}