| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2013 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors 3D & Lines |
| Type | Perpendicularity conditions |
| Difficulty | Moderate -0.3 This is a straightforward multi-part vectors question testing standard techniques: parallel vectors (scalar multiples), perpendicularity (dot product = 0), and unit vectors. All parts are routine applications of basic vector concepts with minimal problem-solving required, making it slightly easier than average but not trivial due to the algebraic manipulation needed in part (ii). |
| Spec | 1.10c Magnitude and direction: of vectors1.10d Vector operations: addition and scalar multiplication1.10f Distance between points: using position vectors |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(p = -6, q = 6\) | B1 B1 | [2] |
| (ii) dot product \(= 0 \rightarrow 3 - 2p + 4q = 0\) → \(p = -1.5\) | M1 A1 | Use of \(x_1x_2 + y_1y_2 + z_1z_2 = 0\) |
| (iii) \(\overrightarrow{AB} = \mathbf{b} - \mathbf{a} = 2\mathbf{i} + 3\mathbf{j} + 6\mathbf{k}\). Unit vector \(= (2\mathbf{i} + 3\mathbf{j} + 6\mathbf{k}) \div 7\) | B1 M1 A1 | not for \(\mathbf{b} - \mathbf{a}\). M1 for division by modulus. \(\checkmark\) on B1 |
| [2] | [3] |
$\overrightarrow{OA} = \mathbf{i} - 2\mathbf{j} + 2\mathbf{k}$, $\overrightarrow{OB} = 3\mathbf{i} + p\mathbf{j} + q\mathbf{k}$
(i) $p = -6, q = 6$ | B1 B1 | [2] |
(ii) dot product $= 0 \rightarrow 3 - 2p + 4q = 0$ → $p = -1.5$ | M1 A1 | Use of $x_1x_2 + y_1y_2 + z_1z_2 = 0$
(iii) $\overrightarrow{AB} = \mathbf{b} - \mathbf{a} = 2\mathbf{i} + 3\mathbf{j} + 6\mathbf{k}$. Unit vector $= (2\mathbf{i} + 3\mathbf{j} + 6\mathbf{k}) \div 7$ | B1 M1 A1 | not for $\mathbf{b} - \mathbf{a}$. M1 for division by modulus. $\checkmark$ on B1
[2] | [3] |
---6 Relative to an origin $O$, the position vectors of points $A$ and $B$ are given by
$$\overrightarrow { O A } = \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k } \quad \text { and } \quad \overrightarrow { O B } = 3 \mathbf { i } + p \mathbf { j } + q \mathbf { k }$$
where $p$ and $q$ are constants.\\
(i) State the values of $p$ and $q$ for which $\overrightarrow { O A }$ is parallel to $\overrightarrow { O B }$.\\
(ii) In the case where $q = 2 p$, find the value of $p$ for which angle $B O A$ is $90 ^ { \circ }$.\\
(iii) In the case where $p = 1$ and $q = 8$, find the unit vector in the direction of $\overrightarrow { A B }$.
\hfill \mbox{\textit{CAIE P1 2013 Q6 [7]}}