| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2009 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Lines & Planes |
| Type | Perpendicularity conditions |
| Difficulty | Standard +0.3 This is a straightforward multi-part vectors question testing standard techniques: angle between vectors using dot product (routine calculation), finding a unit vector and scaling it (direct application), and using perpendicularity condition (dot product = 0, solve linear equation). All parts are textbook exercises requiring only recall and basic algebraic manipulation, making it slightly easier than average. |
| Spec | 1.02u Functions: definition and vocabulary (domain, range, mapping)1.02v Inverse and composite functions: graphs and conditions for existence |
M1: $\vec{OA}= \begin{pmatrix}2\\3\\-6\end{pmatrix}$, $\vec{OB}= \begin{pmatrix}0\\-6\\8\end{pmatrix}$, $\vec{OC} = \begin{pmatrix}-2\\5\\-2\end{pmatrix}$ — Use of $x_1 x_2 + y_1 y_2 + z_1 z_2$
(i) M1: Scalar product = $−18 − 48$ — Linking everything correctly
M1: $−66 = ab\cos\theta$ — Linking everything correctly
M1: $a= \sqrt{7}$ and $b= \sqrt{10}$ — Correct modulus of either $a$ or $b$
A1: $\to$ Angle $AOB = 160.5°$ — co; allow $\pm$
[4]
(ii) B1: $\vec{AC} = \vec{c}-\vec{a}= \begin{pmatrix}-4\\2\\4\end{pmatrix}$ — For modulus and multiplying by "5"
M1: Modulus = $6$
A1: Vector = $\begin{pmatrix}-20\\10\\20\end{pmatrix}$ (or $\begin{pmatrix}-4\\2\\4\end{pmatrix}$ scaled)
[3]
(iii) B1: $\begin{pmatrix}2\\3\\-6\end{pmatrix} + 6p\begin9 Relative to an origin $O$, the position vectors of the points $A , B$ and $C$ are given by
$$\overrightarrow { O A } = \left( \begin{array} { r }
2 \\
3 \\
- 6
\end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r }
0 \\
- 6 \\
8
\end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { r }
- 2 \\
5 \\
- 2
\end{array} \right)$$
(i) Find angle $A O B$.\\
(ii) Find the vector which is in the same direction as $\overrightarrow { A C }$ and has magnitude 30 .\\
(iii) Find the value of the constant $p$ for which $\overrightarrow { O A } + p \overrightarrow { O B }$ is perpendicular to $\overrightarrow { O C }$.
\hfill \mbox{\textit{CAIE P1 2009 Q9 [10]}}