| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Year | 2012 |
| Session | June |
| Marks | 10 |
| Topic | Vectors: Cross Product & Distances |
| Type | Angle between vectors using scalar product |
| Difficulty | Standard +0.3 This is a straightforward multi-part vectors question requiring standard techniques: finding vectors CA and CB, computing their scalar product to find an angle, using the sine formula for triangle area, and verifying a given line equation. All steps are routine applications of formulas with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.10c Magnitude and direction: of vectors1.10f Distance between points: using position vectors4.04a Line equations: 2D and 3D, cartesian and vector forms4.04c Scalar product: calculate and use for angles |
| Answer | Marks | Guidance |
|---|---|---|
| - Attempt \(\cos\theta = \dfrac{\overrightarrow{CA}\cdot\overrightarrow{CB}}{ | \overrightarrow{CA} |
**Part (i)**
- Obtain $\pm 111$ anywhere: M1
- Obtain at least one of $\sqrt{198}$ or $\sqrt{285}$: B1
- Attempt $\cos\theta = \dfrac{\overrightarrow{CA}\cdot\overrightarrow{CB}}{|\overrightarrow{CA}||\overrightarrow{CB}|}$: M1
- Obtain $\dfrac{111}{\sqrt{198}\times\sqrt{285}}$: A1
- Obtain $62.14°$ ($62.14276°$): A1 **[5]**
**Part (ii)**
- Use $0.5$ (their $AC$)(their $CB$)$\sin ACB$: M1
- Obtain 105: A1 **[2]**
**Part (iii)**
- Attempt $\mathbf{b} - \mathbf{a} = \begin{pmatrix}13\\9\\1\end{pmatrix} - \begin{pmatrix}1\\0\\7\end{pmatrix}$ or $\mathbf{a} - \mathbf{b}$: M1
- Obtain $\begin{pmatrix}12\\9\\-6\end{pmatrix} = 3\begin{pmatrix}4\\3\\-2\end{pmatrix}$ or $\begin{pmatrix}-12\\-9\\6\end{pmatrix} = -3\begin{pmatrix}4\\3\\-2\end{pmatrix}$ in column vector form or aef: A1
- Obtain $\mathbf{r} = \mathbf{i} + 0\mathbf{j} + 7\mathbf{k} + \lambda(4\mathbf{i} + 3\mathbf{j} - 2\mathbf{k})$ AG: A1 **[3]**
**[Total: 10]**9 Three points $A , B$ and $C$ have coordinates $( 1,0,7 ) , ( 13,9,1 )$ and $( 2 , - 1 , - 7 )$ respectively.\\
(i) Use a scalar product to find angle $A C B$.\\
(ii) Hence find the area of triangle $A C B$.\\
(iii) Show that a vector equation of the line $A B$ is given by $\mathbf { r } = \mathbf { i } + 7 \mathbf { k } + \lambda ( 4 \mathbf { i } + 3 \mathbf { j } - 2 \mathbf { k } )$, where $\lambda$ is a scalar parameter.
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2012 Q9 [10]}}