| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Year | 2011 |
| Session | June |
| Marks | 7 |
| Topic | Vectors 3D & Lines |
| Type | Angle between two vectors/lines (direct) |
| Difficulty | Standard +0.3 This is a straightforward vector geometry question requiring substitution to find parameters (part i) and then applying the standard dot product formula for angle between lines (part ii). Both parts use routine A-level techniques with no novel insight required, making it slightly easier than average. |
| Spec | 4.04a Line equations: 2D and 3D, cartesian and vector forms4.04c Scalar product: calculate and use for angles |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(2 - 3\lambda = 2 - \mu\) | B1 | Obtain correct eqns |
| \(-3 + \lambda = 4 - 2\mu\) | ||
| Obtain \(\lambda = 1\) \(\mu = 3\) | B1 | |
| Obtain \(a = 3\) and \(b = 1\) | B1 | |
| (ii) \(\begin{pmatrix} -1 \\ -2 \\ 1 \end{pmatrix}\) and \(\begin{pmatrix} -3 \\ 1 \\ 1 \end{pmatrix}\) | M1 | Use correct vectors aef |
| \(\frac{(-3)(-1) + 1(-2) + 1(1)}{(\sqrt{11})(\sqrt{6})}\) | M1 | Use correct dot product formula |
| \(= \frac{2}{\sqrt{66}}\) (= 0.246) | B1 | Find the length of any vector |
| Obtain \(75.7°\) | A1 | [7] |
(i) $2 - 3\lambda = 2 - \mu$ | B1 | Obtain correct eqns
$-3 + \lambda = 4 - 2\mu$ |
Obtain $\lambda = 1$ $\mu = 3$ | B1 |
Obtain $a = 3$ and $b = 1$ | B1 |
(ii) $\begin{pmatrix} -1 \\ -2 \\ 1 \end{pmatrix}$ and $\begin{pmatrix} -3 \\ 1 \\ 1 \end{pmatrix}$ | M1 | Use correct vectors aef
$\frac{(-3)(-1) + 1(-2) + 1(1)}{(\sqrt{11})(\sqrt{6})}$ | M1 | Use correct dot product formula
$= \frac{2}{\sqrt{66}}$ (= 0.246) | B1 | Find the length of any vector
Obtain $75.7°$ | A1 | [7] | Obtain acute answer only cao\begin{enumerate}[label=(\roman*)]
\item Given that the point $(-1, -2, 4)$ lies on both the lines
$$\mathbf{r} = \begin{pmatrix} 2 \\ -3 \\ a \end{pmatrix} + \lambda \begin{pmatrix} -3 \\ 2 \\ 1 \end{pmatrix} \quad \text{and} \quad \mathbf{r} = \begin{pmatrix} 2 \\ 4 \\ b \end{pmatrix} + \mu \begin{pmatrix} -1 \\ -2 \\ 1 \end{pmatrix},$$
find $a$ and $b$. [3]
\item Find the acute angle between the lines. [4]
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2011 Q7 [7]}}