| Exam Board | AQA |
|---|---|
| Module | Further AS Paper 1 (Further AS Paper 1) |
| Session | Specimen |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors Introduction & 2D |
| Type | Angle between two vectors |
| Difficulty | Standard +0.3 This is a straightforward 3D vectors question requiring students to form vector equations of two lines from given coordinates, verify intersection by solving simultaneously, then use the scalar product formula to find the angle between direction vectors. While it involves multiple steps and 3D visualization, all techniques are standard Further Maths AS content with no novel problem-solving required. |
| Spec | 4.04a Line equations: 2D and 3D, cartesian and vector forms4.04c Scalar product: calculate and use for angles4.04e Line intersections: parallel, skew, or intersecting |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Models beams of light as straight lines; \(\mathbf{r}_A = \begin{pmatrix}0\\0\\3\end{pmatrix} + \lambda\begin{pmatrix}30\\125\\20\end{pmatrix}\) | M1, A1 | Models light beams as straight lines using suitable origin; forms correct vector equation, allow one slip |
| \(\mathbf{r}_B = \begin{pmatrix}8\\10\\1\end{pmatrix} + \mu\begin{pmatrix}20\\130\\28\end{pmatrix}\) | A1 | Forms correct vector equation for second line; allow one slip |
| \(30\lambda = 8 + 20\mu\); \(125\lambda = 10 + 130\mu\) | M1 | Forms equations for two components using 'their' model |
| \(\lambda = \frac{3}{5}\) and \(\mu = \frac{1}{2}\) | A1F | Solves 'their' equations correctly |
| \(3 + \frac{3}{5}\times20 = 15\); \(1 + \frac{1}{2}\times28 = 15\) \(\therefore\) Intersect | R1 | Checks with third component and concludes beams intersect; available only if all previous marks awarded |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\begin{pmatrix}30\\125\\20\end{pmatrix}\cdot\begin{pmatrix}20\\130\\28\end{pmatrix} = 17410\) | M1 | Evaluates scalar product for 'their' direction vectors |
| \(\cos\theta = \frac{17410}{\sqrt{30^2+125^2+20^2}\times\sqrt{20^2+130^2+28^2}} = \frac{17410}{\sqrt{16925}\times\sqrt{18084}} = 0.9951\) | M1 | Sets up equation to find angle; FT only if previous M1 awarded |
| \(\theta = 5.6°\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Take account of the width of the beams. | E1 | States appropriate refinement |
## Question 7(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Models beams of light as straight lines; $\mathbf{r}_A = \begin{pmatrix}0\\0\\3\end{pmatrix} + \lambda\begin{pmatrix}30\\125\\20\end{pmatrix}$ | M1, A1 | Models light beams as straight lines using suitable origin; forms correct vector equation, allow one slip |
| $\mathbf{r}_B = \begin{pmatrix}8\\10\\1\end{pmatrix} + \mu\begin{pmatrix}20\\130\\28\end{pmatrix}$ | A1 | Forms correct vector equation for second line; allow one slip |
| $30\lambda = 8 + 20\mu$; $125\lambda = 10 + 130\mu$ | M1 | Forms equations for two components using 'their' model |
| $\lambda = \frac{3}{5}$ and $\mu = \frac{1}{2}$ | A1F | Solves 'their' equations correctly |
| $3 + \frac{3}{5}\times20 = 15$; $1 + \frac{1}{2}\times28 = 15$ $\therefore$ Intersect | R1 | Checks with third component and concludes beams intersect; available only if all previous marks awarded |
---
## Question 7(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\begin{pmatrix}30\\125\\20\end{pmatrix}\cdot\begin{pmatrix}20\\130\\28\end{pmatrix} = 17410$ | M1 | Evaluates scalar product for 'their' direction vectors |
| $\cos\theta = \frac{17410}{\sqrt{30^2+125^2+20^2}\times\sqrt{20^2+130^2+28^2}} = \frac{17410}{\sqrt{16925}\times\sqrt{18084}} = 0.9951$ | M1 | Sets up equation to find angle; FT only if previous M1 awarded |
| $\theta = 5.6°$ | A1 | |
---
## Question 7(c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Take account of the width of the beams. | E1 | States appropriate refinement |7 A lighting engineer is setting up part of a display inside a large building. The diagram shows a plan view of the area in which he is working.
He has two lights, which project narrow beams of light.
One is set up at a point 3 metres above the point $A$ and the beam from this light hits the wall 23 metres above the point $D$.
The other is set up 1 metre above the point $B$ and the beam from this light hits the wall 29 metres above the point $C$.\\
\includegraphics[max width=\textwidth, alt={}, center]{e61d0202-49c9-4ed9-9fa3-f10734e17463-10_776_1301_826_392}
7
\begin{enumerate}[label=(\alph*)]
\item By creating a suitable model, show that the beams of light intersect.
7
\item Find the angle between the two beams of light.\\[0pt]
[3 marks]\\
7
\item State one way in which the model you created in part (a) could be refined.\\[0pt]
[1 mark]
\end{enumerate}
\hfill \mbox{\textit{AQA Further AS Paper 1 Q7 [4]}}