| Exam Board | AQA |
|---|---|
| Module | Further AS Paper 1 (Further AS Paper 1) |
| Year | 2021 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circles |
| Type | Geometric properties with circles |
| Difficulty | Standard +0.3 This is a multi-part question on ellipses and transformations that is slightly easier than average. Parts (a) and (b) are straightforward applications of reflection and scaling transformations. Part (c)(ii) requires finding a translation vector given tangency conditions, which involves basic geometry of ellipses. Part (c)(iii) requires finding a common tangent, which is more involved but still a standard technique. Overall, this tests routine understanding of conic transformations without requiring deep insight. |
| Spec | 1.02w Graph transformations: simple transformations of f(x)1.02x Combinations of transformations: multiple transformations1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.03e Complete the square: find centre and radius of circle1.07m Tangents and normals: gradient and equations |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{y^2}{16} + \frac{x^2}{4} = 1\) | B1 | Correct equation written down |
| Answer | Marks | Guidance |
|---|---|---|
| \(y \rightarrow \frac{y}{2}\) | M1 | Indicates a stretch in any direction |
| stretch, parallel to the \(y\)-axis, scale factor 2 | A1 | Identifies correct transformation |
| Answer | Marks | Guidance |
|---|---|---|
| One loop centred on origin and second loop approximately same shape in first quadrant with positive \(x\) and \(y\) axes as tangents | M1 | Or draws one correct graph with one correct \(x\)-intercept and one correct \(y\)-intercept |
| Two correct graphs with all four intercepts correctly indicated: \(x\)-intercepts at \(\pm4\), \(y\)-intercepts at \(\pm2\) | A1 | Both graphs fully correct with intercepts |
| Answer | Marks | Guidance |
|---|---|---|
| Translation vector containing either 2 or \(-2\) and 4 or \(-4\) | M1 | Follow through their intercepts |
| \(\begin{bmatrix}4\\2\end{bmatrix}\) | A1 | Correct translation vector |
| Answer | Marks | Guidance |
|---|---|---|
| Calculates \(\frac{b}{a}\) or \(\frac{a}{b}\) for their translation vector \(\begin{bmatrix}a\\b\end{bmatrix}\) | M1 | |
| \(m = \frac{2}{4} = \frac{1}{2}\) | A1F | Follow through their part (cii) |
## Question 14(a):
$\frac{y^2}{16} + \frac{x^2}{4} = 1$ | B1 | Correct equation written down
---
## Question 14(b):
$y \rightarrow \frac{y}{2}$ | M1 | Indicates a stretch in any direction
stretch, parallel to the $y$-axis, scale factor 2 | A1 | Identifies correct transformation
---
## Question 14(c)(i):
One loop centred on origin and second loop approximately same shape in first quadrant with positive $x$ and $y$ axes as tangents | M1 | Or draws one correct graph with one correct $x$-intercept and one correct $y$-intercept
Two correct graphs with all four intercepts correctly indicated: $x$-intercepts at $\pm4$, $y$-intercepts at $\pm2$ | A1 | Both graphs fully correct with intercepts
---
## Question 14(c)(ii):
Translation vector containing either 2 or $-2$ **and** 4 or $-4$ | M1 | Follow through their intercepts
$\begin{bmatrix}4\\2\end{bmatrix}$ | A1 | Correct translation vector
---
## Question 14(c)(iii):
Calculates $\frac{b}{a}$ or $\frac{a}{b}$ for their translation vector $\begin{bmatrix}a\\b\end{bmatrix}$ | M1 |
$m = \frac{2}{4} = \frac{1}{2}$ | A1F | Follow through their part (cii)
---14 Curve $C _ { 1 }$ has equation
$$\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 4 } = 1$$
14
\begin{enumerate}[label=(\alph*)]
\item Curve $C _ { 2 }$ is a reflection of $C _ { 1 }$ in the line $y = x$\\
Write down an equation of $C _ { 2 }$\\
14
\item Curve $C _ { 3 }$ is a circle of radius 4 , centred at the origin.\\
Describe a single transformation which maps $C _ { 1 }$ onto $C _ { 3 }$\\
14
\item Curve $C _ { 4 }$ is a translation of $C _ { 1 }$\\
The positive $x$-axis and the positive $y$-axis are tangents to $C _ { 4 }$\\
14 (c) (i) Sketch the graphs of $C _ { 1 }$ and $C _ { 4 }$ on the axes opposite. Indicate the coordinates of the $x$ and $y$ intercepts on your graphs.\\[0pt]
[2 marks]\\
14 (c) (ii) Determine the translation vector.\\[0pt]
[2 marks]\\
14 (c) (iii) The line $y = m x + c$ is a tangent to both $C _ { 1 }$ and $C _ { 4 }$ Find the value of $m$
\end{enumerate}
\hfill \mbox{\textit{AQA Further AS Paper 1 2021 Q14 [9]}}