| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2008 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Conic sections |
| Type | Conic translation and transformation |
| Difficulty | Standard +0.3 This is a straightforward Further Maths question on hyperbolas requiring basic recall of asymptote equations, substitution to find coordinates, and applying a simple translation. All parts are routine applications of standard techniques with no problem-solving insight needed, though it's slightly above average difficulty due to being FP1 content rather than core A-level. |
| Spec | 1.02n Sketch curves: simple equations including polynomials1.02w Graph transformations: simple transformations of f(x)1.02x Combinations of transformations: multiple transformations |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| Asymptotes are \(y = \pm\frac{1}{2}x\) | M1A1 | OE; M1 for \(y = \pm mx\) |
| Subtotal | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| \(x = 4\) substituted into equation | M1 | |
| \(y^2 = 3\) so \(y = \pm\sqrt{3}\) | A1 | Allow NMS |
| Subtotal | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| \(y\)-coords are \(2 \pm \sqrt{3}\) | B1F | ft wrong answer to (b) |
| Subtotal | 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| Hyperbola is \(\frac{x^2}{4} - (y-2)^2 = 1\) | M1A1 | M1A0 if \(y+2\) used |
| Asymptotes are \(y = 2 \pm \frac{1}{2}x\) | B1F | ft wrong gradients in (a) |
| Subtotal | 3 | |
| Total | 8 |
## Question 5(a):
| Working/Answer | Marks | Guidance |
|---|---|---|
| Asymptotes are $y = \pm\frac{1}{2}x$ | M1A1 | OE; M1 for $y = \pm mx$ |
| **Subtotal** | **2** | |
## Question 5(b):
| Working/Answer | Marks | Guidance |
|---|---|---|
| $x = 4$ substituted into equation | M1 | |
| $y^2 = 3$ so $y = \pm\sqrt{3}$ | A1 | Allow NMS |
| **Subtotal** | **2** | |
## Question 5(c)(i):
| Working/Answer | Marks | Guidance |
|---|---|---|
| $y$-coords are $2 \pm \sqrt{3}$ | B1F | ft wrong answer to (b) |
| **Subtotal** | **1** | |
## Question 5(c)(ii):
| Working/Answer | Marks | Guidance |
|---|---|---|
| Hyperbola is $\frac{x^2}{4} - (y-2)^2 = 1$ | M1A1 | M1A0 if $y+2$ used |
| Asymptotes are $y = 2 \pm \frac{1}{2}x$ | B1F | ft wrong gradients in (a) |
| **Subtotal** | **3** | |
| **Total** | **8** | |
---5 The diagram shows the hyperbola
$$\frac { x ^ { 2 } } { 4 } - y ^ { 2 } = 1$$
and its asymptotes.\\
\includegraphics[max width=\textwidth, alt={}, center]{a0a30197-ca11-40d9-9ccd-30281c5e0fb4-03_531_1013_616_516}
\begin{enumerate}[label=(\alph*)]
\item Write down the equations of the two asymptotes.
\item The points on the hyperbola for which $x = 4$ are denoted by $A$ and $B$.
Find, in surd form, the $y$-coordinates of $A$ and $B$.
\item The hyperbola and its asymptotes are translated by two units in the positive $y$ direction.
Write down:
\begin{enumerate}[label=(\roman*)]
\item the $y$-coordinates of the image points of $A$ and $B$ under this translation;
\item the equations of the hyperbola and the asymptotes after the translation.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2008 Q5 [8]}}