| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2009 |
| Session | January |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Conic sections |
| Type | Conic tangent through external point |
| Difficulty | Standard +0.3 This is a structured multi-part question on hyperbolas that guides students through standard techniques: finding asymptotes (routine), sketching (basic), substituting a line equation into the hyperbola (algebraic manipulation), using the discriminant to prove intersection (standard FP1 technique), and finding coordinates. While it involves several steps, each part is straightforward with clear signposting, making it slightly easier than average for an A-level question. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions1.02n Sketch curves: simple equations including polynomials1.02q Use intersection points: of graphs to solve equations |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Asymptotes are \(y = \pm\sqrt{2}\,x\) | M1A1 | M1A0 if correct but not in required form |
| Total | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Asymptotes correct on sketch | B1F | With gradients steeper than 1; ft from \(y=\pm mx\) with \(m>1\) |
| Two branches in roughly correct positions | B1 | Asymptotes \(y=\pm mx\) needed here |
| Approaching asymptotes correctly | B1 | |
| Total | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Elimination of \(y\) | M1 | |
| Clearing denominator correctly | M1 | |
| \(x^2 - 2cx - (c^2+2) = 0\) | m1A1 | Convincingly found (AG) |
| Total | 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Discriminant \(= 8c^2+8\) | B1 | Accept unsimplified |
| \(\ldots > 0\) for all \(c\), hence result | E1 | OE |
| Total | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Solving gives \(x = c \pm \sqrt{2(c^2+1)}\) | M1A1 | |
| \(y = x+c = 2c \pm \sqrt{2(c^2+1)}\) | A1 | Accept \(y = c + \dfrac{2c \pm \sqrt{8c^2+8}}{2}\) |
| Total | 3 | Question Total: 14 |
## Question 9(a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Asymptotes are $y = \pm\sqrt{2}\,x$ | M1A1 | M1A0 if correct but not in required form |
| **Total** | **2** | |
## Question 9(b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Asymptotes correct on sketch | B1F | With gradients steeper than 1; ft from $y=\pm mx$ with $m>1$ |
| Two branches in roughly correct positions | B1 | Asymptotes $y=\pm mx$ needed here |
| Approaching asymptotes correctly | B1 | |
| **Total** | **3** | |
## Question 9(c)(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Elimination of $y$ | M1 | |
| Clearing denominator correctly | M1 | |
| $x^2 - 2cx - (c^2+2) = 0$ | m1A1 | Convincingly found (AG) |
| **Total** | **4** | |
## Question 9(c)(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Discriminant $= 8c^2+8$ | B1 | Accept unsimplified |
| $\ldots > 0$ for all $c$, hence result | E1 | OE |
| **Total** | **2** | |
## Question 9(c)(iii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Solving gives $x = c \pm \sqrt{2(c^2+1)}$ | M1A1 | |
| $y = x+c = 2c \pm \sqrt{2(c^2+1)}$ | A1 | Accept $y = c + \dfrac{2c \pm \sqrt{8c^2+8}}{2}$ |
| **Total** | **3** | **Question Total: 14** |
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**OVERALL TOTAL: 75**9 A hyperbola $H$ has equation
$$x ^ { 2 } - \frac { y ^ { 2 } } { 2 } = 1$$
\begin{enumerate}[label=(\alph*)]
\item Find the equations of the two asymptotes of $H$, giving each answer in the form $y = m x$.
\item Draw a sketch of the two asymptotes of $H$, using roughly equal scales on the two coordinate axes. Using the same axes, sketch the hyperbola $H$.
\item \begin{enumerate}[label=(\roman*)]
\item Show that, if the line $y = x + c$ intersects $H$, the $x$-coordinates of the points of intersection must satisfy the equation
$$x ^ { 2 } - 2 c x - \left( c ^ { 2 } + 2 \right) = 0$$
\item Hence show that the line $y = x + c$ intersects $H$ in two distinct points, whatever the value of $c$.
\item Find, in terms of $c$, the $y$-coordinates of these two points.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2009 Q9 [14]}}