| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2007 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Conic sections |
| Type | Conic tangent through external point |
| Difficulty | Standard +0.3 This is a straightforward hyperbola question testing standard techniques: substitution to find points, sketching, finding tangent equations, and determining tangency conditions. All parts follow routine procedures with no novel insight required. Part (d) involves algebraic manipulation and recognizing a repeated root indicates tangency, which is a standard A-level concept. Slightly easier than average due to the step-by-step guidance and familiar methods. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions1.02n Sketch curves: simple equations including polynomials1.02q Use intersection points: of graphs to solve equations1.03d Circles: equation (x-a)^2+(y-b)^2=r^2 |
| Answer | Marks | Guidance |
|---|---|---|
| \(x = 10 \Rightarrow 4 - \frac{y^2}{9} = 1\) | M1 | |
| \(\Rightarrow y^2 = 27\) | A1 | |
| \(\Rightarrow y = \pm 3\sqrt{3}\) | A1 | 3 marks |
| Answer | Marks | Guidance |
|---|---|---|
| One branch generally correct | B1 | |
| Both branches correct | B1 | |
| Intersections at \((\pm 5, 0)\) | B1 | 3 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Required tangent is \(x = 5\) | B1F | 1 mark |
| Answer | Marks | Guidance |
|---|---|---|
| \(y\) correctly eliminated | M1 | |
| Fractions correctly cleared | m1 | |
| \(16x^2 - 200x + 625 = 0\) | A1 | 3 marks |
| Answer | Marks | Guidance |
|---|---|---|
| \(x = \frac{25}{4}\) | B1 | |
| Equal roots \(\Rightarrow\) tangency | E1 | 2 marks |
### Part (a)
$x = 10 \Rightarrow 4 - \frac{y^2}{9} = 1$ | M1 | |
$\Rightarrow y^2 = 27$ | A1 | |
$\Rightarrow y = \pm 3\sqrt{3}$ | A1 | 3 marks | PI
### Part (b)
One branch generally correct | B1 | | Asymptotes not needed
Both branches correct | B1 | | With implied asymptotes
Intersections at $(\pm 5, 0)$ | B1 | 3 marks |
### Part (c)
Required tangent is $x = 5$ | B1F | 1 mark | ft wrong value in (b)
### Part (d)(i)
$y$ correctly eliminated | M1 | |
Fractions correctly cleared | m1 | |
$16x^2 - 200x + 625 = 0$ | A1 | 3 marks | convincingly shown (AG)
### Part (d)(ii)
$x = \frac{25}{4}$ | B1 | | No need to mention repeated root, but B0 if other values given as well
Equal roots $\Rightarrow$ tangency | E1 | 2 marks | Accept 'it's a tangent'
### **Total for Question 8: 12 marks**
---
## **GRAND TOTAL: 75 marks**8 A curve $C$ has equation
$$\frac { x ^ { 2 } } { 25 } - \frac { y ^ { 2 } } { 9 } = 1$$
\begin{enumerate}[label=(\alph*)]
\item Find the $y$-coordinates of the points on $C$ for which $x = 10$, giving each answer in the form $k \sqrt { 3 }$, where $k$ is an integer.
\item Sketch the curve $C$, indicating the coordinates of any points where the curve intersects the coordinate axes.
\item Write down the equation of the tangent to $C$ at the point where $C$ intersects the positive $x$-axis.
\item \begin{enumerate}[label=(\roman*)]
\item Show that, if the line $y = x - 4$ intersects $C$, the $x$-coordinates of the points of intersection must satisfy the equation
$$16 x ^ { 2 } - 200 x + 625 = 0$$
\item Solve this equation and hence state the relationship between the line $y = x - 4$ and the curve $C$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2007 Q8 [12]}}