| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2005 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Conic sections |
| Type | Conic translation and transformation |
| Difficulty | Moderate -0.5 This is a straightforward ellipse question requiring standard techniques: sketching from standard form (identifying a=3, b=2), substituting x=1 to find intersection points (basic algebra with surds), and applying a horizontal translation (replacing x with x-1). All parts are routine textbook exercises with no problem-solving insight required, making it easier than average but not trivial since it involves Further Maths content and surd manipulation. |
| Spec | 1.02w Graph transformations: simple transformations of f(x)1.03d Circles: equation (x-a)^2+(y-b)^2=r^2 |
| Answer | Marks | Guidance |
|---|---|---|
| Correct shape | B1 | |
| Coordinates \((\pm 3, 0)\), \((0, \pm 2)\) shown | B2,1 (3 marks) | Allow labels on sketch |
| Answer | Marks | Guidance |
|---|---|---|
| Attempt to solve \(\frac{1}{9} + \frac{y^2}{4} = 1\) | M1 | |
| At least one correct root | m1 | Allow decimals; allow \(\sqrt{\frac{32}{9}}\) |
| \(y = \pm\frac{4}{3}\sqrt{2}\) | A1 (3 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Equation is \(\frac{(x-1)^2}{9} + \frac{y^2}{4} = 1\) | M1A1 (2 marks) | M1A0 for e.g. wrong sign |
## Question 2:
**Part (a)**
Correct shape | B1 |
Coordinates $(\pm 3, 0)$, $(0, \pm 2)$ shown | B2,1 (3 marks) | Allow labels on sketch
**Part (b)**
Attempt to solve $\frac{1}{9} + \frac{y^2}{4} = 1$ | M1 |
At least one correct root | m1 | Allow decimals; allow $\sqrt{\frac{32}{9}}$
$y = \pm\frac{4}{3}\sqrt{2}$ | A1 (3 marks) |
**Part (c)**
Equation is $\frac{(x-1)^2}{9} + \frac{y^2}{4} = 1$ | M1A1 (2 marks) | M1A0 for e.g. wrong sign
---2 A curve has equation
$$\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$$
\begin{enumerate}[label=(\alph*)]
\item Sketch the curve, showing the coordinates of the points of intersection with the coordinate axes.
\item Calculate the $y$-coordinates of the points of intersection of the curve with the line $x = 1$. Give your answers in the form $p \sqrt { 2 }$, where $p$ is a rational number.
\item The curve is translated one unit in the positive $x$ direction. Write down the equation of the curve after the translation.
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2005 Q2 [8]}}