| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2007 |
| Session | June |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Straight Lines & Coordinate Geometry |
| Type | Line and curve intersection |
| Difficulty | Standard +0.3 This is a structured multi-part question on ellipse translation and tangency conditions. Parts (a)-(c) involve routine coordinate geometry (finding intercepts, applying translations, substituting to form a quadratic). Part (d) uses the discriminant condition for tangency (b²-4ac=0), which is standard FP1 material. The algebraic manipulation is straightforward, and the question guides students through each step methodically. Slightly easier than average due to its scaffolded nature. |
| Spec | 1.02c Simultaneous equations: two variables by elimination and substitution1.02d Quadratic functions: graphs and discriminant conditions1.02f Solve quadratic equations: including in a function of unknown1.02w Graph transformations: simple transformations of f(x)1.03d Circles: equation (x-a)^2+(y-b)^2=r^2 |
| Answer | Marks | Guidance |
|---|---|---|
| (a) Intersections \(\left(\pm\sqrt{2}, 0\right)\), \((0, \pm 1)\) | B1B1 | Allow B1 for \(\left(\sqrt{2}, 0\right)\), \((0, 1)\) |
| Answer | Marks | Guidance |
|---|---|---|
| (b) Equation is \(\frac{(x-k)^2}{2} + y^2 = 1\) | M1A1 | M1 if only one small error, eg \(x + k\) for \(x - k\) |
| Answer | Marks | Guidance |
|---|---|---|
| (c) Correct elimination of \(y\) | M1 | |
| Correct expansion of squares | M1 | |
| Correct removal of denominator | M1 | |
| Answer convincingly established | A1 | AG |
| Answer | Marks | Guidance |
|---|---|---|
| (d) Tgt \(\Rightarrow 4(k+4)^2 - 12(k^2 + 6) = 0\) | M1 | |
| \(\ldots \Rightarrow k^2 - 4k + 1 = 0\) | m1A1 | OE |
| \(\ldots \Rightarrow k = 2\pm\sqrt{3}\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| (e) Curve to left of line | B1 | |
| Curve to right of line | B2 | Curves must touch the line in approx correct positions |
| SC 1/3 if both curves are incomplete but touch the line correctly |
**(a)** Intersections $\left(\pm\sqrt{2}, 0\right)$, $(0, \pm 1)$ | B1B1 | Allow B1 for $\left(\sqrt{2}, 0\right)$, $(0, 1)$
**Total: 2 marks**
**(b)** Equation is $\frac{(x-k)^2}{2} + y^2 = 1$ | M1A1 | M1 if only one small error, eg $x + k$ for $x - k$
**Total: 2 marks**
**(c)** Correct elimination of $y$ | M1 |
Correct expansion of squares | M1 |
Correct removal of denominator | M1 |
Answer convincingly established | A1 | AG
**Total: 4 marks**
**(d)** Tgt $\Rightarrow 4(k+4)^2 - 12(k^2 + 6) = 0$ | M1 |
$\ldots \Rightarrow k^2 - 4k + 1 = 0$ | m1A1 | OE
$\ldots \Rightarrow k = 2\pm\sqrt{3}$ | A1 |
**Total: 4 marks**
**(e)** Curve to left of line | B1 |
Curve to right of line | B2 | Curves must touch the line in approx correct positions
| | | SC 1/3 if both curves are incomplete but touch the line correctly
**Total: 3 marks**
---
**TOTAL: 75 marks**9 [Figure 3, printed on the insert, is provided for use in this question.]\\
The diagram shows the curve with equation
$$\frac { x ^ { 2 } } { 2 } + y ^ { 2 } = 1$$
and the straight line with equation
$$x + y = 2$$
\includegraphics[max width=\textwidth, alt={}, center]{354cbeda-d84e-433a-8834-a6f20e7e9513-05_805_1499_863_267}
\begin{enumerate}[label=(\alph*)]
\item Write down the exact coordinates of the points where the curve with equation $\frac { x ^ { 2 } } { 2 } + y ^ { 2 } = 1$ intersects the coordinate axes.
\item The curve is translated $k$ units in the positive $x$ direction, where $k$ is a constant. Write down, in terms of $k$, the equation of the curve after this translation.
\item Show that, if the line $x + y = 2$ intersects the translated curve, the $x$-coordinates of the points of intersection must satisfy the equation
$$3 x ^ { 2 } - 2 ( k + 4 ) x + \left( k ^ { 2 } + 6 \right) = 0$$
\item Hence find the two values of $k$ for which the line $x + y = 2$ is a tangent to the translated curve. Give your answer in the form $p \pm \sqrt { q }$, where $p$ and $q$ are integers.
\item On Figure 3, show the translated curves corresponding to these two values of $k$.
\end{table}
\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Figure 2 (for use in Question 5)}
\includegraphics[alt={},max width=\textwidth]{354cbeda-d84e-433a-8834-a6f20e7e9513-10_677_1056_886_466}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Figure 3 (for use in Question 9)}
\includegraphics[alt={},max width=\textwidth]{354cbeda-d84e-433a-8834-a6f20e7e9513-10_798_1488_1891_274}
\end{center}
\end{figure}
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2007 Q9 [15]}}