| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2009 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Straight Lines & Coordinate Geometry |
| Type | Line and curve intersection |
| Difficulty | Standard +0.3 Part (a) is a straightforward coordinate geometry derivation using the two-point form of a line and finding x-intercept. Parts (b)(i-ii) involve substituting into a quartic, solving equations, and numerical approximation. While it spans multiple steps, each component uses standard A-level techniques without requiring novel insight—slightly easier than average due to the guided structure. |
| Spec | 1.03a Straight lines: equation forms y=mx+c, ax+by+c=0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Use of similar triangles or algebra | M1 | Some progress needed |
| Correct relationship established | m1A1 | e.g. \(\dfrac{r-a}{c} = \dfrac{b-a}{c-d}\) |
| Hence result convincingly shown | A1 | AG |
| Total | 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(c = f(a) = 24\), \(d = f(b) = -21\) | B1, B1 | |
| \(r = \dfrac{38}{15}\ (\approx 2.5333)\) | B1F | Allow AWRT 2.53; ft small error |
| Total | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\beta = 20^{\frac{1}{3}} \approx 2.714(4)\) | M1A1 | Allow AWRT 2.71 |
| So \(\beta - r \approx 0.181 \approx 0.18\) (AG) | A1 | Allow only 2dp if earlier values to 3dp |
| Total | 10 |
## Question 7(a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Use of similar triangles or algebra | M1 | Some progress needed |
| Correct relationship established | m1A1 | e.g. $\dfrac{r-a}{c} = \dfrac{b-a}{c-d}$ |
| Hence result convincingly shown | A1 | AG |
| **Total** | **4** | |
## Question 7(b)(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $c = f(a) = 24$, $d = f(b) = -21$ | B1, B1 | |
| $r = \dfrac{38}{15}\ (\approx 2.5333)$ | B1F | Allow AWRT 2.53; ft small error |
| **Total** | **3** | |
## Question 7(b)(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\beta = 20^{\frac{1}{3}} \approx 2.714(4)$ | M1A1 | Allow AWRT 2.71 |
| So $\beta - r \approx 0.181 \approx 0.18$ (AG) | A1 | Allow only 2dp if earlier values to 3dp |
| **Total** | **10** | |
---7 The points $P ( a , c )$ and $Q ( b , d )$ lie on the curve with equation $y = \mathrm { f } ( x )$. The straight line $P Q$ intersects the $x$-axis at the point $R ( r , 0 )$. The curve $y = \mathrm { f } ( x )$ intersects the $x$-axis at the point $S ( \beta , 0 )$.\\
\includegraphics[max width=\textwidth, alt={}, center]{38c2a2c8-84cc-4bd2-b3ad-f9dee59763ba-4_951_971_470_539}
\begin{enumerate}[label=(\alph*)]
\item Show that
$$r = a + c \left( \frac { b - a } { c - d } \right)$$
\item Given that
$$a = 2 , b = 3 \text { and } \mathrm { f } ( x ) = 20 x - x ^ { 4 }$$
\begin{enumerate}[label=(\roman*)]
\item find the value of $r$;
\item show that $\beta - r \approx 0.18$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2009 Q7 [10]}}