| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2011 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Functions |
| Type | Linear transformation to find constants |
| Difficulty | Moderate -0.3 This is a standard logarithmic linearization question requiring students to manipulate logarithms to find constants, plot two points, and read values from a graph. While it involves Further Maths content (FP1), the techniques are routine: taking logs of both sides, identifying m and c from the linear form, and basic graph work. The multi-part structure adds length but not conceptual difficulty—each step follows a well-practiced procedure with no novel problem-solving required. |
| Spec | 1.06h Logarithmic graphs: reduce y=ax^n and y=kb^x to linear form |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Use of one law of logs or exponentials | M1 | |
| \(\lg a = c\) and \(\lg b = m\) | A1 | OE; both needed |
| So \(a = 10^c\) and \(b = 10^m\) | A1 | |
| Total | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Points \((1, 1.08)\), \((5, 1.43)\) plotted | M1A1 | M1 A0 if one point correct |
| Straight line drawn through points | A1F | ft small inaccuracy |
| Total | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Attempt at antilog of \(Y(3)\) | M1 | OE |
| When \(x = 3\), \(Y \approx 1.25\) so \(y \approx 18\) | A1 | Allow AWRT 18 |
| Total | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Attempt at \(a\) as antilog of \(Y\)-intercept | M1 | OE |
| \(a \approx 9.3\) to \(10\) | A1 | AWRT |
| Total | 2 |
## Question 4:
### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Use of one law of logs or exponentials | M1 | |
| $\lg a = c$ and $\lg b = m$ | A1 | OE; both needed |
| So $a = 10^c$ and $b = 10^m$ | A1 | |
| **Total** | **3** | |
### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Points $(1, 1.08)$, $(5, 1.43)$ plotted | M1A1 | M1 A0 if one point correct |
| Straight line drawn through points | A1F | ft small inaccuracy |
| **Total** | **3** | |
### Part (c)(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Attempt at antilog of $Y(3)$ | M1 | OE |
| When $x = 3$, $Y \approx 1.25$ so $y \approx 18$ | A1 | Allow AWRT 18 |
| **Total** | **2** | |
### Part (c)(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Attempt at $a$ as antilog of $Y$-intercept | M1 | OE |
| $a \approx 9.3$ to $10$ | A1 | AWRT |
| **Total** | **2** | |
---4 The variables $x$ and $Y$, where $Y = \log _ { 10 } y$, are related by the equation
$$Y = m x + c$$
where $m$ and $c$ are constants.
\begin{enumerate}[label=(\alph*)]
\item Given that $y = a b ^ { x }$, express $a$ in terms of $c$, and $b$ in terms of $m$.
\item It is given that $y = 12$ when $x = 1$ and that $y = 27$ when $x = 5$.
On the diagram below, draw a linear graph relating $x$ and $Y$.
\item Use your graph to estimate, to two significant figures:
\begin{enumerate}[label=(\roman*)]
\item the value of $y$ when $x = 3$;
\item the value of $a$.\\
\includegraphics[max width=\textwidth, alt={}, center]{7441c4e6-5448-483b-b100-f8076e7e6cd8-3_976_1173_1110_484}
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2011 Q4 [10]}}