| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2009 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Functions |
| Type | Linear transformation to find constants |
| Difficulty | Moderate -0.3 This is a standard FP1 linearisation question requiring log transformation of an exponential relationship and reading values from a graph. Part (a) is routine algebraic manipulation, while part (b) involves straightforward graph reading with inverse operations. Slightly easier than average A-level due to minimal problem-solving and no calculation of constants required. |
| Spec | 1.06h Logarithmic graphs: reduce y=ax^n and y=kb^x to linear form2.02c Scatter diagrams and regression lines |
| Answer | Marks | Guidance |
|---|---|---|
| \(y = ab^x \Rightarrow \log_{10}y = \log_{10}a + x\log_{10}b\) | M1 | Take logs |
| \(Y = \log_{10}a + x\log_{10}b\) | A1 | |
| So \(m = \log_{10}b\), \(c = \log_{10}a\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| At \(x = 2.3\), read \(Y \approx 1.85\) (from graph) | M1 | |
| \(y = 10^{1.85} \approx 70.8\) | A1 | Allow answers in range \([70, 72]\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(y = 80 \Rightarrow Y = \log_{10}80 \approx 1.903\) | M1 | |
| Read \(x \approx 2.2\) from graph | A1 | Allow suitable range |
# Question 4:
## Part (a)
| $y = ab^x \Rightarrow \log_{10}y = \log_{10}a + x\log_{10}b$ | M1 | Take logs |
| $Y = \log_{10}a + x\log_{10}b$ | A1 | |
| So $m = \log_{10}b$, $c = \log_{10}a$ | A1 | |
## Part (b)(i)
| At $x = 2.3$, read $Y \approx 1.85$ (from graph) | M1 | |
| $y = 10^{1.85} \approx 70.8$ | A1 | Allow answers in range $[70, 72]$ |
## Part (b)(ii)
| $y = 80 \Rightarrow Y = \log_{10}80 \approx 1.903$ | M1 | |
| Read $x \approx 2.2$ from graph | A1 | Allow suitable range |
---4 The variables $x$ and $y$ are known to be related by an equation of the form
$$y = a b ^ { x }$$
where $a$ and $b$ are constants.
\begin{enumerate}[label=(\alph*)]
\item Given that $Y = \log _ { 10 } y$, show that $x$ and $Y$ must satisfy an equation of the form
$$Y = m x + c$$
\item The diagram shows the linear graph which has equation $Y = m x + c$.\\
\includegraphics[max width=\textwidth, alt={}, center]{932d4c7e-6514-4543-b1d1-753fca5a08fd-5_744_720_833_699}
Use this graph to calculate:
\begin{enumerate}[label=(\roman*)]
\item an approximate value of $y$ when $x = 2.3$, giving your answer to one decimal place;
\item an approximate value of $x$ when $y = 80$, giving your answer to one decimal place.\\
(You are not required to find the values of $m$ and $c$.)
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2009 Q4 [7]}}