| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2007 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Equations & Modelling |
| Type | log(y) vs x: convert and interpret |
| Difficulty | Moderate -0.3 This is a standard log-linearization question requiring routine application of logarithm laws to transform y = ab^x into Y = mx + c, followed by straightforward graph plotting and reading off intercept/gradient. While it involves multiple steps (4 parts), each step is mechanical with no novel insight required—slightly easier than average due to its cookbook nature. |
| Spec | 1.06h Logarithmic graphs: reduce y=ax^n and y=kb^x to linear form |
| \(x\) | 1 | 2 | 3 | 4 |
| \(y\) | 3.84 | 6.14 | 9.82 | 15.7 |
| Answer | Marks | Guidance |
|---|---|---|
| (a) Values 0.788, 0.992, 1.196 in table | B2,1 | B1 if one correct (or if wrong number of dp given) |
| (b) \(\lg ab^x = \lg a + \lg b^x\) | M1 | |
| \(\lg b^x = x \lg b\) | M1 | |
| So \(Y = (\lg b) x + \lg a\) | A1 | Allow NMS |
| Answer | Marks |
|---|---|
| (c) Four points plotted; ft wrong values in (a) | B1F |
| Good straight line drawn; ft incorrect points | B1F |
| Answer | Marks | Guidance |
|---|---|---|
| (d) \(a = \) antilog of \(y\)-intercept | M1A1 | |
| \(b = \) antilog of gradient | M1A1 | Accept 2.23 to 2.52; Accept 1.58 to 1.62 |
**(a)** Values 0.788, 0.992, 1.196 in table | B2,1 | B1 if one correct (or if wrong number of dp given)
**(b)** $\lg ab^x = \lg a + \lg b^x$ | M1 |
$\lg b^x = x \lg b$ | M1 |
So $Y = (\lg b) x + \lg a$ | A1 | Allow NMS
**Total: 3 marks**
**(c)** Four points plotted; ft wrong values in (a) | B1F |
Good straight line drawn; ft incorrect points | B1F |
**Total: 2 marks**
**(d)** $a = $ antilog of $y$-intercept | M1A1 |
$b = $ antilog of gradient | M1A1 | Accept 2.23 to 2.52; Accept 1.58 to 1.62
**Total: 4 marks**
---5 [Figure 1 and Figure 2, printed on the insert, are provided for use in this question.] 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.
The following approximate values of $x$ and $y$ have been found.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | }
\hline
$x$ & 1 & 2 & 3 & 4 \\
\hline
$y$ & 3.84 & 6.14 & 9.82 & 15.7 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Complete the table in Figure 1, showing values of $x$ and $Y$, where $Y = \log _ { 10 } y$. Give each value of $Y$ to three decimal places.
\item Show that, if $y = a b ^ { x }$, then $x$ and $Y$ must satisfy an equation of the form
$$Y = m x + c$$
\item Draw on Figure 2 a linear graph relating $x$ and $Y$.
\item Hence find estimates for the values of $a$ and $b$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2007 Q5 [11]}}