| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2006 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Laws of Logarithms |
| Type | Model y=ax^b: linearise and find constants from graph/data |
| Difficulty | Moderate -0.5 This is a standard data transformation question requiring logarithmic conversion of tabular data, deriving a linear relationship (straightforward log manipulation), plotting points, and finding gradient from a graph. While it involves multiple parts, each step is routine and follows a well-practiced textbook procedure with no novel problem-solving required. |
| Spec | 1.06h Logarithmic graphs: reduce y=ax^n and y=kb^x to linear form |
| \(x\) | 4 | 17 | 150 | 300 |
| \(y\) | 1.8 | 5.0 | 30 | 50 |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(X\) values 1.23, 2, 2.18; \(Y\) values 0.70, 1.48 | B3, 2, 1 | −1 for each error |
| (b) lg \(y = \) lg \(k + \) lg \(x^n\) | M1 | |
| lg \(x^n = n\) lg \(x\) | M1 | |
| So \(Y = nX + \) lg \(k\) | A1 | |
| (c) Four points plotted | B2, 1√ | B1 if one error here; ft wrong values in (a) (approx collinear) |
| (d) Good straight line drawn | B1√ | B1 if one error here |
| Method for gradient | M1 | |
| Estimate for \(n\) | A1√ | Allow AWRT 0.75 - 0.78; ft grad of candidate's graph |
**(a)** $X$ values 1.23, 2, 2.18; $Y$ values 0.70, 1.48 | B3, 2, 1 | −1 for each error
**(b)** lg $y = $ lg $k + $ lg $x^n$ | M1
lg $x^n = n$ lg $x$ | M1
So $Y = nX + $ lg $k$ | A1 |
**(c)** Four points plotted | B2, 1√ | B1 if one error here; ft wrong values in (a) (approx collinear)
**(d)** Good straight line drawn | B1√ | B1 if one error here
Method for gradient | M1
Estimate for $n$ | A1√ | Allow AWRT 0.75 - 0.78; ft grad of candidate's graph
**Total for Q6: 11 marks**
---6 [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
$$y = k x ^ { n }$$
where $k$ and $n$ are constants.\\
Experimental evidence has provided the following approximate values:
\begin{center}
\begin{tabular}{ | c | c | c | c | c | }
\hline
$x$ & 4 & 17 & 150 & 300 \\
\hline
$y$ & 1.8 & 5.0 & 30 & 50 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Complete the table in Figure 1, showing values of $X$ and $Y$, where
$$X = \log _ { 10 } x \quad \text { and } \quad Y = \log _ { 10 } y$$
Give each value to two decimal places.
\item Show that if $y = k x ^ { n }$, then $X$ and $Y$ must satisfy an equation of the form
$$Y = a X + b$$
\item Draw on Figure 2 a linear graph relating $X$ and $Y$.
\item Find an estimate for the value of $n$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2006 Q6 [11]}}