| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2013 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear regression |
| Type | Linearize non-linear relationships |
| Difficulty | Standard +0.8 This is a Further Maths statistics question requiring students to identify appropriate transformations to linearize data, then use the linearized graph to determine unknown parameters in the original non-linear model. It combines algebraic manipulation with graphical interpretation and requires understanding of logarithmic transformations, placing it moderately above average difficulty. |
| 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 |
|---|---|---|
| \(\log_{10} y = \log_{10} a + \log_{10} x^n\) | M1 | Take logs and apply one log law correctly. PI. Apply a further log law correctly. |
| \(\log_{10} y = \log_{10} a + n\log_{10} x\) | m1 | |
| \(Y = \log_{10} a + nX\) (linear relationship between \(Y\) and \(X\)) | A1 | Correct eqn. with base 10 (or lg or later evidence of use of base 10 if log without base here) |
| Answer | Marks | Guidance |
|---|---|---|
| \(n =\) gradient of line | M1 | Stated or used. Accept \(n = \pm\frac{2}{3}\) OE as evidence |
| \(n = -\dfrac{2}{3}\) | A1 | \(n = -\frac{2}{3}\) (OE 3sf) |
| \(\log_{10} a = 4\) | M1 | Equating c's constant term [must involve a log] in c's (a) eqn. to the \(Y\)-intercept value PI by correct value of \(a\) |
| \(a = 10^4\ (= 10\,000)\) | A1 |
## Question 7:
### Part (a):
$y = ax^n \Rightarrow \log_{10} y = \log_{10} ax^n$
$\log_{10} y = \log_{10} a + \log_{10} x^n$ | M1 | Take logs and apply one log law correctly. PI. Apply a further log law correctly.
$\log_{10} y = \log_{10} a + n\log_{10} x$ | m1 |
$Y = \log_{10} a + nX$ (linear relationship between $Y$ and $X$) | A1 | Correct eqn. with base 10 (or lg or later evidence of use of base 10 if log without base here)
### Part (b):
$n =$ gradient of line | M1 | Stated or used. Accept $n = \pm\frac{2}{3}$ OE as evidence
$n = -\dfrac{2}{3}$ | A1 | $n = -\frac{2}{3}$ (OE 3sf)
$\log_{10} a = 4$ | M1 | Equating c's constant term [must involve a log] in c's (a) eqn. to the $Y$-intercept value PI by correct value of $a$
$a = 10^4\ (= 10\,000)$ | A1 |
---\begin{enumerate}[label=(\alph*)]
\item Show that there is a linear relationship between $Y$ and $X$.
\item The graph of $Y$ against $X$ is shown in the diagram.\\
\includegraphics[max width=\textwidth, alt={}, center]{cf9337b9-b766-4ce5-967c-5d7522e2aa42-4_748_858_849_593}
Find the value of $n$ and the value of $a$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2013 Q7 [7]}}