| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2005 |
| Session | January |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Logarithmic graph for power law |
| Difficulty | Moderate -0.3 This is a standard data linearization exercise requiring students to transform given data points, plot them, and extract constants from gradient and intercept. While it involves cube and square transformations, the method is routine for FP1 students and requires only careful arithmetic and graph reading rather than problem-solving insight. |
| Spec | 1.06h Logarithmic graphs: reduce y=ax^n and y=kb^x to linear form |
| \(x\) | 1.5 | 4.0 | 5.0 | 6.5 | 8.0 |
| \(y\) | 5.0 | 6.3 | 7.0 | 8.0 | 9.0 |
| Answer | Marks | Guidance |
|---|---|---|
| \((X, Y)\) values: \((2.25, 125)\), \((16, 250)\), \((25, 343)\), \((42.25, 512)\), \((64, 729)\) | B2,1 | PI by candidate's graph; ft wrong values |
| Five points accurately plotted | B2,1ft | |
| Reasonable straight line drawn | B1ft (5 marks) | ft errors in plotting |
| Answer | Marks |
|---|---|
| Calculation of gradient of line | M1 |
| Value of \(a\) equal to gradient found | A1 |
| Value of \(b\) = \(y\)-intercept of line | B1 (3 marks) |
## Question 7:
**Part (a)**
$(X, Y)$ values: $(2.25, 125)$, $(16, 250)$, $(25, 343)$, $(42.25, 512)$, $(64, 729)$ | B2,1 | PI by candidate's graph; ft wrong values
Five points accurately plotted | B2,1ft |
Reasonable straight line drawn | B1ft (5 marks) | ft errors in plotting
**Part (b)**
Calculation of gradient of line | M1 |
Value of $a$ equal to gradient found | A1 |
Value of $b$ = $y$-intercept of line | B1 (3 marks) |
---7 [Figure 1, printed on the insert, is provided for use in this question.]\\
The variables $x$ and $y$ are known to be related by an equation of the form
$$y ^ { 3 } = a x ^ { 2 } + b$$
where $a$ and $b$ are constants.
Experimental evidence has provided the following approximate values:
\begin{center}
\begin{tabular}{ | l | l | l | l | l | l | }
\hline
$x$ & 1.5 & 4.0 & 5.0 & 6.5 & 8.0 \\
\hline
$y$ & 5.0 & 6.3 & 7.0 & 8.0 & 9.0 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item On Figure 1, draw a linear graph connecting the variables $X$ and $Y$, where
$$X = x ^ { 2 } \quad \text { and } \quad Y = y ^ { 3 }$$
\item From your graph, find approximate values for the constants $a$ and $b$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2005 Q7}}