| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2010 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Logarithmic graph for power law |
| Difficulty | Moderate -0.8 This is a straightforward data transformation and linear graph question requiring only routine algebraic manipulation (squaring x values), plotting points, drawing a line of best fit, and reading off values. The conceptual demand is low—students simply need to recognize that y = ax² + b becomes linear when plotted against X = x². All steps are mechanical with no problem-solving insight required. |
| Spec | 2.02c Scatter diagrams and regression lines |
| \(\boldsymbol { x }\) | 2 | 4 | 6 | 8 |
| \(\boldsymbol { y }\) | 6.0 | 10.5 | 18.0 | 28.2 |
| \(\boldsymbol { x }\) | 2 | 4 | 6 | 8 |
| \(\boldsymbol { X }\) | ||||
| \(\boldsymbol { y }\) | 6.0 | 10.5 | 18.0 | 28.2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(X = 4, 16, 36, 64\) | B1 | All four values correct |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Correct linear graph plotted through points | B2 | B1 for at least 3 points correctly plotted; B2 for good straight line |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Read off \(X\) when \(y=15\), then \(x = \sqrt{X}\) | M1 | Correct method using graph |
| \(x \approx 4.7\) (accept range ~4.6–4.8) | A1 | Correct reading |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Gradient \(= a \approx 0.36\) (accept ~0.35–0.37) | B1 | Correct gradient |
| Intercept \(= b \approx 5.2\) (accept ~5.0–5.4) | B1 | Correct intercept |
| Method shown for finding both | M1 |
# Question 4:
## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $X = 4, 16, 36, 64$ | B1 | All four values correct |
## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Correct linear graph plotted through points | B2 | B1 for at least 3 points correctly plotted; B2 for good straight line |
## Part (c)(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Read off $X$ when $y=15$, then $x = \sqrt{X}$ | M1 | Correct method using graph |
| $x \approx 4.7$ (accept range ~4.6–4.8) | A1 | Correct reading |
## Part (c)(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Gradient $= a \approx 0.36$ (accept ~0.35–0.37) | B1 | Correct gradient |
| Intercept $= b \approx 5.2$ (accept ~5.0–5.4) | B1 | Correct intercept |
| Method shown for finding both | M1 | |
---4 The variables $x$ and $y$ are related by an equation of the form
$$y = a x ^ { 2 } + b$$
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
$\boldsymbol { x }$ & 2 & 4 & 6 & 8 \\
\hline
$\boldsymbol { y }$ & 6.0 & 10.5 & 18.0 & 28.2 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Complete the table below, showing values of $X$, where $X = x ^ { 2 }$.
\item On the diagram below, draw a linear graph relating $X$ and $y$.
\item Use your graph to find estimates, to two significant figures, for:
\begin{enumerate}[label=(\roman*)]
\item the value of $x$ when $y = 15$;
\item the values of $a$ and $b$.\\
(a)
\begin{center}
\begin{tabular}{ | l | c | c | c | c | }
\hline
$\boldsymbol { x }$ & 2 & 4 & 6 & 8 \\
\hline
$\boldsymbol { X }$ & & & & \\
\hline
$\boldsymbol { y }$ & 6.0 & 10.5 & 18.0 & 28.2 \\
\hline
\end{tabular}
\end{center}
(b)\\
\includegraphics[max width=\textwidth, alt={}, center]{763d89e4-861a-4754-a93c-d0902987673f-05_771_1586_1772_274}
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2010 Q4 [8]}}