| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2008 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Generalised Binomial Theorem and Partial Fractions |
| Type | Partial fractions then binomial expansion |
| Difficulty | Moderate -0.5 This is a straightforward data linearization problem requiring algebraic manipulation to show Y = aX + b, then plotting points and reading off gradient/intercept. The algebra is simple substitution, and the graphical work is routine. Below average difficulty for FP1 as it requires no advanced techniques, just careful arithmetic and basic linear regression concepts. |
| Spec | 1.06h Logarithmic graphs: reduce y=ax^n and y=kb^x to linear form2.02c Scatter diagrams and regression lines |
| \(x\) | 1 | 2 | 3 | 4 |
| \(y\) | 0.40 | 1.43 | 2.40 | 3.35 |
| Answer | Marks | Guidance |
|---|---|---|
| (a) Multiplication by \(x + 2\); \(Y = ax + b\) convincingly shown | M1, A1 | 2 marks |
| (b)(i) \(X = 8, 15, 24\) in table; \(Y = 5.72, 12, 20.1\) in table | B1, B1 | 2 marks |
| (b)(ii) Four points plotted; Reasonable line drawn | B1F, B1F | 2 marks |
| (b)(iii) Method for gradient: \(a = \text{gradient} \approx 0.9\); \(b = Y\text{-intercept} \approx -1.5\) | M1, A1, B1F | 3 marks |
**(a)** Multiplication by $x + 2$; $Y = ax + b$ convincingly shown | M1, A1 | 2 marks | applied to all 3 terms; AG
**(b)(i)** $X = 8, 15, 24$ in table; $Y = 5.72, 12, 20.1$ in table | B1, B1 | 2 marks | Allow correct to 2SF
**(b)(ii)** Four points plotted; Reasonable line drawn | B1F, B1F | 2 marks | ft incorrect values in table; ft incorrect points
**(b)(iii)** Method for gradient: $a = \text{gradient} \approx 0.9$; $b = Y\text{-intercept} \approx -1.5$ | M1, A1, B1F | 3 marks | or algebraic method for $a$ or $b$; Allow from 0.88 to 0.93 incl; Allow from $-2$ to $-1$ inclusive; ft incorrect points/line; NMS B1 for $a$, B1 for $b$
**Total: 9 marks**4 [Figure 1 and Figure 2, printed on the insert, are provided for use in this question.]\\
The variables $x$ and $y$ are related by an equation of the form
$$y = a x + \frac { b } { x + 2 }$$
where $a$ and $b$ are constants.
\begin{enumerate}[label=(\alph*)]
\item The variables $X$ and $Y$ are defined by $X = x ( x + 2 ) , Y = y ( x + 2 )$.
Show that $Y = a X + b$.
\item 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
$y$ & 0.40 & 1.43 & 2.40 & 3.35 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\roman*)]
\item Complete the table in Figure 1, showing values of $X$ and $Y$.
\item Draw on Figure 2 a linear graph relating $X$ and $Y$.
\item Estimate the values of $a$ and $b$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2008 Q4 [9]}}