| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2022 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Topic | Sign Change & Interval Methods |
| Type | Pure Interval Bisection Only |
| Difficulty | Standard +0.3 This is a straightforward application of standard numerical methods (sign change, interval bisection, linear interpolation) with a given function and table. Parts (a)-(c) involve routine substitution and bisection algorithm, while part (d) applies a standard linear interpolation formula. The function is complex but all calculations are guided, requiring no novel insight—slightly easier than average A-level due to its procedural nature. |
| Spec | 1.09a Sign change methods: locate roots1.09b Sign change methods: understand failure cases1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| \(x\) | 1 | 2 | 3 | 4 | 5 |
| \(f(x)\) | 0.5 | \(-0.2885\) | 0.5834 |
$$f(x) = 2x^{-\frac{2}{3}} + \frac{1}{2}x - \frac{1}{3x - 5} - \frac{5}{2} \quad x \neq \frac{5}{3}$$
The table below shows values of $f(x)$ for some values of $x$, with values of $f(x)$ given to 4 decimal places where appropriate.
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$x$ & 1 & 2 & 3 & 4 & 5 \\
\hline
$f(x)$ & 0.5 & & $-0.2885$ & & 0.5834 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Complete the table giving the values to 4 decimal places.
[2]
\end{enumerate}
The equation $f(x) = 0$ has exactly one positive root, $\alpha$.
Using the values in the completed table and explaining your reasoning,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item determine an interval of width one that contains $\alpha$.
[2]
\item Hence use interval bisection twice to obtain an interval of width 0.25 that contains $\alpha$.
[3]
\end{enumerate}
Given also that the equation $f(x) = 0$ has a negative root, $\beta$, in the interval $[-1, -0.5]$
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item use linear interpolation once on this interval to find an approximation for $\beta$.
Give your answer to 3 significant figures.
[3]
\end{enumerate}
\hfill \mbox{\textit{Edexcel F1 2022 Q8 [10]}}