| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Rearrange to iterative form |
| Difficulty | Standard +0.2 This is a standard C3 iteration question covering routine techniques: algebraic rearrangement (straightforward manipulation), applying an iterative formula (direct substitution), and change of sign method for proving accuracy. Part (d) requires recognizing when the formula is undefined (denominator = -1 or negative under square root), but this is a common textbook exercise type. All parts follow predictable patterns with no novel problem-solving required, making it slightly easier than the average A-level question. |
| 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 |
$\text{f}(x) = x^3 + x^2 - 4x - 1$.
The equation f(x) = 0 has only one positive root, $\alpha$.
\begin{enumerate}[label=(\alph*)]
\item Show that f(x) = 0 can be rearranged as
$$x = \sqrt{\frac{4x+1}{x+1}}, \quad x \neq -1.$$ [2]
\end{enumerate}
The iterative formula $x_{n+1} = \sqrt{\frac{4x_n+1}{x_n+1}}$ is used to find an approximation to $\alpha$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Taking $x_1 = 1$, find, to 2 decimal places, the values of $x_2$, $x_3$ and $x_4$. [3]
\item By choosing values of $x$ in a suitable interval, prove that $\alpha = 1.70$, correct to 2 decimal places. [3]
\item Write down a value of $x_1$ for which the iteration formula $x_{n+1} = \sqrt{\frac{4x_n+1}{x_n+1}}$ does not produce a valid value for $x_2$.
Justify your answer. [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C3 Q4 [10]}}