| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2023 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Topic | Newton-Raphson method |
| Type | Newton-Raphson convergence failure |
| Difficulty | Challenging +1.2 This question tests standard Newton-Raphson application with a twist: identifying when it fails (f'(0.25)=0). Part (a)(i) requires differentiation of negative/fractional powers (routine for Further Maths). Part (a)(ii) tests conceptual understanding of N-R failure. Part (a)(iii) and (b) are mechanical applications of N-R and linear interpolation formulas. While it requires multiple techniques and careful calculation, these are all standard Further Pure 1 procedures without requiring novel insight or extended reasoning. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.09a Sign change methods: locate roots1.09d Newton-Raphson method |
4.
$$f ( x ) = 1 - \frac { 1 } { 8 x ^ { 4 } } + \frac { 2 } { 7 \sqrt { x ^ { 7 } } } \quad x > 0$$
The equation $\mathrm { f } ( x ) = 0$ has a single root, $\alpha$, that lies in the interval $[ 0.15,0.25 ]$
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Determine $\mathrm { f } ^ { \prime } ( x )$
\item Explain why 0.25 cannot be used as an initial approximation for $\alpha$ in the Newton-Raphson process.
\item Taking 0.15 as a first approximation to $\alpha$ apply the Newton-Raphson process once to $\mathrm { f } ( x )$ to obtain a second approximation to $\alpha$ Give your answer to 3 decimal places.
\end{enumerate}\item Use linear interpolation once on the interval $[ 0.15,0.25 ]$ to find another approximation to $\alpha$ Give your answer to 3 decimal places.
\end{enumerate}
\hfill \mbox{\textit{Edexcel F1 2023 Q4 [8]}}