| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2024 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Topic | Newton-Raphson method |
| Type | Newton-Raphson with complex derivative required |
| Difficulty | Standard +0.3 This question involves standard application of linear interpolation and one iteration of Newton-Raphson method. While it requires differentiation of a function with a square root term and careful arithmetic, both methods are routine A-level techniques with no conceptual challenges. The multi-part structure and need for accuracy pushes it slightly above average, but it remains a straightforward textbook exercise. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.09d Newton-Raphson method1.09f Trapezium rule: numerical integration |
3.
$$\mathrm { f } ( x ) = x ^ { 3 } - 5 \sqrt { x } - 4 x + 7 \quad x \geqslant 0$$
The equation $\mathrm { f } ( x ) = 0$ has a root $\alpha$ in the interval $[ 0.25,1 ]$
\begin{enumerate}[label=(\alph*)]
\item Use linear interpolation once on the interval [ $0.25,1$ ] to determine an approximation to $\alpha$, giving your answer to 3 decimal places.
The equation $\mathrm { f } ( x ) = 0$ has another root $\beta$ in the interval [1.5, 2.5]
\item Determine $\mathrm { f } ^ { \prime } ( x )$
\item Hence, using $x _ { 0 } = 1.75$ as a first approximation to $\beta$, apply the Newton-Raphson process once to $\mathrm { f } ( x )$ to determine a second approximation to $\beta$, giving your answer to 3 decimal places.
\end{enumerate}
\hfill \mbox{\textit{Edexcel F1 2024 Q3 [7]}}