| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2023 |
| Session | June |
| Marks | 6 |
| Topic | Newton-Raphson method |
| Type | Newton-Raphson with derivative given or simple |
| Difficulty | Standard +0.3 This is a straightforward Further Maths question testing standard techniques: sign change for root existence, one iteration of Newton-Raphson with values provided, and proving uniqueness via monotonicity. All parts are routine applications of A-level methods with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.09a Sign change methods: locate roots1.09d Newton-Raphson method |
You are given that $f(x) = \ln(2x - 5) + 2x^2 - 30$, for $x > 2.5$.
\begin{enumerate}[label=(\roman*)]
\item Show that $f(x) = 0$ has a root $\alpha$ in the interval $[3.5, 4]$. [2]
\end{enumerate}
A student takes 4 as the first approximation to $\alpha$.
Given $f(4) = 3.099$ and $f'(4) = 16.67$ to 4 significant figures,
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{1}
\item apply the Newton-Raphson procedure once to obtain a second approximation for $\alpha$, giving your answer to 3 significant figures. [2]
\item Show that $\alpha$ is the only root of $f(x) = 0$. [2]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2023 Q3 [6]}}