| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2021 |
| Session | June |
| Marks | 6 |
| Topic | Newton-Raphson method |
| Type | Newton-Raphson convergence failure |
| Difficulty | Moderate -0.3 This is a standard Newton-Raphson question with routine tasks: (a) uses the intermediate value theorem with simple substitution, (b) applies the Newton-Raphson formula mechanically with straightforward differentiation, and (c) asks students to identify that f'(-1)=0 causes division by zero. All parts are textbook exercises requiring no problem-solving insight, making it slightly easier than average. |
| Spec | 1.09a Sign change methods: locate roots1.09d Newton-Raphson method1.09e Iterative method failure: convergence conditions |
The equation $x^3 - 3x + 1 = 0$ has three real roots.
\begin{enumerate}[label=(\alph*)]
\item Show that one of the roots lies between $-2$ and $-1$
[2 marks]
\item Taking $x_1 = -2$ as the first approximation to one of the roots, use the Newton-Raphson method to find $x_2$, the second approximation.
[3 marks]
\item Explain why the Newton-Raphson method fails in the case when the first approximation is $x_1 = -1$
[1 mark]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2021 Q2 [6]}}