| Exam Board | WJEC |
|---|---|
| Module | Unit 3 (Unit 3) |
| Year | 2023 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Newton-Raphson method |
| Type | Newton-Raphson convergence failure |
| Difficulty | Standard +0.3 This is a straightforward multi-part question on numerical methods covering standard A-level content: sign changes to count roots, Newton-Raphson iteration (routine application of the formula), and identifying why an iterative method fails (likely due to convergence issues or leaving the domain). All parts require only direct application of learned techniques with no novel 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 |
A function $f$ with domain $(-\infty,\infty)$ is defined by $f(x) = 6x^3 + 35x^2 - 7x - 6$.
\begin{enumerate}[label=(\alph*)]
\item Determine the number of roots of the equation $f(x) = 0$ in the interval $[-1, 1]$. [2]
\item Use the Newton-Raphson method to find a root of the equation $f(x) = 0$.
Starting with $x_0 = 1$,
\begin{enumerate}[label=(\roman*)]
\item write down the value of $x_1$,
\item determine the value of the root correct to one decimal place. [4]
\end{enumerate}
\item It is suggested that another iterative sequence
$$x_{n+1} = \sqrt{\frac{7x_n + 6 - 6x_n^3}{35}},$$
starting with $x_0 = -3$, could be used to find a root of the equation $f(x) = 0$.
Explain why this method fails. [2]
\end{enumerate}
\hfill \mbox{\textit{WJEC Unit 3 2023 Q4 [8]}}