| Exam Board | OCR |
|---|---|
| Module | H240/01 (Pure Mathematics) |
| Year | 2017 |
| Session | Specimen |
| Marks | 9 |
| Topic | Newton-Raphson method |
| Type | Derive Newton-Raphson formula |
| Difficulty | Standard +0.3 This is a standard Newton-Raphson question requiring routine differentiation, algebraic manipulation to derive the formula, calculator work for iterations, and sign-change verification. Part (d) tests understanding of convergence failure (likely due to stationary point or divergence), which is a common textbook scenario. All techniques are straightforward 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 |
The equation $x^3 - x^2 - 5x + 10 = 0$ has exactly one real root $\alpha$.
\begin{enumerate}[label=(\alph*)]
\item Show that the Newton-Raphson iterative formula for finding this root can be written as
$$x_{n+1} = \frac{2x_n^3 - x_n^2 - 10}{3x_n^2 - 2x_n - 5}.$$ [3]
\item Apply the iterative formula in part (a) with initial value $x_1 = -3$ to find $x_2, x_3, x_4$ correct to 4 significant figures. [1]
\item Use a change of sign method to show that $\alpha = -2.533$ is correct to 4 significant figures. [3]
\item Explain why the Newton-Raphson method with initial value $x_1 = -1$ would not converge to $\alpha$. [2]
\end{enumerate}
\hfill \mbox{\textit{OCR H240/01 2017 Q9 [9]}}