| Exam Board | OCR |
|---|---|
| Module | H240/03 (Pure Mathematics and Mechanics) |
| Year | 2022 |
| Session | June |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Topic | Newton-Raphson method |
| Type | Derive equation from calculus condition |
| Difficulty | Standard +0.3 This is a structured multi-part question requiring differentiation using the quotient rule, solving to obtain a cubic equation, sign-change verification, iteration analysis (checking |g'(x)| > 1), and Newton-Raphson application. While it involves several techniques, each part is clearly signposted with standard A-level methods and no novel insight is required—slightly easier than average due to the scaffolding. |
| Spec | 1.07l Derivative of ln(x): and related functions1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates1.09a Sign change methods: locate roots1.09b Sign change methods: understand failure cases1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams1.09d Newton-Raphson method |
\textbf{In this question you must show detailed reasoning.}
\includegraphics{figure_5}
The diagram shows the curve with equation $y = \frac{2x - 3}{4x^2 + 1}$. The tangent to the curve at the point $P$ has gradient 2.
\begin{enumerate}[label=(\alph*)]
\item Show that the $x$-coordinate of $P$ satisfies the equation
$$4x^3 + 3x - 3 = 0.$$ [5]
\item Show by calculation that the $x$-coordinate of $P$ lies between 0.5 and 1. [2]
\item Show that the iteration
$$x_{n+1} = \frac{3 - 4x_n^3}{3}$$
cannot converge to the $x$-coordinate of $P$ whatever starting value is used. [2]
\item Use the Newton-Raphson method, with initial value 0.5, to determine the coordinates of $P$ correct to 5 decimal places. [5]
\end{enumerate}
\hfill \mbox{\textit{OCR H240/03 2022 Q5 [14]}}