| Exam Board | OCR |
|---|---|
| Module | H240/03 (Pure Mathematics and Mechanics) |
| Year | 2021 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Show root in interval |
| Difficulty | Standard +0.3 This is a straightforward numerical methods question requiring routine techniques: substitution to verify a sign change, rearranging an equation to match a given iterative formula form, and applying the iteration. All steps are standard A-level procedures with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.05i Inverse trig functions: arcsin, arccos, arctan domains and graphs1.09a Sign change methods: locate roots1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
The equation $6\arcsin(2x - 1) - x^2 = 0$ has exactly one real root.
\begin{enumerate}[label=(\alph*)]
\item Show by calculation that the root lies between 0.5 and 0.6. [2]
\end{enumerate}
In order to find the root, the iterative formula
$x_{n+1} = p + q\sin(rx_n^2)$,
with initial value $x_0 = 0.5$, is to be used.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Determine the values of the constants $p$, $q$ and $r$. [2]
\item Hence find the root correct to 4 significant figures. Show the result of each step of the iteration process. [2]
\end{enumerate}
\hfill \mbox{\textit{OCR H240/03 2021 Q6 [6]}}