| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Show root in interval |
| Difficulty | Standard +0.3 This is a structured multi-part question on inverse trigonometric functions and iterative methods. Parts (i)-(iii) involve standard transformations, graphical reasoning, and sign-change verification—all routine C3 techniques. Part (iv) requires running an iteration and connecting it to the original equation through algebraic manipulation, which is slightly more demanding but still follows a predictable pattern for this module. Overall, slightly easier than average due to the scaffolded structure and standard methods. |
| Spec | 1.02w Graph transformations: simple transformations of f(x)1.05i Inverse trig functions: arcsin, arccos, arctan domains and graphs1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
\includegraphics{figure_7}
The diagram shows the curve with equation $y = \cos^{-1} x$.
\begin{enumerate}[label=(\roman*)]
\item Sketch the curve with equation $y = 3 \cos^{-1}(x - 1)$, showing the coordinates of the points where the curve meets the axes. [3]
\item By drawing an appropriate straight line on your sketch in part (i), show that the equation $3 \cos^{-1}(x - 1) = x$ has exactly one root. [1]
\item Show by calculation that the root of the equation $3 \cos^{-1}(x - 1) = x$ lies between 1.8 and 1.9. [2]
\item The sequence defined by
$$x_1 = 2, \quad x_{n+1} = 1 + \cos(\frac{1}{3}x_n)$$
converges to a number $\alpha$. Find the value of $\alpha$ correct to 2 decimal places and explain why $\alpha$ is the root of the equation $3 \cos^{-1}(x - 1) = x$. [5]
\end{enumerate}
\hfill \mbox{\textit{OCR C3 Q7 [11]}}