| Exam Board | AQA |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2011 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Show root in interval |
| Difficulty | Standard +0.3 This is a straightforward iterative methods question requiring basic substitution to verify bounds, simple algebraic rearrangement using inverse function properties, and calculator-based iteration. All techniques are standard C3 content with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.05i Inverse trig functions: arcsin, arccos, arctan domains and graphs1.06a Exponential function: a^x and e^x graphs and properties1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
The curve $y = \cos^{-1}(2x - 1)$ intersects the curve $y = e^x$ at a single point where $x = \alpha$.
\begin{enumerate}[label=(\alph*)]
\item Show that $\alpha$ lies between 0.4 and 0.5. [2]
\item Show that the equation $\cos^{-1}(2x - 1) = e^x$ can be written as $x = \frac{1}{2} + \frac{1}{2}\cos(e^x)$. [1]
\item Use the iteration $x_{n+1} = \frac{1}{2} + \frac{1}{2}\cos(e^{x_n})$ with $x_1 = 0.4$ to find the values of $x_2$ and $x_3$, giving your answers to three decimal places. [2]
\end{enumerate}
\hfill \mbox{\textit{AQA C3 2011 Q3 [5]}}