| Exam Board | AQA |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2007 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Solve exponential equation via iteration |
| Difficulty | Moderate -0.3 This is a multi-part question covering standard C3 techniques: Simpson's rule (routine application with given ordinates), showing a root exists (simple substitution), algebraic rearrangement (straightforward logarithm manipulation), and performing two iterations of a given formula. Part (iv) requires drawing a convergence diagram, which is a standard textbook exercise. All parts are procedural with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.09a Sign change methods: locate roots1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams1.09d Newton-Raphson method1.09f Trapezium rule: numerical integration |
4 [Figure 1, printed on the insert, is provided for use in this question.]
\begin{enumerate}[label=(\alph*)]
\item Use Simpson's rule with 5 ordinates (4 strips) to find an approximation to $\int _ { 1 } ^ { 2 } 3 ^ { x } \mathrm {~d} x$, giving your answer to three significant figures.
\item The curve $y = 3 ^ { x }$ intersects the line $y = x + 3$ at the point where $x = \alpha$.
\begin{enumerate}[label=(\roman*)]
\item Show that $\alpha$ lies between 0.5 and 1.5.
\item Show that the equation $3 ^ { x } = x + 3$ can be rearranged into the form
$$x = \frac { \ln ( x + 3 ) } { \ln 3 }$$
\item Use the iteration $x _ { n + 1 } = \frac { \ln \left( x _ { n } + 3 \right) } { \ln 3 }$ with $x _ { 1 } = 0.5$ to find $x _ { 3 }$ to two significant figures.
\item The sketch on Figure 1 shows part of the graphs of $y = \frac { \ln ( x + 3 ) } { \ln 3 }$ and $y = x$, and the position of $x _ { 1 }$.
On Figure 1, draw a cobweb or staircase diagram to show how convergence takes place, indicating the positions of $x _ { 2 }$ and $x _ { 3 }$ on the $x$-axis.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA C3 2007 Q4 [12]}}