| Exam Board | Edexcel |
|---|---|
| Module | C34 (Core Mathematics 3 & 4) |
| Year | 2018 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Solve exponential equation via iteration |
| Difficulty | Standard +0.3 This is a straightforward fixed point iteration question with standard parts: algebraic rearrangement (routine), applying an iteration formula (mechanical calculation), and using change of sign to verify a root to given accuracy (standard C3/C4 technique). All components are textbook exercises requiring no novel insight. |
| Spec | 1.06g Equations with exponentials: solve a^x = b1.09b Sign change methods: understand failure cases1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
3.
$$f ( x ) = 2 ^ { x - 1 } - 4 + 1.5 x \quad x \in \mathbb { R }$$
\begin{enumerate}[label=(\alph*)]
\item Show that the equation $\mathrm { f } ( x ) = 0$ can be written as
$$x = \frac { 1 } { 3 } \left( 8 - 2 ^ { x } \right)$$
The equation $\mathrm { f } ( x ) = 0$ has a root $\alpha$, where $\alpha = 1.6$ to one decimal place.
\item Starting with $x _ { 0 } = 1.6$, use the iteration formula
$$x _ { n + 1 } = \frac { 1 } { 3 } \left( 8 - 2 ^ { x _ { n } } \right)$$
to calculate the values of $x _ { 1 } , x _ { 2 }$ and $x _ { 3 }$, giving your answers to 3 decimal places.
\item By choosing a suitable interval, prove that $\alpha = 1.633$ to 3 decimal places.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C34 2018 Q3 [7]}}