| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2020 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Show convergence to specific root |
| Difficulty | Standard +0.8 This is a multi-part question requiring understanding of fixed point iteration, convergence analysis, and algebraic manipulation. Part (b) requires showing the iterative formula converges to the root by substituting and rearranging x^5 = 2 + x, which demands algebraic insight beyond routine application. The iteration formula itself is non-obvious (Newton-Raphson derived). While the sketching and calculation parts are standard, the convergence proof elevates this above typical A-level fare. |
| Spec | 1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams1.09d Newton-Raphson method |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Sketch a relevant graph, e.g. \(y = x^5\) | B1 | |
| Sketch a second relevant graph, e.g. \(y = x + 2\) and justify the given statement | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State a suitable equation, e.g. \(x = \dfrac{4x^5+2}{5x^4-1}\) | B1 | |
| Rearrange this as \(x^5 = 2 + x\) or commence working *vice versa* | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use the iterative formula correctly at least once | M1 | |
| Obtain final answer 1.267 | A1 | |
| Show sufficient iterations to 5 d.p. to justify 1.267 to 3 d.p., or show there is a sign change in the interval (1.2665, 1.2675) | A1 |
## Question 6(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Sketch a relevant graph, e.g. $y = x^5$ | B1 | |
| Sketch a second relevant graph, e.g. $y = x + 2$ and justify the given statement | B1 | |
## Question 6(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| State a suitable equation, e.g. $x = \dfrac{4x^5+2}{5x^4-1}$ | B1 | |
| Rearrange this as $x^5 = 2 + x$ or commence working *vice versa* | B1 | |
## Question 6(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use the iterative formula correctly at least once | M1 | |
| Obtain final answer 1.267 | A1 | |
| Show sufficient iterations to 5 d.p. to justify 1.267 to 3 d.p., or show there is a sign change in the interval (1.2665, 1.2675) | A1 | |6
\begin{enumerate}[label=(\alph*)]
\item By sketching a suitable pair of graphs, show that the equation $x ^ { 5 } = 2 + x$ has exactly one real root.
\item Show that if a sequence of values given by the iterative formula
$$x _ { n + 1 } = \frac { 4 x _ { n } ^ { 5 } + 2 } { 5 x _ { n } ^ { 4 } - 1 }$$
converges, then it converges to the root of the equation in part (a).
\item Use the iterative formula with initial value $x _ { 1 } = 1.5$ to calculate the root correct to 3 decimal places. Give the result of each iteration to 5 decimal places.\\
$7 \quad$ Let $\mathrm { f } ( x ) = \frac { 2 } { ( 2 x - 1 ) ( 2 x + 1 ) }$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2020 Q6 [7]}}