| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2017 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Compare iteration convergence |
| Difficulty | Standard +0.3 This is a straightforward fixed point iteration question requiring routine application of standard techniques: substituting values to locate a root, then applying two given iterative formulae to determine convergence. The question provides the rearrangements and requires only mechanical calculation with no novel insight or proof of convergence conditions. |
| Spec | 1.09a Sign change methods: locate roots1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams1.09e Iterative method failure: convergence conditions |
| Answer | Marks | Guidance |
|---|---|---|
| 3(i) | Calculate value of a relevant expression or expressions at x = 2 and x = 3 | M1 |
| Complete the argument correctly with correct calculated values | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| 3(ii) | Use an iterative formula correctly at least once | M1 |
| Show that (B) fails to converge | A1 | |
| Using (A), obtain final answer 2.43 | A1 |
| Answer | Marks |
|---|---|
| (2.425, 2.435) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 3:
--- 3(i) ---
3(i) | Calculate value of a relevant expression or expressions at x = 2 and x = 3 | M1
Complete the argument correctly with correct calculated values | A1
2
--- 3(ii) ---
3(ii) | Use an iterative formula correctly at least once | M1
Show that (B) fails to converge | A1
Using (A), obtain final answer 2.43 | A1
Show sufficient iterations to justify 2.43 to 2 d.p., or show there is a sign change in
(2.425, 2.435) | A1
4
Question | Answer | MarksThe equation $x^3 = 3x + 7$ has one real root, denoted by $\alpha$.
\begin{enumerate}[label=(\roman*)]
\item Show by calculation that $\alpha$ lies between 2 and 3. [2]
\end{enumerate}
Two iterative formulae, $A$ and $B$, derived from this equation are as follows:
$$x_{n+1} = (3x_n + 7)^{\frac{1}{3}}, \quad (A)$$
$$x_{n+1} = \frac{x_n^3 - 7}{3}. \quad (B)$$
Each formula is used with initial value $x_1 = 2.5$.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{1}
\item Show that one of these formulae produces a sequence which fails to converge, and use the other formula to calculate $\alpha$ correct to 2 decimal places. Give the result of each iteration to 4 decimal places. [4]
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2017 Q3 [6]}}