| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2016 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Find equation satisfied by limit |
| Difficulty | Moderate -0.3 This is a straightforward iterative formula question requiring mechanical calculation (3-4 iterations with a calculator) and simple algebraic manipulation to find the exact value by setting x_{n+1} = x_n = α. Both parts are standard textbook exercises with no conceptual challenges or novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks |
|---|---|
| (ii) | Use the iterative formula correctly at least once |
| Answer | Marks |
|---|---|
| Obtain exact value 12 1 3 or 312 | M1 |
| Answer | Marks |
|---|---|
| B1 | [3] |
Question 1:
--- 1 (i)
(ii) ---
1 (i)
(ii) | Use the iterative formula correctly at least once
Obtain final answer 2.289
Show sufficient iterations to justify accuracy to 3 dp or show sign change in interval
(2.2885,2.2895)
4
State equation x= + 2x or equivalent
x2 3
Obtain exact value 12 1 3 or 312 | M1
A1
B1
B1
B1 | [3]
[2]The sequence of values given by the iterative formula
$$x_{n+1} = \frac{4}{x_n^2} + \frac{2x_n}{3},$$
with initial value $x_1 = 2$, converges to $\alpha$.
\begin{enumerate}[label=(\roman*)]
\item Use this iterative formula to find $\alpha$ correct to 3 decimal places. Give the result of each iteration to 5 decimal places. [3]
\item State an equation that is satisfied by $\alpha$, and hence find the exact value of $\alpha$. [2]
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2016 Q1 [5]}}