| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2020 |
| 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 fixed point iteration question requiring routine application of the formula (part a) and simple algebraic manipulation to find the limit equation by setting x_{n+1} = x_n = α (part b). The equation solving is standard quadratic work. Slightly easier than average due to its mechanical nature with no conceptual challenges. |
| Spec | 1.04e Sequences: nth term and recurrence relations1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use iteration correctly at least once | M1 | Need to see 3 values including the starting values |
| Obtain final answer \(1.817\) | A1 | Answer required to exactly 4 significant figures |
| Show sufficient iterations to 6 significant figures to justify answer or show sign change in interval \([1.8165, 1.8175]\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State equation \(x = \frac{6+8x}{8+x^2}\) or equivalent using \(\alpha\) | B1 | |
| Obtain \(\sqrt[3]{6}\) or exact equivalent | B1 |
## Question 5(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use iteration correctly at least once | M1 | Need to see 3 values including the starting values |
| Obtain final answer $1.817$ | A1 | Answer required to exactly 4 significant figures |
| Show sufficient iterations to 6 significant figures to justify answer or show sign change in interval $[1.8165, 1.8175]$ | A1 | |
## Question 5(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| State equation $x = \frac{6+8x}{8+x^2}$ or equivalent using $\alpha$ | B1 | |
| Obtain $\sqrt[3]{6}$ or exact equivalent | B1 | |5 The sequence of values given by the iterative formula $x _ { n + 1 } = \frac { 6 + 8 x _ { n } } { 8 + x _ { n } ^ { 2 } }$ with initial value $x _ { 1 } = 2$ converges to $\alpha$.
\begin{enumerate}[label=(\alph*)]
\item Use the iterative formula to find the value of $\alpha$ correct to 4 significant figures. Give the result of each iteration to 6 significant figures.
\item State an equation satisfied by $\alpha$ and hence determine the exact value of $\alpha$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2020 Q5 [5]}}