| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2017 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Find equation satisfied by limit |
| Difficulty | Standard +0.3 This is a straightforward fixed point iteration question requiring (i) numerical iteration with a calculator to find α ≈ 2.xx, then (ii) solving the equation x = (2x² + x + 9)/(x + 1)² algebraically. Both parts are routine applications of standard techniques with no novel insight 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 | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use iteration correctly at least once | M1 | |
| Obtain final answer 2.08 | A1 | |
| Show sufficient iterations to 4 dp to justify answer or show sign change in interval \((2.075,\ 2.085)\) | A1 | |
| Total: | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| State or clearly imply equation \(x = \frac{2x^2+x+9}{(x+1)^2}\) or same equation using \(\alpha\) | B1 | |
| Carry out relevant simplification | M1 | |
| Obtain \(\sqrt[3]{9}\) | A1 | |
| Total: | 3 |
## Question 4(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use iteration correctly at least once | M1 | |
| Obtain final answer 2.08 | A1 | |
| Show sufficient iterations to 4 dp to justify answer or show sign change in interval $(2.075,\ 2.085)$ | A1 | |
| **Total:** | **3** | |
## Question 4(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| State or clearly imply equation $x = \frac{2x^2+x+9}{(x+1)^2}$ or same equation using $\alpha$ | B1 | |
| Carry out relevant simplification | M1 | |
| Obtain $\sqrt[3]{9}$ | A1 | |
| **Total:** | **3** | |4 The sequence of values given by the iterative formula
$$x _ { n + 1 } = \frac { 2 x _ { n } ^ { 2 } + x _ { n } + 9 } { \left( x _ { n } + 1 \right) ^ { 2 } }$$
with $x _ { 1 } = 2$, converges to $\alpha$.\\
(i) Find the value of $\alpha$ correct to 2 decimal places, giving the result of each iteration to 4 decimal places.\\
(ii) Determine the exact value of $\alpha$.\\
\hfill \mbox{\textit{CAIE P2 2017 Q4 [6]}}