| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2002 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Show convergence to specific root |
| Difficulty | Standard +0.3 This is a straightforward fixed-point iteration question requiring routine application of a given formula with x₁=0. Part (iii) involves simple calculator work iterating until convergence to 2 decimal places—no derivation of the iteration formula or convergence analysis is needed. Slightly easier than average due to minimal problem-solving and being purely procedural. |
| Spec | 1.09b Sign change methods: understand failure cases1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Obtain a derivative of the form \(ke^{-x} + he^{-4x}\) where \(kl \neq 0\) | B1 | |
| Obtain correct derivative \(2e^{-x} - 2xe^{-4x}\), or equivalent | B1 | |
| Equate \(\frac{dy}{dx}\) to zero and solve for \(x\) | M1 | |
| Obtain coordinates \((1, 2e^{-1})\) for \(P\) | A1 | 4 |
| (ii) State that \(\frac{1}{y} = 2xe^{-4x}\) and deduce the given answer correctly | B1 | 1 |
| (iii) State or imply that \(x_1 = 0.25\) | B1 | |
| Continue the iteration correctly | M1 | |
| Obtain final answer \(0.36\) after sufficient iterations to justify its accuracy to 2d.p., or after showing there is a sign change in \((0.355, 0.365)\) | A1 | 3 |
**(i)** Obtain a derivative of the form $ke^{-x} + he^{-4x}$ where $kl \neq 0$ | B1 |
Obtain correct derivative $2e^{-x} - 2xe^{-4x}$, or equivalent | B1 |
Equate $\frac{dy}{dx}$ to zero and solve for $x$ | M1 |
Obtain coordinates $(1, 2e^{-1})$ for $P$ | A1 | 4
**(ii)** State that $\frac{1}{y} = 2xe^{-4x}$ and deduce the given answer correctly | B1 | 1
**(iii)** State or imply that $x_1 = 0.25$ | B1 |
Continue the iteration correctly | M1 |
Obtain final answer $0.36$ after sufficient iterations to justify its accuracy to 2d.p., or after showing there is a sign change in $(0.355, 0.365)$ | A1 | 3
---(i) Find the exact coordinates of $P$.\\
(ii) Show that the $x$-coordinates of $Q$ and $R$ satisfy the equation
$$x = \frac { 1 } { 4 } e ^ { x } .$$
(iii) Use the iterative formula
$$x _ { n + 1 } = \frac { 1 } { 4 } e ^ { x _ { n } }$$
with initial value $x _ { 1 } = 0$, to find the $x$-coordinate of $Q$ correct to 2 decimal places, showing the value of each approximation that you calculate.
\hfill \mbox{\textit{CAIE P2 2002 Q5 [8]}}