| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2017 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Find coordinate from gradient condition |
| Difficulty | Standard +0.3 This is a straightforward fixed-point iteration question requiring students to apply a given iterative formula repeatedly until convergence. Part (i) involves algebraic rearrangement (likely from an intersection problem), and part (ii) is mechanical calculation with clear stopping criteria. While it requires understanding of iteration and careful arithmetic, it demands no novel insight or complex problem-solving beyond standard A-level technique. |
| 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 |
|---|---|---|
| Answer | Mark | Guidance |
| Obtain derivative of the form \(ke^{-2x}\) | \*M1 | Condone \(k = 4\) for M1 |
| State or imply gradient of curve at \(P\) is \(-8\) | A1 | |
| Form equation of straight line through \((0, 9)\) with negative gradient | \*DM1 | dep on \*M |
| Obtain \(y = -8x + 9\) or equivalent | A1 | |
| Equate equation of curve and equation of straight line | DM1 | dep on both \*M |
| Rearrange to confirm \(x = \frac{9}{8} - \frac{1}{2}e^{-2x}\) | A1 | |
| Total | 6 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use iterative process correctly at least once | M1 | |
| Obtain final answer \(1.07\) | A1 | |
| Show sufficient iterations to 5 sf to justify answer or show sign change in interval \((1.065, 1.075)\) | A1 | |
| Total | 6 |
## Question 5(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Obtain derivative of the form $ke^{-2x}$ | \*M1 | Condone $k = 4$ for M1 |
| State or imply gradient of curve at $P$ is $-8$ | A1 | |
| Form equation of straight line through $(0, 9)$ with negative gradient | \*DM1 | dep on \*M |
| Obtain $y = -8x + 9$ or equivalent | A1 | |
| Equate equation of curve and equation of straight line | DM1 | dep on both \*M |
| Rearrange to confirm $x = \frac{9}{8} - \frac{1}{2}e^{-2x}$ | A1 | |
| **Total** | **6** | |
## Question 5(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use iterative process correctly at least once | M1 | |
| Obtain final answer $1.07$ | A1 | |
| Show sufficient iterations to 5 sf to justify answer or show sign change in interval $(1.065, 1.075)$ | A1 | |
| **Total** | **6** | |
---(i) Show that the $x$-coordinate of $Q$ satisfies the equation $x = \frac { 9 } { 8 } - \frac { 1 } { 2 } \mathrm { e } ^ { - 2 x }$.\\
(ii) Use an iterative formula based on the equation in part (i) to find the $x$-coordinate of $Q$ correct to 3 significant figures. Give the result of each iteration to 5 significant figures.\\
\hfill \mbox{\textit{CAIE P2 2017 Q5 [9]}}