| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2016 |
| Session | June |
| 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 standard A-level iterative methods question requiring differentiation (product rule with exponential), setting up an equation from a gradient condition, and applying fixed-point iteration. While it involves multiple steps, each component is routine: the product rule is straightforward, the rearrangement is given, and the iteration is mechanical. The 'show that' parts provide scaffolding that reduces problem-solving demand, making this slightly easier than average. |
| Spec | 1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams1.09d Newton-Raphson method |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Use product rule to obtain form \(k_1e^{\frac{1}{3}x} + k_2xe^{\frac{1}{3}x}\) | *M1 | |
| Obtain correct \(6e^{\frac{1}{3}x} + 2xe^{\frac{1}{3}x}\) | A1 | |
| Equate first derivative to 40 and obtain equation without e present, dep *M | DM1 | |
| Confirm \(p = 3\ln\frac{20}{p+3}\) or \(x = 3\ln\frac{20}{x+3}\) | A1 | [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Consider sign of \(p - 3\ln\frac{20}{p+3}\) at 3.3 and 3.5 or equivalent | M1 | |
| Complete argument correctly with appropriate calculations | A1 | [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Carry out iterative process correctly at least once | M1 | |
| Obtain final answer 3.412 | A1 | |
| Show sufficient iterations to justify accuracy to 3 dp or show sign change in interval (3.4115, 3.4125) | B1 | [3] |
## Question 5:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Use product rule to obtain form $k_1e^{\frac{1}{3}x} + k_2xe^{\frac{1}{3}x}$ | *M1 | |
| Obtain correct $6e^{\frac{1}{3}x} + 2xe^{\frac{1}{3}x}$ | A1 | |
| Equate first derivative to 40 and obtain equation without e present, dep *M | DM1 | |
| Confirm $p = 3\ln\frac{20}{p+3}$ or $x = 3\ln\frac{20}{x+3}$ | A1 | [4] |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Consider sign of $p - 3\ln\frac{20}{p+3}$ at 3.3 and 3.5 or equivalent | M1 | |
| Complete argument correctly with appropriate calculations | A1 | [2] |
### Part (iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Carry out iterative process correctly at least once | M1 | |
| Obtain final answer 3.412 | A1 | |
| Show sufficient iterations to justify accuracy to 3 dp or show sign change in interval (3.4115, 3.4125) | B1 | [3] |
---5 The equation of a curve is $y = 6 x \mathrm { e } ^ { \frac { 1 } { 3 } x }$. At the point on the curve with $x$-coordinate $p$, the gradient of the curve is 40 .\\
(i) Show that $p = 3 \ln \left( \frac { 20 } { p + 3 } \right)$.\\
(ii) Show by calculation that $3.3 < p < 3.5$.\\
(iii) Use an iterative formula based on the equation in part (i) to find the value of $p$ correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
\hfill \mbox{\textit{CAIE P2 2016 Q5 [9]}}