| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2019 |
| Session | March |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Rearrange to iterative form |
| Difficulty | Moderate -0.3 This is a straightforward fixed-point iteration question requiring algebraic rearrangement (substituting y=10 and taking logarithms), applying a given iterative formula with a calculator, and finding a derivative using quotient rule. All steps are routine A-level techniques with no novel problem-solving required, making it 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 diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Attempt rearrangement of \(\dfrac{e^{2x}}{4x+1} = 10\) to \(x = ...\) involving ln | M1 | |
| Confirm \(x = \frac{1}{2}\ln(40x + 10)\) | A1 | Answer given; necessary detail needed |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use iteration process correctly at least once | M1 | |
| Obtain final answer 2.316 | A1 | |
| Show sufficient iterations to 6 sf to justify answer or show a sign change in the interval \([2.3155,\ 2.3165]\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use quotient rule (or product rule) to find derivative | M1 | |
| Obtain \(\dfrac{2e^{2x}(4x+1) - 4e^{2x}}{(4x+1)^2}\) or equivalent | A1 | |
| Substitute answer from part (ii) (or more accurate value) into attempt at first derivative | M1 | |
| Obtain 16.1 | A1 |
## Question 5(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Attempt rearrangement of $\dfrac{e^{2x}}{4x+1} = 10$ to $x = ...$ involving ln | M1 | |
| Confirm $x = \frac{1}{2}\ln(40x + 10)$ | A1 | Answer given; necessary detail needed |
## Question 5(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use iteration process correctly at least once | M1 | |
| Obtain final answer 2.316 | A1 | |
| Show sufficient iterations to 6 sf to justify answer or show a sign change in the interval $[2.3155,\ 2.3165]$ | A1 | |
## Question 5(iii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use quotient rule (or product rule) to find derivative | M1 | |
| Obtain $\dfrac{2e^{2x}(4x+1) - 4e^{2x}}{(4x+1)^2}$ or equivalent | A1 | |
| Substitute answer from part (ii) (or more accurate value) into attempt at first derivative | M1 | |
| Obtain 16.1 | A1 | |
---5 The equation of a curve is $y = \frac { \mathrm { e } ^ { 2 x } } { 4 x + 1 }$ and the point $P$ on the curve has $y$-coordinate 10 .\\
(i) Show that the $x$-coordinate of $P$ satisfies the equation $x = \frac { 1 } { 2 } \ln ( 40 x + 10 )$.\\
(ii) Use the iterative formula $x _ { n + 1 } = \frac { 1 } { 2 } \ln \left( 40 x _ { n } + 10 \right)$ with $x _ { 1 } = 2.3$ to find the $x$-coordinate of $P$ correct to 4 significant figures. Give the result of each iteration to 6 significant figures.\\
(iii) Find the gradient of the curve at $P$, giving the answer correct to 3 significant figures.\\
\hfill \mbox{\textit{CAIE P2 2019 Q5 [9]}}