| 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 multi-part question requiring differentiation, solving for gradient conditions, and applying fixed point iteration. Part (i) involves routine differentiation and algebraic manipulation, part (ii) is simple substitution to verify an interval, and part (iii) is mechanical iteration. While it combines several techniques, each step is standard with no novel insight required, making it slightly easier than average. |
| Spec | 1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Differentiate to obtain form \(k_1x + k_2 + k_3\sin\frac{1}{2}x\) | \*M1 | |
| Obtain correct \(2x + 3 - \frac{5}{2}\sin\frac{1}{2}x\) and deduce or imply gradient at \(P\) is 3 | A1 | |
| Equate first derivative to their \(-3\) and rearrange | DM1 | |
| Obtain \(x = \frac{5}{4}\sin\frac{1}{2}x - 3\) | A1 | |
| Total: 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Consider sign of their \(2x + 6 - \frac{5}{2}\sin\frac{1}{2}x\) at \(-4.5\) and \(-4.0\) or equivalent | M1 | |
| Complete argument correctly for correct expression with appropriate calculations | A1 | |
| Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use iteration formula correctly at least once | M1 | |
| Obtain final answer \(-4.11\) | A1 | |
| Show sufficient iterations to justify accuracy to 3 sf or show sign change in interval \((-4.115, -4.105)\) | A1 | |
| Total: 3 |
## Question 7(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Differentiate to obtain form $k_1x + k_2 + k_3\sin\frac{1}{2}x$ | **\*M1** | |
| Obtain correct $2x + 3 - \frac{5}{2}\sin\frac{1}{2}x$ and deduce or imply gradient at $P$ is 3 | **A1** | |
| Equate first derivative to their $-3$ and rearrange | **DM1** | |
| Obtain $x = \frac{5}{4}\sin\frac{1}{2}x - 3$ | **A1** | |
| **Total: 4** | | |
---
## Question 7(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Consider sign of their $2x + 6 - \frac{5}{2}\sin\frac{1}{2}x$ at $-4.5$ and $-4.0$ or equivalent | **M1** | |
| Complete argument correctly for correct expression with appropriate calculations | **A1** | |
| **Total: 2** | | |
---
## Question 7(iii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use iteration formula correctly at least once | **M1** | |
| Obtain final answer $-4.11$ | **A1** | |
| Show sufficient iterations to justify accuracy to 3 sf or show sign change in interval $(-4.115, -4.105)$ | **A1** | |
| **Total: 3** | | |7\\
\includegraphics[max width=\textwidth, alt={}, center]{ebfaafca-3279-44f8-a910-0756ffa25c70-10_542_789_260_676}
The diagram shows the curve
$$y = x ^ { 2 } + 3 x + 1 + 5 \cos \frac { 1 } { 2 } x .$$
The curve crosses the $y$-axis at the point $P$ and the gradient of the curve at $P$ is $m$. The point $Q$ on the curve has $x$-coordinate $q$ and the gradient of the curve at $Q$ is $- m$.\\
(i) Find the value of $m$ and hence show that $q$ satisfies the equation
$$x = a \sin \frac { 1 } { 2 } x + b ,$$
where the values of the constants $a$ and $b$ are to be determined.\\
(ii) Show by calculation that $- 4.5 < q < - 4.0$.\\
(iii) Use an iterative formula based on the equation in part (i) to find the value of $q$ correct to 3 significant figures. Give the result of each iteration to 5 significant figures.\\
\hfill \mbox{\textit{CAIE P2 2017 Q7 [9]}}