| 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 of a polynomial-trigonometric function, solving for gradients at specific points, rearranging to iterative form, sign-change verification, and applying fixed-point iteration. All techniques are standard A-level procedures 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]{da5162f3-b5d5-417f-9b6c-5ae0024f22d9-10_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]}}