| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2010 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Derive equation from area/geometry |
| Difficulty | Standard +0.3 This is a straightforward multi-part question combining standard circle geometry (area of segment formula) with basic fixed-point iteration. Part (i) requires routine application of sector and triangle area formulas, part (ii) is simple substitution, and part (iii) is mechanical iteration with a calculator. No novel insight required, just careful execution of standard techniques. |
| Spec | 1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta1.09b Sign change methods: understand failure cases1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Using the formulae \(\frac{1}{2}r^2\theta\) and \(\frac{1}{2}r^2\sin\theta\), or equivalent, form an equation | M1 | |
| Obtain a correct equation in \(r\) and \(x\) and/or \(x/2\) in any form | A1 | |
| Obtain the given equation correctly | A1 | [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Consider the sign of \(x - (\frac{3}{4}\pi - \sin x)\) at \(x = 1.3\) and \(x = 1.5\), or equivalent | M1 | |
| Complete the argument with correct calculations | A1 | [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Use the iterative formula correctly at least once | M1 | |
| Obtain final answer \(1.38\) | A1 | |
| Show sufficient iterations to at least 4 d.p. to justify accuracy to 2 d.p., or show there is a sign change in the interval \((1.375, 1.385)\) | A1 | [3] |
## Question 6:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Using the formulae $\frac{1}{2}r^2\theta$ and $\frac{1}{2}r^2\sin\theta$, or equivalent, form an equation | M1 | |
| Obtain a correct equation in $r$ and $x$ and/or $x/2$ in any form | A1 | |
| Obtain the given equation correctly | A1 | [3] |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Consider the sign of $x - (\frac{3}{4}\pi - \sin x)$ at $x = 1.3$ and $x = 1.5$, or equivalent | M1 | |
| Complete the argument with correct calculations | A1 | [2] |
### Part (iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Use the iterative formula correctly at least once | M1 | |
| Obtain final answer $1.38$ | A1 | |
| Show sufficient iterations to at least 4 d.p. to justify accuracy to 2 d.p., or show there is a sign change in the interval $(1.375, 1.385)$ | A1 | [3] |
---6\\
\includegraphics[max width=\textwidth, alt={}, center]{a74e4ddf-d254-45f3-bd9a-adf7cd53b3a6-3_380_641_258_751}
The diagram shows a semicircle $A C B$ with centre $O$ and radius $r$. The angle $B O C$ is $x$ radians. The area of the shaded segment is a quarter of the area of the semicircle.\\
(i) Show that $x$ satisfies the equation
$$x = \frac { 3 } { 4 } \pi - \sin x$$
(ii) This equation has one root. Verify by calculation that the root lies between 1.3 and 1.5.\\
(iii) Use the iterative formula
$$x _ { n + 1 } = \frac { 3 } { 4 } \pi - \sin x _ { n }$$
to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
\hfill \mbox{\textit{CAIE P3 2010 Q6 [8]}}