| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2007 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Derive equation from area/geometry |
| Difficulty | Standard +0.3 This is a standard multi-part question combining geometry (sector/triangle areas), equation verification, and routine fixed point iteration. Part (i) requires basic area formulas, parts (ii) and (iv) are computational, and part (iii) is a standard rearrangement proof. All techniques are textbook exercises with no novel insight required, making it slightly easier than average. |
| Spec | 1.02q Use intersection points: of graphs to solve equations1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta1.09a Sign change methods: locate roots1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Using the formulae \(\frac{1}{2}r^2\alpha\) and \(\frac{1}{2}r^2\sin\alpha\), or equivalent, form an equation | M1 | |
| Obtain given equation correctly | A1 | Total: 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| (ii) Consider sign of \(x - 2\sin x\) at \(x = \frac{1}{3}\pi\) and \(x = \frac{2}{3}\pi\), or equivalent | M1 | |
| Complete the argument correctly with appropriate calculations | A1 | Total: 2 marks |
| (iii) State or imply the equation \(x = \frac{1}{4}(x + 4\sin x)\) | B1 | |
| Rearrange this as \(x = 2\sin x\), or work here | B1 | Total: 2 marks |
| (iv) Use the iterative formula correctly at least once | M1 | |
| Obtain final answer 1.90 | A1 | |
| Show sufficient iterations to 4 d.p. to justify its accuracy to 2 d.p., or show there is a sign change in the interval (1.895, 1.905) | A1 | Total: 3 marks |
(i) Using the formulae $\frac{1}{2}r^2\alpha$ and $\frac{1}{2}r^2\sin\alpha$, or equivalent, form an equation | M1 |
Obtain given equation correctly | A1 | **Total: 2 marks**
**Guidance notes:**
- [Allow the use of $OA$ and/or $OB$ for $r$.]
(ii) Consider sign of $x - 2\sin x$ at $x = \frac{1}{3}\pi$ and $x = \frac{2}{3}\pi$, or equivalent | M1 |
Complete the argument correctly with appropriate calculations | A1 | **Total: 2 marks**
(iii) State or imply the equation $x = \frac{1}{4}(x + 4\sin x)$ | B1 |
Rearrange this as $x = 2\sin x$, or work here | B1 | **Total: 2 marks**
(iv) Use the iterative formula correctly at least once | M1 |
Obtain final answer 1.90 | A1 |
Show sufficient iterations to 4 d.p. to justify its accuracy to 2 d.p., or show there is a sign change in the interval (1.895, 1.905) | A1 | **Total: 3 marks**
**Guidance notes:**
- [The final answer 1.9 scores A0.]
---6\\
\includegraphics[max width=\textwidth, alt={}, center]{8580dddb-cc72-4745-9e0f-1ac641c6506d-2_355_601_1562_772}
The diagram shows a sector $A O B$ of a circle with centre $O$ and radius $r$. The angle $A O B$ is $\alpha$ radians, where $0 < \alpha < \pi$. The area of triangle $A O B$ is half the area of the sector.\\
(i) Show that $\alpha$ satisfies the equation
$$x = 2 \sin x$$
(ii) Verify by calculation that $\alpha$ lies between $\frac { 1 } { 2 } \pi$ and $\frac { 2 } { 3 } \pi$.\\
(iii) Show that, if a sequence of values given by the iterative formula
$$x _ { n + 1 } = \frac { 1 } { 3 } \left( x _ { n } + 4 \sin x _ { n } \right)$$
converges, then it converges to a root of the equation in part (i).\\
(iv) Use this iterative formula, with initial value $x _ { 1 } = 1.8$, to find $\alpha$ correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
\hfill \mbox{\textit{CAIE P3 2007 Q6 [9]}}