| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2012 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Derive equation from area/geometry |
| Difficulty | Standard +0.8 This question requires deriving a transcendental equation from geometric area relationships (involving both triangle and sector areas), then applying iterative methods to solve it. Part (i) demands careful geometric reasoning with multiple steps, while part (ii) is more routine iteration. The combination of geometric insight, algebraic manipulation, and numerical methods places this above average difficulty but not at the extreme end. |
| Spec | 1.05i Inverse trig functions: arcsin, arccos, arctan domains and graphs1.05o Trigonometric equations: solve in given intervals1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Using the formula \(\frac{1}{2}r^2\theta\) and \(\frac{1}{2}bh\), form an equation an \(a\) and \(\theta\) | M1 | |
| Obtain given answer | A1 | [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Use the iterative formula correctly at least once | M1 | |
| Obtain answer \(\theta = 1.32\) | A1 | |
| Show sufficient iterations to 4 d.p. to justify 1.32 to 2 d.p., or show there is a sign change in the interval (1.315, 1.325) | A1 | [3] |
**(i)**
| Using the formula $\frac{1}{2}r^2\theta$ and $\frac{1}{2}bh$, form an equation an $a$ and $\theta$ | M1 | |
| Obtain given answer | A1 | [2] |
**(ii)**
| Use the iterative formula correctly at least once | M1 | |
| Obtain answer $\theta = 1.32$ | A1 | |
| Show sufficient iterations to 4 d.p. to justify 1.32 to 2 d.p., or show there is a sign change in the interval (1.315, 1.325) | A1 | [3] |
---2\\
\includegraphics[max width=\textwidth, alt={}, center]{d3f0b201-3004-497a-9b29-30c94d0bec5b-2_300_767_518_689}
In the diagram, $A B C$ is a triangle in which angle $A B C$ is a right angle and $B C = a$. A circular arc, with centre $C$ and radius $a$, joins $B$ and the point $M$ on $A C$. The angle $A C B$ is $\theta$ radians. The area of the sector $C M B$ is equal to one third of the area of the triangle $A B C$.\\
(i) Show that $\theta$ satisfies the equation
$$\tan \theta = 3 \theta .$$
(ii) This equation has one root in the interval $0 < \theta < \frac { 1 } { 2 } \pi$. Use the iterative formula
$$\theta _ { n + 1 } = \tan ^ { - 1 } \left( 3 \theta _ { n } \right)$$
to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
\hfill \mbox{\textit{CAIE P3 2012 Q2 [5]}}