| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2014 |
| 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 fixed point iteration question with standard geometric setup. Part (i) requires basic circle geometry and perimeter equations (routine manipulation), part (ii) is simple substitution, and part (iii) is mechanical iteration with a given formula. The geometric insight needed is minimal, and the iterative process is entirely procedural once the formula is established. Slightly easier than average due to the highly structured nature and lack of conceptual challenges. |
| Spec | 1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks |
|---|---|
| (i) Use correct arc formula and form an equation in \(r\) and \(x\) | M1 |
| Obtain a correct equation in any form | A1 |
| Rearrange in the given form | A1 |
| Total: 3 marks | |
| (ii) Consider sign of a relevant expression at \(x = 1\) and \(x = 1.5\), or compare values of relevant expressions at \(x = 1\) and \(x = 1.5\) | M1 |
| Complete the argument correctly with correct calculated values | A1 |
| Total: 2 marks | |
| (iii) Use the iterative formula correctly at least once | M1 |
| Obtain final answer 1.21 | A1 |
| Show sufficient iterations to 4 d.p. to justify 1.21 to 2 d.p., or show there is a sign change in the interval \((1.205, 1.215)\) | A1 |
| Total: 3 marks |
**(i)** Use correct arc formula and form an equation in $r$ and $x$ | M1 |
Obtain a correct equation in any form | A1 |
Rearrange in the given form | A1 |
| Total: 3 marks |
**(ii)** Consider sign of a relevant expression at $x = 1$ and $x = 1.5$, or compare values of relevant expressions at $x = 1$ and $x = 1.5$ | M1 |
Complete the argument correctly with correct calculated values | A1 |
| Total: 2 marks |
**(iii)** Use the iterative formula correctly at least once | M1 |
Obtain final answer 1.21 | A1 |
Show sufficient iterations to 4 d.p. to justify 1.21 to 2 d.p., or show there is a sign change in the interval $(1.205, 1.215)$ | A1 |
| Total: 3 marks |
---\includegraphics{figure_6}
In the diagram, $A$ is a point on the circumference of a circle with centre $O$ and radius $r$. A circular arc with centre $A$ meets the circumference at $B$ and $C$. The angle $\angle OAB$ is equal to $x$ radians. The shaded region is bounded by $AB$, $AC$ and the circular arc with centre $A$ joining $B$ and $C$. The perimeter of the shaded region is equal to half the circumference of the circle.
\begin{enumerate}[label=(\roman*)]
\item Show that $x = \cos^{-1}\left(\frac{\pi}{4 + 4x}\right)$. [3]
\item Verify by calculation that $x$ lies between 1 and 1.5. [2]
\item Use the iterative formula
$$x_{n+1} = \cos^{-1}\left(\frac{\pi}{4 + 4x_n}\right)$$
to determine the value of $x$ correct to 2 decimal places. Give the result of each iteration to 4 decimal places. [3]
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2014 Q6 [8]}}