| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2002 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Derive equation from area/geometry |
| Difficulty | Challenging +1.2 This is a multi-part question involving geometric setup, equation verification, convergence proof, and iterative calculation. Part (i) requires deriving relationships from arc length and chord length (standard circle geometry), part (ii) is routine substitution, part (iii) requires algebraic manipulation to show the fixed point matches the original equation, and part (iv) is mechanical iteration. While it spans multiple techniques, each step follows standard procedures without requiring novel insight or particularly sophisticated reasoning. |
| Spec | 1.05d 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 |
|---|---|---|
| Content | Mark | Guidance |
| (i) State or obtain a relevant equation e.g. \(2r\alpha = 100\) | B1 | |
| State or obtain a second independent relevant equation e.g. \(2r \sin \alpha = 99\) | B1 | |
| Derive the given equation in \(x\) (or \(\alpha\)) correctly | B1 | Max 3 marks |
| (ii) Calculate ordinates at \(x = 0.1\) and \(x = 0.5\) of a suitable function or pair of functions | M1 | |
| Justify the given statement correctly | A1 | [If calculations are not given but the given statement is justified using correct statements about the signs of a suitable function or the difference between a pair of suitable functions, award B1.] |
| (iii) State \(x = 50\sin x - 48.5x\) or equivalent | B1 | |
| Rearrange thus in the form given in part (i) (or vice versa) | B1 | Max 2 marks |
| (iv) Use the method of iteration at least once with \(0.1 \leq x_0 \leq 0.5\) | M1 | |
| Obtain final answer \(0.245\), showing sufficient iterations to justify its accuracy to 3d.p., or showing a sign change in the interval \((0.2445, 0.2455)\) | A1 | [SR: both the M marks are available if calculations are attempted in degree mode.] |
| Content | Mark | Guidance |
|---------|------|----------|
| **(i)** State or obtain a relevant equation e.g. $2r\alpha = 100$ | B1 | |
| State or obtain a second independent relevant equation e.g. $2r \sin \alpha = 99$ | B1 | |
| Derive the given equation in $x$ (or $\alpha$) correctly | B1 | Max 3 marks |
| **(ii)** Calculate ordinates at $x = 0.1$ and $x = 0.5$ of a suitable function or pair of functions | M1 | |
| Justify the given statement correctly | A1 | [If calculations are not given but the given statement is justified using correct statements about the signs of a suitable function or the difference between a pair of suitable functions, award B1.] | Max 2 marks |
| **(iii)** State $x = 50\sin x - 48.5x$ or equivalent | B1 | |
| Rearrange thus in the form given in part (i) (or vice versa) | B1 | Max 2 marks |
| **(iv)** Use the method of iteration at least once with $0.1 \leq x_0 \leq 0.5$ | M1 | |
| Obtain final answer $0.245$, showing sufficient iterations to justify its accuracy to 3d.p., or showing a sign change in the interval $(0.2445, 0.2455)$ | A1 | [SR: both the M marks are available if calculations are attempted in degree mode.] | Max 2 marks |
---7\\
\includegraphics[max width=\textwidth, alt={}, center]{b89c016e-dc56-48f4-b4c4-b432418e1b28-3_435_672_273_684}
The diagram shows a curved rod $A B$ of length 100 cm which forms an arc of a circle. The end points $A$ and $B$ of the rod are 99 cm apart. The circle has radius $r \mathrm {~cm}$ and the arc $A B$ subtends an angle of $2 \alpha$ radians at $O$, the centre of the circle.\\
(i) Show that $\alpha$ satisfies the equation $\frac { 99 } { 100 } x = \sin x$.\\
(ii) Given that this equation has exactly one root in the interval $0 < x < \frac { 1 } { 2 } \pi$, verify by calculation that this root lies between 0.1 and 0.5.\\
(iii) Show that if the sequence of values given by the iterative formula
$$x _ { n + 1 } = 50 \sin x _ { n } - 48.5 x _ { n }$$
converges, then it converges to a root of the equation in part (i).\\
(iv) Use this iterative formula, with initial value $x _ { 1 } = 0.25$, to find $\alpha$ correct to 3 decimal places, showing the result of each iteration.
\hfill \mbox{\textit{CAIE P3 2002 Q7 [9]}}