| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2004 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Solve trigonometric equation via iteration |
| Difficulty | Standard +0.3 This is a standard multi-part question combining geometry with numerical methods. Part (i) requires setting up area equations (routine for P3), part (ii) is a straightforward graph sketching exercise, and part (iii) applies a given iterative formula mechanically. All techniques are standard textbook material with no novel insight required, making it slightly easier than average. |
| 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 |
|---|---|---|
| (i) Obtain area of \(ONB\) in terms of \(r\) and \(\alpha\) e.g. \(\frac{1}{2}r^2\cos\alpha\sin\alpha\) | B1 | |
| Equate area of triangle in terms of \(r\) and \(\alpha\) to \(\frac{1}{2}\left(\frac{1}{2}r^2\alpha\right)\) or equivalent | M1 | |
| Obtain given form, \(\sin 2\alpha = \alpha\), correctly | A1 | [Allow use of OA and/or OB for \(r\).] |
| Total: 3 marks | ||
| (ii) Make recognisable sketch in one diagram over the given range of two suitable graphs, e.g. \(y = \sin 2x\) and \(y = x\) | B1 | |
| State or imply link between intersections and roots and justify the given answer | B1 | Total: 2 marks |
| [Allow a single graph and its intersection with \(y = 0\) to earn full marks.] | ||
| (iii) Use the iterative formula correctly at least once | M1 | |
| Obtain final answer 0.95 | A1 | |
| Show sufficient iterations to justify its accuracy to 2d.p., or show there is a sign change in (0.945, 0.955) | A1 | [SR: Allow the M mark if calculations are attempted in degree mode.] |
| Total: 3 marks |
**(i)** Obtain area of $ONB$ in terms of $r$ and $\alpha$ e.g. $\frac{1}{2}r^2\cos\alpha\sin\alpha$ | B1 | |
Equate area of triangle in terms of $r$ and $\alpha$ to $\frac{1}{2}\left(\frac{1}{2}r^2\alpha\right)$ or equivalent | M1 | |
Obtain given form, $\sin 2\alpha = \alpha$, correctly | A1 | [Allow use of OA and/or OB for $r$.] |
| | | **Total: 3 marks** |
**(ii)** Make recognisable sketch in one diagram over the given range of two suitable graphs, e.g. $y = \sin 2x$ and $y = x$ | B1 | |
State or imply link between intersections and roots and justify the given answer | B1 | **Total: 2 marks** |
| | | [Allow a single graph and its intersection with $y = 0$ to earn full marks.] |
**(iii)** Use the iterative formula correctly at least once | M1 | |
Obtain final answer 0.95 | A1 | |
Show sufficient iterations to justify its accuracy to 2d.p., or show there is a sign change in (0.945, 0.955) | A1 | [SR: Allow the M mark if calculations are attempted in degree mode.] |
| | | **Total: 3 marks** |5\\
\includegraphics[max width=\textwidth, alt={}, center]{8c533469-393c-4e4c-a6ec-eab1303741e7-2_385_476_1653_836}
The diagram shows a sector $O A B$ of a circle with centre $O$ and radius $r$. The angle $A O B$ is $\alpha$ radians, where $0 < \alpha < \frac { 1 } { 2 } \pi$. The point $N$ on $O A$ is such that $B N$ is perpendicular to $O A$. The area of the triangle $O N B$ is half the area of the sector $O A B$.\\
(i) Show that $\alpha$ satisfies the equation $\sin 2 x = x$.\\
(ii) By sketching a suitable pair of graphs, show that this equation has exactly one root in the interval $0 < x < \frac { 1 } { 2 } \pi$.\\
(iii) Use the iterative formula
$$x _ { n + 1 } = \sin \left( 2 x _ { n } \right)$$
with initial value $x _ { 1 } = 1$, to find $\alpha$ correct to 2 decimal places, showing the result of each iteration.
\hfill \mbox{\textit{CAIE P3 2004 Q5 [8]}}