| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2012 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Derive equation from integral condition |
| Difficulty | Standard +0.3 This is a straightforward multi-part question requiring standard integration to find area R, equating two areas to derive a given equation, and applying an iterative formula. All steps are routine A-level techniques with no novel insight required. The integration is basic (∫cos x dx = sin x), the algebraic manipulation is simple, and the iteration is mechanical calculator work. Slightly easier than average due to the guided structure and standard methods. |
| Spec | 1.05a Sine, cosine, tangent: definitions for all arguments1.08e Area between curve and x-axis: using definite integrals1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Attempt to integrate and use limits \(\theta\) and \(\pi\) | M1 | |
| Obtain \(1 - \sin \theta\) | A1 | [2] |
| (ii) State that area of rectangle \(= \theta \cos \theta\), equate area of rectangle to area of \(R\) and rearrange to given equation | B1 | [1] |
| (iii) Use the iterative formula correctly at least once | M1 | |
| Obtain final answer 0.56 | A1 | |
| Show sufficient iterations to justify its accuracy to 2 d.p. or show there is a sign change in the interval (0.555, 0.565) | B1 | [3] |
(i) Attempt to integrate and use limits $\theta$ and $\pi$ | M1 |
Obtain $1 - \sin \theta$ | A1 | [2]
(ii) State that area of rectangle $= \theta \cos \theta$, equate area of rectangle to area of $R$ and rearrange to given equation | B1 | [1]
(iii) Use the iterative formula correctly at least once | M1 |
Obtain final answer 0.56 | A1 |
Show sufficient iterations to justify its accuracy to 2 d.p. or show there is a sign change in the interval (0.555, 0.565) | B1 | [3]5\\
\includegraphics[max width=\textwidth, alt={}, center]{9e1bd528-e7c4-4936-a05a-dde1d1ace7c2-2_512_775_1318_683}
The diagram shows the curve $y = \cos x$, for $0 \leqslant x \leqslant \frac { 1 } { 2 } \pi$. A rectangle $O A B C$ is drawn, where $B$ is the point on the curve with $x$-coordinate $\theta$, and $A$ and $C$ are on the axes, as shown. The shaded region $R$ is bounded by the curve and by the lines $x = \theta$ and $y = 0$.\\
(i) Find the area of $R$ in terms of $\theta$.\\
(ii) The area of the rectangle $O A B C$ is equal to the area of $R$. Show that
$$\theta = \frac { 1 - \sin \theta } { \cos \theta }$$
(iii) Use the iterative formula $\theta _ { n + 1 } = \frac { 1 - \sin \theta _ { n } } { \cos \theta _ { n } }$, with initial value $\theta _ { 1 } = 0.5$, to determine the value of $\theta$ correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
\hfill \mbox{\textit{CAIE P2 2012 Q5 [6]}}