CAIE P2 2012 November — Question 5 6 marks

Exam BoardCAIE
ModuleP2 (Pure Mathematics 2)
Year2012
SessionNovember
Marks6
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicFixed Point Iteration
TypeDerive equation from integral condition
DifficultyStandard +0.3 This is a straightforward multi-part question requiring standard integration to find area R, equating two areas to derive the given equation, and applying an iterative formula with calculator work. All techniques are routine for P2 level with no novel insight required, making it slightly easier than average.
Spec1.05a 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
5 \includegraphics[max width=\textwidth, alt={}, center]{96a4df57-b3c7-4dbf-9bea-bb00ed6a4a16-2_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 \(y = 0\).
  1. Find the area of \(R\) in terms of \(\theta\).
  2. The area of the rectangle \(O A B C\) is equal to the area of \(R\). Show that $$\theta = \frac { 1 - \sin \theta } { \cos \theta }$$
  3. 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.
AnswerMarks Guidance
(i) Attempt to integrate and use limits \(0\) 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 equationB1 [1]
(iii) Use the iterative formula correctly at least onceM1
Obtain final answer 0.56A1
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 $0$ 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]{96a4df57-b3c7-4dbf-9bea-bb00ed6a4a16-2_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]}}
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