Area between two polar curves

10 In this question you must show detailed reasoning. A region, \(R\), of the floor of an art gallery is to be painted for the purposes of an art installation. A suitable polar coordinate system is set up on the floor of the gallery with units in metres and radians. \(R\) is modelled as being the region enclosed by two curves, \(C _ { 1 }\) and \(C _ { 2 }\). The polar equations of \(C _ { 1 }\) and \(C _ { 2 }\) are $$\begin{array} { l l } C _ { 1 } : r = 5 , & - \frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 1 } { 2 } \pi \\ C _ { 2 } : r = 3 \cosh \theta , & - \frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 1 } { 2 } \pi \end{array}$$ Both curves are shown in the diagram, with \(R\) indicated.
\includegraphics[max width=\textwidth, alt={}, center]{7b2bfb4e-524f-4d1c-ae98-075c7fb404f9-6_1481_821_836_251} The gallery must buy tins of paint to paint \(R\). Each tin of paint can cover an area of \(0.5 \mathrm {~m} ^ { 2 }\).
Determine the smallest number of tins of paint that the gallery must buy in order to be able to paint \(R\) completely.

9. \begin{figure}[h] \end{figure} Figure 1 shows the curve \(C _ { 1 }\) with polar equation \(r = 2 a \sin 2 \theta , 0 \leq \theta \leq \frac { \pi } { 2 }\), and the circle \(C _ { 2 }\) with polar equation \(r = a , 0 \leq \theta \leq 2 \pi\), where \(a\) is a positive constant.

1. Find, in terms of \(a\), the polar coordinates of the points where the curve \(C _ { 1 }\) meets the circle \(C _ { 2 }\). The regions enclosed by the curve \(C _ { 1 }\) and the circle \(C _ { 2 }\) overlap and the common region \(R\) is shaded in Figure 1.
2. Find the area of the shaded region \(R\), giving your answer in the form \(\frac { 1 } { 12 } a ^ { 2 } ( p \pi + q \sqrt { } 3 )\), where \(p\) and \(q\) are integers to be found. \section*{END} \section*{6668/01R Edexcel GCE} \section*{Further Pure Mathematics FP2 (R)} \section*{Advanced/Advanced Subsidiary} \section*{Friday 6 June 2014 - Afternoon} Mathematical Formulae (Pink) Nil Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation or symbolic differentiation/integration, or have retrievable mathematical formulae stored in them. In the boxes above, write your centre number, candidate number, your surname, initials and signature. Check that you have the correct question paper.
   Answer ALL the questions.
   You must write your answer for each question in the space following the question.
   When a calculator is used, the answer should be given to an appropriate degree of accuracy. A booklet 'Mathematical Formulae and Statistical Tables' is provided.
   Full marks may be obtained for answers to ALL questions.
   The marks for the parts of questions are shown in round brackets, e.g. (2).
   There are 8 questions in this question paper. The total mark for this paper is 75 .
   There are 28 pages in this question paper. Any blank pages are indicated. You must ensure that your answers to parts of questions are clearly labelled.
   You must show sufficient working to make your methods clear to the Examiner.
   Answers without working may not gain full credit.
   1. (a) Express \(\frac { 2 } { 4 r ^ { 2 } - 1 }\) in partial fractions.
   2. Hence use the method of differences to show that
   $$\sum _ { r = 1 } ^ { n } \frac { 1 } { 4 r ^ { 2 } - 1 } = \frac { n } { 2 n + 1 }$$
   1. Using algebra, find the set of values of \(x\) for which
   $$3 x - 5 < \frac { 2 } { x }$$
   1. (a) Find the general solution of the differential equation
   $$\frac { \mathrm { d } y } { \mathrm {~d} x } + 2 y \tan x = \mathrm { e } ^ { 4 x } \cos ^ { 2 } x , \quad - \frac { \pi } { 2 } < x < \frac { \pi } { 2 }$$ giving your answer in the form \(y = \mathrm { f } ( x )\).
3. Find the particular solution for which \(y = 1\) at \(x = 0\).
   4. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{3caa00c9-8a77-4987-888d-054eba7712ee-18_311_841_251_331} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} Figure 1 shows the curve \(C\) with polar equation $$r = 2 \cos 2 \theta , \quad 0 \leq \theta \leq \frac { \pi } { 4 }$$ The line \(l\) is parallel to the initial line and is a tangent to \(C\).
   Find an equation of \(l\), giving your answer in the form \(r = \mathrm { f } ( \theta )\).
   5. $$y \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } + 2 y = 0$$
4. Find an expression for \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }\) in terms of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } , \frac { \mathrm {~d} y } { \mathrm {~d} x }\) and \(y\). Given that \(y = 2\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0.5\) at \(x = 0\),
5. find a series solution for \(y\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
   6. The transformation \(T\) maps points from the \(z\)-plane, where \(z = x + \mathrm { i } y\), to the \(w\)-plane, where \(w = u + \mathrm { i } v\). The transformation \(T\) is given by $$w = \frac { z } { \mathrm { i } z + 1 } , \quad z \neq \mathrm { i }$$ The transformation \(T\) maps the line \(l\) in the \(z\)-plane onto the line with equation \(v = - 1\) in the \(w\)-plane.
6. Find a cartesian equation of \(l\) in terms of \(x\) and \(y\). The transformation \(T\) maps the line with equation \(y = \frac { 1 } { 2 }\) in the \(z\)-plane onto the curve \(C\) in the \(w\)-plane.
7. 1. Show that \(C\) is a circle with centre the origin.
   2. Write down a cartesian equation of \(C\) in terms of \(u\) and \(v\).
      7. (a) Use de Moivre's theorem to show that $$\sin 5 \theta \equiv 16 \sin ^ { 5 } \theta - 20 \sin ^ { 3 } \theta + 5 \sin \theta$$
8. Hence find the five distinct solutions of the equation $$16 x ^ { 5 } - 20 x ^ { 3 } + 5 x + \frac { 1 } { 2 } = 0$$ giving your answers to 3 decimal places where necessary.
9. Use the identity given in (a) to find $$\int _ { 0 } ^ { \frac { \pi } { 4 } } \left( 4 \sin ^ { 5 } \theta - 5 \sin ^ { 3 } \theta \right) d \theta$$ expressing your answer in the form \(a \sqrt { } 2 + b\), where \(a\) and \(b\) are rational numbers.
   8. (a) Show that the substitution \(x = \mathrm { e } ^ { z }\) transforms the differential equation
   into the equation
10. Find the general solution of the differential equation (II). the form \(y = \mathrm { f } ( x )\). $$x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 x \frac { \mathrm {~d} y } { \mathrm {~d} x } - 2 y = 3 \ln x , \quad x > 0$$ $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} z ^ { 2 } } + \frac { \mathrm { d } y } { \mathrm {~d} z } - 2 y = 3 z$$
11. Hence obtain the general solution of the differential equation (I) giving your answer in Mathematical Formulae (Pink) \section*{Paper Reference(s)} 6668/01 \section*{Advanced/Advanced Subsidiary} \section*{Friday 6 June 2014 - Afternoon} Time: 1 hour 30 minutes Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation or symbolic differentiation/integration, or have \section*{TOTAL FOR PAPER: 75 MARKS} \section*{\textbackslash section*\{END\}} retrievable mathematical formulae stored in them. Nil In the boxes above, write your centre number, candidate number, your surname, initials and signature. Check that you have the correct question paper.
    Answer ALL the questions.
    You must write your answer for each question in the space following the question.
    When a calculator is used, the answer should be given to an appropriate degree of accuracy. A booklet 'Mathematical Formulae and Statistical Tables' is provided.
    Full marks may be obtained for answers to ALL questions.
    The marks for the parts of questions are shown in round brackets, e.g. (2).
    There are 8 questions in this question paper. The total mark for this paper is 75 .
    There are 28 pages in this question paper. Any blank pages are indicated. You must ensure that your answers to parts of questions are clearly labelled. You must show sufficient working to make your methods clear to the Examiner. Answers without working may not gain full credit. \section*{P44512A} This publication may only be reproduced in accordance with Pearson Education Limited copyright policy. ©2014 Pearson Education Limited.
    1. (a) Express \(\frac { 2 } { ( r + 2 ) ( r + 4 ) }\) in partial fractions.
    2. Hence show that
    $$\sum _ { r = 1 } ^ { n } \frac { 2 } { ( r + 2 ) ( r + 4 ) } = \frac { n ( 7 n + 25 ) } { 12 ( n + 3 ) ( n + 4 ) }$$
    1. Use algebra to find the set of values of \(x\) for which
    $$\left| 3 x ^ { 2 } - 19 x + 20 \right| < 2 x + 2$$ 3. $$y = \sqrt { 8 + \mathrm { e } ^ { x } } , \quad x \in$$ Find the series expansion for \(y\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), giving each coefficient in its simplest form.
    4. (a) Use de Moivre's theorem to show that
12. Hence show that
    \(\_\_\_\_\) v
    s \(x\) for which
    3.
13. $$\begin{aligned} & \qquad \cos 6 \theta = 32 \cos ^ { 6 } \theta - 48 \cos ^ { 4 } \theta + 18 \cos ^ { 2 } \theta - 1 \\ & \text { (b) Hence solve for } 0 \leq \theta \leq \frac { \pi } { 2 } \\ & \qquad 64 \cos ^ { 6 } \theta - 96 \cos ^ { 4 } \theta + 36 \cos ^ { 2 } \theta - 3 = 0 \\ & \text { giving your answers as exact multiples of } \pi \end{aligned}$$
    1. (a) Find the general solution of the differential equation
    $$\begin{aligned} & \qquad \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 10 y = 27 \mathrm { e } ^ { - x } \\ & \text { (b) Find the particular solution that satisfies } y = 0 \text { and } \frac { \mathrm { d } y } { \mathrm {~d} x } = 0 \text { when } x = 0 \text {. } \\ \hline & \end{aligned}$$
    1. The transformation \(T\) from the \(z\)-plane, where \(z = x + \mathrm { i } y\), to the \(w\)-plane, where \(w = u + \mathrm { i } v\), is given by
    $$w = \frac { 4 ( 1 - \mathrm { i } ) z - 8 \mathrm { i } } { 2 ( - 1 + \mathrm { i } ) z - \mathrm { i } } , \quad z \neq \frac { 1 } { 4 } - \frac { 1 } { 4 } \mathrm { i }$$ The transformation \(T\) maps the points on the line \(l\) with equation \(y = x\) in the \(z\)-plane to a circle \(C\) in the \(w\)-plane.
14. Show that $$w = \frac { a x ^ { 2 } + b x i + c } { 16 x ^ { 2 } + 1 }$$ where \(a , b\) and \(c\) are real constants to be found.
15. Hence show that the circle \(C\) has equation $$( u - 3 ) ^ { 2 } + v ^ { 2 } = k ^ { 2 }$$ where \(k\) is a constant to be found.
    7. (a) Show that the substitution \(v = y ^ { - 3 }\) transforms the differential equation $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = 2 x ^ { 4 } y ^ { 4 }$$ into the differential equation $$\frac { \mathrm { d } v } { \mathrm {~d} x } - \frac { 3 v } { x } = - 6 x ^ { 3 }$$
16. By solving differential equation (II), find a general solution of differential equation (I) in the form \(y ^ { 3 } = \mathrm { f } ( x )\).
    8. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{3caa00c9-8a77-4987-888d-054eba7712ee-21_511_684_255_408} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Figure 1 shows a sketch of part of the curve \(C\) with polar equation $$r = 1 + \tan \theta , \quad 0 \leq \theta < \frac { \pi } { 2 }$$ The tangent to the curve \(C\) at the point \(P\) is perpendicular to the initial line.
17. Find the polar coordinates of the point \(P\). The point \(Q\) lies on the curve \(C\), where \(\theta = \frac { \pi } { 3 }\).
    The shaded region \(R\) is bounded by \(O P , O Q\) and the curve \(C\), as shown in Figure 1.
18. Find the exact area of \(R\), giving your answer in the form $$\frac { 1 } { 2 } ( \ln p + \sqrt { q } + r )$$ where \(p , q\) and \(r\) are integers to be found.

4. The curve \(C\) has polar equation \(r = 3 a \cos \theta , - \frac { \pi } { 2 } \leq \frac { \pi } { 2 }\). The curve \(D\) has polar equation \(r = a ( 1 + \cos \theta ) , - \pi \leq \theta < \pi\). Given that \(a\) is a positive constant,

1. sketch, on the same diagram, the graphs of \(C\) and \(D\), indicating where each curve cuts the initial line. The graphs of \(C\) intersect at the pole \(O\) and at the points \(P\) and \(Q\).
2. Find the polar coordinates of \(P\) and \(Q\).
3. Use integration to find the exact value of the area enclosed by the curve \(D\) and the lines \(\theta = 0\) and \(\theta = \frac { \pi } { 3 }\). The region \(R\) contains all points which lie outside \(D\) and inside \(C\). Given that the value of the smaller area enclosed by the curve \(C\) and the line \(\theta = \frac { \pi } { 3 }\) is $$\frac { 3 a ^ { 2 } } { 16 } ( 2 \pi - 3 \sqrt { } 3 )$$
4. show that the area of \(R\) is \(\pi a ^ { 2 }\).
   (4)
   [0pt] [P4 January 2002 Qn 8]

6

1. A circle \(C _ { 1 }\) has cartesian equation \(x ^ { 2 } + ( y - 6 ) ^ { 2 } = 36\). Show that the polar equation of \(C _ { 1 }\) is \(r = 12 \sin \theta\).
2. A curve \(C _ { 2 }\) with polar equation \(r = 2 \sin \theta + 5,0 \leqslant \theta \leqslant 2 \pi\) is shown in the diagram.
   \includegraphics[max width=\textwidth, alt={}, center]{b572aeb5-bcbb-4d50-964c-7f37e223f51d-5_545_837_559_651} Calculate the area bounded by \(C _ { 2 }\).
3. The circle \(C _ { 1 }\) intersects the curve \(C _ { 2 }\) at the points \(P\) and \(Q\). Find, in surd form, the area of the quadrilateral \(O P M Q\), where \(M\) is the centre of the circle and \(O\) is the pole.
   (6 marks)

14 The hyperbola \(H\) has equation \(y ^ { 2 } - x ^ { 2 } = 16\) The circle \(C\) has equation \(x ^ { 2 } + y ^ { 2 } = 32\)
The diagram below shows part of the graph of \(H\) and part of the graph of \(C\).
\includegraphics[max width=\textwidth, alt={}, center]{8f7a5fc0-6936-4aed-a173-e221bf86e4fd-22_825_716_539_662} Show that the shaded region in the first quadrant enclosed by \(H , C\), the \(x\)-axis and the \(y\)-axis has area $$\frac { 16 \pi } { 3 } + 8 \ln \left( \frac { \sqrt { 2 } + \sqrt { 6 } } { 2 } \right)$$

14 The diagram shows the polar curve \(C _ { 1 }\) with equation \(r = 2 \sin \theta\) The diagram also shows part of the polar curve \(C _ { 2 }\) with equation \(r = 1 + \cos 2 \theta\)
\includegraphics[max width=\textwidth, alt={}, center]{b4ba8a08-333d-4efc-a0ed-14fef2d99410-20_378_897_456_954} 14

1. On the diagram above, complete the sketch of \(C _ { 2 }\) 14
2. Show that the area of the region shaded in the diagram is equal to $$k \pi + m \alpha - \sin 2 \alpha + q \sin 4 \alpha$$ where \(\alpha = \sin ^ { - 1 } \left( \frac { \sqrt { 5 } - 1 } { 2 } \right)\), and \(k , m\) and \(q\) are rational numbers.