Area enclosed by polar curve

7 The diagram below shows the curve with polar equation \(r = \sin 3 \theta\) for \(0 \leqslant \theta \leqslant \frac { 1 } { 3 } \pi\).
\includegraphics[max width=\textwidth, alt={}, center]{58e9b480-f561-4a28-b911-7d9d6a80e976-3_385_807_1834_260}

1. Find the values of \(\theta\) at the pole.
2. Find the polar coordinates of the point on the curve where \(r\) takes its maximum value.
3. In this question you must show detailed reasoning. Find the exact area enclosed by the curve.
4. Given that \(\sin 3 \theta = 3 \sin \theta - 4 \sin ^ { 3 } \theta\), find a cartesian equation for the curve.

3 In this question you must show detailed reasoning. The diagram below shows the curve \(r = 2 \cos 4 \theta\) for \(- k \pi \leq \theta \leq k \pi\) where \(k\) is a constant to be determined. Calculate the exact area enclosed by the curve.

6 The equation of a curve in polar coordinates is \(r = \ln ( 1 + \sin \theta )\) for \(\alpha \leqslant \theta \leqslant \beta\) where \(\alpha\) and \(\beta\) are non-negative angles. The curve consists of a single closed loop through the pole.

1. By solving the equation \(r = 0\), determine the smallest possible values of \(\alpha\) and \(\beta\).
2. Find the area enclosed by the curve, giving your answer to 4 significant figures.
3. Hence, by considering the value of \(r\) at \(\theta = \frac { \alpha + \beta } { 2 }\), show that the loop is not circular.

5

1. Sketch the polar curve \(\mathrm { r } = \mathrm { a } ( 1 - \cos \theta ) , 0 \leqslant \theta < 2 \pi\), where \(a\) is a positive constant.
2. Determine the exact area of the region enclosed by the curve.

9 A curve has polar equation \(r = \operatorname { asin } 3 \theta\), for \(0 \leqslant \theta \leqslant \pi\), where \(a\) is a positive constant.

1. Sketch the curve. Indicate the parts of the curve where \(r\) is negative by using a broken line.
2. In this question you must show detailed reasoning. Determine the area of one of the loops of the curve.

5 Fig. 5 shows the curve with polar equation \(r = a ( 3 + 2 \cos \theta )\) for \(- \pi \leqslant \theta \leqslant \pi\), where \(a\) is a constant. \begin{figure}[h] \end{figure}

1. Write down the polar coordinates of the points A and B .
2. Explain why the curve is symmetrical about the initial line.
3. In this question you must show detailed reasoning. Find in terms of \(a\) the exact area of the region enclosed by the curve.

9 A curve has polar equation \(r = a \sin 3 \theta\) for \(- \frac { 1 } { 3 } \pi \leq \theta \leq \frac { 1 } { 3 } \pi\), where \(a\) is a positive constant.

1. Sketch the curve.
2. In this question you must show detailed reasoning. Find, in terms of \(a\) and \(\pi\), the area enclosed by one of the loops of the curve.
3. Obtain the solution to the differential equation $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 3 y = \frac { 1 } { x } , \text { where } x > 0 ,$$ given that \(y = 1\) when \(x = 1\).
4. Deduce that \(y\) decreases as \(x\) increases.

4. (a) Given that \(z ^ { n } + \frac { 1 } { z ^ { n } } = 2 \cos n \theta\), where \(z = \cos \theta + \mathrm { i } \sin \theta\), express \(16 \cos ^ { 4 } \theta\) in the form \(a \cos 4 \theta + b \cos 2 \theta + c\), where \(a , b , c\) are integers whose values are to be determined. [5]
The diagram below shows a sketch of the curve C with polar equation $$r = 3 - 4 \cos ^ { 2 } \theta , \quad \text { where } \frac { \pi } { 6 } \leqslant \theta \leqslant \frac { 5 \pi } { 6 }$$ Initial line
(b) Calculate the area of the region enclosed by the curve \(C\).
(c) Find the exact polar coordinates of the points on \(C\) at which the tangent is perpendicular to the initial line.
\section*{PLEASE DO NOT WRITE ON THIS PAGE}

1. The curve \(C\) has polar equation \(r = 3 ( 2 + \cos \theta ) , 0 \leq \theta \leq \pi\). Determine the area enclosed between \(C\) and the initial line. Give your answer in the form \(\frac { a } { b } \pi\), where \(a\) and \(b\) are positive integers whose values are to be found.
2. Find the three cube roots of the complex number \(2 + 3 \mathrm { i }\), giving your answers in Cartesian form.
3. Find all the roots of the equation

$$\cos \theta + \cos 3 \theta + \cos 5 \theta = 0$$ lying in the interval \([ 0 , \pi ]\). Give all the roots in radians in terms of \(\pi\).

3. \begin{figure}[h] \end{figure} Diagram not to scale Figure 1 shows the design for a table top in the shape of a rectangle \(A B C D\). The length of the table, \(A B\), is 1.2 m . The area inside the closed curve is made of glass and the surrounding area, shown shaded in Figure 1, is made of wood. The perimeter of the glass is modelled by the curve with polar equation $$r = 0.4 + a \cos 2 \theta \quad 0 \leqslant \theta < 2 \pi$$ where \(a\) is a constant.

1. Show that \(a = 0.2\) Hence, given that \(A D = 60 \mathrm {~cm}\),
2. find the area of the wooden part of the table top, giving your answer in \(\mathrm { m } ^ { 2 }\) to 3 significant figures.

9 The diagram below shows the curve \(r = 4 \sin 3 \theta\) for \(0 \leqslant \theta \leqslant \frac { 1 } { 3 } \pi\).
\includegraphics[max width=\textwidth, alt={}, center]{c03cae53-eb00-496b-948f-ccff676bc03c-3_311_775_1713_644}

1. On the diagram in your Printed Answer Booklet, shade the region \(R\) for which $$r \leqslant 4 \sin 3 \theta \text { and } 0 \leqslant \theta \leqslant \frac { 1 } { 6 } \pi .$$ In this question you must show detailed reasoning.
2. Find the exact area of the region \(R\).

2 The equation of the curve shown on the graph is, in polar coordinates, \(r = 3 \sin 2 \theta\) for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
\includegraphics[max width=\textwidth, alt={}, center]{8315a796-0e7d-464f-8604-9fe3ab7af359-2_470_657_913_319}

1. The greatest value of \(r\) on the curve occurs at the point \(P\).
   1. Show that \(\theta = \frac { 1 } { 4 } \pi\) at the point \(P\).
   2. Find the value of \(r\) at the point \(P\).
   3. Mark the point \(P\) on the copy of the graph in the Printed Answer Booklet.
2. In this question you must show detailed reasoning. Find the exact area of the region enclosed by the curve.

16. Figure 1
\includegraphics[max width=\textwidth, alt={}, center]{858e6727-8498-462a-8064-d65254f1fd0f-08_478_938_283_502} Figure 1 shows a sketch of the cardioid \(C\) with equation \(r = a ( 1 + \cos \theta ) , - \pi < \theta \leq \pi\). Also shown are the tangents to \(C\) that are parallel and perpendicular to the initial line. These tangents form a rectangle \(W X Y Z\).

1. Find the area of the finite region, shaded in Fig. 1, bounded by the curve \(C\).
2. Find the polar coordinates of the points \(A\) and \(B\) where \(W Z\) touches the curve \(C\).
3. Hence find the length of \(W X\). Given that the length of \(W Z\) is \(\frac { 3 \sqrt { 3 } a } { 2 }\),
4. find the area of the rectangle \(W X Y Z\).
   (1) A heart-shape is modelled by the cardioid \(C\), where \(a = 10 \mathrm {~cm}\). The heart shape is cut from the rectangular card \(W X Y Z\), shown in Fig. 1.
5. Find a numerical value for the area of card wasted in making this heart shape.
   (2)
   [0pt] [P4 January 2003 Qn 8]

8 The equation of a curve, in polar coordinates, is $$r = 1 + \cos 2 \theta , \quad \text { for } 0 \leqslant \theta < 2 \pi$$

1. State the greatest value of \(r\) and the corresponding values of \(\theta\).
2. Find the equations of the tangents at the pole.
3. Find the exact area enclosed by the curve and the lines \(\theta = 0\) and \(\theta = \frac { 1 } { 2 } \pi\).
4. Find, in simplified form, the cartesian equation of the curve.

4

1. Show that \(( \cos \theta + \sin \theta ) ^ { 2 } = 1 + \sin 2 \theta\).
2. A curve has cartesian equation $$\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 3 } = ( x + y ) ^ { 4 }$$ Given that \(r \geqslant 0\), show that the polar equation of the curve is $$r = 1 + \sin 2 \theta$$
3. The curve with polar equation $$r = 1 + \sin 2 \theta , \quad - \pi \leqslant \theta \leqslant \pi$$ is shown in the diagram.
   \includegraphics[max width=\textwidth, alt={}, center]{f90167c3-2ffd-464a-b2d2-9f86a8d64887-3_389_611_1062_708}
   1. Find the two values of \(\theta\) for which \(r = 0\).
   2. Find the area of one of the loops.

9 Roberto is solving this mathematics problem: The curve \(C _ { 1 }\) has polar equation $$r ^ { 2 } = 9 \sin 2 \theta$$ for all possible values of \(\theta\)
Find the area enclosed by \(C _ { 1 }\) Roberto's solution is as follows: $$\begin{aligned} A & = \frac { 1 } { 2 } \int _ { - \pi } ^ { \pi } 9 \sin 2 \theta \mathrm {~d} \theta \\ & = \left[ - \frac { 9 } { 4 } \cos 2 \theta \right] _ { - \pi } ^ { \pi } \\ & = 0 \end{aligned}$$ 9

1. \(\quad\) Sketch the curve \(C _ { 1 }\) 9
2. Explain what Roberto has done wrong.
   9
3. \(\quad\) Find the area enclosed by \(C _ { 1 }\)
   9
4. \(\quad P\) and \(Q\) are distinct points on \(C _ { 1 }\) for which \(r\) is a maximum. \(P\) is above the initial line. Find the polar coordinates of \(P\) and \(Q\)
   9
5. The matrix \(\mathbf { M } = \left[ \begin{array} { l l } 1 & 2 \\ 0 & 1 \end{array} \right]\) represents the transformation T T maps \(C _ { 1 }\) onto a curve \(C _ { 2 }\)
   9
6. 1. T maps \(P\) onto the point \(P ^ { \prime }\)
      Find the polar coordinates of \(P ^ { \prime }\)
      [0pt] [4 marks]
      9
7. (ii) Find the area enclosed by \(C _ { 2 }\) Fully justify your answer.

1 The equation of the curve shown on the graph is, in polar coordinates, \(r = 3 \sin 2 \theta\) for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
\includegraphics[max width=\textwidth, alt={}, center]{4282a136-2ad0-43ce-929a-3d154a4c4af1-02_481_675_440_264}

1. The greatest value of \(r\) on the curve occurs at the point \(P\).
   1. Show that \(\theta = \frac { 1 } { 4 } \pi\) at the point \(P\).
   2. Find the value of \(r\) at the point \(P\).
   3. Mark the point \(P\) on a copy of the graph.
2. In this question you must show detailed reasoning. Find the exact area of the region enclosed by the curve.

3 The equation of a curve in polar coordinates is \(r = \ln ( 1 + \sin \theta )\) for \(\alpha \leqslant \theta \leqslant \beta\) where \(\alpha\) and \(\beta\) are non-negative angles. The curve consists of a single closed loop through the pole.

1. By solving the equation \(r = 0\), determine the smallest possible values of \(\alpha\) and \(\beta\).
2. Find the area enclosed by the curve, giving your answer to 4 significant figures.
3. Hence, by considering the value of \(r\) at \(\theta = \frac { \alpha + \beta } { 2 }\), show that the loop is not circular.