4.09c Area enclosed: by polar curve

1 A family of curves has polar equation \(r = \cos n \left( \frac { \theta } { n } \right) , 0 \leq \theta < n \pi\), where \(n\) is a positive even integer.

1. (A) Sketch the curve for the cases \(n = 2\) and \(n = 4\).
   (B) State two points which lie on every curve in the family.
   (C) State one other feature common to all the curves.
2. (A) Write down an integral for the length of the curve for the case \(n = 4\).
   (B) Evaluate the integral.
3. (A) Using \(t = \theta\) as the parameter, find a parametric form of the equation of the family of curves.
   (B) Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { \sin t \sin \left( \frac { t } { n } \right) - \cos t \cos \left( \frac { t } { n } \right) } { \sin t \cos \left( \frac { t } { n } \right) + \cos t \sin \left( \frac { t } { n } \right) }\).
4. Hence show that there are \(n + 1\) points where the tangent to the curve is parallel to the \(y\)-axis.
5. By referring to appropriate sketches, show that the result in part (iv) is true in the case \(n = 4\).
6. (A) Create a program to find all the solutions to \(x ^ { 2 } \equiv - 1 ( \bmod p )\) where \(0 \leq x < p\). Write out your program in full in the Printed Answer Booklet.
   (B) Use the program to find the solutions to \(x ^ { 2 } \equiv - 1 ( \bmod p )\) for the primes
   $$\begin{aligned} ( 4 k ) ! & \equiv 1 \times 2 \times 3 \times \ldots \times ( 2 k - 1 ) \times 2 k \times ( 2 k + 1 ) \times ( 2 k + 2 ) \times \ldots \times ( 4 k - 1 ) \times 4 k ( \bmod p ) \\ & \equiv 1 \times 2 \times 3 \times \ldots \times ( 2 k - 1 ) \times 2 k \times ( - 2 k ) \times ( - ( 2 k - 1 ) ) \times \ldots \times ( - 2 ) \times ( - 1 ) ( \bmod p ) \\ & \equiv ( ( 2 k ) ! ) ^ { 2 } ( \bmod p ) \end{aligned}$$ (A) Explain why ( \(2 k + 2\) ) can be written as ( \(- ( 2 k - 1 )\) ) in line ( 2 ).
   (B) Explain how line (3) has been obtained.
   (C) Explain why, if \(p\) is a prime of the form \(p = 4 k + 1\), then \(x ^ { 2 } \equiv - 1 ( \bmod p )\) will have at least one solution.
   (D) Hence find a solution of \(x ^ { 2 } \equiv - 1 ( \bmod 29 )\).
7. (A) Create a program that will find all the positive integers \(n\), where \(n < 1000\), such that \(( n - 1 ) ! \equiv - 1 \left( \bmod n ^ { 2 } \right)\). Write out your program in full.
   (B) State the values of \(n\) obtained.
   (C) A Wilson prime is a prime \(p\) such that \(( p - 1 ) ! \equiv - 1 \left( \bmod p ^ { 2 } \right)\). Write down all the Wilson primes \(p\) where \(p < 1000\).

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.

3. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of two curves \(C _ { 1 }\) and \(C _ { 2 }\) with polar equations $$\begin{array} { l l } C _ { 1 } : r = ( 1 + \sin \theta ) & 0 \leqslant \theta < 2 \pi \\ C _ { 2 } : r = 3 ( 1 - \sin \theta ) & 0 \leqslant \theta < 2 \pi \end{array}$$ The region \(R\) lies inside \(C _ { 1 }\) and outside \(C _ { 2 }\) and is shown shaded in Figure 1.
Show that the area of \(R\) is $$p \sqrt { 3 } - q \pi$$ where \(p\) and \(q\) are integers to be determined.

1. In this question you must show all stages of your working.

\section*{Solutions relying entirely on calculator technology are not acceptable.} \begin{figure}[h] \end{figure} Figure 1 shows the design for a bathing pool.
The pool, \(P\), shown unshaded in Figure 1, is surrounded by a tiled area, \(T\), shown shaded in Figure 1. The tiled area is bounded by the edge of the pool and by a circle, \(C\), with radius 6 m .
The centre of the pool and the centre of the circle are the same point.
The edge of the pool is modelled by the curve with polar equation $$r = 4 - a \sin 3 \theta \quad 0 \leqslant \theta \leqslant 2 \pi$$ where \(a\) is a positive constant.
Given that the shortest distance between the edge of the pool and the circle \(C\) is 0.5 m ,

1. determine the value of \(a\).
2. Hence, using algebraic integration, determine, according to the model, the exact area of \(T\).

4. \begin{figure}[h] \end{figure} The curve \(C\) shown in Figure 1 has polar equation $$r = 4 + \cos 2 \theta \quad 0 \leqslant \theta \leqslant \frac { \pi } { 2 }$$ At the point \(A\) on \(C\), the value of \(r\) is \(\frac { 9 } { 2 }\) The point \(N\) lies on the initial line and \(A N\) is perpendicular to the initial line.
The finite region \(R\), shown shaded in Figure 1, is bounded by the curve \(C\), the initial line and the line \(A N\). Find the exact area of the shaded region \(R\), giving your answer in the form \(p \pi + q \sqrt { 3 }\) where \(p\) and \(q\) are rational numbers to be found.

1. The curve \(C\) has equation

$$r = a ( p + 2 \cos \theta ) \quad 0 \leqslant \theta < 2 \pi$$ where \(a\) and \(p\) are positive constants and \(p > 2\) There are exactly four points on \(C\) where the tangent is perpendicular to the initial line.

1. Show that the range of possible values for \(p\) is $$2 < p < 4$$
2. Sketch the curve with equation $$r = a ( 3 + 2 \cos \theta ) \quad 0 \leqslant \theta < 2 \pi \quad \text { where } a > 0$$ John digs a hole in his garden in order to make a pond.
   The pond has a uniform horizontal cross section that is modelled by the curve with equation $$r = 20 ( 3 + 2 \cos \theta ) \quad 0 \leqslant \theta < 2 \pi$$ where \(r\) is measured in centimetres. The depth of the pond is 90 centimetres.
   Water flows through a hosepipe into the pond at a rate of 50 litres per minute.
   Given that the pond is initially empty,
3. determine how long it will take to completely fill the pond with water using the hosepipe, according to the model. Give your answer to the nearest minute.
4. State a limitation of the model.

7. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve \(C\) with equation $$r = 1 + \tan \theta \quad 0 \leqslant \theta < \frac { \pi } { 3 }$$ Figure 1 also shows the tangent to \(C\) at the point \(A\).
This tangent is perpendicular to the initial line.

1. Use differentiation to prove that the polar coordinates of \(A\) are \(\left( 2 , \frac { \pi } { 4 } \right)\) The finite region \(R\), shown shaded in Figure 1, is bounded by \(C\), the tangent at \(A\) and the initial line.
2. Use calculus to show that the exact area of \(R\) is \(\frac { 1 } { 2 } ( 1 - \ln 2 )\)

1. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with polar equation $$r = 2 \sqrt { \sinh \theta + \cosh \theta } \quad 0 \leqslant \theta \leqslant \pi$$ The region \(R\), shown shaded in Figure 1, is bounded by the initial line, the curve and the line with equation \(\theta = \pi\) Use algebraic integration to determine the exact area of \(R\) giving your answer in the form \(p \mathrm { e } ^ { q } - r\) where \(p , q\) and \(r\) are real numbers to be found.

1. The locus \(C\) is given by

$$| z - 4 | = 4$$ The locus \(D\) is given by $$\arg z = \frac { \pi } { 3 }$$

1. Sketch, on the same Argand diagram, the locus \(C\) and the locus \(D\) The set of points \(A\) is defined by $$A = \{ z \in \mathbb { C } : | z - 4 | \leqslant 4 \} \cap \left\{ z \in \mathbb { C } : 0 \leqslant \arg z \leqslant \frac { \pi } { 3 } \right\}$$
2. Show, by shading on your Argand diagram, the set of points \(A\)
3. Find the area of the region defined by \(A\), giving your answer in the form \(p \pi + q \sqrt { 3 }\) where \(p\) and \(q\) are constants to be determined.

1. (a) (i) Show on an Argand diagram the locus of points given by the values of \(z\) satisfying

$$| z - 4 - 3 \mathbf { i } | = 5$$ Taking the initial line as the positive real axis with the pole at the origin and given that \(\theta \in [ \alpha , \alpha + \pi ]\), where \(\alpha = - \arctan \left( \frac { 4 } { 3 } \right)\),
(ii) show that this locus of points can be represented by the polar curve with equation $$r = 8 \cos \theta + 6 \sin \theta$$ The set of points \(A\) is defined by $$A = \left\{ z : 0 \leqslant \arg z \leqslant \frac { \pi } { 3 } \right\} \cap \{ z : | z - 4 - 3 \mathbf { i } | \leqslant 5 \}$$ (b) (i) Show, by shading on your Argand diagram, the set of points \(A\).
(ii) Find the exact area of the region defined by \(A\), giving your answer in simplest form.

1. A community is concerned about the rising level of pollutant in its local pond and applies a chemical treatment to stop the increase of pollutant.

The concentration, \(x\) parts per million (ppm), of the pollutant in the pond water \(t\) days after the chemical treatment was applied, is modelled by the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { 3 + \cosh t } { 3 x ^ { 2 } \cosh t } - \frac { 1 } { 3 } x \tanh t$$ When the chemical treatment was applied the concentration of pollutant was 3 ppm .

1. Use the iteration formula $$\left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) _ { n } \approx \frac { \left( y _ { n + 1 } - y _ { n } \right) } { h }$$ once to estimate the concentration of the pollutant in the pond water 6 hours after the chemical treatment was applied.
2. Show that the transformation \(u = x ^ { 3 }\) transforms the differential equation (I) into the differential equation $$\frac { \mathrm { d } u } { \mathrm {~d} t } + u \tanh t = 1 + \frac { 3 } { \cosh t }$$
3. Determine the general solution of equation (II)
4. Hence find an equation for the concentration of pollutant in the pond water \(t\) days after the chemical treatment was applied.
5. Find the percentage error of the estimate found in part (a) compared to the value predicted by the model, stating if it is an overestimate or an underestimate.

8. \begin{figure}[h] \end{figure} Figure 1 shows the vertical cross section of a child's spinning top. The point \(A\) is vertically above the point \(B\) and the height of the spinning top is 5 cm . The line \(C D\) is perpendicular to \(A B\) such that \(C D\) is the maximum width of the spinning top.
The spinning top is modelled as the solid of revolution created when part of the curve with polar equation $$r ^ { 2 } = 25 \cos 2 \theta$$ is rotated through \(2 \pi\) radians about the initial line.

1. Show that, according to the model, the surface area of the spinning top is $$k \pi ( 2 - \sqrt { 2 } ) \mathrm { cm } ^ { 2 }$$ where \(k\) is a constant to be determined.
2. Show that, according to the model, the length \(C D\) is \(\frac { 5 \sqrt { 2 } } { 2 } \mathrm {~cm}\).

4. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} Figure 1 shows a sketch of a design for a road speed bump of width 2.35 metres. The speed bump has a uniform cross-section with vertical ends and its length is 30 cm . A side profile of the speed bump is shown in Figure 2. The curve \(C\) shown in Figure 2 is modelled by the polar equation $$r = 30 \left( 1 - \theta ^ { 2 } \right) \quad 0 \leqslant \theta \leqslant 1$$ The units for \(r\) are centimetres and the initial line lies along the road surface, which is assumed to be horizontal. Once the speed bump has been fixed to the road, the visible surfaces of the speed bump are to be painted. Determine, in \(\mathrm { cm } ^ { 2 }\), the area that is to be painted, according to the model.

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 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.

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.

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)

7 A curve \(C\) has polar equation $$r = 6 + 4 \cos \theta , \quad - \pi \leqslant \theta \leqslant \pi$$ The diagram shows a sketch of the curve \(C\), the pole \(O\) and the initial line. \includegraphics[max width=\textwidth, alt={}, center]{0d894ac0-8d96-4182-8454-c306e1fdad8f-4_599_866_612_587}

1. Calculate the area of the region bounded by the curve \(C\).
2. The point \(P\) is the point on the curve \(C\) for which \(\theta = \frac { 2 \pi } { 3 }\). The point \(Q\) is the point on \(C\) for which \(\theta = \pi\).
   Show that \(Q P\) is parallel to the line \(\theta = \frac { \pi } { 2 }\).
3. The line \(P Q\) intersects the curve \(C\) again at a point \(R\). The line \(R O\) intersects \(C\) again at a point \(S\).
   1. Find, in surd form, the length of \(P S\).
   2. Show that the angle \(O P S\) is a right angle.

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.

7. \begin{figure}[h] \end{figure} Figure 4 shows a circle with radius \(r _ { 1 }\) and a circle with radius \(r _ { 2 }\) The circles touch externally at a single point above the \(x\)-axis.
Both circles also have the \(x\)-axis as a tangent.

1. Show that the horizontal distance between the centres of the circles, \(d\), is given by $$d ^ { 2 } = 4 r _ { 1 } r _ { 2 }$$ The finite region \(R\), shown shaded in Figure 4, is bounded by the \(x\)-axis and minor arcs of the two circles. Given that \(r _ { 1 } \geqslant r _ { 2 }\)
2. show that the area of \(R\) is given by $$\left( r _ { 1 } + r _ { 2 } \right) \sqrt { r _ { 1 } r _ { 2 } } - \frac { 1 } { 2 } \left( r _ { 1 } ^ { 2 } - r _ { 2 } ^ { 2 } \right) \theta - \frac { 1 } { 2 } \pi r _ { 2 } ^ { 2 }$$ where \(\cos \theta = \frac { r _ { 1 } - r _ { 2 } } { r _ { 1 } + r _ { 2 } }\) Question 7 continues on the next page.
   
   \includegraphics[max width=\textwidth, alt={}]{a8e9db6b-dfad-4278-82d8-a8fa5ba61008-27_2269_53_306_36}
   \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a8e9db6b-dfad-4278-82d8-a8fa5ba61008-27_759_1378_269_347} \captionsetup{labelformat=empty} \caption{Figure 5}
   \end{figure} A sequence of circles, \(C _ { 1 } , C _ { 2 } , C _ { 3 } , \ldots\) with radii \(r _ { 1 } , r _ { 2 } , r _ { 3 } , \ldots\) respectively, is constructed such that
   • each circle is tangential to and above the \(x\)-axis
   • the first circle, \(C _ { 1 }\), has centre \(( 0,1 )\)
   • each successive circle touches the preceding one externally at a single point
   • the horizontal distances between the centres of successive circles form a geometric sequence with first term 2 and common ratio \(\frac { 1 } { \sqrt { 3 } }\)
   The first few circles in the sequence are shown in Figure 5.
3. 1. Determine the value of \(r _ { 3 }\)
   2. Show that, for \(n \geqslant 1 , r _ { n + 2 } = k r _ { n }\) where \(k\) is a constant to be determined.
   3. Hence show that, for \(n \geqslant 1 , r _ { 2 n } = r _ { 2 n - 1 }\) The region bounded between \(C _ { n } , C _ { n + 1 }\) and the \(x\)-axis is \(R _ { n }\) The total area, \(A\), bounded above the \(x\)-axis and under all the circles is the sum of the areas of all these regions.
4. Determine the value of \(A\), giving the answer in simplest form. \section*{Paper reference} \section*{Advanced Extension Award Mathematics} Insert for questions 5, 6 and 7
   Do not write on this insert.
   5. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a8e9db6b-dfad-4278-82d8-a8fa5ba61008-34_298_1040_212_516} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure} Figure 2 shows a sketch of a hexagon \(O A B C D E\) where
   Given that \(\overrightarrow { O A } = \mathbf { a }\) and \(\overrightarrow { O E } = \mathbf { e }\) determine as simplified expressions in terms of \(\mathbf { a }\) and \(\mathbf { e }\)
   1. \(\overrightarrow { A B }\)
   2. \(\overrightarrow { O D }\) The point \(R\) divides \(A B\) internally in the ratio \(1 : 2\)
   3. Determine \(\overrightarrow { R C }\) as a simplified expression in terms of \(\mathbf { a }\) and \(\mathbf { e }\) The line through the points \(R\) and \(C\) meets the line through the points \(O\) and \(D\) at the point \(X\).
   4. Determine \(\overrightarrow { O X }\) in the form \(\lambda \mathbf { a } + \mu \mathbf { e }\), where \(\lambda\) and \(\mu\) are real values in simplest form.
      6. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{a8e9db6b-dfad-4278-82d8-a8fa5ba61008-35_236_1363_205_351} \captionsetup{labelformat=empty} \caption{Figure 3}
      \end{figure} Figure 3 shows a block \(A\) with mass \(4 m\) and a block \(B\) with mass \(5 m\).
      Block \(A\) is at rest on a rough plane inclined at an angle \(\alpha\) to the horizontal.
      Block \(B\) is at rest on a rough plane inclined at an angle \(\beta\) to the horizontal.
      The blocks are connected by a light inextensible string which passes over a small smooth pulley at the top of each plane. A small smooth ring \(C\), of mass \(8 m\), is threaded on the string between the pulleys so that \(A , B\) and \(C\) all lie in the same vertical plane. The part of the string between \(A\) and its pulley lies along a line of greatest slope of the plane of angle \(\alpha\). The part of the string between \(B\) and its pulley lies along a line of greatest slope of the plane of angle \(\beta\). The angle between the vertical and the string between each pulley and the ring \(C\) is \(\gamma\).
      The two blocks, \(A\) and \(B\), are modelled as particles.
      Given that
      7. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{a8e9db6b-dfad-4278-82d8-a8fa5ba61008-36_721_1771_205_146} \captionsetup{labelformat=empty} \caption{Figure 4}
      \end{figure} Figure 4 shows a circle with radius \(r _ { 1 }\) and a circle with radius \(r _ { 2 }\) The circles touch externally at a single point above the \(x\)-axis.
      Both circles also have the \(x\)-axis as a tangent.
   5. Show that the horizontal distance between the centres of the circles, \(d\), is given by $$d ^ { 2 } = 4 r _ { 1 } r _ { 2 }$$ The finite region \(R\), shown shaded in Figure 4, is bounded by the \(x\)-axis and minor arcs of the two circles. Given that \(r _ { 1 } \geqslant r _ { 2 }\)
   6. show that the area of \(R\) is given by $$\left( r _ { 1 } + r _ { 2 } \right) \sqrt { r _ { 1 } r _ { 2 } } - \frac { 1 } { 2 } \left( r _ { 1 } ^ { 2 } - r _ { 2 } ^ { 2 } \right) \theta - \frac { 1 } { 2 } \pi r _ { 2 } ^ { 2 }$$ where \(\cos \theta = \frac { r _ { 1 } - r _ { 2 } } { r _ { 1 } + r _ { 2 } }\) Question 7 continues on the next page. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{a8e9db6b-dfad-4278-82d8-a8fa5ba61008-37_761_1376_210_349} \captionsetup{labelformat=empty} \caption{Figure 5}
      \end{figure} A sequence of circles, \(C _ { 1 } , C _ { 2 } , C _ { 3 } , \ldots\) with radii \(r _ { 1 } , r _ { 2 } , r _ { 3 } , \ldots\) respectively, is constructed such that
      The first few circles in the sequence are shown in Figure 5.
   7. 1. Determine the value of \(r _ { 3 }\)
      2. Show that, for \(n \geqslant 1 , r _ { n + 2 } = k r _ { n }\) where \(k\) is a constant to be determined.
      3. Hence show that, for \(n \geqslant 1 , r _ { 2 n } = r _ { 2 n - 1 }\) The region bounded between \(C _ { n } , C _ { n + 1 }\) and the \(x\)-axis is \(R _ { n }\) The total area, \(A\), bounded above the \(x\)-axis and under all the circles is the sum of the areas of all these regions.
   8. Determine the value of \(A\), giving the answer in simplest form.

17 The curve \(C _ { 1 }\) has polar equation \(r = 2 a ( 1 + \sin \theta )\) for \(- \pi < \theta \leq \pi\) where \(a\) is a positive constant. \includegraphics[max width=\textwidth, alt={}, center]{f7e7c21b-6e72-4c20-92fc-ba0336a11136-22_469_830_402_605} The point \(M\) lies on \(C _ { 1 }\) and the initial line.
17

1. Write down, in terms of \(a\), the polar coordinates of \(M\) 17
2. \(\quad N\) is the point on \(C _ { 1 }\) that is furthest from the pole \(O\) Find, in terms of \(a\), the polar coordinates of \(N\) 17
3. The curve \(C _ { 2 }\) has polar equation \(r = 3 a\) for \(- \pi < \theta \leq \pi\) \(C _ { 2 }\) intersects \(C _ { 1 }\) at points \(P\) and \(Q\) Show that the area of triangle \(N P Q\) can be written in the form $$m \sqrt { 3 } a ^ { 2 }$$ where \(m\) is a rational number to be determined.
   17
4. On the initial line below, sketch the graph of \(r = 2 a ( 1 + \cos \theta )\) for \(- \pi < \theta \leq \pi\) Include the polar coordinates, in terms of \(a\), of any intersection points with the initial line.
   [0pt] [2 marks] \includegraphics[max width=\textwidth, alt={}, center]{f7e7c21b-6e72-4c20-92fc-ba0336a11136-24_65_657_1425_991} \includegraphics[max width=\textwidth, alt={}, center]{f7e7c21b-6e72-4c20-92fc-ba0336a11136-25_2492_1721_217_150}

9

1. The line \(L\) has polar equation $$r = \frac { 7 } { 4 } \sec \theta \quad \left( - \frac { \pi } { 2 } < \theta < \frac { \pi } { 2 } \right)$$ Show that \(L\) is perpendicular to the initial line.
   9
2. The curve \(C\) has polar equation $$r = 3 + \cos \theta \quad ( - \pi < \theta \leq \pi )$$ Find the polar coordinates of the points of intersection of \(L\) and \(C\) Fully justify your answer.
   9
3. The region \(R\) is the set of points such that
   and $$r > \frac { 7 } { 4 } \sec \theta$$ Find the exact area of \(R\) $$r < 3 + \cos \theta$$ Find the exact area of \(R\) [0pt] [7 marks]

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.