4.09b Sketch polar curves: r = f(theta)

7 The diagram below shows the curve with polar equation \(r = a ( 1 - 2 \sin \theta )\) for \(0 \leqslant \theta \leqslant 2 \pi\), where \(a\) is a positive constant. \includegraphics[max width=\textwidth, alt={}, center]{76631941-3cd5-4b3e-a7e4-27b8f991975a-4_634_865_486_239} The curve crosses the initial line at A , and the points B and C are the lowest points on the two loops.

1. Find the values of \(r\) and \(\theta\) at the points A , B and C .
2. Find the set of values of \(\theta\) for the points on the inner loop (shown in the diagram with a broken line).

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.

14 A curve has polar equation \(\mathrm { r } = \mathrm { a } ( \cos \theta + 2 \sin \theta )\), where \(a\) is a positive constant and \(0 \leqslant \theta \leqslant \pi\).

1. Determine the polar coordinates of the point on the curve which is furthest from the pole.
2. 1. Show that the curve is a circle whose radius should be specified.
   2. Write down the polar coordinates of the centre of the circle.

1 A family of curves is given by the parametric equations \(x ( t ) = \cos ( t ) - \frac { \cos ( ( m + 1 ) t ) } { m + 1 }\) and \(y ( t ) = \sin ( t ) - \frac { \sin ( ( m + 1 ) t ) } { m + 1 }\) where \(0 \leqslant t < 2 \pi\) and \(m\) is a positive integer.

1. 1. Sketch the curves in the cases \(m = 3 , m = 4\) and \(m = 5\) on separate axes in the Printed Answer Booklet.
   2. State one common feature of these three curves.
   3. State a feature for the case \(m = 4\) which is absent in the cases \(m = 3\) and \(m = 5\).
2. 1. Determine, in terms of \(m\), the values of \(t\) for which \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 0\) but \(\frac { \mathrm { d } y } { \mathrm {~d} t } \neq 0\).
   2. Describe the tangent to the curve at the points corresponding to such values of \(t\).
3. 1. Show that the curve lies between the circle centred at the origin with radius $$1 - \frac { 1 } { m + 1 }$$ and the circle centred at the origin with radius $$1 + \frac { 1 } { m + 1 }$$
   2. Hence, or otherwise, show that the area \(A\) bounded by the curve satisfies $$\frac { m ^ { 2 } \pi } { ( m + 1 ) ^ { 2 } } < A < \frac { ( m + 2 ) ^ { 2 } \pi } { ( m + 1 ) ^ { 2 } }$$
   3. Find the limit of the area bounded by the curve as \(m\) tends to infinity.
4. The arc length of a curve defined by parametric equations \(x ( t )\) and \(y ( t )\) between points corresponding to \(t = c\) and \(t = d\), where \(c < d\), is $$\int _ { c } ^ { d } \sqrt { \left( \frac { \mathrm {~d} x } { \mathrm {~d} t } \right) ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} t } \right) ^ { 2 } } \mathrm {~d} t$$ Use this to show that the length of the curve is independent of \(m\).

1

1. A family of curves is given by the equation $$x ^ { 2 } + y ^ { 2 } + 2 a x y = 1 ( * )$$ where the parameter \(a\) is a real number.
   You may find it helpful to use a slider (for \(a\) ) to investigate this family of curves.
   1. On the axes in the Printed Answer Booklet, sketch the curve in each of the cases
      • \(a = 0\)
      • \(a = 0.5\)
      • \(a = 2\)
      • State a feature of the curve for the cases \(a = 0 , a = 0.5\) that is not a feature of the curve in the case \(a = 2\).
      • In the case \(a = 1\), the curve consists of two straight lines. Determine the equations of these lines.
      • 1. Find an equation of the curve (*) in polar form.
        2. Hence, or otherwise, find, in exact form, the area bounded by the curve, the positive part of the \(x\)-axis and the positive part of the \(y\)-axis, in the case \(a = 2\).
2. In this part of the question \(m\) is any real number.
Describing all possible cases, determine the pairs of values \(a\) and \(m\) for which the curve with equation (*) intersects the straight line given by \(y = m x\).

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 \(T\).

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. (a) Sketch the polar curve \(C\), with equation

$$r = 3 + \sqrt { 5 } \cos \theta \quad 0 \leqslant \theta \leqslant 2 \pi$$ On your sketch clearly label the pole, the initial line and the value of \(r\) at the point where the curve intersects the initial line. The tangent to \(C\) at the point \(A\), where \(0 < \theta < \frac { \pi } { 2 }\), is parallel to the initial line.
(b) Use calculus to show that at \(A\) $$\cos \theta = \frac { 1 } { \sqrt { 5 } }$$ (c) Hence determine the value of \(r\) at \(A\).

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.

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

1. (a) A curve has polar equation \(r = 2 - \sec \theta\). Show that the cartesian equation of the curve can be written in the form $$y ^ { 2 } = \left( \frac { 2 x } { x + 1 } \right) ^ { 2 } - x ^ { 2 }$$ The figure shows a sketch of part of the curve with equation \(y ^ { 2 } = \left( \frac { 2 x } { x + 1 } \right) ^ { 2 } - x ^ { 2 }\). \includegraphics[max width=\textwidth, alt={}, center]{9d2db858-9c4d-4281-8e8d-9fb5cb11b8ca-4_681_695_667_685}
   (b) Explain why the curve is symmetrical in the \(x\)-axis.
   (c) The line \(x = a\) is an asymptote of the curve. State the value of \(a\).
2. The enclosed loop shown in the figure is rotated through \(180 ^ { \circ }\) about the \(x\)-axis. Find the exact volume of the solid formed. \section*{END OF QUESTION PAPER}

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.

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.

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}

8 The curve \(C\) has the polar equation $$r = 4 - 2 \cos \theta \quad - \pi < \theta \leq \pi$$ 8

1. Verify that the point with polar coordinates \(\left( 3 , \frac { \pi } { 3 } \right)\) lies on \(C\) 8
2. Find the exact polar coordinates of the point on \(C\) which is furthest from the pole, \(O\) [3 marks]
   8
3. Find the exact Cartesian coordinates of the point on \(C\) where \(\theta\) is \(\frac { \pi } { 6 }\)

16 The curve \(C\) has the polar equation $$r = \frac { 2 } { \sqrt { \cos ^ { 2 } \theta + 4 \sin ^ { 2 } \theta } } \quad - \pi < \theta \leq \pi$$ 16

1. Show that the Cartesian equation of \(C\) can be written as $$\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$$ where \(a\) and \(b\) are positive integers to be determined.
   [0pt] [4 marks]
   16
2. Hence sketch the graph of \(C\) on the axes below. Indicate the value of any intercepts of the curve with the axes. \includegraphics[max width=\textwidth, alt={}, center]{47b12ae4-ca3f-472c-9d15-2ef17a2a4d87-23_1122_1121_452_447}

3 Find the equations of the asymptotes of the curve \(x ^ { 2 } - 3 y ^ { 2 } = 1\) Circle your answer.
[0pt] [1 mark] $$y = \pm 3 x \quad y = \pm \frac { 1 } { 3 } x \quad y = \pm \sqrt { 3 } x \quad y = \pm \frac { 1 } { \sqrt { 3 } } x$$ Turn over for the next question \(\mathbf { 4 } \quad \mathbf { A } = \left[ \begin{array} { l l } 1 & 2 \\ 1 & k \end{array} \right] \quad \mathbf { B } = \left[ \begin{array} { c c } - 1 & 0 \\ 0 & 1 \end{array} \right]\)

8 A curve has polar equation \(r = 3 + 2 \cos \theta\), where \(0 \leq \theta < 2 \pi\) 8

1. 1. State the maximum and minimum values of \(r\).
      [0pt] [2 marks]
      L
      8
      1. (ii) Sketch the curve. \includegraphics[max width=\textwidth, alt={}, center]{e61d0202-49c9-4ed9-9fa3-f10734e17463-12_77_832_2037_651} 8
   2. The curve \(r = 3 + 2 \cos \theta\) intersects the curve with polar equation \(r = 8 \cos ^ { 2 } \theta\), where \(0 \leq \theta < 2 \pi\) Find all of the points of intersection of the two curves in the form \([ r , \theta ]\).
      [0pt] [6 marks]