Polar coordinates

8 The curve \(C\) has polar equation \(r = \theta ^ { 2 } + 2 \theta\) for \(0 \leq \theta \leq 3\).

1. Find the area of the region enclosed by \(C\) and the half-lines \(\theta = 0\) and \(\theta = 3\).
2. Determine the length of \(C\).

5 The curve \(C\) has polar equation \(r = 1 + 2 \cos \theta\). Sketch the curve for \(- \frac { 2 } { 3 } \pi \leqslant \theta < \frac { 2 } { 3 } \pi\). Find the area bounded by \(C\) and the half-lines \(\theta = - \frac { 1 } { 3 } \pi , \theta = \frac { 1 } { 3 } \pi\).

5 The curve \(C\) has polar equation \(r = \operatorname { acot } \left( \frac { 1 } { 3 } \pi - \theta \right)\), where \(a\) is a positive constant and \(0 \leqslant \theta \leqslant \frac { 1 } { 6 } \pi\). It is given that the greatest distance of a point on \(C\) from the pole is \(2 \sqrt { 3 }\).

1. Sketch \(C\) and show that \(a = 2\).
2. Find the exact value of the area of the region bounded by \(C\), the initial line and the half-line \(\theta = \frac { 1 } { 6 } \pi\).
3. Show that \(C\) has Cartesian equation \(2 ( x + y \sqrt { 3 } ) = ( x \sqrt { 3 } - y ) \sqrt { x ^ { 2 } + y ^ { 2 } }\).

1 Find the area of the region enclosed by the curve with polar equation \(r = 2 ( 1 + \cos \theta )\), for \(0 \leqslant \theta < 2 \pi\).

5. \begin{figure}[h] \end{figure} The curve \(C\), shown in Figure 1, has polar equation, \(r = 2 + \sin 3 \theta , 0 \leqslant \theta \leqslant \frac { \pi } { 2 }\) Use integration to calculate the exact value of the area enclosed by \(C\), the line \(\theta = 0\) and the line \(\theta = \frac { \pi } { 2 }\).

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.

7. \begin{figure}[h] \end{figure} Figure 1 shows the two curves given by the polar equations $$\begin{array} { l l } r = \sqrt { 3 } \sin \theta , & 0 \leqslant \theta \leqslant \pi \\ r = 1 + \cos \theta , & 0 \leqslant \theta \leqslant \pi \end{array}$$

1. Verify that the curves intersect at the point \(P\) with polar coordinates \(\left( \frac { 3 } { 2 } , \frac { \pi } { 3 } \right)\). The region \(R\), bounded by the two curves, is shown shaded in Figure 1.
2. Use calculus to find the exact area of \(R\), giving your answer in the form \(a ( \pi - \sqrt { 3 } )\), where \(a\) is a constant to be found.

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{figure_14}

1. On the diagram above, complete the sketch of \(C_2\) [2 marks]
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. [9 marks]

The curve \(C\) has polar equation $$r = 1 + 2 \cos \theta, \quad 0 \leq \theta \leq \frac{\pi}{2}.$$ At the point \(P\) on \(C\), the tangent to \(C\) is parallel to the initial line. Given that \(O\) is the pole, find the exact length of the line \(OP\). [7]

1. The curve \(C\), with pole \(O\), has polar equation

$$r = 1 + \cos \theta , \quad 0 \leqslant \theta \leqslant \frac { \pi } { 2 }$$ At the point \(A\) on \(C\), the tangent to \(C\) is parallel to the initial line.

1. Find the polar coordinates of \(A\).
2. Find the finite area enclosed by the initial line, the line \(O A\) and the curve \(C\), giving your answer in the form \(a \pi + b \sqrt { 3 }\), where \(a\) and \(b\) are rational constants to be found.

4. A curve \(C\) has polar equation \(r ^ { 2 } = a ^ { 2 } \cos 2 \theta , 0 \leq \theta \leq \frac { \pi } { 4 }\). The line \(l\) is parallel to the initial line, and \(l\) is the tangent to \(C\) at
above. above.

1. 1. Show that, for any point on \(C , r ^ { 2 } \sin ^ { 2 } \theta\) can be expressed in terms of \(\sin \theta\) and \(a\) only. (1)
   2. Hence, using differentiation, show that the polar coordinates of \(P\) are \(\left( \frac { a } { \sqrt { 2 } } , \frac { \pi } { 6 } \right)\).(6) \includegraphics[max width=\textwidth, alt={}, center]{2352f367-ddf9-4770-ace5-b561b0fbabbb-1_298_725_2163_1169} The shaded region \(R\), shown in the figure above, is bounded by \(C\), the line \(l\) and the half-line with equation \(\theta = \frac { \pi } { 2 }\).
2. Show that the area of \(R\) is \(\frac { a ^ { 2 } } { 16 } ( 3 \sqrt { 3 } - 4 )\).

Sketch the polar graph of $$r = \sinh \theta + \cosh \theta$$ for \(0 \leq \theta \leq 2\pi\) \includegraphics{figure_11} [3 marks]

The line \(L\) has polar equation $$r = \frac{k}{\sin \theta}$$ where \(k\) is a positive constant.

1. Sketch \(L\). [1 mark]
2. State the minimum distance between \(L\) and the point \(O\). [1 mark]

7 The curve \(C\) has polar equation \(r ^ { 2 } = \sin 2 \theta \cos \theta\), for \(0 \leqslant \theta \leqslant \pi\).

1. Sketch \(C\) and state the equation of the line of symmetry.
2. Find a Cartesian equation for \(C\).
   
3. Find the total area enclosed by \(C\).
4. Find the greatest distance of a point on \(C\) from the pole. \includegraphics[max width=\textwidth, alt={}, center]{7eb2abb1-68f4-4cc8-8314-f436906d6c4e-16_2718_36_141_2011} If you use the following page to complete the answer to any question, the question number must be clearly shown.
   

   \includegraphics[max width=\textwidth, alt={}]{7eb2abb1-68f4-4cc8-8314-f436906d6c4e-18_436_29_143_2014}

   \includegraphics[max width=\textwidth, alt={}]{7eb2abb1-68f4-4cc8-8314-f436906d6c4e-18_436_29_714_2014}

   \includegraphics[max width=\textwidth, alt={}]{7eb2abb1-68f4-4cc8-8314-f436906d6c4e-18_438_29_1283_2014}

   \includegraphics[max width=\textwidth, alt={}]{7eb2abb1-68f4-4cc8-8314-f436906d6c4e-18_436_29_1852_2014}

   \includegraphics[max width=\textwidth, alt={}]{7eb2abb1-68f4-4cc8-8314-f436906d6c4e-18_436_29_2423_2014}

2 A curve has polar equation \(r ( 1 - \sin \theta ) = 4\). Find its cartesian equation in the form \(y = \mathrm { f } ( x )\).

Point \(P\) has polar coordinates \(\left(2, \frac{2\pi}{3}\right)\). Which of the following are the Cartesian coordinates of \(P\)? Circle your answer. [1 mark] \((1, -\sqrt{3})\) \quad \((-\sqrt{3}, 1)\) \quad \((\sqrt{3}, -1)\) \quad \((-1, \sqrt{3})\)

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 \(y = m x\).

The polar equation of the circle \(C\) is $$r = a(\cos \theta + \sin \theta)$$ Find, in terms of \(a\), the radius of \(C\). Fully justify your answer. [4 marks]

The curve C has polar equation $$r^2 \sin 2\theta = 4$$ Find a Cartesian equation for C. Circle your answer. [1 mark] \(y = 2x\) \quad \(y = \frac{x}{2}\) \quad \(y = \frac{2}{x}\) \quad \(y = 4x\)

Fig. 5 shows a circle with centre C \((a, 0)\) and radius \(a\). B is the point \((0, 1)\). The line BC intersects the circle at P and Q. P is above the \(x\)-axis and Q is below. \includegraphics{figure_5}

1. Show that, in the case \(a = 1\), P has coordinates \(\left(1 - \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)\). Write down the coordinates of Q. [3]
2. Show that, for all positive values of \(a\), the coordinates of P are $$x = a\left(1 - \frac{a}{\sqrt{a^2 + 1}}\right), \quad y = \frac{a}{\sqrt{a^2 + 1}} \quad (*)$$ Write down the coordinates of Q in a similar form. [4] Now let the variable point P be defined by the parametric equations (*) for all values of the parameter \(a\), positive, zero and negative. Let Q be defined for all \(a\) by your answer in part (ii).
3. Using your calculator, sketch the locus of P as \(a\) varies. State what happens to P as \(a \to \infty\) and as \(a \to -\infty\). Show algebraically that this locus has an asymptote at \(y = -1\). On the same axes, sketch, as a dotted line, the locus of Q as \(a\) varies. [8] (The single curve made up of these two loci and including the point B is called a right strophoid.)
4. State, with a reason, the size of the angle POQ in Fig. 5. What does this indicate about the angle at which a right strophoid crosses itself? [3]

1. Find the exact area enclosed by the curve.
2. Show that the greatest value of \(r\) on the curve is \(\sqrt { \frac { \sqrt { 3 } } { 2 } } \mathrm { e } ^ { \frac { 1 } { 6 } }\).

10 The curve \(C\) has polar equation \(r = 2 \sin \theta ( 1 - \cos \theta )\), for \(0 \leqslant \theta \leqslant \pi\). Find \(\frac { \mathrm { d } r } { \mathrm {~d} \theta }\) and hence find the polar coordinates of the point of \(C\) that is furthest from the pole. Sketch \(C\). Find the exact area of the sector from \(\theta = 0\) to \(\theta = \frac { 1 } { 4 } \pi\).

6 The curve \(C\) has polar equation \(r ^ { 2 } = \tan ^ { - 1 } \left( \frac { 1 } { 2 } \theta \right)\), where \(0 \leqslant \theta \leqslant 2\).

1. Sketch \(C\) and state, in exact form, the greatest distance of a point on \(C\) from the pole.
2. Find the exact value of the area of the region bounded by \(C\) and the half-line \(\theta = 2\).
   Now consider the part of \(C\) where \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
3. Show that, at the point furthest from the half-line \(\theta = \frac { 1 } { 2 } \pi\), $$\left( \theta ^ { 2 } + 4 \right) \tan ^ { - 1 } \left( \frac { 1 } { 2 } \theta \right) \sin \theta - \cos \theta = 0$$ and verify that this equation has a root between 0.6 and 0.7 . \(7 \quad\) The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { l l l } 1 & 2 & 3 \\ 4 & k & 6 \\ 7 & 8 & 9 \end{array} \right)\).
   1. Find the set of values of \(k\) for which \(\mathbf { A }\) is non-singular.
   2. Given that \(\mathbf { A }\) is non-singular, find, in terms of \(k\), the entries in the top row of \(\mathbf { A } ^ { - 1 }\).
   3. Given that \(\mathbf { B } = \left( \begin{array} { l l l } 1 & 0 & 0 \\ 0 & 1 & 0 \end{array} \right)\), give an example of a matrix \(\mathbf { C }\) such that \(\mathbf { B A C } = \left( \begin{array} { l l } 2 & 1 \\ k & 4 \end{array} \right)\).
   4. Find the set of values of \(k\) for which the transformation in the \(x - y\) plane represented by \(\left( \begin{array} { l l } 2 & 1 \\ k & 4 \end{array} \right)\) has two distinct invariant lines through the origin.
      If you use the following page to complete the answer to any question, the question number must be clearly shown.

1 The curve \(C\) has polar equation \(r = \mathrm { e } ^ { \frac { 3 } { 4 } \theta }\) for \(0 \leqslant \theta \leqslant \alpha\).
Given that the length of \(C\) is \(s\), find \(\alpha\) in terms of \(s\).

1 The curve \(C\) has polar equation \(r = \mathrm { e } ^ { \frac { 3 } { 4 } \theta }\) for \(0 \leqslant \theta \leqslant \alpha\).
Given that the length of \(C\) is \(s\), find \(\alpha\) in terms of \(s\).

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 \(p < 1000\).

\includegraphics{figure_1} Figure 1 shows a closed curve \(C\) with equation $$r = 3(\cos 2\theta)^{\frac{1}{2}}, \quad \text{where } -\frac{\pi}{4} \leq \theta \leq \frac{\pi}{4}, \frac{3\pi}{4} \leq \theta \leq \frac{5\pi}{4}.$$ The lines \(PQ\), \(SR\), \(PS\) and \(QR\) are tangents to \(C\), where \(PQ\) and \(SR\) are parallel to the initial line and \(PS\) and \(QR\) are perpendicular to the initial line. The point \(O\) is the pole.

1. Find the total area enclosed by the curve \(C\), shown unshaded inside the rectangle in Figure 1. [4]
2. Find the total area of the region bounded by the curve \(C\) and the four tangents, shown shaded in Figure 1. [9]

\includegraphics{figure_1} The curve \(C\) has polar equation $$r = \theta^{\frac{1}{2}}e^{\theta/\pi},$$ where \(0 \leq \theta \leq \pi\). The area of the finite region bounded by \(C\) and the line \(\theta = \beta\) is \(\pi\) (see diagram). Show that $$\beta = (\pi \ln 3)^{\frac{1}{2}}.$$ [6]

4 The curves \(C _ { 1 }\) and \(C _ { 2 }\) have polar equations $$r = \theta + 2 \quad \text { and } \quad r = \theta ^ { 2 }$$ respectively, where \(0 \leqslant \theta \leqslant \pi\).

1. Find the polar coordinates of the point of intersection of \(C _ { 1 }\) and \(C _ { 2 }\).
2. Sketch \(C _ { 1 }\) and \(C _ { 2 }\) on the same diagram.
3. Find the area bounded by \(C _ { 1 } , C _ { 2 }\) and the line \(\theta = 0\).

\includegraphics{figure_1} Figure 1 shows a sketch of the curve with polar equation $$r = a + 3\cos \theta, \quad a > 0, \quad 0 \leq \theta < 2\pi.$$ The area enclosed by the curve is \(\frac{10\pi}{2}\). Find the value of \(a\). [8]

3 The diagram shows a sketch of a curve \(C\), the pole \(O\) and the initial line. \includegraphics[max width=\textwidth, alt={}, center]{c4bce668-61f1-4be0-97ee-c635df7e1fc6-2_380_735_1827_648} The polar equation of \(C\) is $$r = 2 \sqrt { 1 + \tan \theta } , \quad - \frac { \pi } { 4 } \leqslant \theta \leqslant \frac { \pi } { 4 }$$ Show that the area of the shaded region, bounded by the curve \(C\) and the initial line, is \(\frac { \pi } { 2 } - \ln 2\).
(4 marks)

5 The curve \(C\) has polar equation \(r = \operatorname { atan } \theta\), where \(a\) is a positive constant and \(0 \leqslant \theta \leqslant \frac { 1 } { 4 } \pi\).

1. Sketch \(C\) and state the greatest distance of a point on \(C\) from the pole.
2. Find the exact value of the area of the region bounded by \(C\) and the half-line \(\theta = \frac { 1 } { 4 } \pi\).
3. Show that \(C\) has Cartesian equation \(\mathrm { y } = \frac { \mathrm { x } ^ { 2 } } { \sqrt { \mathrm { a } ^ { 2 } - \mathrm { x } ^ { 2 } } }\).
4. Using your answer to part (b), deduce the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } a \sqrt { 2 } } \frac { x ^ { 2 } } { \sqrt { a ^ { 2 } - x ^ { 2 } } } d x\).

7 \includegraphics[max width=\textwidth, alt={}, center]{b2cefbd6-6e89-495a-9f42-60f76c8c5975-4_556_524_255_813} The diagram shows the circular cross-section of a uniform cylindrical log with centre \(O\) and radius 20 cm . The points \(A , X\) and \(B\) lie on the circumference of the cross-section and \(A B = 32 \mathrm {~cm}\).

1. Show that angle \(A O B = 1.855\) radians, correct to 3 decimal places.
2. Find the area of the sector \(A X B O\). The section \(A X B C D\), where \(A B C D\) is a rectangle with \(A D = 18 \mathrm {~cm}\), is removed.
3. Find the area of the new cross-section (shown shaded in the diagram).

6 The curve \(C\) has Cartesian equation \(x ^ { 2 } + x y + y ^ { 2 } = a\), where \(a\) is a positive constant.

1. Show that the polar equation of \(C\) is \(r ^ { 2 } = \frac { 2 a } { 2 + \sin 2 \theta }\).
2. Sketch the part of \(C\) for \(0 \leqslant \theta \leqslant \frac { 1 } { 4 } \pi\). The region \(R\) is enclosed by this part of \(C\), the initial line and the half-line \(\theta = \frac { 1 } { 4 } \pi\).
3. It is given that \(\sin 2 \theta\) may be expressed as \(\frac { 2 \tan \theta } { 1 + \tan ^ { 2 } \theta }\). Use this result to show that the area of \(R\) is $$\frac { 1 } { 2 } a \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \frac { 1 + \tan ^ { 2 } \theta } { 1 + \tan \theta + \tan ^ { 2 } \theta } \mathrm {~d} \theta$$ and use the substitution \(t = \tan \theta\) to find the exact value of this area.

7 \includegraphics[max width=\textwidth, alt={}, center]{71fe6352-e0dc-4c3a-8b54-99709a1782ca-3_744_675_255_735} The diagram shows a metal plate \(A B C D E F\) which has been made by removing the two shaded regions from a circle of radius 10 cm and centre \(O\). The parallel edges \(A B\) and \(E D\) are both of length 12 cm .

1. Show that angle \(D O E\) is 1.287 radians, correct to 4 significant figures.
2. Find the perimeter of the metal plate.
3. Find the area of the metal plate.

6 \includegraphics[max width=\textwidth, alt={}, center]{08729aab-586b-4210-94c9-77b1f6b1d873-3_597_417_274_865} In the diagram, the circle has centre \(O\) and radius 5 cm . The points \(P\) and \(Q\) lie on the circle, and the arc length \(P Q\) is 9 cm . The tangents to the circle at \(P\) and \(Q\) meet at the point \(T\). Calculate

1. angle \(P O Q\) in radians,
2. the length of \(P T\),
3. the area of the shaded region.