Edexcel FP2 (Further Pure Mathematics 2)

1. (a) Express \(\frac { 1 } { r ( r + 2 ) }\) in partial fractions.
   (b) Hence show that \(\sum _ { r = 1 } ^ { n } \frac { 4 } { r ( r + 2 ) } = \frac { n ( 3 n + 5 ) } { ( n + 1 ) ( n + 2 ) }\).
2. Solve the equation

$$z ^ { 3 } = 4 \sqrt { } 2 - 4 \sqrt { } 2 i ,$$ giving your answers in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\), where \(- \pi < \theta \leq \pi\).
3. Find the general solution of the differential equation $$\sin x \frac { \mathrm {~d} y } { \mathrm {~d} x } - y \cos x = \sin 2 x \sin x$$ giving your answer in the form \(y = \mathrm { f } ( x )\).
4. \begin{figure}[h] \end{figure} 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 \(\frac { 107 } { 2 } \pi\).
Find the value of \(a\).
5. $$y = \sec ^ { 2 } x$$ (a) Show that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 6 \sec ^ { 4 } x - 4 \sec ^ { 2 } x\).
(b) Find a Taylor series expansion of \(\sec ^ { 2 } x\) in ascending powers of \(\left( x - \frac { \pi } { 4 } \right)\), up to and including the term in \(\left( x - \frac { \pi } { 4 } \right) ^ { 3 }\).
6. A transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac { Z } { Z + \mathrm { i } } , \quad Z \neq - \mathrm { i }$$ The circle with equation \(| z | = 3\) is mapped by \(T\) onto the curve \(C\).
(a) Show that \(C\) is a circle and find its centre and radius. The region \(| z | < 3\) in the \(z\)-plane is mapped by \(T\) onto the region \(R\) in the \(w\)-plane.
(b) Shade the region \(R\) on an Argand diagram.
7. (a) Sketch the graph of \(y = \left| x ^ { 2 } - a ^ { 2 } \right|\), where \(a > 1\), showing the coordinates of the points where the graph meets the axes.
(b) Solve \(\left| x ^ { 2 } - a ^ { 2 } \right| = a ^ { 2 } - x , a > 1\).
(c) Find the set of values of \(x\) for which \(\left| x ^ { 2 } - a ^ { 2 } \right| > a ^ { 2 } - x , a > 1\).
8. $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 5 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 6 x = 2 \mathrm { e } ^ { - t }$$ Given that \(x = 0\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 2\) at \(t = 0\),
(a) find \(x\) in terms of \(t\). The solution to part (a) is used to represent the motion of a particle \(P\) on the \(x\)-axis. At time \(t\) seconds, where \(t > 0 , P\) is \(x\) metres from the origin \(O\).
(b) Show that the maximum distance between \(O\) and \(P\) is \(\frac { 2 \sqrt { 3 } } { 9 } \mathrm {~m}\) and justify that this distance is a maximum.
(7) \section*{TOTAL FOR PAPER: 75 MARKS} \section*{6668/01} \section*{Further Pure Mathematics FP2 Advanced Subsidiary} \section*{Thursday 24 June 2010 - Morning} 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, differentiation and integration, or have retrievable mathematical formulas stored in them. Write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Further Pure Mathematics FP2), the paper reference (6668), your surname, initials and signature. A booklet 'Mathematical Formulae and Statistical Tables' is provided.
Full marks may be obtained for answers to ALL questions.
There are 8 questions in this question paper. The total mark for this paper is 75 . 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 \(\frac { 3 } { ( 3 r - 1 ) ( 3 r + 2 ) }\) in partial fractions.
   (b) Using your answer to part (a) and the method of differences, show that

$$\sum _ { r = 1 } ^ { n } \frac { 3 } { ( 3 r - 1 ) ( 3 r + 2 ) } = \frac { 3 n } { 2 ( 3 n + 2 ) }$$ (c) Evaluate \(\sum _ { r = 100 } ^ { 1000 } \frac { 3 } { ( 3 r - 1 ) ( 3 r + 2 ) }\), giving your answer to 3 significant figures.

4. \begin{figure}[h] \end{figure} 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 \(\frac { 107 } { 2 } \pi\).
Find the value of \(a\).

5. $$y = \sec ^ { 2 } x$$

1. Show that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 6 \sec ^ { 4 } x - 4 \sec ^ { 2 } x\).
2. Find a Taylor series expansion of \(\sec ^ { 2 } x\) in ascending powers of \(\left( x - \frac { \pi } { 4 } \right)\), up to and including the term in \(\left( x - \frac { \pi } { 4 } \right) ^ { 3 }\).

6. A transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac { Z } { Z + \mathrm { i } } , \quad Z \neq - \mathrm { i }$$ The circle with equation \(| z | = 3\) is mapped by \(T\) onto the curve \(C\).

1. Show that \(C\) is a circle and find its centre and radius. The region \(| z | < 3\) in the \(z\)-plane is mapped by \(T\) onto the region \(R\) in the \(w\)-plane.
2. Shade the region \(R\) on an Argand diagram.

7. (a) Sketch the graph of \(y = \left| x ^ { 2 } - a ^ { 2 } \right|\), where \(a > 1\), showing the coordinates of the points where the graph meets the axes.
(b) Solve \(\left| x ^ { 2 } - a ^ { 2 } \right| = a ^ { 2 } - x , a > 1\).
(c) Find the set of values of \(x\) for which \(\left| x ^ { 2 } - a ^ { 2 } \right| > a ^ { 2 } - x , a > 1\).

8. $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 5 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 6 x = 2 \mathrm { e } ^ { - t }$$ Given that \(x = 0\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 2\) at \(t = 0\),

1. find \(x\) in terms of \(t\). The solution to part (a) is used to represent the motion of a particle \(P\) on the \(x\)-axis. At time \(t\) seconds, where \(t > 0 , P\) is \(x\) metres from the origin \(O\).
2. Show that the maximum distance between \(O\) and \(P\) is \(\frac { 2 \sqrt { 3 } } { 9 } \mathrm {~m}\) and justify that this distance is a maximum.
   (7) \section*{TOTAL FOR PAPER: 75 MARKS} \section*{6668/01} \section*{Further Pure Mathematics FP2 Advanced Subsidiary} \section*{Thursday 24 June 2010 - Morning} 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, differentiation and integration, or have retrievable mathematical formulas stored in them. Write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Further Pure Mathematics FP2), the paper reference (6668), your surname, initials and signature. A booklet 'Mathematical Formulae and Statistical Tables' is provided.
   Full marks may be obtained for answers to ALL questions.
   There are 8 questions in this question paper. The total mark for this paper is 75 . 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 \(\frac { 3 } { ( 3 r - 1 ) ( 3 r + 2 ) }\) in partial fractions.
   2. Using your answer to part (a) and the method of differences, show that
   $$\sum _ { r = 1 } ^ { n } \frac { 3 } { ( 3 r - 1 ) ( 3 r + 2 ) } = \frac { 3 n } { 2 ( 3 n + 2 ) }$$
3. Evaluate \(\sum _ { r = 100 } ^ { 1000 } \frac { 3 } { ( 3 r - 1 ) ( 3 r + 2 ) }\), giving your answer to 3 significant figures.
   2. The displacement \(x\) metres of a particle at time \(t\) seconds is given by the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + x + \cos x = 0$$ When \(t = 0 , x = 0\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { 1 } { 2 }\).
   Find a Taylor series solution for \(x\) in ascending powers of \(t\), up to and including the term in \(t ^ { 3 }\).
   3. (a) Find the set of values of \(x\) for which $$x + 4 > \frac { 2 } { x + 3 } .$$
4. Deduce, or otherwise find, the values of \(x\) for which $$x + 4 > \frac { 2 } { | x + 3 | }$$ 4. $$z = - 8 + ( 8 \sqrt { } 3 ) \mathrm { i }$$
5. Find the modulus of \(z\) and the argument of \(z\). Using de Moivre's theorem,
6. find \(z ^ { 3 }\),
7. find the values of \(w\) such that \(w ^ { 4 } = z\), giving your answers in the form \(a + \mathrm { i } b\), where \(a , b \in \mathbb { R }\).
   5. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{3caa00c9-8a77-4987-888d-054eba7712ee-04_339_488_687_511} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} Figure 1 shows the curves given by the polar equations $$\begin{array} { c c } r = 2 , & 0 \leq \theta \leq \frac { \pi } { 2 }
   \text { and } r = 1.5 + \sin 3 \theta , & 0 \leq \theta \leq \frac { \pi } { 2 } \end{array}$$
8. Find the coordinates of the points where the curves intersect. The region \(S\), between the curves, for which \(r > 2\) and for which \(r < ( 1.5 + \sin 3 \theta )\), is shown shaded in Figure 1.
9. Find, by integration, the area of the shaded region \(S\), giving your answer in the form \(a \pi + b \sqrt { } 3\), where \(a\) and \(b\) are simplified fractions.
   6. A complex number \(z\) is represented by the point \(P\) in the Argand diagram.
10. Given that \(| z - 6 | = | z |\), sketch the locus of \(P\).
11. Find the complex numbers \(z\) which satisfy both \(| z - 6 | = | z |\) and \(| z - 3 - 4 \mathrm { i } | = 5\). The transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by \(w = \frac { 30 } { z }\).
12. Show that \(T\) maps \(| z - 6 | = | z |\) onto a circle in the \(w\)-plane and give the cartesian equation of this circle.
    7. (a) Show that the transformation \(z = y ^ { \frac { 1 } { 2 } }\) transforms the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } - 4 y \tan x = 2 y ^ { \frac { 1 } { 2 } }$$ into the differential equation $$\frac { \mathrm { d } z } { \mathrm {~d} x } - 2 z \tan x = 1$$
13. Solve the differential equation (II) to find \(z\) as a function of \(x\).
14. Hence obtain the general solution of the differential equation (I).
    8. (a) Find the value of \(\lambda\) for which \(y = \lambda x \sin 5 x\) is a particular integral of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 25 y = 3 \cos 5 x$$
15. Using your answer to part (a), find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 25 y = 3 \cos 5 x$$ Given that at \(x = 0 , y = 0\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 5\),
16. find the particular solution of this differential equation, giving your solution in the form \(y = \mathrm { f } ( x )\).
17. Sketch the curve with equation \(y = \mathrm { f } ( x )\) for \(0 \leq x \leq \pi\).

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.