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.