Edexcel FP2 (Further Pure Mathematics 2) 2013 June

1. (a) Express \(\frac { 2 } { ( 2 r + 1 ) ( 2 r + 3 ) }\) in partial fractions.
   (b) Using your answer to (a), find, in terms of \(n\),

$$\sum _ { r = 1 } ^ { n } \frac { 3 } { ( 2 r + 1 ) ( 2 r + 3 ) }$$ Give your answer as a single fraction in its simplest form.

2. $$z = 5 \sqrt { } 3 - 5 i$$ Find

1. \(| z |\),
2. \(\arg ( z )\), in terms of \(\pi\). $$w = 2 \left( \cos \frac { \pi } { 4 } + i \sin \frac { \pi } { 4 } \right)$$ Find
3. \(\left| \frac { w } { z } \right|\),
4. \(\quad \arg \left( \frac { w } { z } \right)\), in terms of \(\pi\).

3. $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 y - \sin x = 0$$ Given that \(y = \frac { 1 } { 2 }\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 8 }\) at \(x = 0\), find a series expansion for \(y\) in terms of \(x\), up to and including the term in \(x ^ { 3 }\).

4. (a) Given that $$z = r ( \cos \theta + \mathrm { i } \sin \theta ) , \quad r \in \mathbb { R }$$ prove, by induction, that \(z ^ { n } = r ^ { n } ( \cos n \theta + \mathrm { i } \sin n \theta ) , \quad n \in \mathbb { Z } ^ { + }\) $$w = 3 \left( \cos \frac { 3 \pi } { 4 } + i \sin \frac { 3 \pi } { 4 } \right)$$ (b) Find the exact value of \(w ^ { 5 }\), giving your answer in the form \(a + \mathrm { i } b\), where \(a , b \in \mathbb { R }\).

1. (a) Find the general solution of the differential equation
   (b) Find the particular solution for which \(y = 5\) at \(x = 1\), giving your answer in the form \(y = \mathrm { f } ( x )\).

$$x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y = 4 x ^ { 2 }$$ (c) (i) Find the exact values of the coordinates of the turning points of the curve with equation \(y = \mathrm { f } ( x )\), making your method clear.
(ii) Sketch the curve with equation \(y = \mathrm { f } ( x )\), showing the coordinates of the turning points.

1. (a) Use algebra to find the exact solutions of the equation

$$\left| 2 x ^ { 2 } + 6 x - 5 \right| = 5 - 2 x$$ (b) On the same diagram, sketch the curve with equation \(y = \left| 2 x ^ { 2 } + 6 x - 5 \right|\) and the line with equation \(y = 5 - 2 x\), showing the \(x\)-coordinates of the points where the line crosses the curve.
(c) Find the set of values of \(x\) for which $$\left| 2 x ^ { 2 } + 6 x - 5 \right| > 5 - 2 x$$

1. (a) Show that the transformation \(y = x v\) transforms the equation

$$4 x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 8 x \frac { \mathrm {~d} y } { \mathrm {~d} x } + \left( 8 + 4 x ^ { 2 } \right) y = x ^ { 4 }$$ into the equation $$4 \frac { \mathrm {~d} ^ { 2 } v } { \mathrm {~d} x ^ { 2 } } + 4 v = x$$ (b) Solve the differential equation (II) to find \(v\) as a function of \(x\).
(c) Hence state the general solution of the differential equation (I).

8. \begin{figure}[h] \end{figure} Figure 1 shows a curve \(C\) with polar equation \(r = a \sin 2 \theta , 0 \leqslant \theta \leqslant \frac { \pi } { 2 }\), and a half-line \(l\).
The half-line \(l\) meets \(C\) at the pole \(O\) and at the point \(P\). The tangent to \(C\) at \(P\) is parallel to the initial line. The polar coordinates of \(P\) are \(( R , \phi )\).

1. Show that \(\cos \phi = \frac { 1 } { \sqrt { 3 } }\)
2. Find the exact value of \(R\). The region \(S\), shown shaded in Figure 1, is bounded by \(C\) and \(l\).
3. Use calculus to show that the exact area of \(S\) is $$\frac { 1 } { 36 } a ^ { 2 } \left( 9 \arccos \left( \frac { 1 } { \sqrt { 3 } } \right) + \sqrt { 2 } \right)$$