Edexcel FP2 (Further Pure Mathematics 2) 2013 June

1. $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + x \frac { \mathrm {~d} y } { \mathrm {~d} x } = 2 \cos x$$

1. Find \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }\) in terms of \(x , \frac { \mathrm {~d} y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\). At \(x = 0 , y = 1\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3\)
2. Find the value of \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }\) at \(x = 0\)
3. Express \(y\) as a series in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).

2. (a) Sketch, on the same axes,

1. \(y = | 2 x - 3 |\)
2. \(y = 4 - x ^ { 2 }\)
   (b) Find the set of values of \(x\) for which $$4 - x ^ { 2 } > | 2 x - 3 |$$

3. $$f ( x ) = \ln ( 1 + \sin k x )$$ where \(k\) is a constant, \(x \in \mathbb { R }\) and \(- \frac { \pi } { 2 } < k x < \frac { 3 \pi } { 2 }\)

1. Find f \({ } ^ { \prime } ( x )\)
2. Show that \(\mathrm { f } ^ { \prime \prime } ( x ) = \frac { - k ^ { 2 } } { 1 + \sin k x }\)
3. Find the Maclaurin series of \(\mathrm { f } ( x )\), in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).

4. Find the general solution of the differential equation $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } + ( 1 + x \cot x ) y = \sin x , \quad 0 < x < \pi$$ giving your answer in the form \(y = \mathrm { f } ( x )\).

5. (a) Express \(\frac { 2 } { r ( r + 1 ) ( r + 2 ) }\) in partial fractions.
(b) Using your answer to part (a) and the method of differences, show that $$\sum _ { r = 1 } ^ { n } \frac { 2 } { r ( r + 1 ) ( r + 2 ) } = \frac { n ( n + 3 ) } { 2 ( n + 1 ) ( n + 2 ) }$$

6. Solve the equation $$z ^ { 5 } = - 16 \sqrt { } 3 + 16 i$$ giving your answers in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\), where \(r > 0\) and \(- \pi < \theta < \pi\).

7. (a) Find the value of the constant \(\lambda\) for which \(y = \lambda x \mathrm { e } ^ { 2 x }\) is a particular integral of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 4 y = 6 \mathrm { e } ^ { 2 x }$$ (b) Hence, or otherwise, find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 4 y = 6 \mathrm { e } ^ { 2 x }$$

8. A complex number \(z\) is represented by the point \(P\) on an Argand diagram.

1. Given that \(| z | = 1\), sketch the locus of \(P\). The transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac { z + 7 \mathrm { i } } { z - 2 \mathrm { i } }$$
2. Show that \(T\) maps \(| z | = 1\) onto a circle in the \(w\)-plane.
3. Show that this circle has its centre at \(w = - 5\) and find its radius.

9. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curves given by the polar equations $$r = 1 \text { and } r = 2 - 2 \sin \theta$$

1. Find the coordinates of the points where the curves intersect. The region \(S\), between the curves, for which \(r < 1\) and for which \(r < 2 - 2 \sin \theta\), is shown shaded in Figure 1.
2. 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 rational numbers.