Edexcel FP2 (Further Pure Mathematics 2) 2013 June

1. A transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by

$$w = \frac { z + 2 \mathrm { i } } { \mathrm { i } z } \quad z \neq 0$$ The transformation maps points on the real axis in the \(z\)-plane onto a line in the \(w\)-plane. Find an equation of this line.

2. Use algebra to find the set of values of \(x\) for which $$\frac { 6 x } { 3 - x } > \frac { 1 } { x + 1 }$$

3. (a) Express \(\frac { 2 } { ( r + 1 ) ( r + 3 ) }\) in partial fractions.
(b) Hence show that $$\sum _ { r = 1 } ^ { n } \frac { 2 } { ( r + 1 ) ( r + 3 ) } = \frac { n ( 5 n + 13 ) } { 6 ( n + 2 ) ( n + 3 ) }$$ (c) Evaluate \(\sum _ { r = 10 } ^ { 100 } \frac { 2 } { ( r + 1 ) ( r + 3 ) }\), giving your answer to 3 significant figures.

1. Given that

$$y \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + \left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) ^ { 2 } + 5 y = 0$$

1. find \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }\) 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 \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2\) at \(x = 0\)
2. find a series solution for \(y\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).

1. (a) Find, in the form \(y = \mathrm { f } ( x )\), the general solution of the equation

$$\frac { \mathrm { d } y } { \mathrm {~d} x } + 2 y \tan x = \sin 2 x , \quad 0 < x < \frac { \pi } { 2 }$$ Given that \(y = 2\) at \(x = \frac { \pi } { 3 }\)
(b) find the value of \(y\) at \(x = \frac { \pi } { 6 }\), giving your answer in the form \(a + k \ln b\), where \(a\) and \(b\) are integers and \(k\) is rational.

6. The complex number \(z = \mathrm { e } ^ { \mathrm { i } \theta }\), where \(\theta\) is real.

1. Use de Moivre's theorem to show that $$z ^ { n } + \frac { 1 } { z ^ { n } } = 2 \cos n \theta$$ where \(n\) is a positive integer.
2. Show that $$\cos ^ { 5 } \theta = \frac { 1 } { 16 } ( \cos 5 \theta + 5 \cos 3 \theta + 10 \cos \theta )$$
3. Hence find all the solutions of $$\cos 5 \theta + 5 \cos 3 \theta + 12 \cos \theta = 0$$ in the interval \(0 \leqslant \theta < 2 \pi\)

1. (a) Find the value of \(\lambda\) for which \(\lambda t ^ { 2 } \mathrm { e } ^ { 3 t }\) is a particular integral of the differential equation

$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } - 6 \frac { \mathrm {~d} y } { \mathrm {~d} t } + 9 y = 6 \mathrm { e } ^ { 3 t } , \quad t \geqslant 0$$ (b) Hence find the general solution of this differential equation. Given that when \(t = 0 , y = 5\) and \(\frac { \mathrm { d } y } { \mathrm {~d} t } = 4\)
(c) find the particular solution of this differential equation, giving your solution in the form \(y = \mathrm { f } ( t )\).

8. \begin{figure}[h] \end{figure} Figure 1 shows a closed curve \(C\) with equation $$r = 3 ( \cos 2 \theta ) ^ { \frac { 1 } { 2 } } , \quad \text { where } - \frac { \pi } { 4 } < \theta \leqslant \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } < \theta \leqslant \frac { 5 \pi } { 4 }$$ The lines \(P Q , S R , P S\) and \(Q R\) are tangents to \(C\), where \(P Q\) and \(S R\) are parallel to the initial line and \(P S\) and \(Q R\) 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.
2. Find the total area of the region bounded by the curve \(C\) and the four tangents, shown shaded in Figure 1.