Edexcel F2 (Further Pure Mathematics 2) 2018 June

1. Use algebra to find the set of values of \(x\) for which

$$\frac { 1 } { x - 2 } > \frac { 2 } { x }$$

1. (a) Find the general solution of the differential equation

$$\left( x ^ { 2 } + 1 \right) \frac { \mathrm { d } y } { \mathrm {~d} x } + x y - x = 0$$ giving your answer in the form \(y = \mathrm { f } ( x )\).
(b) Find the particular solution for which \(y = 2\) when \(x = 3\)

3. $$2 \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + \frac { \mathrm { d } y } { \mathrm {~d} x } - x y = 1$$

1. Show that $$\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } } = \frac { 1 } { 2 } \left( a \frac { \mathrm {~d} y } { \mathrm {~d} x } + b x \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + c \frac { \mathrm {~d} ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } \right)$$ where \(a , b\) and \(c\) are constants to be found. Given that \(y = 1\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1\) when \(x = 2\)
2. find a series solution for \(y\) in ascending powers of ( \(x - 2\) ), up to and including the term in \(( x - 2 ) ^ { 4 }\). Write each term in its simplest form.
3. Use the solution to part (b) to find an approximate value for \(y\) when \(x = 2.1\), giving your answer to 3 decimal places.

4. A complex number \(z\) is represented by the point \(P\) in an Argand diagram. Given that $$| z + i | = 1$$

1. sketch the locus of \(P\). The transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac { 3 \mathrm { i } z - 2 } { z + \mathrm { i } } , \quad z \neq - \mathrm { i }$$
2. Given that \(T\) maps \(| z + i | = 1\) to a circle \(C\) in the \(w\)-plane, find a cartesian equation of \(C\).

1. (a) Express \(\frac { 4 r + 2 } { r ( r + 1 ) ( r + 2 ) }\) in partial fractions.
   (b) Hence, using the method of differences, prove that

$$\sum _ { r = 1 } ^ { n } \frac { 4 r + 2 } { r ( r + 1 ) ( r + 2 ) } = \frac { n ( a n + b ) } { 2 ( n + 1 ) ( n + 2 ) }$$ where \(a\) and \(b\) are constants to be found.

1. (a) Show that the transformation \(x = \mathrm { e } ^ { t }\) transforms the differential equation

$$x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 3 x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 3 y = x ^ { 2 } \quad x > 0$$ into the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } - 4 \frac { \mathrm {~d} y } { \mathrm {~d} t } + 3 y = \mathrm { e } ^ { 2 t }$$ (b) Find the general solution of the differential equation (II), expressing \(y\) as a function of \(t\).
(c) Hence find the general solution of the differential equation (I).

7.(a)Use de Moivre's theorem to show that $$\cos 7 \theta \equiv 64 \cos ^ { 7 } \theta - 112 \cos ^ { 5 } \theta + 56 \cos ^ { 3 } \theta - 7 \cos \theta$$ (b)Hence find the four distinct roots of the equation $$64 x ^ { 7 } - 112 x ^ { 5 } + 56 x ^ { 3 } - 7 x + 1 = 0$$ giving your answers to 3 decimal places where necessary.

8. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curves with polar equations $$\begin{array} { l l } r = 2 \sin \theta & 0 \leqslant \theta \leqslant \pi \\ r = 1.5 - \sin \theta & 0 \leqslant \theta \leqslant 2 \pi \end{array}$$ The curves intersect at the points \(P\) and \(Q\).

1. Find the polar coordinates of the point \(P\) and the polar coordinates of the point \(Q\). The region \(R\), shown shaded in Figure 1, is enclosed by the two curves.
2. Find the exact area of \(R\), giving your answer in the form \(p \pi + q \sqrt { 3 }\), where \(p\) and \(q\) are rational numbers to be found.