Edexcel F2 (Further Pure Mathematics 2) 2021 June

1. (a) Express \(\frac { 2 } { r \left( r ^ { 2 } - 1 \right) }\) in partial fractions.
   (b) Hence find, in terms of \(n\),

$$\sum _ { r = 2 } ^ { n } \frac { 1 } { r \left( r ^ { 2 } - 1 \right) }$$ Give your answer in the form $$\frac { n ^ { 2 } + A n + B } { C n ( n + 1 ) }$$ where \(A\), \(B\) and \(C\) are constants to be found.

2. The transformation \(T\) from the \(z\)-plane, where \(z = x + \mathrm { i } y\), to the \(w\)-plane, where \(w = u + \mathrm { i } v\), is given by $$w = \frac { z + 2 } { z - \mathrm { i } } \quad z \neq \mathrm { i }$$ The transformation \(T\) maps the circle \(| z | = 2\) in the \(z\)-plane onto a circle \(C\) in the \(w\)-plane.
Find (i) the centre of \(C\),
(ii) the radius of \(C\).

1. The curve \(C\), with pole \(O\), has polar equation

$$r = 1 + \cos \theta , \quad 0 \leqslant \theta \leqslant \frac { \pi } { 2 }$$ At the point \(A\) on \(C\), the tangent to \(C\) is parallel to the initial line.

1. Find the polar coordinates of \(A\).
2. Find the finite area enclosed by the initial line, the line \(O A\) and the curve \(C\), giving your answer in the form \(a \pi + b \sqrt { 3 }\), where \(a\) and \(b\) are rational constants to be found.

4. Given that $$y \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 4 \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } + 3 y = 0$$

1. show that $$\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } = \frac { 28 } { y ^ { 2 } } \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 3 } - \frac { 24 } { y } \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right)$$ Given also that \(y = 8\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1\) at \(x = 0\)
2. find a series solution for \(y\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), simplifying the coefficients where possible.

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

$$\left| 2 x ^ { 2 } + x - 3 \right| > 3 ( 1 - x )$$ [Solutions based entirely on graphical or numerical methods are not acceptable.] \includegraphics[max width=\textwidth, alt={}, center]{0d44aec7-a6e8-47fc-a215-7c8c4790e93f-21_2647_1840_118_111}

6. (a) Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 6 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 8 y = 2 x ^ { 2 } + x$$ (b) Find the particular solution of this differential equation for which \(y = 1\) and $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 0 \text { when } x = 0$$

1. (a) Use de Moivre's theorem to show that

$$\tan 4 \theta \equiv \frac { 4 \tan \theta - 4 \tan ^ { 3 } \theta } { 1 - 6 \tan ^ { 2 } \theta + \tan ^ { 4 } \theta }$$ (b) Use the identity given in part (a) to find the 2 positive roots of $$x ^ { 4 } + 2 x ^ { 3 } - 6 x ^ { 2 } - 2 x + 1 = 0$$ giving your answers to 3 significant figures.
\includegraphics[max width=\textwidth, alt={}, center]{0d44aec7-a6e8-47fc-a215-7c8c4790e93f-29_2255_50_314_35}

8. (a) Show that the substitution \(v = y ^ { - 2 }\) transforms the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + 6 x y = 3 x \mathrm { e } ^ { x ^ { 2 } } y ^ { 3 } \quad x > 0$$ into the differential equation $$\frac { \mathrm { d } v } { \mathrm {~d} x } - 12 v x = - 6 x \mathrm { e } ^ { x ^ { 2 } } \quad x > 0$$ (b) Hence find the general solution of the differential equation (I), giving your answer in the form \(y ^ { 2 } = \mathrm { f } ( x )\).