Edexcel F2 (Further Pure Mathematics 2) 2022 June

1. Given that

$$\frac { 2 n + 1 } { n ^ { 2 } ( n + 1 ) ^ { 2 } } \equiv \frac { A } { n ^ { 2 } } + \frac { B } { ( n + 1 ) ^ { 2 } }$$

1. determine the value of \(A\) and the value of \(B\)
2. Hence show that, for \(n \geqslant 5\) $$\sum _ { r = 5 } ^ { n } \frac { 2 r + 1 } { r ^ { 2 } ( r + 1 ) ^ { 2 } } = \frac { n ^ { 2 } + a n + b } { c ( n + 1 ) ^ { 2 } }$$ where \(a\), \(b\) and \(c\) are integers to be determined.

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

$$x - 5 < \frac { 9 } { x + 3 }$$ (b) Hence, or otherwise, determine the set of values of \(x\) for which $$x - 5 < \frac { 9 } { | x + 3 | }$$

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

$$w = \frac { z } { z + 4 \mathrm { i } } \quad z \neq - 4 \mathrm { i }$$ The circle with equation \(| z | = 3\) is mapped by \(T\) onto the circle \(C\) Determine

1. a Cartesian equation of \(C\)
2. the centre and radius of \(C\)

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

$$\frac { \mathrm { d } y } { \mathrm {~d} x } - 3 y \tan x = \mathrm { e } ^ { 4 x } \sec ^ { 3 } x$$ giving your answer in the form \(y = \mathrm { f } ( x )\)
(b) Determine the particular solution for which \(y = 4\) at \(x = 0\)

1. Given that

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

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

6. \begin{figure}[h] \end{figure} The curve shown in Figure 1 has polar equation $$r = 4 a ( 1 + \cos \theta ) \quad 0 \leqslant \theta < \pi$$ where \(a\) is a positive constant.
The tangent to the curve at the point \(A\) is parallel to the initial line.

1. Show that the polar coordinates of \(A\) are \(\left( 6 a , \frac { \pi } { 3 } \right)\) The point \(B\) lies on the curve such that angle \(A O B = \frac { \pi } { 6 }\)
   The finite region \(R\), shown shaded in Figure 1, is bounded by the line \(A B\) and the curve.
2. Use calculus to determine the area of the shaded region \(R\), giving your answer in the form \(a ^ { 2 } ( n \pi + p \sqrt { 3 } + q )\), where \(n , p\) and \(q\) are integers.

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

$$3 \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - \frac { 6 } { x } \frac { \mathrm {~d} y } { \mathrm {~d} x } + \frac { 6 y } { x ^ { 2 } } + 3 y = x ^ { 2 } \quad x \neq 0$$ into the equation $$3 \frac { \mathrm {~d} ^ { 2 } v } { \mathrm {~d} x ^ { 2 } } + 3 v = x$$ (b) Hence obtain the general solution of the differential equation (I), giving your answer in the form \(y = \mathrm { f } ( x )\)

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

$$\sin 5 \theta \equiv 16 \sin ^ { 5 } \theta - 20 \sin ^ { 3 } \theta + 5 \sin \theta$$ (b) Hence determine the five distinct solutions of the equation $$16 x ^ { 5 } - 20 x ^ { 3 } + 5 x + \frac { 1 } { 5 } = 0$$ giving your answers to 3 decimal places.
(c) Use the identity given in part (a) to show that $$\int _ { 0 } ^ { \frac { \pi } { 4 } } \left( 4 \sin ^ { 5 } \theta - 5 \sin ^ { 3 } \theta - 6 \sin \theta \right) \mathrm { d } \theta = a \sqrt { 2 } + b$$ where \(a\) and \(b\) are rational numbers to be determined.