Edexcel F2 (Further Pure Mathematics 2) 2023 January

1. Given that \(y = \ln ( 5 + 3 x )\)
   1. determine, in simplest form, \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }\)
   2. Hence determine the Maclaurin series expansion of \(\ln ( 5 + 3 x )\), in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), giving each coefficient in simplest form.
   3. Hence write down the Maclaurin series expansion of \(\ln ( 5 - 3 x )\), in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), giving each coefficient in simplest form.
   4. Use the answers to parts (b) and (c) to determine the first 2 non-zero terms, in ascending powers of \(x\), of the Maclaurin series expansion of
   $$\ln \left( \frac { 5 + 3 x } { 5 - 3 x } \right)$$

1. (a) Express

$$\frac { 1 } { ( 2 n - 1 ) ( 2 n + 1 ) ( 2 n + 3 ) }$$ in partial fractions.
(b) Hence, using the method of differences, show that for all integer values of \(n\), $$\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 2 r - 1 ) ( 2 r + 1 ) ( 2 r + 3 ) } = \frac { n ( n + 2 ) } { a ( 2 n + b ) ( 2 n + c ) }$$ where \(a\), \(b\) and \(c\) are integers to be determined.

1. (a) Show that the transformation \(y = \frac { 1 } { z }\) transforms the differential equation

$$x ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + x y = 2 y ^ { 2 }$$ into the differential equation $$\frac { \mathrm { d } z } { \mathrm {~d} x } - \frac { z } { x } = - \frac { 2 } { x ^ { 2 } }$$ (b) Solve differential equation (II) to determine \(z\) in terms of \(x\).
(c) Hence determine the particular solution of differential equation (I) for which \(y = - \frac { 3 } { 8 }\) at \(x = 3\) Give your answer in the form \(y = \mathrm { f } ( x )\).

4. $$\frac { \mathrm { d } y } { \mathrm {~d} x } = y ^ { 2 } - x$$

1. Show that $$\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } } = A y \frac { \mathrm {~d} ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } + B \frac { \mathrm {~d} y } { \mathrm {~d} x } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$$ where \(A\) and \(B\) are integers to be determined. Given that \(y = 1\) at \(x = - 1\)
2. determine the Taylor series solution for \(y\), in ascending powers of \(( x + 1 )\) up to and including the term in \(( x + 1 ) ^ { 4 }\), giving each coefficient in simplest form.

1. In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.
Use algebra to determine the set of values of \(x\) for which $$\frac { x ^ { 2 } - 9 } { | x + 8 | } > 6 - 2 x$$

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

Given that $$| z - 2 i | = | z - 3 |$$

1. sketch the locus of \(P\). You do not need to find the coordinates of any intercepts. The transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac { \mathrm { i } z } { z - 2 \mathrm { i } } \quad z \neq 2 \mathrm { i }$$ Given that \(T\) maps \(| z - 2 i | = | z - 3 |\) to a circle \(C\) in the \(w\)-plane,
2. find the equation of \(C\), giving your answer in the form $$| w - ( p + q \mathrm { i } ) | = r$$ where \(p , q\) and \(r\) are real numbers to be determined.

1. In this question you must show all stages of your working.

\section*{Solutions relying entirely on calculator technology are not acceptable.}

1. Use de Moivre's theorem to show that $$\cos 5 x \equiv \cos x \left( a \sin ^ { 4 } x + b \sin ^ { 2 } x + c \right)$$ where \(a\), \(b\) and \(c\) are integers to be determined.
2. Hence solve, for \(0 < \theta < \frac { \pi } { 2 }\) $$\cos 5 \theta = \sin 2 \theta \sin \theta - \cos \theta$$ giving your answers to 3 decimal places.

8. \begin{figure}[h] \end{figure} The curve \(C\) shown in Figure 1 has polar equation $$r = 1 - \sin \theta \quad 0 \leqslant \theta < \frac { \pi } { 2 }$$ The point \(P\) lies on \(C\), such that the tangent to \(C\) at \(P\) is parallel to the initial line.

1. Use calculus to determine the polar coordinates of \(P\) The finite region \(R\), shown shaded in Figure 1, is bounded by
   • the line with equation \(\theta = \frac { \pi } { 2 }\)
   • the tangent to \(C\) at \(P\)
   • part of the curve \(C\)
   • the initial line
   • Use algebraic integration to show that the area of \(R\) is
   $$\frac { 1 } { 32 } ( a \pi + b \sqrt { 3 } + c )$$ where \(a\), \(b\) and \(c\) are integers to be determined.

1. (a) Given that \(x = t ^ { \frac { 1 } { 2 } }\), determine, in terms of \(y\) and \(t\),
   1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
   2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\)
      (b) Hence show that the transformation \(x = t ^ { \frac { 1 } { 2 } }\), where \(t > 0\), transforms the differential equation
   $$x \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - \left( 6 x ^ { 2 } + 1 \right) \frac { \mathrm { d } y } { \mathrm {~d} x } + 9 x ^ { 3 } y = x ^ { 5 }$$ into the differential equation $$4 \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } - 12 \frac { \mathrm {~d} y } { \mathrm {~d} t } + 9 y = t$$ (c) Solve differential equation (II) to determine a general solution for \(y\) in terms of \(t\).
   (d) Hence determine the general solution of differential equation (I).