Edexcel F2 (Further Pure Mathematics 2) 2024 June

1. The complex number \(z = x + i y\) satisfies the equation

$$| z - 3 - 4 i | = | z + 1 + i |$$

1. Determine an equation for the locus of \(z\) giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.
2. Shade, on an Argand diagram, the region defined by $$| z - 3 - 4 i | \leqslant | z + 1 + i |$$ You do not need to determine the coordinates of any intercepts on the coordinate axes.

2. $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } - y ^ { 3 } = 4$$

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

1. (a) Express

$$\frac { 1 } { ( n + 3 ) ( n + 5 ) }$$ in partial fractions.
(b) Hence, using the method of differences, show that for all positive integer values of \(n\), $$\sum _ { r = 1 } ^ { n } \frac { 1 } { ( r + 3 ) ( r + 5 ) } = \frac { n ( p n + q ) } { 40 ( n + 4 ) ( n + 5 ) }$$ where \(p\) and \(q\) are integers to be determined.
(c) Use the answer to part (b) to determine, as a simplified fraction, the value of $$\frac { 1 } { 9 \times 11 } + \frac { 1 } { 10 \times 12 } + \ldots + \frac { 1 } { 24 \times 26 }$$

1. (a) Show that the substitution \(y ^ { 2 } = \frac { 1 } { t }\) transforms the differential equation

$$\frac { \mathrm { d } y } { \mathrm {~d} x } + y = x y ^ { 3 }$$ into the differential equation $$\frac { \mathrm { d } t } { \mathrm {~d} x } - 2 t = - 2 x$$ (b) Solve differential equation (II) and determine \(y ^ { 2 }\) in terms of \(x\).

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 values of \(x\) for which $$\frac { x + 1 } { ( x - 3 ) ( x + 2 ) } \leqslant 1 - \frac { 2 } { x - 3 }$$

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

$$w = \frac { z - \mathrm { i } } { z + 1 } \quad z \neq - 1$$ Given that \(T\) maps the imaginary axis in the \(z\)-plane to the circle \(C\) in the \(w\)-plane, determine (i) the coordinates of the centre of \(C\)
(ii) the radius of \(C\)

1. Given that \(y = \mathrm { e } ^ { x } \sin x\)
   1. show that
   $$\frac { \mathrm { d } ^ { 6 } y } { \mathrm {~d} x ^ { 6 } } = k \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$$ where \(k\) is a constant to be determined.
2. Hence determine the first 5 non-zero terms in the Maclaurin series expansion for \(y\), giving each coefficient in simplest form.

1. (a) Given that \(t = \ln x\), where \(x > 0\), show that

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

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

Solutions relying entirely on calculator technology are not acceptable.

1. Use De Moivre's theorem to show that $$\cos 6 \theta \equiv 32 \cos ^ { 6 } \theta - 48 \cos ^ { 4 } \theta + 18 \cos ^ { 2 } \theta - 1$$
2. Hence determine the smallest positive root of the equation $$48 x ^ { 6 } - 72 x ^ { 4 } + 27 x ^ { 2 } - 1 = 0$$ giving your answer to 3 decimal places.

10. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve \(C\) with polar equation $$r = 1 + \cos \theta \quad 0 \leqslant \theta \leqslant \pi$$ and the line \(l\) with polar equation $$r = k \sec \theta \quad 0 \leqslant \theta < \frac { \pi } { 2 }$$ where \(k\) is a positive constant.
Given that

• \(\quad C\) and \(l\) intersect at the point \(P\)
• \(O P = 1 + \frac { \sqrt { 3 } } { 2 }\)
  1. determine the exact value of \(k\).

The finite region \(R\), shown shaded in Figure 1, is bounded by \(C\), the initial line and \(l\).

Use algebraic integration to show that the area of \(R\) is $$p \pi + q \sqrt { 3 } + r$$ where \(p , q\) and \(r\) are simplified rational numbers to be determined.