Edexcel F2 (Further Pure Mathematics 2) 2022 January

1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
   1. Express the complex number
   $$- 4 - 4 \sqrt { 3 } i$$ in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\)
2. Solve the equation $$z ^ { 3 } + 4 + 4 \sqrt { 3 } i = 0$$ giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\)

2. Determine the general solution of the differential equation $$2 \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 5 \frac { \mathrm {~d} y } { \mathrm {~d} x } - 3 y = 2 \mathrm { e } ^ { 3 x }$$

3. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve \(C _ { 1 }\) with equation $$y = \frac { 4 x } { 4 - | x | }$$ and the curve \(C _ { 2 }\) with equation $$y = x ^ { 2 } - 8 x$$ For \(x > 0 , C _ { 1 }\) has equation \(y = \frac { 4 x } { 4 - x }\)

1. Use algebra to show that \(C _ { 1 }\) touches \(C _ { 2 }\) at a point \(P\), stating the coordinates of \(P\)
2. Hence or otherwise, using algebra, solve the inequality $$x ^ { 2 } - 8 x > \frac { 4 x } { 4 - | x | }$$

4. \begin{figure}[h] \end{figure} Figure 2 shows part of the curve with polar equation $$r = 4 - \frac { 3 } { 2 } \cos 6 \theta \quad 0 \leqslant \theta < 2 \pi$$

1. Sketch, on the polar grid in Figure 2,
   1. the rest of the curve with equation $$r = 4 - \frac { 3 } { 2 } \cos 6 \theta \quad 0 \leqslant \theta < 2 \pi$$
   2. the polar curve with equation $$r = 1$$ $$0 \leqslant \theta < 2 \pi$$ A spare copy of the grid is given on page 15. In part (b) you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
2. Determine the exact area enclosed between the two curves defined in part (a). Only use this grid if you need to redraw your answer to part (a) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{0d458344-42cb-48d1-90b3-e071df8ea7bb-15_901_1042_1651_532} \captionsetup{labelformat=empty} \caption{Copy of Figure 2}
   \end{figure}

5. $$y = \sqrt { 4 + \ln x } \quad x > \frac { 1 } { 2 }$$

1. Show that $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = - \frac { 9 + 2 \ln x } { 4 x ^ { 2 } ( 4 + \ln x ) ^ { \frac { 3 } { 2 } } }$$
2. Hence, or otherwise, determine the Taylor series expansion about \(x = 1\) for \(y\), in ascending powers of ( \(x - 1\) ), up to and including the term in \(( x - 1 ) ^ { 2 }\), giving each coefficient in simplest form.

6. Given that \(A > B > 0\), by letting \(x = \arctan A\) and \(y = \arctan B\)

1. prove that $$\arctan A - \arctan B = \arctan \left( \frac { A - B } { 1 + A B } \right)$$
2. Show that when \(A = r + 2\) and \(B = r\) $$\frac { A - B } { 1 + A B } = \frac { 2 } { ( 1 + r ) ^ { 2 } }$$
3. Hence, using the method of differences, show that $$\sum _ { r = 1 } ^ { n } \arctan \frac { 2 } { ( 1 + r ) ^ { 2 } } = \arctan ( n + p ) + \arctan ( n + q ) - \arctan 2 - \frac { \pi } { 4 }$$ where \(p\) and \(q\) are integers to be determined.
4. Hence, making your reasoning clear, determine $$\sum _ { r = 1 } ^ { \infty } \arctan \left( \frac { 2 } { ( 1 + r ) ^ { 2 } } \right)$$ giving the answer in the form \(k \pi - \arctan 2\), where \(k\) is a constant.

7. A transformation from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac { ( 1 + \mathrm { i } ) z + 2 ( 1 - \mathrm { i } ) } { z - \mathrm { i } } \quad z \neq \mathrm { i }$$ The transformation maps points on the imaginary axis in the \(z\)-plane onto a line in the \(w\)-plane.

1. Find an equation for this line. The transformation maps points on the real axis in the \(z\)-plane onto a circle in the \(w\)-plane.
2. Find the centre and radius of this circle.

1. (a) Show that the transformation \(v = y - 2 x\) transforms the differential equation

$$\frac { \mathrm { d } y } { \mathrm {~d} x } + 2 y x ( y - 4 x ) = 2 - 8 x ^ { 3 }$$ into the differential equation $$\frac { \mathrm { d } v } { \mathrm {~d} x } = - 2 x v ^ { 2 }$$ (b) Solve the differential equation (II) to determine \(v\) as a function of \(x\)
(c) Hence obtain the general solution of the differential equation (I).
(d) Sketch the solution curve that passes through the point \(( - 1 , - 1 )\). On your sketch show clearly the equation of any horizontal or vertical asymptotes.
You do not need to find the coordinates of any intercepts with the coordinate axes or the coordinates of any stationary points.