Edexcel F2 (Further Pure Mathematics 2) Specimen

4. $$z = - 8 + ( 8 \sqrt { } 3 ) \mathrm { i }$$

1. Find the modulus of \(z\) and the argument of \(z\). Using de Moivre's theorem,
2. find \(z ^ { 3 }\),
3. find the values of \(w\) such that \(w ^ { 4 } = z\), giving your answers in the form \(a + \mathrm { i } b\), where \(a , b \in \mathbb { R }\).

5. \begin{figure}[h] \end{figure} Figure 1 shows the curves given by the polar equations $$r = 2 , \quad 0 \leqslant \theta \leqslant \frac { \pi } { 2 }$$ and \(\quad r = 1.5 + \sin 3 \theta , \quad 0 \leqslant \theta \leqslant \frac { \pi } { 2 }\).

1. Find the coordinates of the points where the curves intersect. The region \(S\), between the curves, for which \(r > 2\) and for which \(r < ( 1.5 + \sin 3 \theta )\), is shown shaded in Figure 1.
2. Find, by integration, the area of the shaded region \(S\), giving your answer in the form \(a \pi + b \sqrt { 3 }\), where \(a\) and \(b\) are simplified fractions. $$\left[ \begin{array} { l l l } \text { Leave }
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1. A complex number \(z\) is represented by the point \(P\) in the Argand diagram.
   1. Given that \(| z - 6 | = | z |\), sketch the locus of \(P\).
   2. Find the complex numbers \(z\) which satisfy both \(| z - 6 | = | z |\) and \(| z - 3 - 4 \mathrm { i } | = 5\).
   The transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by \(w = \frac { 30 } { z }\).
2. Show that \(T\) maps \(| z - 6 | = | z |\) onto a circle in the \(w\)-plane and give the cartesian equation of this circle.

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

$$\frac { \mathrm { d } y } { \mathrm {~d} x } - 4 y \tan x = 2 y ^ { \frac { 1 } { 2 } }$$ into the differential equation $$\frac { \mathrm { d } z } { \mathrm {~d} x } - 2 z \tan x = 1$$ (b) Solve the differential equation (II) to find \(z\) as a function of \(x\).
(c) Hence obtain the general solution of the differential equation (I).
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1. (a) Find the value of \(\lambda\) for which \(y = \lambda x \sin 5 x\) is a particular integral of the differential equation

$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 25 y = 3 \cos 5 x$$ (b) Using your answer to part (a), find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 25 y = 3 \cos 5 x$$ Given that at \(x = 0 , y = 0\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 5\),
(c) find the particular solution of this differential equation, giving your solution in the form \(y = \mathrm { f } ( x )\).
(d) Sketch the curve with equation \(y = \mathrm { f } ( x )\) for \(0 \leqslant x \leqslant \pi\).