Edexcel F2 (Further Pure Mathematics 2) 2017 June

1. Solve the equation

$$z ^ { 5 } = 32$$ Give your answers in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\), where \(r > 0\) and \(0 \leqslant \theta < 2 \pi\)

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

$$\frac { x - 4 } { ( x + 3 ) } \leqslant \frac { 5 } { x ( x + 3 ) }$$

3. (a) Show that \(r ^ { 3 } - ( r - 1 ) ^ { 3 } \equiv 3 r ^ { 2 } - 3 r + 1\)
(b) Hence prove by the method of differences that, for \(n \in \mathbb { Z } ^ { + }\) $$\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { n ( n + 1 ) ( 2 n + 1 ) } { 6 }$$ [You may use \(\sum _ { r = 1 } ^ { n } r = \frac { n ( n + 1 ) } { 2 }\) without proof.]

4. $$y = 3 \mathrm { e } ^ { - x } \cos 3 x + A \mathrm { e } ^ { - x } \sin 3 x$$ is a particular integral of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 10 y = 40 \mathrm { e } ^ { - x } \sin 3 x$$ where \(A\) is a constant.

1. Find the value of \(A\).
2. Hence find the general solution of this differential equation.
3. Find the particular solution of this differential equation for which both \(y = 3\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3\) at \(x = 0\)

5. $$y = \mathrm { e } ^ { \cos ^ { 2 } x }$$

1. Show that $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \mathrm { e } ^ { \cos ^ { 2 } x } \left( \sin ^ { 2 } 2 x - 2 \cos 2 x \right)$$
2. Hence find the Maclaurin series expansion of \(\mathrm { e } ^ { \cos ^ { 2 } x }\) up to and including the term in \(x ^ { 2 }\)

1. Find the general solution of the differential equation

$$\cos x \frac { \mathrm {~d} y } { \mathrm {~d} x } + y \sin x = \left( \cos ^ { 2 } x \right) \ln x , \quad 0 < x < \frac { \pi } { 2 }$$ Give your answer in the form \(y = \mathrm { f } ( x )\).

7. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve \(C\) with polar equation $$r = 4 \cos 2 \theta , \quad - \frac { \pi } { 4 } \leqslant \theta \leqslant \frac { \pi } { 4 } \text { and } \frac { 3 \pi } { 4 } \leqslant \theta \leqslant \frac { 5 \pi } { 4 }$$ The lines \(P Q , Q R , R S\) and \(S P\) are tangents to \(C\), where \(Q R\) and \(S P\) are parallel to the initial line and \(P Q\) and \(R S\) are perpendicular to the initial line.

1. Find the polar coordinates of the points where the tangent SP touches the curve. Give the values of \(\theta\) to 3 significant figures.
2. Find the exact area of the finite region bounded by the curve \(C\), shown unshaded in Figure 1.
3. Find the area enclosed by the rectangle \(P Q R S\) but outside the curve \(C\), shown shaded in Figure 1.

1. (a) Use de Moivre's theorem to
   1. show that
   $$\cos 5 \theta \equiv \cos ^ { 5 } \theta - 10 \cos ^ { 3 } \theta \sin ^ { 2 } \theta + 5 \cos \theta \sin ^ { 4 } \theta$$
2. find an expression for \(\sin 5 \theta\) in terms of \(\cos \theta\) and \(\sin \theta\)
   (b) Hence show that $$\tan 5 \theta = \frac { t ^ { 5 } - 10 t ^ { 3 } + 5 t } { 5 t ^ { 4 } - 10 t ^ { 2 } + 1 }$$ where \(t = \tan \theta\) and \(\cos 5 \theta \neq 0\)
   (c) Hence find a quadratic equation whose roots \(\operatorname { are } ^ { 2 } \tan ^ { 2 } \frac { \pi } { 5 }\) and \(\tan ^ { 2 } \frac { 2 \pi } { 5 }\) Give your answer in the form \(a x ^ { 2 } + b x + c = 0\) where \(a , b\) and \(c\) are integers to be found.
   (d) Deduce that \(\tan \frac { \pi } { 5 } \tan \frac { 2 \pi } { 5 } = \sqrt { 5 }\)
   

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