Edexcel F2 (Further Pure Mathematics 2) 2014 June

1. (a) Show that

$$\frac { 1 } { ( r + 1 ) ( r + 2 ) ( r + 3 ) } \equiv \frac { 1 } { 2 ( r + 1 ) ( r + 2 ) } - \frac { 1 } { 2 ( r + 2 ) ( r + 3 ) }$$ (b) Hence, or otherwise, find $$\sum _ { r = 1 } ^ { n } \frac { 1 } { ( r + 1 ) ( r + 2 ) ( r + 3 ) }$$ giving your answer as a single fraction in its simplest form.

2. Use algebra to find the set of values of \(x\) for which $$\frac { 6 } { x - 3 } \leqslant x + 2$$

3. Solve the equation $$z ^ { 5 } = 16 - 16 \mathrm { i } \sqrt { 3 }$$ giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\) where \(\theta\) is in terms of \(\pi\) and \(0 \leqslant \theta < 2 \pi\).

4. A transformation from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac { z } { z + 3 } , \quad z \neq - 3$$ Under this transformation, the circle \(| z | = 2\) in the \(z\)-plane is mapped onto a circle \(C\) in the \(w\)-plane. Determine the centre and the radius of the circle \(C\).

5. $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y = 0$$

1. Show that $$\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } } = \left( a x ^ { 2 } + b \right) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$$ where \(a\) and \(b\) are constants to be found. Given that \(y = 1\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3\) at \(x = 0\)
2. find a series solution for \(y\) in ascending powers of \(x\) up to and including the term in \(x ^ { 4 }\)
3. use your series to estimate the value of \(y\) at \(x = - 0.2\), giving your answer to four decimal places.

6. $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } + ( 1 - 2 x ) y = x , \quad x > 0$$ Find the general solution of the differential equation, giving your answer in the form \(y = \mathrm { f } ( x )\).

7. The point \(P\) represents a complex number \(z\) on an Argand diagram, where $$| z + 1 | = | 2 z - 1 |$$ and the point \(Q\) represents a complex number \(w\) on the Argand diagram, where $$| w | = | w - 1 + \mathrm { i } |$$ Find the exact coordinates of the points where the locus of \(P\) intersects the locus of \(Q\).

8. (a) Show that the substitution \(x = \mathrm { e } ^ { t }\) transforms the differential equation $$x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 5 x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 13 y = 0 , \quad x > 0$$ into the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } + 4 \frac { \mathrm {~d} y } { \mathrm {~d} t } + 13 y = 0$$ (b) Hence find the general solution of the differential equation (I).

9. \begin{figure}[h] \end{figure} Figure 1 shows the curve \(C _ { 1 }\) with polar equation \(r = 2 a \sin 2 \theta , 0 \leqslant \theta \leqslant \frac { \pi } { 2 }\), and the circle \(C _ { 2 }\) with polar equation \(r = a , 0 \leqslant \theta \leqslant 2 \pi\), where \(a\) is a positive constant.

1. Find, in terms of \(a\), the polar coordinates of the points where the curve \(C _ { 1 }\) meets the circle \(C _ { 2 }\) The regions enclosed by the curve \(C _ { 1 }\) and the circle \(C _ { 2 }\) overlap and the common region \(R\) is shaded in Figure 1.
2. Find the area of the shaded region \(R\), giving your answer in the form \(\frac { 1 } { 12 } a ^ { 2 } ( p \pi + q \sqrt { 3 } )\), where \(p\) and \(q\) are integers to be found.