Edexcel FP2 (Further Pure Mathematics 2) 2018 June

1. (a) Express \(\frac { 1 } { ( r + 3 ) ( r + 4 ) }\) in partial fractions.
   (b) Hence, using the method of differences, show that

$$\sum _ { r = 1 } ^ { n } \frac { 1 } { ( r + 3 ) ( r + 4 ) } = \frac { n } { a ( n + a ) }$$ where \(a\) is a constant to be found.
(c) Find the exact value of \(\sum _ { r = 15 } ^ { 30 } \frac { 1 } { ( r + 3 ) ( r + 4 ) }\) uestion 1 continued
\includegraphics[max width=\textwidth, alt={}, center]{5aa7f449-215b-4a21-9fdc-df55d26abc9d-05_29_40_182_1914} \includegraphics[max width=\textwidth, alt={}, center]{5aa7f449-215b-4a21-9fdc-df55d26abc9d-05_33_37_201_1914}
□ D D D " "

2. A transformation from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac { 1 - \mathrm { i } z } { z } , \quad z \neq 0$$ The transformation maps points on the real axis in the \(z\)-plane onto the line \(l\) in the \(w\)-plane.
Find an equation of the line \(l\).

3. (a) By writing \(\frac { \pi } { 12 } = \frac { \pi } { 3 } - \frac { \pi } { 4 }\), show that

1. \(\sin \left( \frac { \pi } { 12 } \right) = \frac { 1 } { 4 } ( \sqrt { 6 } - \sqrt { 2 } )\)
2. \(\cos \left( \frac { \pi } { 12 } \right) = \frac { 1 } { 4 } ( \sqrt { 6 } + \sqrt { 2 } )\)
   (b) Hence find the exact values of \(z\) for which $$z ^ { 4 } = 4 \left( \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 } \right)$$ Give your answers in the form \(z = a + i b\) where \(a , b \in \mathbb { R }\)

4. Use algebra to find the set of values of \(x\) for which $$\left| x ^ { 2 } - 2 \right| > 4 x$$

5. $$y \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 3 x \frac { \mathrm {~d} y } { \mathrm {~d} x } - 3 y ^ { 2 } = 0$$ Given that at \(x = 0 , y = 2\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1\)

1. show that, at \(x = 0 , \frac { \mathrm {~d} ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } = \frac { 3 } { 2 }\)
2. Find a series solution for \(y\) up to and including the term in \(x ^ { 3 }\)

6. (a) Find the general solution of the differential equation $$6 \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 5 \frac { \mathrm {~d} y } { \mathrm {~d} x } - 6 y = x - 6 x ^ { 2 }$$ (b) Find the particular solution for which \(y = 0\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 } { 2 }\) when \(x = 0\)

7. \begin{figure}[h] \end{figure} The curve \(C\) shown in Figure 1 has polar equation $$r = 2 + \sqrt { 3 } \cos \theta , \quad 0 \leqslant \theta < 2 \pi$$ The tangent to \(C\) at the point \(P\) is parallel to the initial line.

1. Show that \(O P = \frac { 1 } { 2 } ( 3 + \sqrt { 7 } )\)
2. Find the exact area enclosed by the curve \(C\).

8. (a) Using the substitution \(t = x ^ { 2 }\), or otherwise, find $$\int 2 x ^ { 5 } \mathrm { e } ^ { - x ^ { 2 } } \mathrm {~d} x$$ (b) Hence find the general solution of the differential equation $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 4 y = 2 x ^ { 2 } \mathrm { e } ^ { - x ^ { 2 } }$$ giving your answer in the form \(y = \mathrm { f } ( x )\). Given that \(y = 0\) when \(x = 1\)
(c) find the particular solution of this differential equation, giving your solution in the form \(y = \mathrm { f } ( x )\).