Edexcel F2 (Further Pure Mathematics 2) 2016 June

1. (a) Express \(\frac { 1 } { 4 r ^ { 2 } - 1 }\) in partial fractions.
   (b) Hence prove that

$$\sum _ { r = 1 } ^ { n } \frac { 1 } { 4 r ^ { 2 } - 1 } = \frac { n } { 2 n + 1 }$$ (c) Find the exact value of $$\sum _ { r = 9 } ^ { 25 } \frac { 5 } { 4 r ^ { 2 } - 1 }$$

2. Use algebra to find the set of values of \(x\) for which $$\left| x ^ { 2 } - 9 \right| < | 1 - 2 x |$$

2. Find, in terms of \(k\), where \(k\) is a positive integer, the general solution of the differential equation

$$( 1 + x ) \frac { \mathrm { d } y } { \mathrm {~d} x } + k y = x ^ { \frac { 1 } { 2 } } ( 1 + x ) ^ { 2 - k } , \quad x > 0$$ giving your answer in the form \(y = \mathrm { f } ( x )\).
(6)

4. $$f ( x ) = \sin \left( \frac { 3 } { 2 } x \right)$$

1. Find the Taylor series expansion for \(\mathrm { f } ( x )\) about \(\frac { \pi } { 3 }\) in ascending powers of \(\left( x - \frac { \pi } { 3 } \right)\) up to and including the term in \(\left( x - \frac { \pi } { 3 } \right) ^ { 4 }\)
2. Hence obtain an estimate of \(\sin \frac { 1 } { 2 }\), giving your answer to 4 decimal places.

5. The transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac { 2 z - 1 } { z + 3 } , \quad z \neq - 3$$ The circle in the \(z\)-plane with equation \(x ^ { 2 } + y ^ { 2 } = 1\), where \(z = x + \mathrm { i } y\), is mapped by \(T\) onto the circle \(C\) in the \(w\)-plane. Find the centre and the radius of \(C\).

6. (a) Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 3 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y = 3 x ^ { 2 } + 2 x + 1$$ (9)
(b) Find the particular solution of this differential equation for which \(y = 0\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) when \(x = 0\)
(5)

7. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curves \(C _ { 1 }\) and \(C _ { 2 }\) with polar equations $$\begin{array} { l l } C _ { 1 } : r = \frac { 3 } { 2 } \cos \theta , & 0 \leqslant \theta \leqslant \frac { \pi } { 2 }
C _ { 2 } : r = 3 \sqrt { 3 } - \frac { 9 } { 2 } \cos \theta , & 0 \leqslant \theta \leqslant \frac { \pi } { 2 } \end{array}$$ The curves intersect at the point \(P\).

1. Find the polar coordinates of \(P\). The region \(R\), shown shaded in Figure 1, is enclosed by the curves \(C _ { 1 }\) and \(C _ { 2 }\) and the initial line.
2. Find the exact area of \(R\), giving your answer in the form \(p \pi + q \sqrt { 3 }\) where \(p\) and \(q\) are rational numbers to be found.

8. (a) Use de Moivre's theorem to show that $$\cos ^ { 5 } \theta \equiv p \cos 5 \theta + q \cos 3 \theta + r \cos \theta$$ where \(p , q\) and \(r\) are rational numbers to be found.
(b) Hence, showing all your working, find the exact value of $$\int _ { \frac { \pi } { 6 } } ^ { \frac { \pi } { 3 } } \cos ^ { 5 } \theta \mathrm {~d} \theta$$

9. The complex number \(z\) is represented by the point \(P\) in an Argand diagram. Given that \(\arg \left( \frac { z - 5 } { z - 2 } \right) = \frac { \pi } { 4 }\)

1. sketch the locus of \(P\) as \(z\) varies,
2. find the exact maximum value of \(| z |\).
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