Edexcel F2 (Further Pure Mathematics 2) 2020 June

2. (a) Write \(\frac { 3 r + 1 } { r ( r - 1 ) ( r + 1 ) }\) in partial fractions.
(b) Hence find $$\sum _ { r = 2 } ^ { n } \frac { 3 r + 1 } { r ( r - 1 ) ( r + 1 ) } \quad n \geqslant 2$$ giving your answer in the form $$\frac { a n ^ { 2 } + b n + c } { 2 n ( n + 1 ) }$$ where \(a\), \(b\) and \(c\) are integers to be determined.
(c) Hence determine the exact value of $$\sum _ { r = 15 } ^ { 20 } \frac { 3 r + 1 } { r ( r - 1 ) ( r + 1 ) }$$

3. Use algebra to obtain the set of values of \(x\) for which $$\left| \frac { x ^ { 2 } + 3 x + 10 } { x + 2 } \right| < 7 - x$$

4. (a) Express the complex number \(18 \sqrt { 3 } - 18 \mathrm { i }\) in the form $$r ( \cos \theta + \mathrm { i } \sin \theta ) \quad - \pi < \theta \leqslant \pi$$ (b) Solve the equation $$z ^ { 4 } = 18 \sqrt { 3 } - 18 i$$ giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\) where \(- \pi < \theta \leqslant \pi\)

5. The transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac { z - 3 \mathrm { i } } { z + 2 \mathrm { i } } \quad z \neq - 2 \mathrm { i }$$ The circle with equation \(| z | = 1\) in the \(z\)-plane is mapped by \(T\) onto the circle \(C\) in the \(w\)-plane. Determine

1. the centre of \(C\),
2. the radius of \(C\).
   

6. Obtain the general solution of the equation $$x ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + ( x \cot x + 2 ) x y = 4 \sin x \quad 0 < x < \pi$$ Give your answer in the form \(y = \mathrm { f } ( x )\)
(8)

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

1. Determine the polar coordinates of \(A\). The point \(B\) on the curve has polar coordinates \(\quad a ( 2 + \sqrt { 3 } ) , \frac { \pi } { 6 }\) The finite region \(R\), shown shaded in Figure 1, is bounded by the curve \(C\) and the line \(A B\).
2. Use calculus to determine the exact area of the shaded region \(R\). Give your answer in the form $$\frac { a ^ { 2 } } { 4 } ( d \pi - e + f \sqrt { 3 } )$$ where \(d , e\) and \(f\) are integers.

8. (a) Show that the transformation \(x = \mathrm { e } ^ { u }\) transforms the differential equation $$x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 3 x \frac { \mathrm {~d} y } { \mathrm {~d} x } - 8 y = 4 \ln x \quad x > 0$$ into the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} u ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} u } - 8 y = 4 u$$ (b) Determine the general solution of differential equation (II), expressing \(y\) as a function of \(u\).
(c) Hence obtain the general solution of differential equation (I).