Edexcel F2 (Further Pure Mathematics 2) 2021 January

1. The transformation \(T\) from the \(z\)-plane, where \(z = x + \mathrm { i } y\), to the \(w\)-plane, where \(w = u + \mathrm { i } v\), is given by

$$w = \frac { z + p \mathrm { i } } { \mathrm { i } z + 3 } \quad z \neq 3 \mathrm { i } \quad p \in \mathbb { Z }$$ The point representing \(\mathrm { i } ( 1 + \sqrt { 3 } )\) is invariant under \(T\).
Determine the value of \(p\).

2. (a) Show that, for \(r > 0\) $$\frac { r + 2 } { r ( r + 1 ) } - \frac { r + 3 } { ( r + 1 ) ( r + 2 ) } = \frac { r + 4 } { r ( r + 1 ) ( r + 2 ) }$$ (b) Hence show that $$\sum _ { r = 1 } ^ { n } \frac { r + 4 } { r ( r + 1 ) ( r + 2 ) } = \frac { n ( a n + b ) } { c ( n + 1 ) ( n + 2 ) }$$ where \(a\), \(b\) and \(c\) are integers to be determined.

3. Use algebra to obtain the set of values of \(x\) for which $$\left| x ^ { 2 } + x - 2 \right| < \frac { 1 } { 2 } ( x + 5 )$$

4. (a) Show that the substitution \(y ^ { 2 } = \frac { 1 } { z }\) transforms the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + 2 y = 3 x y ^ { 3 } \quad y \neq 0$$ into the differential equation $$\frac { \mathrm { d } z } { \mathrm {~d} x } - 4 z = - 6 x$$ (b) Obtain the general solution of differential equation (II).
(c) Hence obtain the general solution of differential equation (I), giving your answer in the form \(y ^ { 2 } = \mathrm { f } ( x )\)

5. Given that $$\left( 2 - x ^ { 2 } \right) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 5 x \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } = 3 y$$

1. show that $$\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } = \frac { 1 } { \left( 2 - x ^ { 2 } \right) } \left( 2 x \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } \left( 1 - 5 \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) - 5 \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } + 3 \frac { \mathrm {~d} y } { \mathrm {~d} x } \right)$$ Given also that \(y = 3\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 4 }\) at \(x = 0\)
2. obtain a series solution for \(y\) in ascending powers of \(x\) with simplified coefficients, up to and including the term in \(x ^ { 3 }\)
   

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

7. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of curve \(C\) with polar equation $$r = 3 \sin 2 \theta \quad 0 \leqslant \theta \leqslant \frac { \pi } { 2 }$$ The point \(P\) on \(C\) has polar coordinates \(( R , \phi )\). The tangent to \(C\) at \(P\) is perpendicular to the initial line.

1. Show that \(\tan \phi = \frac { 1 } { \sqrt { 2 } }\)
2. Determine the exact value of \(R\). The region \(S\), shown shaded in Figure 1, is bounded by \(C\) and the line \(O P\), where \(O\) is the pole.
3. Use calculus to show that the exact area of \(S\) is $$p \arctan \frac { 1 } { \sqrt { 2 } } + q \sqrt { 2 }$$ where \(p\) and \(q\) are constants to be determined. Solutions relying entirely on calculator technology are not acceptable.

8. Given that \(z = e ^ { \mathrm { i } \theta }\)

1. show that \(z ^ { n } + \frac { 1 } { z ^ { n } } = 2 \cos n \theta\)
   where \(n\) is a positive integer.
2. Show that $$\cos ^ { 6 } \theta = \frac { 1 } { 32 } ( \cos 6 \theta + 6 \cos 4 \theta + 15 \cos 2 \theta + 10 )$$
3. Hence solve the equation $$\cos 6 \theta + 6 \cos 4 \theta + 15 \cos 2 \theta = 0 \quad 0 \leqslant \theta \leqslant \pi$$ Give your answers to 3 significant figures.
4. Use calculus to determine the exact value of $$\int _ { 0 } ^ { \frac { \pi } { 3 } } \left( 32 \cos ^ { 6 } \theta - 4 \cos ^ { 2 } \theta \right) d \theta$$ Solutions relying entirely on calculator technology are not acceptable.