Edexcel FP2 (Further Pure Mathematics 2) 2002 June

1. Find the set of values for which

$$| x - 1 | > 6 x - 1$$

1. Find the general solution of the differential equation \(t \frac { \mathrm {~d} v } { \mathrm {~d} t } - v = t , t > 0\) and hence show that the solution can be written in the form \(v = t ( \ln t + c )\), where \(c\) is an arbitrary cnst.
2. This differential equation is used to model the motion of a particle which has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\). When \(t = 2\) the speed of the particle is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find, to 3 sf , the speed of the particle when \(t = 4\).

1. Show that \(y = \frac { 1 } { 2 } x ^ { 2 } \mathrm { e } ^ { x }\) is a solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = \mathrm { e } ^ { x }$$
2. Solve the differential equation \(\quad \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = \mathrm { e } ^ { x }\).
   given that at \(x = 0 , y = 1\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2\).

4. The curve \(C\) has polar equation \(r = 3 a \cos \theta , - \frac { \pi } { 2 } \leq \frac { \pi } { 2 }\). The curve \(D\) has polar equation \(r = a ( 1 + \cos \theta ) , - \pi \leq \theta < \pi\). Given that \(a\) is a positive constant, (a) sketch, on the same diagram, the graphs of \(C\) and \(D\), indicating where each curve cuts the initial line. The graphs of \(C\) intersect at the pole \(O\) and at the points \(P\) and \(Q\).
(b) Find the polar coordinates of \(P\) and \(Q\).
(c) Use integration to find the exact area enclosed by the curve \(D\) and the lines \(\theta = 0\) and \(\theta = \frac { \pi } { 3 }\) The region \(R\) contains all points which lie outside \(D\) and inside \(C\).
Given that the value of the smaller area enclosed by the curve \(C\) and the line \(\theta = \frac { \pi } { 3 }\) is $$\frac { 3 a ^ { 2 } } { 16 } ( 2 \pi - 3 \sqrt { } 3 )$$ (d) show that the area of \(R\) is \(\pi a ^ { 2 }\).

5. Using algebra, find the set of values of \(x\) for which \(2 x - 5 > \frac { 3 } { x }\).

6. (a) Find the general solution of the differential equation $$\cos x \frac { \mathrm {~d} y } { \mathrm {~d} x } + ( \sin x ) y = \cos ^ { 3 } x$$ (b) Show that, for \(0 \leq x \leq 2 \pi\), there are two points on the \(x\)-axis through which all the solution curves for this differential equation pass.
(c) Sketch the graph, for \(0 \leq x \leq 2 \pi\), of the particular solution for which \(y = 0\) at \(x = 0\).

7. (a) Find the general solution of the differential equation $$2 \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } + 7 \frac { \mathrm {~d} y } { \mathrm {~d} t } + 3 y = 3 t ^ { 2 } + 11 t$$ (b) Find the particular solution of this differential equation for which \(y = 1\) and \(\frac { \mathrm { d } y } { \mathrm {~d} t } = 1\) when \(t = 0\).
(c) For this particular solution, calculate the value of \(y\) when \(t = 1\).

8. \section*{Figure 1} The curve \(C\) shown in Fig. 1 has polar equation $$r = a ( 3 + \sqrt { 5 } \cos \theta ) , \quad - \pi \leq \theta < \pi .$$ \includegraphics[max width=\textwidth, alt={}, center]{6d92bf8a-df0d-421c-8246-8160f5921ee6-2_460_792_1503_970}

1. Find the polar coordinates of the points \(P\) and \(Q\) where the tangents to \(C\) are parallel to the initial line. (6) The curve \(C\) represents the perimeter of the surface of a swimming pool. The direct distance from \(P\) to \(Q\) is 20 m.
2. Calculate the value of \(a\).
3. Find the area of the surface of the pool. (6)

9. (a) The point \(P\) represents a complex number \(z\) in an Argand diagram. Given that $$| z - 2 i | = 2 | z + i |$$

1. find a cartesian equation for the locus of \(P\), simplifying your answer.
2. sketch the locus of \(P\).
   (b) A transformation \(T\) from the \(z\)-plane to the \(w\)-plane is a translation \(- 7 + 11\) i followed by an enlargement with centre the origin and scale factor 3 . Write down the transformation \(T\) in the form $$w = a z + b , \quad a , b \in \mathbb { C }$$

10. $$y \frac { d ^ { 2 } y } { d x ^ { 2 } } + \left( \frac { d y } { d x } \right) ^ { 2 } + y = 0$$

1. Find an expression for \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }\). Given that \(y = 1\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1\) at \(x = 0\),
2. find the series solution for \(y\), in ascending powers of \(x\), up to an including the term in \(x ^ { 3 }\).
3. Comment on whether it would be sensible to use your series solution to give estimates for \(y\) at \(x = 0.2\) and at \(x = 50\).