Edexcel FP2 (Further Pure Mathematics 2) 2005 June

1. Sketch the graph of \(y = | x - 2 a |\), given that \(a > 0\).
2. Solve \(| x - 2 a | > 2 x + a\), where \(a > 0\).
   (3)(Total 5 marks)

Find the general solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + 2 y \cot 2 x = \sin x , \quad 0 < x < \frac { \pi } { 2 }$$ giving your answer in the form \(y = \mathrm { f } ( x )\).
(Total 7 marks)

3. (a) Show that the transformation \(y = x v\) transforms the equation $$\begin{array} { l l } x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 x \frac { \mathrm {~d} y } { \mathrm {~d} x } + \left( 2 + 9 x ^ { 2 } \right) y = x ^ { 5 } ,
\text { into the equation } & \frac { \mathrm { d } ^ { 2 } v } { \mathrm {~d} x ^ { 2 } } + 9 v = x ^ { 2 } . \end{array}$$I (b) Solve the differential equation II to find \(v\) as a function of \(x\).
(c) Hence state the general solution of the differential equation I.
(1)(Total 12 marks)

4. The curve \(C\) has polar equation \(\quad r = 6 \cos \theta , \quad - \frac { \pi } { 2 } \leq \theta < \frac { \pi } { 2 }\), and the line \(D\) has polar equation \(\quad r = 3 \sec \left( \frac { \pi } { 3 } - \theta \right) , \quad - \frac { \pi } { 6 } < \theta < \frac { 5 \pi } { 6 }\).

1. Find a cartesian equation of \(C\) and a cartesian equation of \(D\).
2. Sketch on the same diagram the graphs of \(C\) and \(D\), indicating where each cuts the initial line. The graphs of \(C\) and \(D\) intersect at the points \(P\) and \(Q\).
3. Find the polar coordinates of \(P\) and \(Q\).
   (5)(Total 13 marks)

5. Find the general solution of the differential equation $$( x + 1 ) \frac { \mathrm { d } y } { \mathrm {~d} x } + 2 y = \frac { 1 } { x } , \quad x > 0 .$$ giving your answer in the form \(y = \mathrm { f } ( x )\).
(7)(Total 7 marks)

6. (a) On the same diagram, sketch the graphs of \(y = \left| x ^ { 2 } - 4 \right|\) and \(y = | 2 x - 1 |\), showing the coordinates of the points where the graphs meet the axes.
(b) Solve \(\left| x ^ { 2 } - 4 \right| = | 2 x - 1 |\), giving your answers in surd form where appropriate.
(c) Hence, or otherwise, find the set of values of \(x\) for which \(\left| x ^ { 2 } - 4 \right| > | 2 x - 1 |\).
(3)(Total 12 marks)

7. (a) Find the general solution of the differential equation $$2 \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 5 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 2 x = 2 t + 9$$ (b) Find the particular solution of this differential equation for which \(x = 3\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = - 1\) when \(t = 0\). The particular solution in part (b) is used to model the motion of a particle \(P\) on the \(x\)-axis. At time \(t\) seconds \(( t \geq 0 ) , P\) is \(x\) metres from the origin \(O\).
(c) Show that the minimum distance between \(O\) and \(P\) is \(\frac { 1 } { 2 } ( 5 + \ln 2 ) \mathrm { m }\) and justify that the distance is a minimum.
(4)(Total 14 marks)

8. The curve \(C\) which passes through \(O\) has polar equation $$r = 4 a ( 1 + \cos \theta ) , \quad - \pi < \theta \leq \pi .$$ The line \(l\) has polar equation $$r = 3 a \sec \theta , \quad - \frac { \pi } { 2 } < \theta < \frac { \pi } { 2 } .$$ The line \(l\) cuts \(C\) at the points \(P\) and \(Q\), as shown in the diagram.

1. Prove that \(P Q = 6 \sqrt { } 3 a\). The region \(R\), shown shaded in the diagram, is bounded by \(l\) and \(C\).
2. Use calculus to find the exact area of \(R\).
   \includegraphics[max width=\textwidth, alt={}, center]{d9aa1f75-ef35-4bf0-85c2-dff36872ca46-2_714_778_1959_1153}

9. A complex number \(z\) is represented by the point \(P\) in the Argand diagram. Given that $$| z - 3 \mathrm { i } | = 3$$

1. sketch the locus of \(P\).
2. Find the complex number \(z\) which satisfies both \(| z - 3 i | = 3\) and \(\arg ( z - 3 i ) = \frac { 3 } { 4 } \pi\). The transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac { 2 \mathrm { i } } { z }$$
3. Show that \(T\) maps \(| z - 3 i | = 3\) to a line in the \(w\)-plane, and give the cartesian equation of this line.
   (5)(Total 11 marks)

10. (a) Given that \(z = e ^ { \mathrm { i } \theta }\), show that $$z ^ { n } - \frac { 1 } { z ^ { n } } = 2 \mathrm { i } \sin n \theta$$ where \(n\) is a positive integer.
(b) Show that $$\sin ^ { 5 } \theta = \frac { 1 } { 16 } ( \sin 5 \theta - 5 \sin 3 \theta + 10 \sin \theta )$$ (c) Hence solve, in the interval \(0 \leq \theta < 2 \pi\), $$\sin 5 \theta - 5 \sin 3 \theta + 6 \sin \theta = 0$$ (5)(Total 12 marks)

11. The variable \(y\) satisfies the differential equation $$4 \left( 1 + x ^ { 2 } \right) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 x \frac { \mathrm {~d} y } { \mathrm {~d} x } = y$$ At \(x = 0 , y = 1\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 2 }\).

1. Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at \(x = 0\).
   (1) (c) Find the value of \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }\) at \(x = 0\)
2. Express \(y\) as a series, in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
3. Find the value that the series gives for \(y\) at \(x = 0.1\), giving your answer to 5 decimal places.
   (1)(Total 14 marks)