Edexcel FP2 (Further Pure Mathematics 2) 2004 June

1. Show that \(( r + 1 ) ^ { 3 } - ( r - 1 ) ^ { 3 } \equiv A r ^ { 2 } + B\), where \(A\) and \(B\) are constants to be found.
2. Prove by the method of differences that \(\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 ) , n > 1\).
   (6)(Total 8 marks)

$$\frac { \mathrm { d } y } { \mathrm {~d} x } + y \left( 1 + \frac { 3 } { x } \right) = \frac { 1 } { x ^ { 2 } } , \quad x > 0$$

1. Verify that \(x ^ { 3 } \mathrm { e } ^ { x }\) is an integrating factor for the differential equation.
2. Find the general solution of the differential equation.
3. Given that \(y = 1\) at \(x = 1\), find \(y\) at \(x = 2\).
   (3)(Total 10 marks)

3. (a) Sketch, on the same axes, the graph of \(y = | ( x - 2 ) ( x - 4 ) |\), and the line with equation \(y = 6 - 2 x\).
(b) Find the exact values of \(x\) for which \(| ( x - 2 ) ( x - 4 ) | = 6 - 2 x\).
(c) Hence solve the inequality \(| ( x - 2 ) ( x - 4 ) | < 6 - 2 x\).
(2)(Total 11 marks)

4. $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 5 y = 65 \sin 2 x , x > 0$$

1. Find the general solution of the differential equation.
2. Show that for large values of \(x\) this general solution may be approximated by a sine function and find this sine function.
   (3)(Total 12 marks)

5. (a) Sketch the curve with polar equation \(\quad r = 3 \cos 2 \theta , \quad - \frac { \pi } { 4 } \leq \theta < \frac { \pi } { 4 }\)
(b) Find the area of the smaller finite region enclosed between the curve and the half-line $$\theta = \frac { \pi } { 6 }$$ (c) Find the exact distance between the two tangents which are parallel to the initial line.
(8)(Total 16 marks)

6. Find the complete set of values of \(x\) for which $$\left| x ^ { 2 } - 2 \right| > 2 x$$

1. (a) Find the general solution of the differential equation

$$\frac { \mathrm { d } y } { \mathrm {~d} x } + 2 y = x$$ Given that \(y = 1\) at \(x = 0\),
(b) find the exact values of the coordinates of the minimum point of the particular solution curve,
(c) draw a sketch of this particular solution curve.

8. (a) Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} t } + 2 y = 2 \mathrm { e } ^ { - t }$$ (b) Find the particular solution that satisfies \(y = 1\) and \(\frac { \mathrm { d } y } { \mathrm {~d} t } = 1\) at \(t = 0\).
(6)(Total 12 marks)

9. The diagram is a sketch of the two curves
\(C _ { 1 }\) and \(C _ { 2 }\) with polar equations
\(C _ { 1 } : r = 3 a ( 1 - \cos \theta ) , - \pi \leq \theta < \pi\)
\(\mathrm { C } _ { 2 } : r = a ( 1 + \cos \theta ) , - \pi \leq \theta < \pi\).
\includegraphics[max width=\textwidth, alt={}, center]{8646b60a-3822-4d41-8978-1ccad1e216d6-2_318_776_1567_1082} The curves meet at the pole \(O\), and at the points \(A\) and \(B\).

1. Find, in terms of \(a\), the polar coordinates of the points \(A\) and \(B\).
2. Show that the length of the line \(A B\) is \(\frac { 3 \sqrt { } 3 } { 2 } a\). The region inside \(C _ { 2 }\) and outside \(C _ { 1 }\) is shown shaded in the diagram above.
3. Find, in terms of \(a\), the area of this region. A badge is designed which has the shape of the shaded region.
   Given that the length of the line \(A B\) is 4.5 cm ,
4. calculate the area of this badge, giving your answer to three significant figures.
   (Total 16 marks)

10. Given that \(y = \tan x\),

1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x } , \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) and \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }\).
2. Find the Taylor series expansion of \(\tan x\) in ascending powers of \(\left( x - \frac { \pi } { 4 } \right)\) up to and including the term in \(\left( x - \frac { \pi } { 4 } \right) ^ { 3 }\).
3. Hence show that \(\tan \frac { 3 \pi } { 10 } \approx 1 + \frac { \pi } { 10 } + \frac { \pi ^ { 2 } } { 200 } + \frac { \pi ^ { 3 } } { 3000 }\).

11. (b) Hence find the Maclaurin series expansion of \(\mathrm { e } ^ { x } \cos x\), in ascending powers of \(x\), up to and including the term in \(x ^ { 4 }\).
(Total 11 marks)

12. The transformation \(T\) from the complex \(z\)-plane to the complex \(w\)-plane is given by $$w = \frac { z + 1 } { z + \mathrm { i } } , \quad z \neq - \mathrm { i }$$

1. Show that \(T\) maps points on the half-line \(\arg ( z ) = \frac { \pi } { 4 }\) in the \(z\)-plane into points on the circle \(| w | = 1\) in the \(w\)-plane.
2. Find the image under \(T\) in the \(w\)-plane of the circle \(| Z | = 1\) in the \(z\)-plane.
3. Sketch on separate diagrams the circle \(| \mathbf { Z } | = 1\) in the \(z\)-plane and its image under \(T\) in the \(w\)-plane.
4. Mark on your sketches the point \(P\), where \(z = \mathrm { i }\), and its image \(Q\) under \(T\) in the \(w\)-plane.