Edexcel FP2 (Further Pure Mathematics 2) 2003 June

1. (i) (a) On the same Argand diagram sketch the loci given by the following equations.

$$| z - 1 | = 1 , \quad , , \arg ( z + 1 ) = \frac { \pi } { 12 } , \quad , \arg ( z + 1 ) = \frac { \pi } { 2 }$$ (b) Shade on your diagram the region for which $$| z - 1 | \leq 1 \quad \text { and } \quad \frac { \pi } { 12 } \leq \arg ( z + 1 ) \leq \frac { \pi } { 2 }$$ (ii) (a) Show that the transformation \(\quad w = \frac { z - 1 } { z } , \quad z \neq 0\), $$\text { maps } | z - 1 | = 1 \text { in the } \boldsymbol { z } \text {-plane onto } | w | = | w - 1 | \text { in the } \boldsymbol { w } \text {-plane. }$$ The region \(| z - 1 | \leq 1\) in the \(z\)-plane is mapped onto the region \(T\) in the \(w\)-plane.
(b) Shade the region \(T\) on an Argand diagram.

2. (a) Use de Moivre's theorem to show that $$\cos 5 \theta = 16 \cos ^ { 5 } \theta - 20 \cos ^ { 3 } \theta + 5 \cos \theta$$ (b) Hence find 3 distinct solutions of the equation \(16 x ^ { 5 } - 20 x ^ { 3 } + 5 x + 1 = 0\), giving your answers to 3 decimal places where appropriate.

3. $$\frac { \mathrm { d } y } { \mathrm {~d} x } = x ^ { 2 } - y ^ { 2 } , \quad y = 1 \text { at } x = 0 \text {. (I) }$$ (b) By differentiating (I) twice with respect to \(x\), show that $$\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } + 2 y \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } - 2 = 0$$ (c) Hence, for (I), find the series solution for \(\boldsymbol { y }\) in ascending powers of \(\boldsymbol { x }\) up to and including the term in \(\boldsymbol { x } ^ { \mathbf { 3 } }\). (4)

4. (a) Express as a simplified single fraction \(\frac { 1 } { ( r - 1 ) ^ { 2 } } - \frac { 1 } { r ^ { 2 } }\).
(b) Hence prove, by the method of differences, that \(\quad \sum _ { r = 2 } ^ { n } \frac { 2 r - 1 } { r ^ { 2 } ( r - 1 ) ^ { 2 } } = 1 - \frac { 1 } { n ^ { 2 } }\).

5. Solve the inequality \(\frac { 1 } { 2 x + 1 } > \frac { x } { 3 x - 2 }\).

6. (a) Using the substitution \(t = x ^ { 2 }\), or otherwise, find $$\int x ^ { 3 } \mathrm { e } ^ { - x ^ { 2 } } \mathrm {~d} x$$ (b) Find the general solution of the differential equation $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 3 y = x \mathrm { e } ^ { - x ^ { 2 } } , \quad x > 0$$

7. \begin{figure}[h] \end{figure} A logo is designed which consists of two overlapping closed curves. The polar equations of these curves are \(r = \boldsymbol { a } ( \mathbf { 3 } + \mathbf { 2 } \cos \boldsymbol { \theta } )\) and $$r = a ( 5 - 2 \cos \theta ) , \quad 0 \leq \theta < 2 \pi .$$ Figure 1 is a sketch (not to scale) of these two curves.

1. Write down the polar corrdinates of the points \(A\) and \(B\) where the curves meet the initial line.(2)
2. Find the polar coordinates of the points \(\boldsymbol { C }\) and \(\boldsymbol { D }\) where the two curves meet. (4)
3. Show that the area of the overlapping region, which is shaded in the figure, is $$\frac { a ^ { 2 } } { 3 } ( 49 \pi - 48 \sqrt { } 3 )$$

8. $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } - 6 \frac { \mathrm {~d} y } { \mathrm {~d} t } + 9 y = 4 \mathrm { e } ^ { 3 t } , \quad t \geq 0 .$$

1. Show that \(K t ^ { 2 } e ^ { 3 t }\) is a particular integral of the differential equation, where \(K\) is a constant to be found.
2. Find the general solution of the differential equation. (3) Given that a particular solution satisfies \(\boldsymbol { y } = 3\) and \(\frac { \mathrm { d } y } { \mathrm {~d} t } = 1\) when \(\boldsymbol { t } = \mathbf { 0 }\),
3. find this solution.(4) Another particular solution which satisfies \(\boldsymbol { y } = \mathbf { 1 }\) and \(\frac { \mathrm { d } y } { \mathrm {~d} t } = \mathbf { 0 }\) when \(\boldsymbol { t } = \mathbf { 0 }\), has equation $$y = \left( 1 - 3 t + 2 t ^ { 2 } \right) \mathrm { e } ^ { 3 t }$$
4. For this particular solution draw a sketch graph of \(y\) against \(t\), showing where the graph crosses the \(t\)-axis. Determine also the coordinates of the minimum of the point on the sketch graph.

9. $$z = 4 \left( \cos \frac { \pi } { 4 } + i \sin \frac { \pi } { 4 } \right) , \text { and } \boldsymbol { w } = 3 \left( \cos \frac { 2 \pi } { 3 } + i \sin \frac { 2 \pi } { 3 } \right)$$ Express zw in the form \(r ( \cos \theta + \mathrm { i } \sin \theta ) , r > 0 , - \pi < \theta < \pi\).

10. (a) Sketch, on the same axes, the graphs with equation \(y = | 2 x - 3 |\), and the line with equation \(y = 5 x - 1\).
(b) Solve the inequality \(| 2 x - 3 | < 5 x - 1\).

11. (a) Express \(\frac { 2 } { ( r + 1 ) ( r + 3 ) }\) in partial fractions.
(b) Hence prove that \(\sum _ { r = 1 } ^ { n } \frac { 2 } { ( r + 1 ) ( r + 3 ) } \equiv \frac { n ( 5 n + 13 ) } { 6 ( n + 2 ) ( n + 3 ) }\).

12. (a) Use the substitution \(y = v x\) to transform the equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { ( 4 x + y ) ( x + y ) } { x ^ { 2 } } , x > 0$$ into the equation $$x \frac { \mathrm {~d} v } { \mathrm {~d} x } = ( 2 + v ) ^ { 2 }$$ (b) Solve the differential equation II to find \(\boldsymbol { v }\) as a function of \(\boldsymbol { x }\)
(c) Hence show that \(\quad y = - 2 x - \frac { x } { \ln x + c }\), where \(c\) is an arbitrary constant, is a general solution of the differential equation I.

13. Given that \(z = 3 - 3 i\) express, in the form \(a + i b\), where \(a\) and \(b\) are real numbers,

1. \(z ^ { 2 }\),
   (2)
2. \(\frac { 1 } { z }\).
   (2)
3. Find the exact value of each of \(| z | , \left| z ^ { 2 } \right|\) and \(\left| \frac { 1 } { z } \right|\).
   (2) The complex numbers \(z , z ^ { 2 }\) and \(\frac { 1 } { z }\) are represented by the points \(A , B\) and \(C\) respectively on an Argand diagram. The real number 1 is represented by the point \(D\), and \(O\) is the origin.
4. Show the points \(A , B , C\) and \(D\) on an Argand diagram.
5. Prove that \(\triangle O A B\) is similar to \(\triangle O C D\).

14. (a) Find the value of \(\lambda\) for which \(\lambda x \cos 3 x\) is a particular integral of the differential equation $$\frac { d ^ { 2 } y } { d x ^ { 2 } } + 9 y = - 12 \sin 3 x$$ (b) Hence find the general solution of this differential equation.(4) The particular solution of the differential equation for which \(\boldsymbol { y } = \mathbf { 1 }\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathbf { 2 }\) at \(\boldsymbol { x } = \mathbf { 0 }\), is \(\boldsymbol { y } = \mathbf { g } ( \boldsymbol { x } )\).
(c) Find \(\mathrm { g } ( x )\).
(d) Sketch the graph of \(y = g ( x ) , 0 \leq x \leq \pi\).
(2) \section*{15.} \section*{Figure 1} Figure 1 shows a sketch of the cardioid \(C\) with equation \(r = a ( 1 + \cos \theta ) , - \pi < \theta \leq \pi\). Also shown are the tangents to \(C\) that are parallel and perpendicular to the initial line. These tangents form a rectangle WXYZ.
\includegraphics[max width=\textwidth, alt={}, center]{141c7b1b-4236-4433-84af-04fa9baa3d96-5_407_782_315_1142}
(a) Find the area of the finite region, shaded in Fig. 1, bounded by the curve \(C\).
(b) Find the polar coordinates of the points \(A\) and \(B\) where \(W Z\) touches the curve \(C\).
(c) Hence find the length of \(W X\). Given that the length of \(\boldsymbol { W } \boldsymbol { Z }\) is \(\frac { 3 \sqrt { 3 } a } { 2 }\),
(d) find the area of the rectangle \(W X Y Z\). A heart-shape is modelled by the cardioid \(C\), where \(\boldsymbol { a } = \mathbf { 1 0 ~ c m }\). The heart shape is cut from the rectangular card WXYZ, shown in Fig. 1.
(e) Find a numerical value for the area of card wasted in making this heart shape.
8. A transformation \(T\) from the \(z\)-plane to the \(w\)-plane is defined by $$w = \frac { z + 1 } { i z - 1 } , \quad z \neq - i$$ where \(z = x + \mathrm { i } y , w = u + \mathrm { i } v\) and \(x , y , u\) and \(v\) are real.
\(T\) transforms the circle \(| z | = 1\) in the \(z\)-plane onto a straight line \(L\) in the \(w\)-plane.
(a) Find an equation of \(L\) giving your answer in terms of \(u\) and \(v\).
(b) Show that \(T\) transforms the line \(\operatorname { Im } z = 0\) in the \(z\)-plane onto a circle \(C\) in the \(w\)-plane, giving the centre and radius of this circle.
(c) On a single Argand diagram sketch \(L\) and \(C\). Question: Solve $$x ^ { 5 } = - ( 9 \sqrt { 3 } ) i$$