Edexcel FP2 (Further Pure Mathematics 2) 2006 January

Find the set of values of \(x\) for which \(\frac { x ^ { 2 } } { x - 2 } > 2 x\).
(Total 6 marks)

1. Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 5 x = 0$$
2. Given that \(x = 1\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 1\) at \(t = 0\), find the particular solution of the differential equation, giving your answer in the form \(x = \mathrm { f } ( t )\).
3. Sketch the curve with equation \(x = \mathrm { f } ( t ) , 0 \leq t \leq \pi\), showing the coordinates, as multiples of \(\pi\), of the points where the curve cuts the \(x\)-axis.
   (4)(Total 13 marks)

3. (a) Show that the substitution \(y = v x\) transforms the differential equation $$\frac { d y } { d x } = \frac { 3 x - 4 y } { 4 x + 3 y }$$ into the differential equation $$x \frac { \mathrm {~d} v } { \mathrm {~d} x } = - \frac { 3 v ^ { 2 } + 8 v - 3 } { 3 v + 4 }$$ (b) By solving differential equation (II), find a general solution of differential equation (I). (5)
(c) Given that \(y = 7\) at \(x = 1\), show that the particular solution of differential equation (I) can be written as $$( 3 y - x ) ( y + 3 x ) = 200$$ (5)(Total 14 marks)

4. A curve \(C\) has polar equation \(r ^ { 2 } = a ^ { 2 } \cos 2 \theta , 0 \leq \theta \leq \frac { \pi } { 4 }\). The line \(l\) is parallel to the initial line, and \(l\) is the tangent to \(C\) at
above. above.

1. 1. Show that, for any point on \(C , r ^ { 2 } \sin ^ { 2 } \theta\) can be expressed in terms of \(\sin \theta\) and \(a\) only. (1)
   2. Hence, using differentiation, show that the polar coordinates of \(P\) are \(\left( \frac { a } { \sqrt { 2 } } , \frac { \pi } { 6 } \right)\).(6)
      \includegraphics[max width=\textwidth, alt={}, center]{2352f367-ddf9-4770-ace5-b561b0fbabbb-1_298_725_2163_1169} The shaded region \(R\), shown in the figure above, is bounded by \(C\), the line \(l\) and the half-line with equation
      \(\theta = \frac { \pi } { 2 }\).
2. Show that the area of \(R\) is \(\frac { a ^ { 2 } } { 16 } ( 3 \sqrt { 3 } - 4 )\).

5. Solve the equation \(z ^ { 5 } = \mathrm { i }\)
giving your answers in the form \(\cos \theta + \mathrm { i } \sin \theta\).
(Total 5 marks)

7. $$( 1 + 2 x ) \frac { \mathrm { d } y } { \mathrm {~d} x } = x + 4 y ^ { 2 }$$

1. Show that $$( 1 + 2 x ) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 1 + 2 ( 4 y - 1 ) \frac { \mathrm { d } y } { \mathrm {~d} x }$$
2. Differentiate equation 1 with respect to \(x\) to obtain an equation involving $$\frac { \mathrm { d } ^ { 3 } } { \mathrm {~d} x ^ { 3 } } , \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } , \frac { \mathrm {~d} y } { \mathrm {~d} x } , \quad x \text { and } y .$$ Given that \(y = \frac { 1 } { 2 }\) at \(x = 0\),
3. find a series solution for \(y\), in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
   (6)(Total 11 marks)

8. In the Argand diagram the point \(P\) represents the complex number \(z\). Given that arg \(\left( \frac { z - 2 \mathrm { i } } { z + 2 } \right) = \frac { \pi } { 2 }\),

1. sketch the locus of \(P\),
2. deduce the value of \(| \mathrm { z } + 1 - \mathrm { i } |\). The transformation \(T\) from the \(z\)-plane to the \(w\)-plane is defined by $$w = \frac { 2 ( 1 + \mathrm { i } ) } { z + 2 } , \quad z \neq - 2$$
3. Show that the locus of \(P\) in the \(z\)-plane is mapped to part of a straight line in the \(w\)-plane, and show this in an Argand diagram.
   (6)(Total 12 marks)