Edexcel FP2 (Further Pure Mathematics 2) 2006 June

1. Given that \(3 x \sin 2 x\) is a particular integral of the differential equation

$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 y = k \cos 2 x$$ where \(k\) is a constant,

1. calculate the value of \(k\),
2. find the particular solution of the differential equation for which at \(x = 0 , y = 2\), and for which at \(x = \frac { \pi } { 4 } , y = \frac { \pi } { 2 }\).
   (4)(Total 8 marks)

2. Given that for all real values of \(r , \quad ( 2 r + 1 ) ^ { 3 } - ( 2 r - 1 ) ^ { 3 } = A r ^ { 2 } + B\), where \(A\) and \(B\) are constants,

1. find the value of \(A\) and the value of \(B\).
2. Hence, or otherwise, prove that \(\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )\).
3. Calculate \(\sum _ { r = 1 } ^ { 40 } ( 3 r - 1 ) ^ { 2 }\).
   (3)(Total 10 marks)

3. (a) Use algebra to find the exact solutions of the equation $$\left| 2 x ^ { 2 } + x - 6 \right| = 6 - 3 x$$ (b) On the same diagram, sketch the curve with equation \(y = \left| 2 x ^ { 2 } + x - 6 \right|\) and the line with equation \(y = 6 - 3 x\).
(c) Find the set of values of \(x\) for which $$\left| 2 x ^ { 2 } + x - 6 \right| > 6 - 3 x$$ (3)(Total 12 marks)

4. During an industrial process, the mass of salt, \(S \mathrm {~kg}\), dissolved in a liquid \(t\) minutes after the process begins is modelled by the differential equation $$\frac { \mathrm { d } S } { \mathrm {~d} t } + \frac { 2 S } { 120 - t } = \frac { 1 } { 4 } , \quad 0 \leq t < 120$$ Given that \(S = 6\) when \(t = 0\),

1. find \(S\) in terms of \(t\),
2. calculate the maximum mass of salt that the model predicts will be dissolved in the liquid at any one time during the process.
   (4)(Total 12 marks)

5. (a) Find the Taylor expansion of \(\cos 2 x\) in ascending powers of \(\left( x - \frac { \pi } { 4 } \right)\) up to and including the term in \(\left( x - \frac { \pi } { 4 } \right) ^ { 5 }\).
(b) Use your answer to (a) to obtain an estimate of \(\cos 2\), giving your answer to 6 decimal places.
(3)(Total 8 marks)

6. (a) Use de Moivre's theorem to show that \(\boldsymbol { \operatorname { s i n } } 5 \boldsymbol { \theta } = \boldsymbol { \operatorname { s i n } } \boldsymbol { \theta } \left( \mathbf { 1 6 } \mathbf { c o s } ^ { 4 } \boldsymbol { \theta } - \mathbf { 1 2 } \boldsymbol { \operatorname { c o s } } ^ { 2 } \boldsymbol { \theta } + \mathbf { 1 } \right)\).
(b) Hence, or otherwise, solve, for \(0 \leq \theta < \pi\) $$\sin 5 \theta + \cos \theta \sin 2 \theta = 0$$ (6)(Total 11 marks)

7. $$\frac { \mathrm { d } ^ { 2 x } } { \mathrm {~d} t ^ { 2 } } + 3 \sin x = 0 . \quad \text { At } t = 0 , \quad x = 0 \quad \text { and } \quad \frac { \mathrm { d } x } { \mathrm {~d} t } = 0.4$$ (b) Find a series solution for \(x\), in ascending powers of \(t\), up to and including the term in \(t ^ { 3 }\).
(c) Use your answer to (b) to obtain an estimate of \(x\) at \(t = 0.3\).
(2)(Total 11 marks)

8. The point \(P\) represents a complex number \(z\) on an Argand diagram, where $$| z - 6 + 3 i | = 3 | z + 2 - i |$$

1. Show that the locus of \(P\) is a circle, giving the coordinates of the centre and the radius of this circle. The point \(Q\) represents a complex number \(z\) on an Argand diagram, where $$\tan [ \arg ( z + 6 ) ] = \frac { 1 } { 2 }$$
2. On the same Argand diagram, sketch the locus of \(P\) and the locus of \(Q\).
3. On your diagram, shade the region which satisfies both $$| z - 6 + 3 \mathrm { i } | > 3 | z + 2 - \mathrm { i } | \text { and } \tan [ \arg ( z + 6 ) ] > \frac { 1 } { 2 }$$ (2)(Total 14 marks)