AQA FP3 (Further Pure Mathematics 3) 2012 January

1 The function \(y ( x )\) satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { f } ( x , y )$$ where $$\mathrm { f } ( x , y ) = \frac { y - x } { y ^ { 2 } + x }$$ and $$y ( 1 ) = 2$$

1. Use the Euler formula $$y _ { r + 1 } = y _ { r } + h \mathrm { f } \left( x _ { r } , y _ { r } \right)$$ with \(h = 0.1\), to obtain an approximation to \(y ( 1.1 )\).
2. Use the formula $$y _ { r + 1 } = y _ { r - 1 } + 2 h \mathrm { f } \left( x _ { r } , y _ { r } \right)$$ with your answer to part (a), to obtain an approximation to \(y ( 1.2 )\), giving your answer to three decimal places.

2 Find $$\lim _ { x \rightarrow 0 } \left[ \frac { \sqrt { 4 + x } - 2 } { x + x ^ { 2 } } \right]$$ (3 marks)

3 Solve the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 10 y = 26 \mathrm { e } ^ { x }$$ given that \(y = 5\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 11\) when \(x = 0\). Give your answer in the form \(y = \mathrm { f } ( x )\).
(10 marks)

4

1. By using an integrating factor, find the general solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { 2 } { x } y = \ln x$$
2. Hence, given that \(y \rightarrow 0\) as \(x \rightarrow 0\), find the value of \(y\) when \(x = 1\).

5

1. Explain why \(\int _ { \frac { 1 } { 2 } } ^ { \infty } \frac { x ( 1 - 2 x ) } { x ^ { 2 } + 3 \mathrm { e } ^ { 4 x } } \mathrm {~d} x\) is an improper integral.
   (1 mark)
2. By using the substitution \(u = x ^ { 2 } \mathrm { e } ^ { - 4 x } + 3\), find $$\int \frac { x ( 1 - 2 x ) } { x ^ { 2 } + 3 \mathrm { e } ^ { 4 x } } \mathrm {~d} x$$
3. Hence evaluate \(\int _ { \frac { 1 } { 2 } } ^ { \infty } \frac { x ( 1 - 2 x ) } { x ^ { 2 } + 3 \mathrm { e } ^ { 4 x } } \mathrm {~d} x\), showing the limiting process used.

6

1. Given that \(y = \ln \cos 2 x\), find \(\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } }\).
2. Use Maclaurin's theorem to show that the first two non-zero terms in the expansion, in ascending powers of \(x\), of \(\ln \cos 2 x\) are \(- 2 x ^ { 2 } - \frac { 4 } { 3 } x ^ { 4 }\).
3. Hence find the first two non-zero terms in the expansion, in ascending powers of \(x\), of \(\ln \sec ^ { 2 } 2 x\).

7 It is given that, for \(x \neq 0 , y\) satisfies the differential equation $$x \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 ( 3 x + 1 ) \frac { \mathrm { d } y } { \mathrm {~d} x } + 3 y ( 3 x + 2 ) = 18 x$$

1. Show that the substitution \(u = x y\) transforms this differential equation into $$\frac { \mathrm { d } ^ { 2 } u } { \mathrm {~d} x ^ { 2 } } + 6 \frac { \mathrm {~d} u } { \mathrm {~d} x } + 9 u = 18 x$$
2. Hence find the general solution of the differential equation $$x \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 ( 3 x + 1 ) \frac { \mathrm { d } y } { \mathrm {~d} x } + 3 y ( 3 x + 2 ) = 18 x$$ giving your answer in the form \(y = \mathrm { f } ( x )\).
   (8 marks)

8 The diagram shows a sketch of the curve \(C\) with polar equation $$r = 3 + 2 \cos \theta , \quad 0 \leqslant \theta \leqslant 2 \pi$$ \includegraphics[max width=\textwidth, alt={}, center]{80c4336c-b0ca-46de-a871-812e6923f7f2-4_463_668_1468_699}

1. Find the area of the region bounded by the curve \(C\).
2. A circle, whose cartesian equation is \(( x - 4 ) ^ { 2 } + y ^ { 2 } = 16\), intersects the curve \(C\) at the points \(A\) and \(B\).
   1. Find, in surd form, the length of \(A B\).
   2. Find the perimeter of the segment \(A O B\) of the circle, where \(O\) is the pole.