AQA FP3 (Further Pure Mathematics 3) 2006 June

1 It is given that \(y\) satisfies the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 5 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 4 y = 8 x - 10 - 10 \cos 2 x$$

1. Show that \(y = 2 x + \sin 2 x\) is a particular integral of the given differential equation.
2. Find the general solution of the differential equation.
3. Hence express \(y\) in terms of \(x\), given that \(y = 2\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) when \(x = 0\).

2 The function \(y ( x )\) satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { f } ( x , y )$$ where $$\mathrm { f } ( x , y ) = \frac { x ^ { 2 } + y ^ { 2 } } { x y }$$ 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 improved Euler formula $$y _ { r + 1 } = y _ { r } + \frac { 1 } { 2 } \left( k _ { 1 } + k _ { 2 } \right)$$ where \(k _ { 1 } = h \mathrm { f } \left( x _ { r } , y _ { r } \right)\) and \(k _ { 2 } = h \mathrm { f } \left( x _ { r } + h , y _ { r } + k _ { 1 } \right)\) and \(h = 0.1\), to obtain an approximation to \(y ( 1.1 )\), giving your answer to four decimal places.

3

1. Show that \(\sin x\) is an integrating factor for the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + ( \cot x ) y = 2 \cos x$$
2. Solve this differential equation, given that \(y = 2\) when \(x = \frac { \pi } { 2 }\).

4 The diagram shows the curve \(C\) with polar equation $$r = 6 ( 1 - \cos \theta ) , \quad 0 \leqslant \theta < 2 \pi$$ \includegraphics[max width=\textwidth, alt={}, center]{06ae13de-5cf3-421d-ac7a-ee9f74b653be-3_552_903_922_550}

1. Find the area of the region bounded by the curve \(C\).
2. The circle with cartesian equation \(x ^ { 2 } + y ^ { 2 } = 9\) intersects the curve \(C\) at the points \(A\) and \(B\).
   1. Find the polar coordinates of \(A\) and \(B\).
   2. Find, in surd form, the length of \(A B\).

5

1. Show that \(\lim _ { a \rightarrow \infty } \left( \frac { 3 a + 2 } { 2 a + 3 } \right) = \frac { 3 } { 2 }\).
2. Evaluate \(\int _ { 1 } ^ { \infty } \left( \frac { 3 } { 3 x + 2 } - \frac { 2 } { 2 x + 3 } \right) \mathrm { d } x\), giving your answer in the form \(\ln k\), where \(k\) is a rational number.
   (5 marks)

6

1. Show that the substitution $$u = \frac { \mathrm { d } y } { \mathrm {~d} x } + 2 y$$ transforms the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 4 y = \mathrm { e } ^ { - 2 x }$$ into $$\frac { \mathrm { d } u } { \mathrm {~d} x } + 2 u = \mathrm { e } ^ { - 2 x }$$ (4 marks)
2. By using an integrating factor, or otherwise, find the general solution of $$\frac { \mathrm { d } u } { \mathrm {~d} x } + 2 u = \mathrm { e } ^ { - 2 x }$$ giving your answer in the form \(u = \mathrm { f } ( x )\).
3. Hence find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 4 y = \mathrm { e } ^ { - 2 x }$$ giving your answer in the form \(y = \mathrm { g } ( x )\).

7

1. 1. Write down the first three terms of the binomial expansion of \(( 1 + y ) ^ { - 1 }\), in ascending powers of \(y\).
   2. By using the expansion $$\cos x = 1 - \frac { x ^ { 2 } } { 2 ! } + \frac { x ^ { 4 } } { 4 ! } - \ldots$$ and your answer to part (a)(i), or otherwise, show that the first three non-zero terms in the expansion, in ascending powers of \(x\), of \(\sec x\) are $$1 + \frac { x ^ { 2 } } { 2 } + \frac { 5 x ^ { 4 } } { 24 }$$
2. By using Maclaurin's theorem, or otherwise, show that the first two non-zero terms in the expansion, in ascending powers of \(x\), of \(\tan x\) are $$x + \frac { x ^ { 3 } } { 3 }$$
3. Hence find \(\lim _ { x \rightarrow 0 } \left( \frac { x \tan 2 x } { \sec x - 1 } \right)\).