AQA FP3 (Further Pure Mathematics 3) 2015 June

1 It is given that \(y ( x )\) satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { f } ( x , y )$$ where $$\mathrm { f } ( x , y ) = \frac { x + y ^ { 2 } } { x }$$ and $$y ( 2 ) = 5$$

1. Use the Euler formula $$y _ { r + 1 } = y _ { r } + h \mathrm { f } \left( x _ { r } , y _ { r } \right)$$ with \(h = 0.05\), to obtain an approximation to \(y ( 2.05 )\).
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 \(y ( 2.1 )\), giving your answer to three significant figures.
   [0pt] [3 marks]

2 By using an integrating factor, find the solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + ( \tan x ) y = \tan ^ { 3 } x \sec x$$ given that \(y = 2\) when \(x = \frac { \pi } { 3 }\).
[0pt] [9 marks]

3

1. 1. Write down the expansion of \(\ln ( 1 + 2 x )\) in ascending powers of \(x\) up to and including the term in \(x ^ { 4 }\).
   2. Hence, or otherwise, find the first two non-zero terms in the expansion of $$\ln [ ( 1 + 2 x ) ( 1 - 2 x ) ]$$ in ascending powers of \(x\) and state the range of values of \(x\) for which the expansion is valid.
2. Find \(\lim _ { x \rightarrow 0 } \left[ \frac { 3 x - x \sqrt { 9 + x } } { \ln [ ( 1 + 2 x ) ( 1 - 2 x ) ] } \right]\).

4

1. Explain why \(\int _ { 2 } ^ { \infty } ( x - 2 ) \mathrm { e } ^ { - 2 x } \mathrm {~d} x\) is an improper integral.
2. Evaluate \(\int _ { 2 } ^ { \infty } ( x - 2 ) \mathrm { e } ^ { - 2 x } \mathrm {~d} x\), showing the limiting process used.

5

1. Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 6 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 9 y = 36 \sin 3 x$$
2. It is given that \(y = \mathrm { f } ( x )\) is the solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 6 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 9 y = 36 \sin 3 x$$ such that \(\mathrm { f } ( 0 ) = 0\) and \(\mathrm { f } ^ { \prime } ( 0 ) = 0\).
   1. Show that \(f ^ { \prime \prime } ( 0 ) = 0\).
   2. Find the first two non-zero terms in the expansion, in ascending powers of \(x\), of \(\mathrm { f } ( x )\).
      [0pt] [3 marks]

6 A differential equation is given by $$4 \sqrt { x ^ { 5 } } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + ( 2 \sqrt { x } ) y = \sqrt { x } ( \ln x ) ^ { 2 } + 5 , \quad x > 0$$

1. Show that the substitution \(x = \mathrm { e } ^ { 2 t }\) transforms this differential equation into $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } - 2 \frac { \mathrm {~d} y } { \mathrm {~d} t } + 2 y = 4 t ^ { 2 } + 5 \mathrm { e } ^ { - t }$$
2. Hence find the general solution of the differential equation $$4 \sqrt { x ^ { 5 } } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + ( 2 \sqrt { x } ) y = \sqrt { x } ( \ln x ) ^ { 2 } + 5 , \quad x > 0$$
   \includegraphics[max width=\textwidth, alt={}]{7b4a1237-bb28-4cba-84b1-35fa91d87408-14_1634_1709_1071_153}

7 The diagram shows the sketch of a curve \(C _ { 1 }\).
\includegraphics[max width=\textwidth, alt={}, center]{7b4a1237-bb28-4cba-84b1-35fa91d87408-18_362_734_360_635} The polar equation of the curve \(C _ { 1 }\) is $$r = 1 + \cos 2 \theta , \quad - \frac { \pi } { 2 } \leqslant \theta \leqslant \frac { \pi } { 2 }$$

1. Find the area of the region bounded by the curve \(C _ { 1 }\).
2. The curve \(C _ { 2 }\) whose polar equation is $$r = 1 + \sin \theta , \quad - \frac { \pi } { 2 } \leqslant \theta \leqslant \frac { \pi } { 2 }$$ intersects the curve \(C _ { 1 }\) at the pole \(O\) and at the point \(A\). The straight line drawn through \(A\) parallel to the initial line intersects \(C _ { 1 }\) again at the point \(B\).
   1. Find the polar coordinates of \(A\).
   2. Show that the length of \(O B\) is \(\frac { 1 } { 4 } ( \sqrt { 13 } + 1 )\).
   3. Find the length of \(A B\), giving your answer to three significant figures. \includegraphics[max width=\textwidth, alt={}, center]{7b4a1237-bb28-4cba-84b1-35fa91d87408-22_2486_1728_221_141}
      \includegraphics[max width=\textwidth, alt={}, center]{7b4a1237-bb28-4cba-84b1-35fa91d87408-23_2486_1728_221_141}
      \includegraphics[max width=\textwidth, alt={}, center]{7b4a1237-bb28-4cba-84b1-35fa91d87408-24_2488_1728_219_141}