Edexcel AEA (Advanced Extension Award) 2014 June

1．The function f is given by $$\mathrm { f } ( x ) = \ln ( 2 x - 5 ) , \quad x > 2.5$$ （a）Find \(\mathrm { f } ^ { - 1 } ( x )\) ． The function g has domain \(x > 2\) and $$\operatorname { fg } ( x ) = \ln \left( \frac { x + 10 } { x - 2 } \right) , \quad x > 2$$ （b）Find \(\mathrm { g } ( x )\) and simplify your answer．

2．Given that $$3 \sin ^ { 2 } x + 2 \sin x = 6 \cos x + 9 \sin x \cos x$$ and that \(- 90 ^ { \circ } < x < 90 ^ { \circ }\) ， find the possible values of \(\tan x\) ．
（Total 6 marks）

3．（a）On separate diagrams sketch the curves with the following equations．On each sketch you should mark the coordinates of the points where the curve crosses the coordinate axes．
（i）\(y = x ^ { 2 } - 2 x - 3\)
（ii）\(y = x ^ { 2 } - 2 | x | - 3\)
（iii）\(y = x ^ { 2 } - x - | x | - 3\)
（b）Solve the equation $$x ^ { 2 } - x - | x | - 3 = x + | x |$$

1. Given that

$$( 1 + x ) ^ { n } = 1 + \sum _ { r = 1 } ^ { \infty } \frac { n ( n - 1 ) \ldots ( n - r + 1 ) } { 1 \times 2 \times \ldots \times r } x ^ { r } \quad ( | x | < 1 , x \in \mathbb { R } , n \in \mathbb { R } )$$

1. show that $$( 1 - x ) ^ { - \frac { 1 } { 2 } } = \sum _ { r = 0 } ^ { \infty } \binom { 2 r } { r } \left( \frac { x } { 4 } \right) ^ { r }$$
2. show that \(\left( 9 - 4 x ^ { 2 } \right) ^ { - \frac { 1 } { 2 } }\) can be written in the form \(\sum _ { r = 0 } ^ { \infty } \binom { 2 r } { r } \frac { x ^ { 2 r } } { 3 ^ { q } }\) and give \(q\) in terms of \(r\).
3. Find \(\sum _ { r = 1 } ^ { \infty } \binom { 2 r } { r } \times \frac { 2 r } { 9 } \times \left( \frac { x } { 3 } \right) ^ { 2 r - 1 }\)
4. Hence find the exact value of $$\sum _ { r = 1 } ^ { \infty } \binom { 2 r } { r } \times \frac { 2 r \sqrt { } 5 } { 9 } \times \frac { 1 } { 5 ^ { r } }$$ giving your answer as a rational number.

5. The square-based pyramid \(P\) has vertices \(A , B , C , D\) and \(E\). The position vectors of \(A , B , C\) and \(D\) are \(\mathbf { a , b , c }\) and \(\mathbf { d }\) respectively where $$\mathbf { a } = \left( \begin{array} { r } - 2
3
- 1 \end{array} \right) , \quad \mathbf { b } = \left( \begin{array} { r } 5
8
- 6 \end{array} \right) , \quad \mathbf { c } = \left( \begin{array} { l } 2
5
3 \end{array} \right) , \quad \mathbf { d } = \left( \begin{array} { l }

6
1
1 \end{array} \right)$$

1. Find the vectors \(\overrightarrow { A B } , \overrightarrow { A C } , \overrightarrow { A D } , \overrightarrow { B C } , \overrightarrow { B D }\) and \(\overrightarrow { C D }\).
2. Find
   1. the length of a side of the square base of \(P\),
   2. the cosine of the angle between one of the slanting edges of \(P\) and its base,
   3. the height of \(P\),
   4. the position vector of \(E\). A second pyramid, identical to \(P\), is attached by its square base to the base of \(P\) to form an octahedron.
3. Find the position vector of the other vertex of this octahedron.
   6. (i) A curve with equation \(y = \mathrm { f } ( x )\) has \(\mathrm { f } ( x ) \geqslant 0\) for \(x \geqslant a\) and $$A = \int _ { a } ^ { b } \mathrm { f } ( x ) \mathrm { d } x \quad \text { and } \quad V = \pi \int _ { a } ^ { b } [ \mathrm { f } ( x ) ] ^ { 2 } \mathrm {~d} x$$ where \(a\) and \(b\) are constants with \(b > a\). Use integration by substitution to show that for the positive constants \(r\) and \(h\) $$\pi \int _ { a + h } ^ { b + h } [ r + \mathrm { f } ( x - h ) ] ^ { 2 } \mathrm {~d} x = \pi r ^ { 2 } ( b - a ) + 2 \pi r A + V$$ (ii) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{65b3fb2e-c603-48a0-b4ad-0f78603c203c-5_492_1038_799_504} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} Figure 1 shows part of the curve \(C\) with equation \(y = 4 + \frac { 2 } { \sqrt { 3 } \cos x + \sin x }\)
   This curve has asymptotes \(x = m\) and \(x = n\) and crosses the \(y\)-axis at \(( 0 , p )\).

1. Find the value of \(p\), the value of \(m\) and the value of \(n\).
2. Show that the equation of \(C\) can be written in the form \(y = r + \mathrm { f } ( x - h )\) and specify the function f and the constants \(r\) and \(h\). The region bounded by \(C\), the \(x\)-axis and the lines \(x = \frac { \pi } { 6 }\) and \(x = \frac { \pi } { 3 }\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
3. Find the volume of the solid formed.

8
- 6 \end{array} \right) , \quad \mathbf { c } = \left( \begin{array} { l } 2
5
3 \end{array} \right) , \quad \mathbf { d } = \left( \begin{array} { l } 6
1
1 \end{array} \right)$$

1. Find the vectors \(\overrightarrow { A B } , \overrightarrow { A C } , \overrightarrow { A D } , \overrightarrow { B C } , \overrightarrow { B D }\) and \(\overrightarrow { C D }\).
2. Find
   1. the length of a side of the square base of \(P\),
   2. the cosine of the angle between one of the slanting edges of \(P\) and its base,
   3. the height of \(P\),
   4. the position vector of \(E\). A second pyramid, identical to \(P\), is attached by its square base to the base of \(P\) to form an octahedron.
3. Find the position vector of the other vertex of this octahedron.
   6. (i) A curve with equation \(y = \mathrm { f } ( x )\) has \(\mathrm { f } ( x ) \geqslant 0\) for \(x \geqslant a\) and $$A = \int _ { a } ^ { b } \mathrm { f } ( x ) \mathrm { d } x \quad \text { and } \quad V = \pi \int _ { a } ^ { b } [ \mathrm { f } ( x ) ] ^ { 2 } \mathrm {~d} x$$ where \(a\) and \(b\) are constants with \(b > a\). Use integration by substitution to show that for the positive constants \(r\) and \(h\) $$\pi \int _ { a + h } ^ { b + h } [ r + \mathrm { f } ( x - h ) ] ^ { 2 } \mathrm {~d} x = \pi r ^ { 2 } ( b - a ) + 2 \pi r A + V$$ (ii) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{65b3fb2e-c603-48a0-b4ad-0f78603c203c-5_492_1038_799_504} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} Figure 1 shows part of the curve \(C\) with equation \(y = 4 + \frac { 2 } { \sqrt { 3 } \cos x + \sin x }\)
   This curve has asymptotes \(x = m\) and \(x = n\) and crosses the \(y\)-axis at \(( 0 , p )\).
4. Find the value of \(p\), the value of \(m\) and the value of \(n\).
5. Show that the equation of \(C\) can be written in the form \(y = r + \mathrm { f } ( x - h )\) and specify the function f and the constants \(r\) and \(h\). The region bounded by \(C\), the \(x\)-axis and the lines \(x = \frac { \pi } { 6 }\) and \(x = \frac { \pi } { 3 }\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
6. Find the volume of the solid formed.
   7. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{65b3fb2e-c603-48a0-b4ad-0f78603c203c-6_631_974_201_548} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure} A circular tower stands in a large horizontal field of grass．A goat is attached to one end of a string and the other end of the string is attached to the fixed point \(O\) at the base of the tower．Taking the point \(O\) as the origin（ 0,0 ），the centre of the base of the tower is at the point \(T ( 0,1 )\) ．The radius of the base of the tower is 1 ．The string has length \(\pi\) and you may ignore the size of the goat．The curve \(C\) represents the edge of the region that the goat can reach as shown in Figure 2.
   （a）Write down the equation of \(C\) for \(y < 0\) ． When the goat is at the point \(G ( x , y )\) ，with \(x > 0\) and \(y > 0\) ，as shown in Figure 2 ，the string lies along \(O A G\) where \(O A\) is an arc of the circle with angle \(O T A = \theta\) radians and \(A G\) is a tangent to the circle at \(A\) ．
   （b）With the aid of a suitable diagram show that $$\begin{aligned} & x = \sin \theta + ( \pi - \theta ) \cos \theta
   & y = 1 - \cos \theta + ( \pi - \theta ) \sin \theta \end{aligned}$$ （c）By considering \(\int y \frac { \mathrm {~d} x } { \mathrm {~d} \theta } \mathrm {~d} \theta\) ，show that the area between \(C\) ，the positive \(x\)－axis and the positive \(y\)－axis can be expressed in the form $$\int _ { 0 } ^ { \pi } u \sin u \mathrm {~d} u + \int _ { 0 } ^ { \pi } u ^ { 2 } \sin ^ { 2 } u \mathrm {~d} u + \int _ { 0 } ^ { \pi } u \sin u \cos u \mathrm {~d} u$$ （d）Show that \(\int _ { 0 } ^ { \pi } u ^ { 2 } \sin ^ { 2 } u \mathrm {~d} u = \frac { \pi ^ { 3 } } { 6 } + \int _ { 0 } ^ { \pi } u \sin u \cos u \mathrm {~d} u\)
   （e）Hence find the area of grass that can be reached by the goat．