Edexcel AEA (Advanced Extension Award) 2017 Specimen

1．（a）For \(| y | < 1\) ，write down the binomial series expansion of \(( 1 - y ) ^ { - 2 }\) in ascending powers of \(y\) up to and including the term in \(y ^ { 3 }\)
（b）Show that when it is convergent，the series $$1 + \frac { 2 x } { x + 3 } + \frac { 3 x ^ { 2 } } { ( x + 3 ) ^ { 2 } } + \ldots + \frac { r x ^ { r - 1 } } { ( x + 3 ) ^ { r - 1 } } + \ldots$$ can be written in the form \(( 1 + a x ) ^ { n }\) ，where \(a\) and \(n\) are constants to be found．
（c）Find the set of values of \(x\) for which the series in part（b）is convergent．

2．（a）On separate diagrams，sketch the curves with the following equations．On each sketch you should label the exact coordinates of the points where the curve meets the coordinate axes．
（i）\(y = 8 + 2 x - x ^ { 2 }\)
（ii）\(y = 8 + 2 | x | - x ^ { 2 }\)
（iii）\(y = 8 + x + | x | - x ^ { 2 }\)
（b）Find the values of \(x\) for which $$\left| 8 + x + | x | - x ^ { 2 } \right| = 8 + 2 | x | - x ^ { 2 }$$

3. \begin{figure}[h] \end{figure} Figure 1 shows a regular pentagon \(O A B C D\). The vectors \(\mathbf { p }\) and \(\mathbf { q }\) are defined by \(\mathbf { p } = \overrightarrow { O A }\) and \(\mathbf { q } = \overrightarrow { O D }\) respectively. Let \(k\) be the number such that \(\overrightarrow { D B } = k \overrightarrow { O A }\).

1. Write down \(\overrightarrow { A C }\) in terms of \(\mathbf { p } , \mathbf { q }\) and \(k\) as appropriate.
2. Show that \(\overrightarrow { C D } = - \mathbf { p } - \frac { 1 } { k } \mathbf { q }\)
3. Hence find the value of \(k\) By considering triangle \(D B C\), or otherwise,
4. find the exact value of \(\sin 54 ^ { \circ }\)

4. \begin{figure}[h] \end{figure} A particle of weight \(W\) lies on a rough plane．The plane is inclined to the horizontal at an angle \(\alpha\) where \(\tan \alpha = \frac { 3 } { 4 }\) ．The coefficient of friction between the particle and the plane is \(\frac { 1 } { 2 }\) The particle is held in equilibrium by a force of magnitude 1.2 W ．The force makes an angle \(\theta\) with the plane，where \(0 < \theta < \pi\) ，and acts in a vertical plane containing a line of greatest slope of the plane，as shown in Figure 2.
（a）Find the value of \(\theta\) for which there is no frictional force acting on the particle． The minimum value of \(\theta\) for the particle to remain in equilibrium is \(\beta\)
（b）Show that $$\beta = \arccos \left( \frac { \sqrt { 5 } } { 3 } \right) - \arctan \left( \frac { 1 } { 2 } \right)$$ （c）Find the range of values of \(\theta\) for which the particle remains in equilibrium with the frictional force acting up the plane．

5. \begin{figure}[h] \end{figure} Show that the area of the finite region between the curves \(y = \tan ^ { 2 } x\) and \(y = 4 \cos 2 x - 1\) in the interval \(- \frac { \pi } { 2 } < x < \frac { \pi } { 2 }\), shown shaded in Figure 3, is given by $$2 \sqrt { 2 \sqrt { 3 } } - 2 \sqrt { 2 \sqrt { 3 } - 3 }$$

6．（i）Eden，who is confused about the laws of logarithms，states that $$\left( \log _ { 5 } p \right) ^ { 2 } = \log _ { 5 } \left( p ^ { 2 } \right)$$ and \(\log _ { 5 } ( q - p ) = \log _ { 5 } q - \log _ { 5 } p\)
However，there is a value of \(p\) and a value of \(q\) for which both statements are correct．
Determine these values．
（ii）（a）Let \(r \in \mathbb { R } ^ { + } , r \neq 1\) ．Prove that $$\log _ { r } A = \log _ { r ^ { 2 } } B \Rightarrow A ^ { 2 } = B$$ （b）Solve $$\log _ { 4 } \left( 3 x ^ { 3 } + 26 x ^ { 2 } + 40 x \right) = 2 + \log _ { 2 } ( x + 2 )$$

7. \begin{figure}[h] \end{figure} A circular tower of radius 1 metre stands in a large horizontal field of grass．A goat is attached to one end of a rope and the other end of the rope is attached to a fixed point \(O\) at the base of the tower．The goat cannot enter the tower． Taking the point \(O\) as the origin（ 0,0 ），the centre of the base of the tower is at the point \(T ( 0,1 )\) ，where the unit of length is the metre． The rope has length \(\pi\) metres and you may ignore the size of the goat．
The curve \(C\) shown in Figure 4 represents the edge of the region that the goat can reach．
（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 4 ，the rope 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, in the first quadrant, 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.