Edexcel AEA (Advanced Extension Award) 2016 June

1．The function f is given by $$\mathrm { f } ( x ) = x ^ { 2 } - 4 x + 9 \quad x \in \mathbb { R } , x \geqslant 3$$ （a）Find the range of f ． The function g is given by $$\operatorname { g } ( x ) = \frac { 10 } { x + 1 } \quad x \in \mathbb { R } , x \geqslant 4$$ （b）Find an expression for \(\operatorname { gf } ( x )\) ．
（c）Find the domain and range of gf．

2．Find the value of $$\arccos \left( \frac { 1 } { \sqrt { 2 } } \right) + \arcsin \left( \frac { 1 } { 3 } \right) + 2 \arctan \left( \frac { 1 } { \sqrt { 2 } } \right)$$ Give your answer as a multiple of \(\pi\) ． $$\text { (arccos } x \text { is an alternative notion for } \cos ^ { - 1 } x \text { etc.) }$$

3．The points \(A , B , C , D\) and \(E\) are five of the vertices of a rectangular cuboid and \(A E\) is a diagonal of the cuboid．With respect to a fixed origin \(O\) ，the position vectors of \(A , B , C\) and \(D\) are \(\mathbf { a , b , c }\) and d respectively，where $$\mathbf { a } = \left( \begin{array} { c } 1
2
- 1 \end{array} \right) , \quad \mathbf { b } = \left( \begin{array} { c } 0
- 3
- 8 \end{array} \right) , \quad \mathbf { c } = \left( \begin{array} { c }

4
- 1
- 10 \end{array} \right) \text { and } \mathbf { d } = \left( \begin{array} { c } - 4
2
- 11 \end{array} \right)$$ （a）Find the position vector of \(E\) ． The volume of a tetrahedron is given by the formula $$\text { volume } = \frac { 1 } { 3 } ( \text { area of base } ) \times ( \text { height } )$$ （b）Find the volume of the tetrahedron \(A B C D\) ． 4．（a）Given that \(x > 0 , y > 0 , x \neq 1\) and \(n > 0\) ，show that $$\log _ { x } y = \log _ { x ^ { n } } y ^ { n }$$ （b）Solve the following，leaving your answers in the form \(2 ^ { p }\) ，where \(p\) is a rational number．
（i） \(\log _ { 2 } u + \log _ { 4 } u ^ { 2 } + \log _ { 8 } u ^ { 3 } + \log _ { 16 } u ^ { 4 } = 5\)
（ii） \(\log _ { 2 } v + \log _ { 4 } v + \log _ { 8 } v + \log _ { 16 } v = 5\)
（iii） \(\log _ { 4 } w ^ { 2 } + \frac { 3 \log _ { 8 } 64 } { \log _ { 2 } w } = 5\)

5．（a）Show that $$\sum _ { r = 0 } ^ { n } x ^ { - r } = \frac { x } { x - 1 } - \frac { x ^ { - n } } { x - 1 } \quad \text { where } x \neq 0 \text { and } x \neq 1$$ （b）Hence find an expression in terms of \(x\) and \(n\) for \(\sum _ { r = 0 } ^ { n } r x ^ { - ( r + 1 ) }\) for \(x \neq 0\) and \(x \neq 1\)
Simplify your answer．
（c）Find \(\sum _ { r = 0 } ^ { n } \left( \frac { 3 + 5 r } { 2 ^ { r } } \right)\) Give your answer in the form \(a - \frac { b + c n } { 2 ^ { n } }\) ，where \(a , b\) and \(c\) are integers．

6.
\includegraphics[max width=\textwidth, alt={}, center]{0214eebf-93f2-4338-9222-443000115225-4_346_1040_303_548} \section*{Figure 1} Figure 1 shows a sketch of the curve \(C _ { 1 }\) with equation $$y = \cos ( \cos x ) \sin x \quad \text { for } \quad 0 \leqslant x \leqslant \pi$$ （a）Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
（b）Hence verify that the turning point is at \(x = \frac { \pi } { 2 }\) and find the \(y\) coordinate of this point．
（c）Find the area of the region bounded by \(C _ { 1 }\) and the positive \(x\)－axis between \(x = 0\) and \(x = \pi\) Figure 2 shows a sketch of the curve \(C _ { 1 }\) and the curve \(C _ { 2 }\) with equation $$y = \sin ( \cos x ) \sin x \quad \text { for } \quad 0 \leqslant x \leqslant \pi$$ \begin{figure}[h] \end{figure} The curves \(C _ { 1 }\) and \(C _ { 2 }\) intersect at the origin and the point \(A ( a , b )\) ，where \(a < \pi\)
（d）Find \(a\) and \(b\) ，giving \(b\) in a form not involving trigonometric functions．
（e）Find the area of the shaded region between \(C _ { 1 }\) and \(C _ { 2 }\)

7. (a) Find the set of values of \(k\) for which the equation $$\frac { x ^ { 2 } + 3 x + 8 } { x ^ { 2 } + x - 2 } = k$$ has no real roots. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of the curve \(C _ { 1 }\) with equation \(y = \mathrm { f } ( x )\) where \(\mathrm { f } ( x ) = \frac { x ^ { 2 } + 3 x + 8 } { x ^ { 2 } + x - 2 }\) The curve has asymptotes \(x = a , x = b\) and \(y = c\), where \(a , b\) and \(c\) are integers.
(b) Find the value of \(a\), the value of \(b\) and the value of \(c\).
(c) Find the coordinates of the points of intersection of \(C _ { 1 }\) with the line \(y = 2\)
(d) Find all the integer pairs \(( r , s )\) that satisfy \(s = \frac { r ^ { 2 } + 3 r + 8 } { r ^ { 2 } + r - 2 }\) The curve \(C _ { 2 }\) has equation \(y = \mathrm { g } ( x )\) where \(\mathrm { g } ( x ) = \frac { 2 x ^ { 2 } - 4 x + 6 } { x ^ { 2 } - 3 x }\)
(e) Show that, for suitable integers \(m\) and \(n , \mathrm {~g} ( x )\) can be written in the form \(\mathrm { f } ( x + m ) + n\).
(f) Sketch \(C _ { 2 }\) showing any asymptotes and stating their equations.