Edexcel AEA (Advanced Extension Award) 2012 June

1．The function f is given by $$\mathrm { f } ( x ) = x ^ { 2 } - 2 x + 6 , \quad x \geqslant 0$$ （a）Find the range of \(f\) ． The function \(g\) is given by $$\mathrm { g } ( x ) = 3 + \sqrt { } ( x + 4 ) , \quad x \geqslant 2$$ （b）Find \(\operatorname { gf } ( x )\) ．
（c）Find the domain and range of gf．

2．（a）Show that $$\sin 3 x = 3 \sin x - 4 \sin ^ { 3 } x$$ Hence find
（b） \(\int \cos x ( 6 \sin x - 2 \sin 3 x ) ^ { \frac { 2 } { 3 } } \mathrm {~d} x\)
（c） \(\int ( 3 \sin 2 x - 2 \sin 3 x \cos x ) ^ { \frac { 1 } { 3 } } \mathrm {~d} x\)

3．The angle \(\theta , 0 < \theta < \frac { \pi } { 2 }\) ，satisfies $$\tan \theta \tan 2 \theta = \sum _ { r = 0 } ^ { \infty } 2 \cos ^ { r } 2 \theta$$ （a）Show that \(\tan \theta = 3 ^ { p }\) ，where \(p\) is a rational number to be found．
（b）Hence show that \(\frac { \pi } { 6 } < \theta < \frac { \pi } { 4 }\)

4. $$\mathbf { a } = \left( \begin{array} { r } - 3
1
4 \end{array} \right) , \quad \mathbf { b } = \left( \begin{array} { r } 5
- 2
9 \end{array} \right) , \quad \mathbf { c } = \left( \begin{array} { r } 8
- 4
3 \end{array} \right)$$ The points \(A , B\) and \(C\) with position vectors \(\mathbf { a } , \mathbf { b }\) and \(\mathbf { c }\) ，respectively，are 3 vertices of a cube．
（a）Find the volume of the cube． The points \(P , Q\) and \(R\) are vertices of a second cube with \(\overrightarrow { P Q } = \left( \begin{array} { l } 3
4
\alpha \end{array} \right) , \overrightarrow { P R } = \left( \begin{array} { l } 7
1
0 \end{array} \right)\) and \(\alpha\) a positive constant．
（b）Given that angle \(Q P R = 60 ^ { \circ }\) ，find the value of \(\alpha\) ．
（c）Find the length of a diagonal of the second cube．

5．［In this question the values of \(a , x\) ，and \(n\) are such that \(a\) and \(x\) are positive real numbers，with \(a > 1 , x \neq a , x \neq 1\) and \(n\) is an integer with \(n > 1\) ］ Sam was confused about the rules of logarithms and thought that $$\log _ { a } x ^ { n } = \left( \log _ { a } x \right) ^ { n }$$ （a）Given that \(x\) satisfies statement（1）find \(x\) in terms of \(a\) and \(n\) ． Sam also thought that $$\log _ { a } x + \log _ { a } x ^ { 2 } + \ldots + \log _ { a } x ^ { n } = \log _ { a } x + \left( \log _ { a } x \right) ^ { 2 } + \ldots + \left( \log _ { a } x \right) ^ { n }$$ （b）For \(n = 3 , x _ { 1 }\) and \(x _ { 2 } \left( x _ { 1 } > x _ { 2 } \right)\) are the two values of \(x\) that satisfy statement（2）．
（i）Find，in terms of \(a\) ，an expression for \(x _ { 1 }\) and an expression for \(x _ { 2 }\) ．
（ii）Find the exact value of \(\log _ { a } \left( \frac { x _ { 1 } } { x _ { 2 } } \right)\) ．
（c）Show that if \(\log _ { a } x\) satisfies statement（2）then $$2 \left( \log _ { a } x \right) ^ { n } - n ( n + 1 ) \log _ { a } x + \left( n ^ { 2 } + n - 2 \right) = 0$$

6. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = ( x + a ) ( x - b ) ^ { 2 }\), where \(a\) and \(b\) are positive constants. The curve cuts the \(x\)-axis at \(P\) and has a maximum point at \(S\) and a minimum point at \(Q\).

1. Write down the coordinates of \(P\) and \(Q\) in terms of \(a\) and \(b\).
2. Show that \(G\), the area of the shaded region between the curve \(P S Q\) and the \(x\)-axis, is given by \(G = \frac { ( a + b ) ^ { 4 } } { 12 }\). The rectangle \(P Q R S T\) has \(R S T\) parallel to \(Q P\) and both \(P T\) and \(Q R\) are parallel to the \(y\)-axis.
3. Show that \(\frac { G } { \text { Area of } P Q R S T } = k\), where \(k\) is a constant independent of \(a\) and \(b\) and find the value of \(k\).

7． \(\left[ \arccos x \right.\) and \(\arctan x\) are alternative notation for \(\cos ^ { - 1 } x\) and \(\tan ^ { - 1 } x\) respectively \(]\) \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve \(C _ { 1 }\) with equation \(y = \cos ( \cos x ) , 0 \leqslant x < 2 \pi\) ．
The curve has turning points at \(( 0 , \cos 1 ) , P , Q\) and \(R\) as shown in Figure 2.
（a）Find the coordinates of the points \(P , Q\) and \(R\) ． The curve \(C _ { 2 }\) has equation \(y = \sin ( \cos x ) , 0 \leqslant x < 2 \pi\) ．The curves \(C _ { 1 }\) and \(C _ { 2 }\) intersect at the points \(S\) and \(T\) ．
（b）Copy Figure 2 and on this diagram sketch \(C _ { 2 }\) stating the coordinates of the minimum point on \(C _ { 2 }\) and the points where \(C _ { 2 }\) meets or crosses the coordinate axes． The coordinates of \(S\) are \(( \alpha , d )\) where \(0 < \alpha < \pi\) ．
（c）Show that \(\alpha = \arccos \left( \frac { \pi } { 4 } \right)\) ．
（d）Find the value of \(d\) in surd form and write down the coordinates of \(T\) ． The tangent to \(C _ { 1 }\) at the point \(S\) has gradient \(\tan \beta\) ．
（e）Show that \(\beta = \arctan \sqrt { } \left( \frac { 16 - \pi ^ { 2 } } { 32 } \right)\) ．
（f）Find，in terms of \(\beta\) ，the obtuse angle between the tangent to \(C _ { 1 }\) at \(S\) and the tangent to \(C _ { 2 }\) at \(S\) ．