Edexcel AEA (Advanced Extension Award) 2008 June

1．The first and second terms of an arithmetic series are 200 and 197.5 respectively．
The sum to \(n\) terms of the series is \(S _ { n }\) ． Find the largest positive value of \(S _ { n }\) ．

2．The points \(( x , y )\) on the curve \(C\) satisfy $$( x + 1 ) ( x + 2 ) \frac { d y } { d x } = x y$$ The line with equation \(y = 2 x + 5\) is the tangent to \(C\) at a point \(P\) ．
（a）Find the coordinates of \(P\) ．
（b）Find the equation of \(C\) ，giving your answer in the form \(y = \mathrm { f } ( x )\) ．

3．（a）Prove that \(\tan 15 ^ { \circ } = 2 - \sqrt { 3 }\)
（b）Solve，for \(0 \leqslant \theta < 360 ^ { \circ }\) ， $$\sin \left( \theta + 60 ^ { \circ } \right) \sin \left( \theta - 60 ^ { \circ } \right) = ( 1 - \sqrt { } 3 ) \cos ^ { 2 } \theta$$ Figure 1 shows a sketch of the curve \(C\) with equation $$y = \cos x \ln ( \sec x ) , \quad - \frac { \pi } { 2 } < x < \frac { \pi } { 2 } .$$ The points \(A\) and \(B\) are maximum points on \(C\) ．
（a）Find the coordinates of \(B\) in terms of e ． The finite region \(R\) lies between \(C\) and the line \(A B\) ．
（b）Show that the area of \(R\) is $$\frac { 2 } { \mathrm { e } } \arccos \left( \frac { 1 } { \mathrm { e } } \right) + 2 \ln \left( \mathrm { e } + \sqrt { } \left( \mathrm { e } ^ { 2 } - 1 \right) \right) - \frac { 4 } { \mathrm { e } } \sqrt { } \left( \mathrm { e } ^ { 2 } - 1 \right) .$$ \(\left[ \arccos x \right.\) is an alternative notation for \(\left. \cos ^ { - 1 } x \right]\)

5. (i) Anna, who is confused about the rules for logarithms, states that $$\left( \log _ { 3 } p \right) ^ { 2 } = \log _ { 3 } \left( p ^ { 2 } \right)$$ and \(\quad \log _ { 3 } ( p + q ) = \log _ { 3 } p + \log _ { 3 } q\).
However, there is a value for \(p\) and a value for \(q\) for which both statements are correct.
Find the value of \(p\) and the value of \(q\).
(ii) Solve $$\frac { \log _ { 3 } \left( 3 x ^ { 3 } - 23 x ^ { 2 } + 40 x \right) } { \log _ { 3 } 9 } = 0.5 + \log _ { 3 } ( 3 x - 8 ) .$$

6. $$\mathrm { f } ( x ) = \frac { a x + b } { x + 2 } ; \quad x \in \mathbb { R } , x \neq - 2$$ where \(a\) and \(b\) are constants and \(b > 0\) ．
（a）Find \(f ^ { - 1 } ( x )\) ．
（b）Hence，or otherwise，find the value of \(a\) so that \(\operatorname { ff } ( x ) = x\) ． The curve \(C\) has equation \(y = \mathrm { f } ( x )\) and \(\mathrm { f } ( x )\) satisfies \(\mathrm { ff } ( x ) = x\) ．
（c）On separate axes sketch
（i）\(y = \mathrm { f } ( x )\) ，
（ii）\(y = \mathrm { f } ( x - 2 ) + 2\) ． On each sketch you should indicate the equations of any asymptotes and the coordinates，in terms of \(b\) ，of any intersections with the axes． The normal to \(C\) at the point \(P\) has equation \(y = 4 x - 39\) ．The normal to \(C\) at the point \(Q\) has equation \(y = 4 x + k\) ，where \(k\) is a constant．
（d）By considering the images of the normals to \(C\) on the curve with equation \(y = \mathrm { f } ( x - 2 ) + 2\) ，or otherwise，find the value of \(k\) ．

7. Relative to a fixed origin \(O\), the position vectors of the points \(A , B\) and \(C\) are $$\overrightarrow { O A } = - 3 \mathbf { i } + \mathbf { j } - 9 \mathbf { k } , \quad \overrightarrow { O B } = \mathbf { i } - \mathbf { k } , \quad \overrightarrow { O C } = 5 \mathbf { i } + 2 \mathbf { j } - 5 \mathbf { k } \text { respectively. }$$

1. Find the cosine of angle \(A B C\). The line \(L\) is the angle bisector of angle \(A B C\).
2. Show that an equation of \(L\) is \(\mathbf { r } = \mathbf { i } - \mathbf { k } + t ( \mathbf { i } + 2 \mathbf { j } - 7 \mathbf { k } )\).
3. Show that \(| \overrightarrow { A B } | = | \overrightarrow { A C } |\). The circle \(S\) lies inside triangle \(A B C\) and each side of the triangle is a tangent to \(S\).
4. Find the position vector of the centre of \(S\).
5. Find the radius of \(S\).