Edexcel AEA (Advanced Extension Award) 2009 June

1. (a) On the same diagram, sketch

$$y = ( x + 1 ) ( 2 - x ) \quad \text { and } \quad y = x ^ { 2 } - 2 | x |$$ Mark clearly the coordinates of the points where these curves cross the coordinate axes.
(b) Find the \(x\)-coordinates of the points of intersection of these two curves.

2. The curve \(C\) has equation \(y = x ^ { \sin x } , \quad x > 0\).

1. Find the equation of the tangent to \(C\) at the point where \(x = \frac { \pi } { 2 }\).
2. Prove that this tangent touches \(C\) at infinitely many points.

3. (a) Solve, for \(0 \leqslant \theta < 2 \pi\), $$\sin \left( \frac { \pi } { 3 } - \theta \right) = \frac { 1 } { \sqrt { } 3 } \cos \theta$$ (b) Find the value of \(x\) for which $$\begin{aligned} & \arcsin ( 1 - 2 x ) = \frac { \pi } { 3 } - \arcsin x , \quad 0 < x < 0.5
& { \left[ \arcsin x \text { is an alternative notation for } \sin ^ { - 1 } x \right] } \end{aligned}$$

1. (a) The function \(\mathrm { f } ( x )\) has \(\mathrm { f } ^ { \prime } ( x ) = \frac { \mathrm { u } ( x ) } { \mathrm { v } ( x ) }\). Given that \(\mathrm { f } ^ { \prime } ( k ) = 0\), show that \(\mathrm { f } ^ { \prime \prime } ( k ) = \frac { \mathrm { u } ^ { \prime } ( k ) } { \mathrm { v } ( k ) }\).

\begin{figure}[h] \end{figure} (b) The curve \(C\) with equation $$y = \frac { 2 x ^ { 2 } + 3 } { x ^ { 2 } - 1 }$$ crosses the \(y\)-axis at the point \(A\). Figure 1 shows a sketch of \(C\) together with its 3 asymptotes.

1. Find the coordinates of the point \(A\).
2. Find the equations of the asymptotes of \(C\). The point \(P ( a , b ) , a > 0\) and \(b > 0\), lies on \(C\). The point \(Q\) also lies on \(C\) with \(P Q\) parallel to the \(x\)-axis and \(A P = A Q\).
3. Show that the area of triangle \(P A Q\) is given by \(\frac { 5 a ^ { 3 } } { a ^ { 2 } - 1 }\).
4. Find, as \(a\) varies, the minimum area of triangle \(P A Q\), giving your answer in its simplest form.

5．（a）The sides of the triangle \(A B C\) have lengths \(B C = a , A C = b\) and \(A B = c\) ，where \(a < b < c\) ．The sizes of the angles \(A , B\) and \(C\) form an arithmetic sequence．
（i）Show that the area of triangle \(A B C\) is \(a c \frac { \sqrt { 3 } } { 4 }\) ． Given that \(a = 2\) and \(\sin A = \frac { \sqrt { } 15 } { 5 }\) ，find
（ii）the value of \(b\) ，
（iii）the value of \(c\) ．
（b）The internal angles of an \(n\)－sided polygon form an arithmetic sequence with first term \(143 ^ { \circ }\) and common difference \(2 ^ { \circ }\) ． Given that all of the internal angles are less than \(180 ^ { \circ }\) ，find the value of \(n\) ．

6. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve \(C\) with parametric equations $$x = 2 \sin t , \quad y = \ln ( \sec t ) , \quad 0 \leqslant t < \frac { \pi } { 2 }$$ The tangent to \(C\) at the point \(P\) ，where \(t = \frac { \pi } { 3 }\) ，cuts the \(x\)－axis at \(A\) ．
（a）Show that the \(x\)－coordinate of \(A\) is \(\frac { \sqrt { } 3 } { 3 } ( 3 - \ln 2 )\) ． The shaded region \(R\) lies between \(C\) ，the positive \(x\)－axis and the tangent \(A P\) as shown in Figure 2 ．
（b）Show that the area of \(R\) is \(\sqrt { 3 } ( 1 + \ln 2 ) - 2 \ln ( 2 + \sqrt { 3 } ) - \frac { \sqrt { 3 } } { 6 } ( \ln 2 ) ^ { 2 }\) ．

7．Relative to a fixed origin \(O\) the points \(A , B\) and \(C\) have position vectors $$\mathbf { a } = - \mathbf { i } + \frac { 4 } { 3 } \mathbf { j } + 7 \mathbf { k } , \quad \mathbf { b } = 4 \mathbf { i } + \frac { 4 } { 3 } \mathbf { j } + 2 \mathbf { k } \text { and } \mathbf { c } = 6 \mathbf { i } + \frac { 16 } { 3 } \mathbf { j } + 2 \mathbf { k } \text { respectively. }$$ （a）Find the cosine of angle \(A B C\) ． The quadrilateral \(A B C D\) is a kite \(K\) ．
（b）Find the area of \(K\) ． A circle is drawn inside \(K\) so that it touches each of the 4 sides of \(K\) ．
（c）Find the radius of the circle，giving your answer in the form \(p \sqrt { } ( q ) - q \sqrt { } ( p )\) ，where \(p\) and \(q\) are positive integers．
（d）Find the position vector of the point \(D\) ．
（Total 18 marks）