Edexcel AEA (Advanced Extension Award) 2007 June

1．（a）Write down the binomial expansion of \(\frac { 1 } { ( 1 - y ) ^ { 2 } } , | y | < 1\) ，in ascending powers of \(y\) up to and including the term in \(y ^ { 3 }\) ．
（b）Hence，or otherwise，show that $$\frac { 1 } { 4 } \operatorname { cosec } ^ { 4 } \left( \frac { \theta } { 2 } \right) = 1 + 2 \cos \theta + 3 \cos ^ { 2 } \theta + 4 \cos ^ { 3 } \theta + \ldots + ( r + 1 ) \cos ^ { r } \theta + \ldots$$ and state the values of \(\theta\) for which this result is not valid．
（4）
Find
（c） $$\begin{aligned} & 1 + \frac { 2 } { 2 } + \frac { 3 } { 2 ^ { 2 } } + \frac { 4 } { 2 ^ { 3 } } + \ldots + \frac { ( r + 1 ) } { 2 ^ { r } } + \ldots
& 1 - \frac { 2 } { 2 } + \frac { 3 } { 2 ^ { 2 } } - \frac { 4 } { 2 ^ { 3 } } + \ldots + ( - 1 ) ^ { r } \frac { ( r + 1 ) } { 2 ^ { r } } + \ldots \end{aligned}$$ （d）

2．（a）On the same diagram，sketch \(y = x\) and \(y = \sqrt { } x\) ，for \(x \geq 0\) ，and mark clearly the coordinates of the points of intersection of the two graphs．
（b）With reference to your sketch，explain why there exists a value \(a\) of \(x ( a > 1 )\) such that $$\int _ { 0 } ^ { a } x \mathrm {~d} x = \int _ { 0 } ^ { a } \sqrt { } x \mathrm {~d} x$$ （c）Find the exact value of \(a\) ．
（d）Hence，or otherwise，find a non－constant function \(\mathrm { f } ( x )\) and a constant \(b ( b \neq 0 )\) such that $$\int _ { - b } ^ { b } \mathrm { f } ( x ) \mathrm { d } x = \int _ { - b } ^ { b } \sqrt { } [ \mathrm { f } ( x ) ] \mathrm { d } x$$

3．（a）Solve，for \(0 \leq x < 2 \pi\) ， $$\cos x + \cos 2 x = 0$$ （b）Find the exact value of \(x , x \geq 0\) ，for which $$\arccos x + \arccos 2 x = \frac { \pi } { 2 }$$ ［ \(\arccos x\) is an alternative notation for \(\cos ^ { - 1 } x\) ．］

4．The function \(\mathrm { h } ( x )\) has domain \(\mathbb { R }\) and range \(\mathrm { h } ( x ) > 0\) ，and satisfies $$\sqrt { \int \mathrm { h } ( x ) \mathrm { d } x } = \int \sqrt { \mathrm { h } ( x ) } \mathrm { d } x$$ （a）By substituting \(\mathrm { h } ( x ) = \left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) ^ { 2 }\) ，show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 ( y + c ) ,$$ where \(c\) is constant．
（b）Hence find a general expression for \(y\) in terms of \(x\) ．
（c）Given that \(\mathrm { h } ( 0 ) = 1\) ，find \(\mathrm { h } ( x )\) ．

5. \begin{figure}[h] \end{figure} Figure 1 shows part of a sequence \(S _ { 1 } , S _ { 2 } , S _ { 3 } , \ldots\), of model snowflakes. The first term \(S _ { 1 }\) consists of a single square of side \(a\). To obtain \(S _ { 2 }\), the middle third of each edge is replaced with a new square, of side \(\frac { a } { 3 }\), as shown in Figure 1 . Subsequent terms are obtained by replacing the middle third of each external edge of a new square formed in the previous snowflake, by a square \(\frac { 1 } { 3 }\) of the size, as illustrated by \(S _ { 3 }\) in Figure 1.

1. Deduce that to form \(S _ { 4 } , 36\) new squares of side \(\frac { a } { 27 }\) must be added to \(S _ { 3 }\).
2. Show that the perimeters of \(S _ { 2 }\) and \(S _ { 3 }\) are \(\frac { 20 a } { 3 }\) and \(\frac { 28 a } { 3 }\) respectively.
3. Find the perimeter of \(S _ { n }\).
4. Describe what happens to the perimeter of \(S _ { n }\) as \(n\) increases.
5. Find the areas of \(S _ { 1 } , S _ { 2 }\) and \(S _ { 3 }\).
6. Find the smallest value of the constant \(S\) such that the area of \(S _ { n } < S\), for all values of \(n\).
   \includegraphics[max width=\textwidth, alt={}, center]{f2290882-b9a4-43ec-a38f-c44d46477242-5_590_1041_283_588} Figure 2 shows a sketch of the curve \(C\) with equation \(y = \tan \frac { t } { 2 } , \quad 0 \leq t \leq \frac { \pi } { 2 }\).
   The point \(P\) on \(C\) has coordinates \(\left( x , \tan \frac { x } { 2 } \right)\).
   The vertices of rectangle \(R\) are at \(( x , 0 ) , \left( \frac { x } { 2 } , 0 \right) , \left( \frac { x } { 2 } , \tan \frac { x } { 2 } \right)\) and \(\left( x , \tan \frac { x } { 2 } \right)\) as shown in Figure 2.

1. Find an expression, in terms of \(x\), for the area \(A\) of \(R\).
2. Show that \(\frac { \mathrm { d } A } { \mathrm {~d} x } = \frac { 1 } { 4 } ( \pi - 2 x - 2 \sin x ) \sec ^ { 2 } \frac { x } { 2 }\).
3. Prove that the maximum value of \(A\) occurs when \(\frac { \pi } { 4 } < x < \frac { \pi } { 3 }\).
4. Prove that \(\tan \frac { \pi } { 8 } = \sqrt { } 2 - 1\).
5. Show that the maximum value of \(A > \frac { \pi } { 4 } ( \sqrt { } 2 - 1 )\).

7．The points \(O , P\) and \(Q\) lie on a circle \(C\) with diameter \(O Q\) ．The position vectors of \(P\) and \(Q\) ， relative to \(O\) ，are \(\mathbf { p }\) and \(\mathbf { q }\) respectively．
（a）Prove that \(\mathbf { p } . \mathbf { q } = | \mathbf { p } | ^ { 2 }\) ． \begin{figure}[h] \end{figure} The point \(R\) also lies on \(C\) and \(O P Q R\) is a kite \(K\) as shown in Figure 3．The point \(S\) has position vector，relative to \(O\) ，of \(\lambda \mathbf { q }\) ，where \(\lambda\) is a constant．Given that \(\mathbf { p } = \mathbf { i } + 2 \mathbf { j } - \mathbf { k } , \mathbf { q } = 2 \mathbf { i } + \mathbf { j } - 2 \mathbf { k }\) and that \(O Q\) is perpendicular to \(P S\) ，find
（b）the value of \(\lambda\) ，
（c）the position vector of \(R\) ，
（d）the area of \(K\) ． Another circle \(C _ { 1 }\) is drawn inside \(K\) so that the 4 sides of the kite are each tangents to \(C _ { 1 }\) ．
（e）Find the radius of \(C _ { 1 }\) giving your answer in the form \(( \sqrt { } 2 - 1 ) \sqrt { } n\) ，where \(n\) is an integer． A second kite \(K _ { 1 }\) is similar to \(K\) and is drawn inside \(C _ { 1 }\) ．
（f）Find that area of \(K _ { 1 }\) ．