Edexcel AEA (Advanced Extension Award) 2005 June

1．A point \(P\) lies on the curve with equation $$x ^ { 2 } + y ^ { 2 } - 6 x + 8 y = 24$$ Find the greatest and least possible values of the length \(O P\) ，where \(O\) is the origin．

2．Solve，for \(0 < \theta < 2 \pi\) ， $$\sin 2 \theta + \cos 2 \theta + 1 = \sqrt { 6 } \cos \theta$$ giving your answers in terms of \(\pi\) ．

3．Given that $$\frac { \mathrm { d } } { \mathrm {~d} x } ( u \sqrt { } x ) = \frac { \mathrm { d } u } { \mathrm {~d} x } \times \frac { \mathrm { d } ( \sqrt { } x ) } { \mathrm { d } x } , \quad 0 < x < \frac { 1 } { 2 }$$ where \(u\) is a function of \(x\) ，and that \(u = 4\) when \(x = \frac { 3 } { 8 }\) ，find \(u\) in terms of \(x\) ．
（9）

4．A rectangle \(A B C D\) is drawn so that \(A\) and \(B\) lie on the \(x\)－axis，and \(C\) and \(D\) lie on the curve with equation \(y = \cos x , - \frac { \pi } { 2 } < x < \frac { \pi } { 2 }\) ．The point \(A\) has coordinates \(( p , 0 )\) ，where \(0 < p < \frac { \pi } { 2 }\) ．
（a）Find an expression，in terms of \(p\) ，for the area of this rectangle． The maximum area of \(A B C D\) is \(S\) and occurs when \(p = \alpha\) ．Show that
（b）\(\frac { \pi } { 4 } < \alpha < 1\) ，
（c）\(S = \frac { 2 \alpha ^ { 2 } } { \sqrt { } \left( 1 + \alpha ^ { 2 } \right) }\) ，
（d）\(\frac { \pi ^ { 2 } } { 2 \sqrt { } \left( 16 + \pi ^ { 2 } \right) } < S < \sqrt { } 2\) ．

5．The point \(A\) has position vector \(7 \mathbf { i } + 2 \mathbf { j } - 7 \mathbf { k }\) and the point \(B\) has position vector \(12 \mathbf { i } + 3 \mathbf { j } - 15 \mathbf { k }\) ．
（a）Find a vector for the line \(L _ { 1 }\) which passes through \(A\) and \(B\) ． The line \(L _ { 2 }\) has vector equation $$\mathbf { r } = - 4 \mathbf { i } + 12 \mathbf { k } + \mu ( \mathbf { i } - 3 \mathbf { k } )$$ （b）Show that \(L _ { 2 }\) passes through the origin \(O\) ．
（c）Show that \(L _ { 1 }\) and \(L _ { 2 }\) intersect at a point \(C\) and find the position vector of \(C\) ．
（d）Find the cosine of \(\angle O C A\) ．
（e）Hence，or otherwise，find the shortest distance from \(O\) to \(L _ { 1 }\) ．
（f）Show that \(| \overrightarrow { C O } | = | \overrightarrow { A B } |\) ．
（g）Find a vector equation for the line which bisects \(\angle O C A\) ．
\includegraphics[max width=\textwidth, alt={}, center]{f9d3e02c-cef2-435b-9cda-76c43fcac575-4_922_1054_279_586} Figure 1 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = x \left( 12 - x ^ { 2 } \right) .$$ The curve cuts the \(x\)-axis at the points \(P , O\) and \(R\), and \(Q\) is the maximum point.

1. Find the coordinates of the points \(P , Q\) and \(R\).
2. Sketch, on separate diagrams, the graphs of
   1. \(y = \mathrm { f } ( 2 x )\),
   2. \(y = \mathrm { f } ( | x | + 1 )\),
      indicating on each sketch the coordinates of any maximum points and the intersections with the \(x\)-axis.
      (6) \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{f9d3e02c-cef2-435b-9cda-76c43fcac575-5_1015_1464_232_337}
      \end{figure} Figure 2 shows a sketch of part of the curve \(C\), with equation \(y = \mathrm { f } ( x - v ) + w\), where \(v\) and \(w\) are constants. The \(x\)-axis is a tangent to \(C\) at the minimum point \(T\), and \(C\) intersects the \(y\)-axis at \(S\). The line joining \(S\) to the maximum point \(U\) is parallel to the \(x\)-axis.
3. Find the value of \(v\) and the value of \(w\) and hence find the roots of the equation $$f ( x - v ) + w = 0$$

1. (a) Use the substitution \(x = \sec \theta\) to show that

$$\int \sqrt { } \left( x ^ { 2 } - 1 \right) d x$$ can be written as $$\int \sec \theta \tan ^ { 2 } \theta \mathrm {~d} \theta$$ (3)
(b) Use integration by parts to show that $$\int \sec \theta \tan ^ { 2 } \theta \mathrm {~d} \theta = \frac { 1 } { 2 } [ \sec \theta \tan \theta - \ln | \sec \theta + \tan \theta | ] + \text { constant. }$$ (c) Evaluate \(\int _ { 0 } ^ { \frac { \pi } { 4 } } \sin x \sqrt { } ( \cos 2 x ) \mathrm { d } x\).