Edexcel AEA (Advanced Extension Award) 2003 June

1. \begin{figure}[h] \end{figure} The point \(A\) is a distance 1 unit from the fixed origin \(O\) ．Its position vector is \(\mathbf { a } = \frac { 1 } { \sqrt { 2 } } ( \mathbf { i } + \mathbf { j } )\) ． The point \(B\) has position vector \(\mathbf { a } + \mathbf { j }\) ，as shown in Figure 1. By considering \(\triangle O A B\) ，prove that \(\tan \frac { 3 \pi } { 8 } = 1 + \sqrt { } 2\) ．

2．Find the values of \(\tan \theta\) such that $$2 \sin ^ { 2 } \theta - \sin \theta \sec \theta = 2 \sin 2 \theta - 2 .$$

3. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of a part of the curve \(C\) with parametric equations $$x = t ^ { 3 } , y = t ^ { 2 } .$$ The tangent at the point \(P ( 8,4 )\) cuts \(C\) at the point \(Q\) ．
Find the area of the shaded region between \(P Q\) and \(C\) ．

4. $$f ( x ) = \frac { 1 - 3 x } { \left( 1 + 3 x ^ { 2 } \right) ( 1 - x ) ^ { 2 } } , x \neq 1$$ （a）Find the constants \(A , B , C\) and \(D\) such that $$\mathrm { f } ( x ) \equiv \frac { A x + B } { 1 + 3 x ^ { 2 } } + \frac { C } { 1 - x } + \frac { D } { ( 1 - x ) ^ { 2 } }$$ （b）Find a series expansion for \(\mathrm { f } ( x )\) in ascending powers of \(x\) ，up to and including the term in \(x ^ { 4 }\) ．
（c）Find an equation of the tangent to the curve with equation \(y = \mathrm { f } ( x )\) at the point where \(x = 0\) ．

5．The function \(f\) is given by $$f ( x ) = \frac { 1 } { \lambda } \left( x ^ { 2 } - 4 \right) \left( x ^ { 2 } - 25 \right)$$ where \(x\) is real and \(\lambda\) is a positive integer．
（a）Sketch the graph of \(y = \mathrm { f } ( x )\) showing clearly where the graph crosses the coordinate axes．
（b）Find，in terms of \(\lambda\) ，the range of f ．
（c）Find the sets of positive integers \(k\) and \(\lambda\) such that the equation $$k = | \mathrm { f } ( x ) |$$ has exactly \(k\) distinct real roots．

6．（a）Show that $$\sqrt { 2 + \sqrt { 3 } } - \sqrt { 2 - \sqrt { 3 } } = \sqrt { 2 }$$ （b）Hence prove that $$\log _ { \frac { 1 } { 8 } } ( \sqrt { 2 + \sqrt { 3 } } - \sqrt { 2 - \sqrt { 3 } } ) = - \frac { 1 } { 6 } .$$ （c）Find all possible pairs of integers \(a\) and \(n\) such that $$\log _ { \frac { 1 } { n } } ( \sqrt { a + \sqrt { 15 } } - \sqrt { a - \sqrt { 15 } } ) = - \frac { 1 } { 2 } .$$

7. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of part of the curve \(C\) with question $$y = \mathrm { e } ^ { - x } \sin x , \quad x \geq 0 .$$ （a）Find the coordinates of the points \(P , Q\) and \(R\) where \(C\) cuts the positive axis．
（b）Use integration by parts to show that $$\int \mathrm { e } ^ { - x } \sin x \mathrm {~d} x = - \frac { 1 } { 2 } \mathrm { e } ^ { - x } ( \sin x + \cos x ) + \text { constant }$$ The terms of the sequence \(A _ { 1 } , A _ { 2 } , \ldots , A _ { n } , \ldots\) represent areas between \(C\) and the \(x\)－axis for successive portions of \(C\) where \(y\) is positive．The area represented by \(A _ { 1 }\) and \(A _ { 2 }\) are shown in Figure 3.
（c）Find an expression for \(A _ { n }\) in terms of \(n\) and \(\pi\) ．
（6）
(d) Show that \(A _ { 1 } + A _ { 2 } + \ldots + A _ { n } + \ldots\) is a geometric series with sum to infinity $$\frac { \mathrm { e } ^ { \pi } } { 2 \left( \mathrm { e } ^ { \pi } - 1 \right) } .$$ (e) Given that $$\int _ { 0 } ^ { \infty } \mathrm { e } ^ { - x } \sin x \mathrm {~d} x = \frac { 1 } { 2 }$$ find the exact value of $$\int _ { 0 } ^ { \infty } \left| e ^ { - x } \sin x \right| d x$$ and simplify your answer. END