Edexcel AEA (Advanced Extension Award) 2011 June

1．Solve for \(0 \leqslant \theta \leqslant 180 ^ { \circ }\) $$\tan \left( \theta + 35 ^ { \circ } \right) = \cot \left( \theta - 53 ^ { \circ } \right)$$

2．Given that $$\int _ { 0 } ^ { \frac { \pi } { 2 } } \left( 1 + \tan \left[ \frac { 1 } { 2 } x \right] \right) ^ { 2 } \mathrm {~d} x = a + \ln b$$ find the value of \(a\) and the value of \(b\) ．

3．A sequence \(\left\{ u _ { n } \right\}\) is given by $$\begin{aligned} u _ { 1 } & = k & &
u _ { 2 n } & = u _ { 2 n - 1 } \times p & & n \geqslant 1
u _ { 2 n + 1 } & = u _ { 2 n } \times q & & n \geqslant 1 \end{aligned}$$ where \(k , p\) and \(q\) are positive constants with \(p q \neq 1\)
（a）Write down the first 6 terms of this sequence．
（b）Show that \(\sum _ { r = 1 } ^ { 2 n } u _ { r } = \frac { k ( 1 + p ) \left( 1 - ( p q ) ^ { n } \right) } { 1 - p q }\) In part（c） \([ x ]\) means the integer part of \(x\) ，so for example \([ 2.73 ] = 2 , [ 4 ] = 4\) and \([ 0 ] = 0\)
（c）Find \(\sum _ { r = 1 } ^ { \infty } 6 \times \left( \frac { 4 } { 3 } \right) ^ { \left[ \frac { r } { 2 } \right] } \times \left( \frac { 3 } { 5 } \right) ^ { \left[ \frac { r - 1 } { 2 } \right] }\)

4. The curve \(C\) has parametric equations $$\begin{gathered} x = \cos ^ { 2 } t
y = \cos t \sin t \end{gathered}$$ where \(0 \leqslant t < \pi\)

1. Show that \(C\) is a circle and find its centre and its radius. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{12d3f92f-8464-4ba1-93a2-c7b841e3d3de-3_668_750_726_660} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} Figure 1 shows a sketch of \(C\). The point \(P\), with coordinates \(\left( \cos ^ { 2 } \alpha , \cos \alpha \sin \alpha \right) , \quad 0 < \alpha < \frac { \pi } { 2 }\), lies on \(C\). The rectangle \(R\) has one side on the \(x\)-axis, one side on the \(y\)-axis and \(O P\) as a diagonal, where \(O\) is the origin.
2. Show that the area of \(R\) is \(\sin \alpha \cos ^ { 3 } \alpha\)
3. Find the maximum area of \(R\), as \(\alpha\) varies.

5. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve \(C\) with equation \(y = \frac { x ^ { 2 } - 2 } { x ^ { 2 } - 4 }\) and \(x \neq \pm 2\).
The curve cuts the \(y\)-axis at \(U\).

1. Write down the coordinates of the point \(U\). The point \(P\) with \(x\)-coordinate \(a ( a \neq 0 )\) lies on \(C\).
2. Show that the normal to \(C\) at \(P\) cuts the \(y\)-axis at the point $$\left( 0 , \left[ \frac { a ^ { 2 } - 2 } { a ^ { 2 } - 4 } - \frac { \left( a ^ { 2 } - 4 \right) ^ { 2 } } { 4 } \right] \right)$$ The circle \(E\), with centre on the \(y\)-axis, touches all three branches of \(C\).
3. 1. Show that $$\left[ \frac { a ^ { 2 } } { 2 \left( a ^ { 2 } - 4 \right) } - \frac { \left( a ^ { 2 } - 4 \right) ^ { 2 } } { 4 } \right] ^ { 2 } = a ^ { 2 } + \frac { \left( a ^ { 2 } - 4 \right) ^ { 4 } } { 16 }$$
   2. Hence, show that $$\left( a ^ { 2 } - 4 \right) ^ { 2 } = 1$$
   3. Find the centre and radius of \(E\).

6．The line \(L\) has equation $$\mathbf { r } = \left( \begin{array} { r } 13
- 3
- 8 \end{array} \right) + t \left( \begin{array} { r } - 5
3
4 \end{array} \right)$$ The point \(P\) has position vector \(\left( \begin{array} { r } - 7
2
7 \end{array} \right)\) ．
The point \(P ^ { \prime }\) is the reflection of \(P\) in \(L\) ．
（a）Find the position vector of \(P ^ { \prime }\) ．
（b）Show that the point \(A\) with position vector \(\left( \begin{array} { r } - 7
9
8 \end{array} \right)\) lies on \(L\) ．
（c）Show that angle \(P A P ^ { \prime } = 120 ^ { \circ }\) ． \begin{figure}[h] \end{figure} The point \(B\) lies on \(L\) and \(A P B P ^ { \prime }\) forms a kite as shown in Figure 3.
The area of the kite is \(50 \sqrt { } 3\)
（d）Find the position vector of the point \(B\) ．
（e）Show that angle \(B P A = 90 ^ { \circ }\) ． The circle \(C\) passes through the points \(A , P , P ^ { \prime }\) and \(B\) ．
（f）Find the position vector of the centre of \(C\) ．

13
- 3
- 8 \end{array} \right) + t \left( \begin{array} { r } - 5
3
4 \end{array} \right)$$ The point \(P\) has position vector \(\left( \begin{array} { r } - 7
2
7 \end{array} \right)\) ．
The point \(P ^ { \prime }\) is the reflection of \(P\) in \(L\) ．
（a）Find the position vector of \(P ^ { \prime }\) ．
（b）Show that the point \(A\) with position vector \(\left( \begin{array} { r } - 7
9
8 \end{array} \right)\) lies on \(L\) ．
（c）Show that angle \(P A P ^ { \prime } = 120 ^ { \circ }\) ． \begin{figure}[h] \end{figure} The point \(B\) lies on \(L\) and \(A P B P ^ { \prime }\) forms a kite as shown in Figure 3.
The area of the kite is \(50 \sqrt { } 3\)
（d）Find the position vector of the point \(B\) ．
（e）Show that angle \(B P A = 90 ^ { \circ }\) ． The circle \(C\) passes through the points \(A , P , P ^ { \prime }\) and \(B\) ．
（f）Find the position vector of the centre of \(C\) ．
7.
\includegraphics[max width=\textwidth, alt={}, center]{12d3f92f-8464-4ba1-93a2-c7b841e3d3de-6_675_1145_237_459} \section*{Figure 4} （a）Figure 4 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\) ，where $$\mathrm { f } ( x ) = \frac { x ^ { 2 } - 5 } { 3 - x } , \quad x \in \mathbb { R } , x \neq 3$$ The curve has a minimum at the point \(A\) ，with \(x\)－coordinate \(\alpha\) ，and a maximum at the point \(B\) ， with \(x\)－coordinate \(\beta\) ． Find the value of \(\alpha\) ，the value of \(\beta\) and the \(y\)－coordinates of the points \(A\) and \(B\) ．
(b) The functions g and h are defined as follows $$\begin{array} { l l } \mathrm { g } : x \rightarrow x + p & x \in \mathbb { R }
\mathrm {~h} : x \rightarrow | x | & x \in \mathbb { R } \end{array}$$ where \(p\) is a constant. \begin{figure}[h] \end{figure} Figure 5 shows a sketch of the curve with equation \(y = \mathrm { h } ( \mathrm { fg } ( x ) + q ) , x \in \mathbb { R } , x \neq 0\), where \(q\) is a constant. The curve is symmetric about the \(y\)-axis and has minimum points at \(C\) and \(D\).

1. Find the value of \(p\) and the value of \(q\).
2. Write down the coordinates of \(D\).
   (c) The function \(m\) is given by $$\mathrm { m } ( x ) = \frac { x ^ { 2 } - 5 } { 3 - x } , \quad x \in \mathbb { R } , x \leqslant \alpha$$ where \(\alpha\) is the \(x\)-coordinate of \(A\) as found in part (a).
3. Find \(\mathrm { m } ^ { - 1 }\)
4. Write down the domain of \(\mathrm { m } ^ { - 1 }\)
5. Find the value of \(t\) such that \(\mathrm { m } ( t ) = \mathrm { m } ^ { - 1 } ( t )\)