Edexcel AEA (Advanced Extension Award) 2022 June

1. $$\mathrm { f } ( x ) = x ^ { \left( x ^ { 2 } \right) } \quad x > 0$$ Use logarithms to find the \(x\) coordinate of the stationary point of the curve with equation \(y = \mathrm { f } ( x )\) ．

2. \begin{figure}[h] \end{figure} Figure 1 shows a regular hexagon \(O P Q R S T\).
The vectors \(\mathbf { p }\) and \(\mathbf { q }\) are defined by \(\mathbf { p } = \overrightarrow { O P }\) and \(\mathbf { q } = \overrightarrow { O Q }\)
Forces, in Newtons, \(\mathbf { F } _ { P } = ( \overrightarrow { O P } ) , \mathbf { F } _ { Q } = 2 \times ( \overrightarrow { O Q } ) , \mathbf { F } _ { R } = 3 \times ( \overrightarrow { O R } ) , \mathbf { F } _ { S } = 4 \times ( \overrightarrow { O S } )\) and \(\mathbf { F } _ { T } = 5 \times ( \overrightarrow { O T } )\) are applied to a particle.

1. Find, in terms of \(\mathbf { p }\) and \(\mathbf { q }\), the resultant force on the particle. The magnitude of the acceleration of the particle due to these forces is \(13 \mathrm {~ms} ^ { - 2 }\)
   Given that the mass of the particle is 3 kg ,
2. find \(| \mathbf { p } |\)
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3．（a）Use the formulae for \(\sin ( A \pm B )\) and \(\cos ( A \pm B )\) to prove that \(\tan \left( 90 ^ { \circ } - \theta \right) \equiv \cot \theta\)
（b）Solve for \(0 < \theta < 360 ^ { \circ }\) $$2 - \sec ^ { 2 } \left( \theta + 11 ^ { \circ } \right) = 2 \tan \left( \theta + 11 ^ { \circ } \right) \tan \left( \theta - 34 ^ { \circ } \right)$$ Give each answer as an integer in degrees．

4．Given that \(\mathrm { f } ( x ) = \mathrm { e } ^ { x ^ { 3 } - 2 x }\)
（a）find \(\mathrm { f } ^ { \prime } ( x )\) The curves \(C _ { 1 }\) and \(C _ { 2 }\) are defined by the functions g and h respectively，where $$\begin{array} { l l } \mathrm { g } ( x ) = 8 x ^ { 3 } \mathrm { e } ^ { x ^ { 3 } - 2 x } & x \in \mathbb { R } , x > 0
\mathrm {~h} ( x ) = \left( 3 x ^ { 5 } + 4 x \right) \mathrm { e } ^ { x ^ { 3 } - 2 x } & x \in \mathbb { R } , x > 0 \end{array}$$ （b）Find the \(x\) coordinates of the points of intersection of \(C _ { 1 }\) and \(C _ { 2 }\) Given that \(C _ { 1 }\) lies above \(C _ { 2 }\) between these points of intersection，
（c）find the area of the region bounded by the curves between these two points．
Give your answer in the form \(A + B \mathrm { e } ^ { C }\) where \(A , B\) ，and \(C\) are exact real numbers to be found．

1. An aeroplane leaves a runway and moves with a constant speed of \(V \mathrm {~km} / \mathrm { h }\) due north along a straight path inclined at an angle \(\arctan \left( \frac { 3 } { 4 } \right)\) to the horizontal.

A light aircraft is moving due north in a straight horizontal line in the same vertical plane as the aeroplane, at a height of 3 km above the runway. The light aircraft is travelling with a constant speed of \(2 V \mathrm {~km} / \mathrm { h }\).
At the moment the aeroplane leaves the runway, the light aircraft is at a horizontal distance \(d \mathrm {~km}\) behind the aeroplane. Both aircraft continue to move with the same trajectories due north.

1. Show that the distance, \(D \mathrm {~km}\), between the two aircraft \(t\) hours after the aeroplane leaves the runway satisfies $$D ^ { 2 } = \left( \frac { 6 } { 5 } V t - d \right) ^ { 2 } + \left( \frac { 3 } { 5 } V t - 3 \right) ^ { 2 }$$ Given that the distance between the two aircraft is never less than 2 km ,
2. find the range of possible values for \(d\).

6. \begin{figure}[h] \end{figure} Figure 2 shows the first few iterations in the construction of a curve, \(L\).
Starting with a straight line \(L _ { 0 }\) of length 4 , the middle half of this line is replaced by three sides of a trapezium above \(L _ { 0 }\) as shown, such that the length of each of these sides is \(\frac { 1 } { 4 }\) of the length of \(L _ { 0 }\) After the first iteration each line segment has length one.
In subsequent iterations, each line segment parallel to \(L _ { 0 }\) similarly has its middle half replaced by three sides of a trapezium above that line segment, with each side \(\frac { 1 } { 4 }\) the length of that line segment. Line segments in \(L _ { n }\) are either parallel to \(L _ { 0 }\) or are sloped.

1. Show that the length of \(L _ { 2 }\) is \(\frac { 23 } { 4 }\)
2. Write down the number of
   1. line segments in \(L _ { n }\) that are parallel to \(L _ { 0 }\)
   2. sloped line segments in \(L _ { 2 }\) that are not in \(L _ { 1 }\)
   3. new sloped line segments that are created by the ( \(n + 1\) )th iteration.
3. Hence find the length of \(L _ { n }\) as \(n \rightarrow \infty\) The area enclosed between \(L _ { 0 }\) and \(L _ { n }\) is \(A _ { n }\)
4. Find the value of \(A _ { 1 }\)
5. Find, in terms of \(n\), an expression for \(A _ { n + 1 } - A _ { n }\)
6. Hence find the value of \(A _ { n }\) as \(n \rightarrow \infty\) The same construction as described above is applied externally to the three sides of an equilateral triangle of side length \(a\).
   Given that the limit of the area of the resulting shape is \(26 \sqrt { 3 }\)
7. find the value of \(a\).

7．A circle \(C\) has centre \(X ( a , b )\) and radius \(r\) ．
A line \(l\) has equation \(y = m x + c\)
（a）Show that the \(x\) coordinates of the points where \(C\) and \(l\) intersect satisfy $$\left( m ^ { 2 } + 1 \right) x ^ { 2 } - 2 ( a - m ( c - b ) ) x + a ^ { 2 } + ( c - b ) ^ { 2 } - r ^ { 2 } = 0$$ Given that \(l\) is a tangent to \(C\) ，
（b）show that $$c = b - m a \pm r \sqrt { m ^ { 2 } + 1 }$$ The circle \(C _ { 1 }\) has equation $$x ^ { 2 } + y ^ { 2 } - 16 = 0$$ and the circle \(C _ { 2 }\) has equation $$x ^ { 2 } + y ^ { 2 } - 20 x - 10 y + 89 = 0$$ （c）Find the equations of any lines that are normal to both \(C _ { 1 }\) and \(C _ { 2 }\) ，justifying your answer．
（d）Find the equations of all lines that are a tangent to both \(C _ { 1 }\) and \(C _ { 2 }\)
［You may find the following Pythagorean triple helpful in this part： \(7 ^ { 2 } + 24 ^ { 2 } = 25 ^ { 2 }\) ］