Questions - Edexcel AEA - A-Level Maths

Edexcel AEA 2017 Specimen Q3

12 marks Challenging +1.8

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{05b21c5d-5958-4267-b1e6-3d1ed20d5609-08_609_631_264_724} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure}

\includegraphics[max width=\textwidth, alt={}]{05b21c5d-5958-4267-b1e6-3d1ed20d5609-08_172_168_781_1548}

Figure 1 shows a regular pentagon $O A B C D$. The vectors $\mathbf { p }$ and $\mathbf { q }$ are defined by $\mathbf { p } = \overrightarrow { O A }$ and $\mathbf { q } = \overrightarrow { O D }$ respectively. Let $k$ be the number such that $\overrightarrow { D B } = k \overrightarrow { O A }$.

Write down $\overrightarrow { A C }$ in terms of $\mathbf { p } , \mathbf { q }$ and $k$ as appropriate.
Show that $\overrightarrow { C D } = - \mathbf { p } - \frac { 1 } { k } \mathbf { q }$
Hence find the value of $k$ By considering triangle $D B C$, or otherwise,
find the exact value of $\sin 54 ^ { \circ }$

Edexcel AEA 2017 Specimen Q4

13 marks Challenging +1.8

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{05b21c5d-5958-4267-b1e6-3d1ed20d5609-12_428_897_251_593} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A particle of weight $W$ lies on a rough plane.The plane is inclined to the horizontal at an angle $\alpha$ where $\tan \alpha = \frac { 3 } { 4 }$ .The coefficient of friction between the particle and the plane is $\frac { 1 } { 2 }$ The particle is held in equilibrium by a force of magnitude 1.2 W .The force makes an angle $\theta$ with the plane,where $0 < \theta < \pi$ ,and acts in a vertical plane containing a line of greatest slope of the plane,as shown in Figure 2.

Find the value of $\theta$ for which there is no frictional force acting on the particle. The minimum value of $\theta$ for the particle to remain in equilibrium is $\beta$
Show that $$\beta = \arccos \left( \frac { \sqrt { 5 } } { 3 } \right) - \arctan \left( \frac { 1 } { 2 } \right)$$
Find the range of values of $\theta$ for which the particle remains in equilibrium with the frictional force acting up the plane.

Edexcel AEA 2017 Specimen Q5

13 marks Challenging +1.8

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{05b21c5d-5958-4267-b1e6-3d1ed20d5609-16_745_862_258_667} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Show that the area of the finite region between the curves $y = \tan ^ { 2 } x$ and $y = 4 \cos 2 x - 1$ in the interval $- \frac { \pi } { 2 } < x < \frac { \pi } { 2 }$, shown shaded in Figure 3, is given by $$2 \sqrt { 2 \sqrt { 3 } } - 2 \sqrt { 2 \sqrt { 3 } - 3 }$$

\includegraphics[max width=\textwidth, alt={}]{05b21c5d-5958-4267-b1e6-3d1ed20d5609-16_2255_51_315_1987}

Edexcel AEA 2017 Specimen Q6

18 marks Challenging +1.2

6.(i)Eden,who is confused about the laws of logarithms,states that $$\left( \log _ { 5 } p \right) ^ { 2 } = \log _ { 5 } \left( p ^ { 2 } \right)$$ and $\log _ { 5 } ( q - p ) = \log _ { 5 } q - \log _ { 5 } p$ However,there is a value of $p$ and a value of $q$ for which both statements are correct.
Determine these values.
(ii)(a)Let $r \in \mathbb { R } ^ { + } , r \neq 1$ .Prove that $$\log _ { r } A = \log _ { r ^ { 2 } } B \Rightarrow A ^ { 2 } = B$$ (b)Solve $$\log _ { 4 } \left( 3 x ^ { 3 } + 26 x ^ { 2 } + 40 x \right) = 2 + \log _ { 2 } ( x + 2 )$$

\includegraphics[max width=\textwidth, alt={}]{05b21c5d-5958-4267-b1e6-3d1ed20d5609-20_2261_53_317_1977}

Edexcel AEA 2017 Specimen Q7

25 marks Challenging +1.8

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{05b21c5d-5958-4267-b1e6-3d1ed20d5609-25_670_682_301_694} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} A circular tower of radius 1 metre stands in a large horizontal field of grass.A goat is attached to one end of a rope and the other end of the rope is attached to a fixed point $O$ at the base of the tower.The goat cannot enter the tower. Taking the point $O$ as the origin( 0,0 ),the centre of the base of the tower is at the point $T ( 0,1 )$ ,where the unit of length is the metre. The rope has length $\pi$ metres and you may ignore the size of the goat.
The curve $C$ shown in Figure 4 represents the edge of the region that the goat can reach.

Write down the equation of $C$ for $y < 0$ When the goat is at the point $G ( x , y )$ ,with $x > 0$ and $y > 0$ ,as shown in Figure 4 ,the rope lies along $O A G$ where $O A$ is an arc of the circle with angle $O T A = \theta$ radians and $A G$ is a tangent to the circle at $A$ .
With the aid of a suitable diagram show that $$\begin{aligned} & x = \sin \theta + ( \pi - \theta ) \cos \theta \\ & y = 1 - \cos \theta + ( \pi - \theta ) \sin \theta \end{aligned}$$
By considering $\int y \frac { \mathrm {~d} x } { \mathrm {~d} \theta } \mathrm {~d} \theta$, show that the area, in the first quadrant, between $C$, the positive $x$-axis and the positive $y$-axis can be expressed in the form $$\int _ { 0 } ^ { \pi } u \sin u \mathrm {~d} u + \int _ { 0 } ^ { \pi } u ^ { 2 } \sin ^ { 2 } u \mathrm {~d} u + \int _ { 0 } ^ { \pi } u \sin u \cos u \mathrm {~d} u$$
Show that $\int _ { 0 } ^ { \pi } u ^ { 2 } \sin ^ { 2 } u \mathrm {~d} u = \frac { \pi ^ { 3 } } { 6 } + \int _ { 0 } ^ { \pi } u \sin u \cos u \mathrm {~d} u$
Hence find the area of grass that can be reached by the goat.

Edexcel AEA 2012 June Q4

11 marks Challenging +1.8

4. $$\mathbf { a } = \left( \begin{array} { r } - 3 \\ 1 \\ 4 \end{array} \right) , \quad \mathbf { b } = \left( \begin{array} { r } 5 \\ - 2 \\ 9 \end{array} \right) , \quad \mathbf { c } = \left( \begin{array} { r } 8 \\ - 4 \\ 3 \end{array} \right)$$ The points $A , B$ and $C$ with position vectors $\mathbf { a } , \mathbf { b }$ and $\mathbf { c }$ ,respectively,are 3 vertices of a cube.

Find the volume of the cube. The points $P , Q$ and $R$ are vertices of a second cube with $\overrightarrow { P Q } = \left( \begin{array} { l } 3 \\ 4 \\ \alpha \end{array} \right) , \overrightarrow { P R } = \left( \begin{array} { l } 7 \\ 1 \\ 0 \end{array} \right)$ and $\alpha$ a positive constant.
Given that angle $Q P R = 60 ^ { \circ }$ ,find the value of $\alpha$ .
Find the length of a diagonal of the second cube.

Edexcel AEA 2012 June Q5

14 marks Challenging +1.8

5.[In this question the values of $a , x$ ,and $n$ are such that $a$ and $x$ are positive real numbers,with $a > 1 , x \neq a , x \neq 1$ and $n$ is an integer with $n > 1$ ] Sam was confused about the rules of logarithms and thought that $$\log _ { a } x ^ { n } = \left( \log _ { a } x \right) ^ { n }$$

Given that $x$ satisfies statement(1)find $x$ in terms of $a$ and $n$ . Sam also thought that $$\log _ { a } x + \log _ { a } x ^ { 2 } + \ldots + \log _ { a } x ^ { n } = \log _ { a } x + \left( \log _ { a } x \right) ^ { 2 } + \ldots + \left( \log _ { a } x \right) ^ { n }$$
For $n = 3 , x _ { 1 }$ and $x _ { 2 } \left( x _ { 1 } > x _ { 2 } \right)$ are the two values of $x$ that satisfy statement(2).
1. Find,in terms of $a$ ,an expression for $x _ { 1 }$ and an expression for $x _ { 2 }$ .
2. Find the exact value of $\log _ { a } \left( \frac { x _ { 1 } } { x _ { 2 } } \right)$ .
Show that if $\log _ { a } x$ satisfies statement(2)then $$2 \left( \log _ { a } x \right) ^ { n } - n ( n + 1 ) \log _ { a } x + \left( n ^ { 2 } + n - 2 \right) = 0$$

Edexcel AEA 2012 June Q7

24 marks Hard +2.3

7. $\left[ \arccos x \right.$ and $\arctan x$ are alternative notation for $\cos ^ { - 1 } x$ and $\tan ^ { - 1 } x$ respectively $]$ \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{fc5d0d07-b750-4646-bdcb-419a290200c9-5_387_935_322_566} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a sketch of the curve $C _ { 1 }$ with equation $y = \cos ( \cos x ) , 0 \leqslant x < 2 \pi$ .
The curve has turning points at $( 0 , \cos 1 ) , P , Q$ and $R$ as shown in Figure 2.

Find the coordinates of the points $P , Q$ and $R$ . The curve $C _ { 2 }$ has equation $y = \sin ( \cos x ) , 0 \leqslant x < 2 \pi$ .The curves $C _ { 1 }$ and $C _ { 2 }$ intersect at the points $S$ and $T$ .
Copy Figure 2 and on this diagram sketch $C _ { 2 }$ stating the coordinates of the minimum point on $C _ { 2 }$ and the points where $C _ { 2 }$ meets or crosses the coordinate axes. The coordinates of $S$ are $( \alpha , d )$ where $0 < \alpha < \pi$ .
Show that $\alpha = \arccos \left( \frac { \pi } { 4 } \right)$ .
Find the value of $d$ in surd form and write down the coordinates of $T$ . The tangent to $C _ { 1 }$ at the point $S$ has gradient $\tan \beta$ .
Show that $\beta = \arctan \sqrt { } \left( \frac { 16 - \pi ^ { 2 } } { 32 } \right)$ .
Find,in terms of $\beta$ ,the obtuse angle between the tangent to $C _ { 1 }$ at $S$ and the tangent to $C _ { 2 }$ at $S$ .

Edexcel AEA 2005 June Q6

19 marks Challenging +1.8

Find the coordinates of the points $P , Q$ and $R$.
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.
Find the value of $v$ and the value of $w$ and hence find the roots of the equation $$f ( x - v ) + w = 0$$

Edexcel AEA 2006 June Q6

15 marks Challenging +1.2

Show that the point $P ( 1,0 )$ lies on $C$ .
Find the coordinates of the point $Q$ .
Find the area of the shaded region between $C$ and the line $P Q$ .

Edexcel AEA 2007 June Q6

17 marks Hard +2.3

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

Edexcel AEA 2024 June Q1

7 marks Challenging +1.2

1.In the binomial expansion of $$( 1 - 8 x ) ^ { p } \quad | x | < \frac { 1 } { 8 }$$ where $p$ is a positive constant,
-the sum of the coefficient of $x$ and the coefficient of $x ^ { 2 }$ is equal to the coefficient of $x ^ { 3 }$ -the coefficient of $x ^ { 2 }$ is positive
Determine the value of $p$ . \includegraphics[max width=\textwidth, alt={}, center]{a8e9db6b-dfad-4278-82d8-a8fa5ba61008-02_2264_56_315_1977}

Edexcel AEA 2024 June Q2

6 marks Challenging +1.8

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{a8e9db6b-dfad-4278-82d8-a8fa5ba61008-04_904_826_255_623} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows the curve defined by the equation $$y ^ { 2 } + 3 y - 6 \sin y = 4 - x ^ { 2 }$$ The point $P ( x , y )$ lies on the curve.
The distance from the origin,$O$ ,to $P$ is $D$ .

Write down an equation for $D ^ { 2 }$ in terms of $y$ only.
Hence determine the minimum value of $D$ giving your answer in simplest form. \includegraphics[max width=\textwidth, alt={}, center]{a8e9db6b-dfad-4278-82d8-a8fa5ba61008-04_2266_53_312_1977}

Edexcel AEA 2024 June Q3

14 marks Challenging +1.8

3.(i)Determine the value of $k$ such that $$\arctan \frac { 1 } { 2 } - \arctan \frac { 1 } { 3 } = \arctan k$$ (ii)(a)Prove that $$\cos 3 A \equiv 4 \cos ^ { 3 } A - 3 \cos A$$ Given that $a = \cos 20 ^ { \circ }$ (b)write down,in terms of $a$ ,an expression for $\cos 40 ^ { \circ }$ (c)determine,in terms of $a$ ,a simplified expression for $\cos 80 ^ { \circ }$ (d)Use part(a)to show that $$4 a ^ { 3 } - 3 a = \frac { 1 } { 2 }$$ (e)Hence,or otherwise,show that $$\cos 20 ^ { \circ } \cos 40 ^ { \circ } \cos 80 ^ { \circ } = \frac { 1 } { 8 }$$

Edexcel AEA 2024 June Q4

16 marks Challenging +1.8

4.(a)Use the substitution $x = \sqrt { 3 } \tan u$ to show that $$\int \frac { 1 } { 3 + x ^ { 2 } } \mathrm {~d} x = p \arctan ( p x ) + c$$ where $p$ is a real constant to be determined and $c$ is an arbitrary constant.
(b)Use the substitution $x = \frac { 3 u + 3 } { u - 3 }$ to determine the exact value of $I$ where $$I = \int _ { - 3 } ^ { 1 } \frac { \ln ( 3 - x ) } { 3 + x ^ { 2 } } \mathrm {~d} x$$ giving your answer in simplest form. \includegraphics[max width=\textwidth, alt={}, center]{a8e9db6b-dfad-4278-82d8-a8fa5ba61008-10_2264_47_314_1984}

Edexcel AEA 2024 June Q5

15 marks Challenging +1.8

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{a8e9db6b-dfad-4278-82d8-a8fa5ba61008-14_300_1043_251_513} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a sketch of a hexagon $O A B C D E$ where
-the interior angle at $O$ and at $C$ are each $60 ^ { \circ }$ -the interior angle at each of the other vertices is $150 ^ { \circ }$ -$O A = O E = B C = C D$ -$A B = E D = 3 \times O A$ Given that $\overrightarrow { O A } = \mathbf { a }$ and $\overrightarrow { O E } = \mathbf { e }$ determine as simplified expressions in terms of $\mathbf { a }$ and $\mathbf { e }$

$\overrightarrow { A B }$
$\overrightarrow { O D }$ The point $R$ divides $A B$ internally in the ratio $1 : 2$
Determine $\overrightarrow { R C }$ as a simplified expression in terms of $\mathbf { a }$ and $\mathbf { e }$ The line through the points $R$ and $C$ meets the line through the points $O$ and $D$ at the point $X$ .
Determine $\overrightarrow { O X }$ in the form $\lambda \mathbf { a } + \mu \mathbf { e }$ ,where $\lambda$ and $\mu$ are real values in simplest form.

Edexcel AEA 2024 June Q6

18 marks Hard +2.3

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{a8e9db6b-dfad-4278-82d8-a8fa5ba61008-20_234_1357_244_354} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Figure 3 shows a block $A$ with mass $4 m$ and a block $B$ with mass $5 m$.
Block $A$ is at rest on a rough plane inclined at an angle $\alpha$ to the horizontal.
Block $B$ is at rest on a rough plane inclined at an angle $\beta$ to the horizontal.
The blocks are connected by a light inextensible string which passes over a small smooth pulley at the top of each plane. A small smooth ring $C$, of mass $8 m$, is threaded on the string between the pulleys so that $A , B$ and $C$ all lie in the same vertical plane. The part of the string between $A$ and its pulley lies along a line of greatest slope of the plane of angle $\alpha$. The part of the string between $B$ and its pulley lies along a line of greatest slope of the plane of angle $\beta$. The angle between the vertical and the string between each pulley and the ring $C$ is $\gamma$.
The two blocks, $A$ and $B$, are modelled as particles.
Given that

$\tan \alpha = \frac { 5 } { 12 }$ and $\tan \beta = \frac { 7 } { 24 }$ and $\tan \gamma = \frac { 3 } { 4 }$
the coefficient of friction, $\mu$, is the same between each block and its plane
one of the blocks is on the point of sliding up its plane
the tension in the string is $T$
1. determine, in terms of $m$ and $g$, an expression for $T$,
2. draw a diagram showing the forces on block $A$, clearly labelling each of the forces acting on the block,
3. determine the value of $\mu$, giving a justification for your answer. \includegraphics[max width=\textwidth, alt={}, center]{a8e9db6b-dfad-4278-82d8-a8fa5ba61008-20_2266_50_312_1978}

Edexcel AEA 2024 June Q7

24 marks Hard +2.3

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{a8e9db6b-dfad-4278-82d8-a8fa5ba61008-26_725_1773_242_146} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} Figure 4 shows a circle with radius $r _ { 1 }$ and a circle with radius $r _ { 2 }$ The circles touch externally at a single point above the $x$-axis.
Both circles also have the $x$-axis as a tangent.

Show that the horizontal distance between the centres of the circles, $d$, is given by $$d ^ { 2 } = 4 r _ { 1 } r _ { 2 }$$ The finite region $R$, shown shaded in Figure 4, is bounded by the $x$-axis and minor arcs of the two circles. Given that $r _ { 1 } \geqslant r _ { 2 }$
show that the area of $R$ is given by $$\left( r _ { 1 } + r _ { 2 } \right) \sqrt { r _ { 1 } r _ { 2 } } - \frac { 1 } { 2 } \left( r _ { 1 } ^ { 2 } - r _ { 2 } ^ { 2 } \right) \theta - \frac { 1 } { 2 } \pi r _ { 2 } ^ { 2 }$$ where $\cos \theta = \frac { r _ { 1 } - r _ { 2 } } { r _ { 1 } + r _ { 2 } }$ Question 7 continues on the next page.

\includegraphics[max width=\textwidth, alt={}]{a8e9db6b-dfad-4278-82d8-a8fa5ba61008-27_2269_53_306_36}
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a8e9db6b-dfad-4278-82d8-a8fa5ba61008-27_759_1378_269_347} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} A sequence of circles, $C _ { 1 } , C _ { 2 } , C _ { 3 } , \ldots$ with radii $r _ { 1 } , r _ { 2 } , r _ { 3 } , \ldots$ respectively, is constructed such that
- each circle is tangential to and above the $x$-axis
- the first circle, $C _ { 1 }$, has centre $( 0,1 )$
- each successive circle touches the preceding one externally at a single point
- the horizontal distances between the centres of successive circles form a geometric sequence with first term 2 and common ratio $\frac { 1 } { \sqrt { 3 } }$
The first few circles in the sequence are shown in Figure 5.
1. Determine the value of $r _ { 3 }$
2. Show that, for $n \geqslant 1 , r _ { n + 2 } = k r _ { n }$ where $k$ is a constant to be determined.
3. Hence show that, for $n \geqslant 1 , r _ { 2 n } = r _ { 2 n - 1 }$ The region bounded between $C _ { n } , C _ { n + 1 }$ and the $x$-axis is $R _ { n }$ The total area, $A$, bounded above the $x$-axis and under all the circles is the sum of the areas of all these regions.
Determine the value of $A$, giving the answer in simplest form. \section*{Paper reference} \section*{Advanced Extension Award Mathematics} Insert for questions 5, 6 and 7
Do not write on this insert.
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a8e9db6b-dfad-4278-82d8-a8fa5ba61008-34_298_1040_212_516} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of a hexagon $O A B C D E$ where
Given that $\overrightarrow { O A } = \mathbf { a }$ and $\overrightarrow { O E } = \mathbf { e }$ determine as simplified expressions in terms of $\mathbf { a }$ and $\mathbf { e }$
1. $\overrightarrow { A B }$
2. $\overrightarrow { O D }$ The point $R$ divides $A B$ internally in the ratio $1 : 2$
3. Determine $\overrightarrow { R C }$ as a simplified expression in terms of $\mathbf { a }$ and $\mathbf { e }$ The line through the points $R$ and $C$ meets the line through the points $O$ and $D$ at the point $X$.
4. Determine $\overrightarrow { O X }$ in the form $\lambda \mathbf { a } + \mu \mathbf { e }$, where $\lambda$ and $\mu$ are real values in simplest form.
  6. \begin{figure}[h]
  \includegraphics[alt={},max width=\textwidth]{a8e9db6b-dfad-4278-82d8-a8fa5ba61008-35_236_1363_205_351} \captionsetup{labelformat=empty} \caption{Figure 3}
  \end{figure} Figure 3 shows a block $A$ with mass $4 m$ and a block $B$ with mass $5 m$.
  Block $A$ is at rest on a rough plane inclined at an angle $\alpha$ to the horizontal.
  Block $B$ is at rest on a rough plane inclined at an angle $\beta$ to the horizontal.
  The blocks are connected by a light inextensible string which passes over a small smooth pulley at the top of each plane. A small smooth ring $C$, of mass $8 m$, is threaded on the string between the pulleys so that $A , B$ and $C$ all lie in the same vertical plane. The part of the string between $A$ and its pulley lies along a line of greatest slope of the plane of angle $\alpha$. The part of the string between $B$ and its pulley lies along a line of greatest slope of the plane of angle $\beta$. The angle between the vertical and the string between each pulley and the ring $C$ is $\gamma$.
  The two blocks, $A$ and $B$, are modelled as particles.
  Given that
  7. \begin{figure}[h]
  \includegraphics[alt={},max width=\textwidth]{a8e9db6b-dfad-4278-82d8-a8fa5ba61008-36_721_1771_205_146} \captionsetup{labelformat=empty} \caption{Figure 4}
  \end{figure} Figure 4 shows a circle with radius $r _ { 1 }$ and a circle with radius $r _ { 2 }$ The circles touch externally at a single point above the $x$-axis.
  Both circles also have the $x$-axis as a tangent.
5. Show that the horizontal distance between the centres of the circles, $d$, is given by $$d ^ { 2 } = 4 r _ { 1 } r _ { 2 }$$ The finite region $R$, shown shaded in Figure 4, is bounded by the $x$-axis and minor arcs of the two circles. Given that $r _ { 1 } \geqslant r _ { 2 }$
6. show that the area of $R$ is given by $$\left( r _ { 1 } + r _ { 2 } \right) \sqrt { r _ { 1 } r _ { 2 } } - \frac { 1 } { 2 } \left( r _ { 1 } ^ { 2 } - r _ { 2 } ^ { 2 } \right) \theta - \frac { 1 } { 2 } \pi r _ { 2 } ^ { 2 }$$ where $\cos \theta = \frac { r _ { 1 } - r _ { 2 } } { r _ { 1 } + r _ { 2 } }$ Question 7 continues on the next page. \begin{figure}[h]
  \includegraphics[alt={},max width=\textwidth]{a8e9db6b-dfad-4278-82d8-a8fa5ba61008-37_761_1376_210_349} \captionsetup{labelformat=empty} \caption{Figure 5}
  \end{figure} A sequence of circles, $C _ { 1 } , C _ { 2 } , C _ { 3 } , \ldots$ with radii $r _ { 1 } , r _ { 2 } , r _ { 3 } , \ldots$ respectively, is constructed such that
  The first few circles in the sequence are shown in Figure 5.
7. 1. Determine the value of $r _ { 3 }$
  2. Show that, for $n \geqslant 1 , r _ { n + 2 } = k r _ { n }$ where $k$ is a constant to be determined.
  3. Hence show that, for $n \geqslant 1 , r _ { 2 n } = r _ { 2 n - 1 }$ The region bounded between $C _ { n } , C _ { n + 1 }$ and the $x$-axis is $R _ { n }$ The total area, $A$, bounded above the $x$-axis and under all the circles is the sum of the areas of all these regions.
8. Determine the value of $A$, giving the answer in simplest form.

Edexcel AEA 2018 June Q1

5 marks Challenging +1.2

1.(a)Show that $\sqrt { \frac { 1 + x } { 1 - x } }$ can be written in the form $\frac { 1 + x } { \sqrt { 1 - x ^ { 2 } } }$ for $| x | < 1$ (b)Hence,or otherwise,find the expansion,in ascending powers of $x$ up to and including the term in $x ^ { 5 }$ ,of $\sqrt { \frac { 1 + x } { 1 - x } }$

Edexcel AEA 2018 June Q2

7 marks Challenging +1.8

2.Solve,for $0 \leqslant x \leqslant 360 ^ { \circ }$ $$\sin 47 ^ { \circ } \cos ^ { 3 } x + \cos 47 ^ { \circ } \sin x \cos ^ { 2 } x = \frac { 1 } { 2 } \cos ^ { 2 } x$$

Edexcel AEA 2018 June Q3

10 marks Challenging +1.2

3.The lines $L _ { 1 }$ and $L _ { 2 }$ have the equations $$L _ { 1 } : \mathbf { r } = \left( \begin{array} { l } 1 \\ 0 \\ 9 \end{array} \right) + s \left( \begin{array} { l } 2 \\ p \\ 6 \end{array} \right) \quad \text { and } \quad L _ { 2 } : \mathbf { r } = \left( \begin{array} { r } - 15 \\ 12 \\ - 9 \end{array} \right) + t \left( \begin{array} { r } 4 \\ - 5 \\ 2 \end{array} \right)$$ where $p$ is a constant.
The acute angle between $L _ { 1 }$ and $L _ { 2 }$ is $\theta$ where $\cos \theta = \frac { \sqrt { 5 } } { 3 }$

Find the value of $p$ . The line $L _ { 3 }$ has equation $\mathbf { r } = \left( \begin{array} { r } - 15 \\ 12 \\ - 9 \end{array} \right) + u \left( \begin{array} { r } 8 \\ - 6 \\ - 5 \end{array} \right)$ and the lines $L _ { 3 }$ and $L _ { 2 }$ intersect at the point $A$ .
The point $B$ on $L _ { 2 }$ has position vector $\left( \begin{array} { r } 5 \\ - 13 \\ 1 \end{array} \right)$ and point $C$ lies on $L _ { 3 }$ such that $A B D C$ is a rhombus.
Find the two possible position vectors of $D$ .

Edexcel AEA 2018 June Q4

13 marks Challenging +1.2

4.A curve $C$ has equation $y = \mathrm { f } ( x )$ where $x \in \mathbb { R }$ and f is a one-one function.

Describe a single transformation that transforms $C$ to the curve with equation $y = - \mathrm { f } ( - x )$ . The curve $C$ is reflected in the line with equation $y = - x$ to give the curve $V$ . The equation of $V$ is $y = \mathrm { g } ( x )$ .
Explain why $\mathrm { g } ^ { - 1 } ( x ) = - \mathrm { f } ( - x )$ . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2a7c2530-a93c-4a26-bc37-c20c0f40c8f2-3_793_979_819_633} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve $C$ with equation $y = \mathrm { f } ( x )$ where $$\mathrm { f } ( x ) = \frac { 3 ( x - 1 ) } { x - 2 } \quad x \in \mathbb { R } , x \neq 2$$ The curve has asymptotes with equations $x = p$ and $y = q$ and $C$ crosses the $x$-axis at the point $A$ and the $y$-axis at the point $B$ .
Write down the value of $p$ and the value of $q$ .
Write down the coordinates of the point $A$ and the coordinates of the point $B$ . Given the definition of $\mathrm { g } ( x )$ in part(b),
find the function g .
Solve $\mathrm { g } ^ { - 1 } \mathrm { f } ( x ) = x$

Edexcel AEA 2018 June Q5

14 marks Challenging +1.8

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{2a7c2530-a93c-4a26-bc37-c20c0f40c8f2-4_484_581_287_843} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows part of the curve $T$ with equation $y = \cos 2 x$ and the circle $C _ { 1 }$ that touches $T$ at $\left( \frac { \pi } { 4 } , 0 \right)$ and $\left( \frac { 3 \pi } { 4 } , 0 \right)$ .

Find the radius of $C _ { 1 }$ \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2a7c2530-a93c-4a26-bc37-c20c0f40c8f2-4_486_586_1199_841} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of part of $T$ and part of a circle $C _ { 2 }$ that touches $T$ at the point $P$ with coordinates $\left( \frac { \pi } { 2 } , - 1 \right)$ .For values of $x$ close to $\frac { \pi } { 2 }$ the curve $T$ lies inside $C _ { 2 }$ as shown in Figure 3.
Without doing any calculation,explain why the value of $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ for $C _ { 2 }$ at $P$ is less than the value of $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ for $T$ at $P$ . The radius of $C _ { 2 }$ is $r$ .
Use the result from part(b)to find a value of $k$ such that $r > k$ . Given that $C _ { 2 }$ cuts $T$ at the point $( 0,1 )$ ,
find the value of $r$ .

Edexcel AEA 2018 June Q6

17 marks Challenging +1.8

6. (a) Use the substitution $u = \sqrt { t }$ to show that $$\int _ { 1 } ^ { x } \frac { \ln t } { \sqrt { t } } \mathrm {~d} t = 4 - 4 \sqrt { x } + 2 \sqrt { x } \ln x \quad x \geqslant 1$$ (b) The function g is such that $$\int _ { 1 } ^ { x } \mathrm {~g} ( t ) \mathrm { d } t = x - \sqrt { x } \ln x - 1 \quad x \geqslant 1$$

Use differentiation to find the function g .
Evaluate $\int _ { 4 } ^ { 16 } \mathrm {~g} ( t ) \mathrm { d } t$ and simplify your answer.
(c) Find the value of $x$ (where $x > 1$ ) that gives the maximum value of $$\int _ { x } ^ { x + 1 } \frac { \ln t } { 2 ^ { t } } \mathrm {~d} t$$

Edexcel AEA 2018 June Q7

27 marks Hard +2.3

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{2a7c2530-a93c-4a26-bc37-c20c0f40c8f2-6_559_923_292_670} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} Figure 4 shows a shape $S ( \theta )$ made up of five line segments $A B , B C , C D , D E$ and $E A$ . The lengths of the sides are $A B = B C = 5 \mathrm {~cm} , C D = E A = 3 \mathrm {~cm}$ and $D E = 7 \mathrm {~cm}$ . Angle $B A E =$ angle $B C D = \theta$ radians. The length of each line segment always remains the same but the value of $\theta$ can be varied so that different symmetrical shapes can be formed,with the added restriction that none of the line segments cross.

Sketch $S ( \pi )$ ,labelling the vertices clearly. The shape $S ( \phi )$ is a trapezium.
Sketch $S ( \phi )$ and calculate the value of $\phi$ . The smallest possible value for $\theta$ is $\alpha$ ,where $\alpha > 0$ ,and the largest possible value for $\theta$ is $\beta$ , where $\beta > \pi$ .
Show that $\alpha = \arccos \left( \frac { 29 } { 40 } \right) \cdot \left[ \arccos ( x ) \right.$ is an alternative notation for $\left. \cos ^ { - 1 } ( x ) \right]$
Find the value of $\beta$ . The area,in $\mathrm { cm } ^ { 2 }$ ,of shape $S ( \theta )$ is $R ( \theta )$ .
Show that for $\alpha \leqslant \theta < \pi$ $$R ( \theta ) = 15 \sin \theta + \frac { 7 } { 4 } \sqrt { 87 - 120 \cos \theta }$$ Given that this formula for $R ( \theta )$ holds for $\alpha \leqslant \theta \leqslant \beta$
show that $R ( \theta )$ has only one stationary point and that this occurs when $\theta = \frac { 2 \pi } { 3 }$
find the maximum and minimum values of $R ( \theta )$. FOR STYLE, CLARITY AND PRESENTATION: 7 MARKS TOTAL FOR PAPER: 100 MARKS
END

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