Questions - Edexcel AEA - A-Level Maths

Edexcel AEA 2013 June Q6

6．（a）Starting from $[ \mathrm { f } ( x ) - \lambda \mathrm { g } ( x ) ] ^ { 2 } \geqslant 0$ show that $\lambda$ satisfies the quadratic inequality $$\left( \int _ { a } ^ { b } [ \operatorname { g } ( x ) ] ^ { 2 } \mathrm {~d} x \right) \lambda ^ { 2 } - 2 \left( \int _ { a } ^ { b } \mathrm { f } ( x ) \mathrm { g } ( x ) \mathrm { d } x \right) \lambda + \int _ { a } ^ { b } [ \mathrm { f } ( x ) ] ^ { 2 } \mathrm {~d} x \geqslant 0$$ where $a$ and $b$ are constants and $\lambda$ can take any real value．
（2）
（b）Hence prove that $$\left[ \int _ { a } ^ { b } \mathrm { f } ( x ) \mathrm { g } ( x ) \mathrm { d } x \right] ^ { 2 } \leqslant \left[ \int _ { a } ^ { b } [ \mathrm { f } ( x ) ] ^ { 2 } \mathrm {~d} x \right] \times \left[ \int _ { a } ^ { b } [ \mathrm {~g} ( x ) ] ^ { 2 } \mathrm {~d} x \right]$$ （c）By letting $\mathrm { f } ( x ) = 1$ and $\mathrm { g } ( x ) = \left( 1 + x ^ { 3 } \right) ^ { \frac { 1 } { 2 } }$ show that $$\int _ { - 1 } ^ { 2 } \left( 1 + x ^ { 3 } \right) ^ { \frac { 1 } { 2 } } \mathrm {~d} x \leqslant \frac { 9 } { 2 }$$ （d）Show that $\int _ { - 1 } ^ { 2 } x ^ { 2 } \left( 1 + x ^ { 3 } \right) ^ { \frac { 1 } { 4 } } \mathrm {~d} x = \frac { 12 \sqrt { } 3 } { 5 }$
（e）Hence show that $$\frac { 144 } { 55 } \leqslant \int _ { - 1 } ^ { 2 } \left( 1 + x ^ { 3 } \right) ^ { \frac { 1 } { 2 } } \mathrm {~d} x$$

Edexcel AEA 2013 June Q7

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8bd0bc33-e69e-4e51-aae7-288810c5db07-6_643_1374_173_351} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of the curve $C _ { 1 }$ with equation $y = \mathrm { f } ( x )$ where $$\mathrm { f } ( x ) = \frac { x } { 3 } + \frac { 12 } { x } \quad x \neq 0$$ The lines $x = 0$ and $y = \frac { x } { 3 }$ are asymptotes to $C _ { 1 }$. The point $A$ on $C _ { 1 }$ is a minimum and the point $B$ on $C _ { 1 }$ is a maximum.

Find the coordinates of $A$ and $B$. There is a normal to $C _ { 1 }$, which does not intersect $C _ { 1 }$ a second time, that has equation $x = k$, where $k > 0$.
Write down the value of $k$. The point $P ( \alpha , \beta ) , \alpha > 0$ and $\alpha \neq k$, lies on $C _ { 1 }$. The normal to $C _ { 1 }$ at $P$ does not intersect $C _ { 1 }$ a second time.
Find the value of $\alpha$, leaving your answer in simplified surd form.
Find the equation of this normal. The curve $C _ { 2 }$ has equation $y = | \mathrm { f } ( x ) |$
Sketch $C _ { 2 }$ stating the coordinates of any turning points and the equations of any asymptotes. The line with equation $y = m x + 1$ does not touch or intersect $C _ { 2 }$.
Find the set of possible values for $m$.

Edexcel AEA 2014 June Q1

1．The function f is given by $$\mathrm { f } ( x ) = \ln ( 2 x - 5 ) , \quad x > 2.5$$ （a）Find $\mathrm { f } ^ { - 1 } ( x )$ ． The function g has domain $x > 2$ and $$\operatorname { fg } ( x ) = \ln \left( \frac { x + 10 } { x - 2 } \right) , \quad x > 2$$ （b）Find $\mathrm { g } ( x )$ and simplify your answer．

Edexcel AEA 2014 June Q2

2．Given that $$3 \sin ^ { 2 } x + 2 \sin x = 6 \cos x + 9 \sin x \cos x$$ and that $- 90 ^ { \circ } < x < 90 ^ { \circ }$ ， find the possible values of $\tan x$ ．
（Total 6 marks）

Edexcel AEA 2014 June Q3

3．（a）On separate diagrams sketch the curves with the following equations．On each sketch you should mark the coordinates of the points where the curve crosses the coordinate axes．
（i）$y = x ^ { 2 } - 2 x - 3$
（ii）$y = x ^ { 2 } - 2 | x | - 3$
（iii）$y = x ^ { 2 } - x - | x | - 3$
（b）Solve the equation $$x ^ { 2 } - x - | x | - 3 = x + | x |$$

Edexcel AEA 2014 June Q4

Given that

$$( 1 + x ) ^ { n } = 1 + \sum _ { r = 1 } ^ { \infty } \frac { n ( n - 1 ) \ldots ( n - r + 1 ) } { 1 \times 2 \times \ldots \times r } x ^ { r } \quad ( | x | < 1 , x \in \mathbb { R } , n \in \mathbb { R } )$$

show that $$( 1 - x ) ^ { - \frac { 1 } { 2 } } = \sum _ { r = 0 } ^ { \infty } \binom { 2 r } { r } \left( \frac { x } { 4 } \right) ^ { r }$$
show that $\left( 9 - 4 x ^ { 2 } \right) ^ { - \frac { 1 } { 2 } }$ can be written in the form $\sum _ { r = 0 } ^ { \infty } \binom { 2 r } { r } \frac { x ^ { 2 r } } { 3 ^ { q } }$ and give $q$ in terms of $r$.
Find $\sum _ { r = 1 } ^ { \infty } \binom { 2 r } { r } \times \frac { 2 r } { 9 } \times \left( \frac { x } { 3 } \right) ^ { 2 r - 1 }$
Hence find the exact value of $$\sum _ { r = 1 } ^ { \infty } \binom { 2 r } { r } \times \frac { 2 r \sqrt { } 5 } { 9 } \times \frac { 1 } { 5 ^ { r } }$$ giving your answer as a rational number.

Edexcel AEA 2014 June Q5

5. The square-based pyramid $P$ has vertices $A , B , C , D$ and $E$. The position vectors of $A , B , C$ and $D$ are $\mathbf { a , b , c }$ and $\mathbf { d }$ respectively where $$\mathbf { a } = \left( \begin{array} { r } - 2
3
- 1 \end{array} \right) , \quad \mathbf { b } = \left( \begin{array} { r } 5
8
- 6 \end{array} \right) , \quad \mathbf { c } = \left( \begin{array} { l } 2
5
3 \end{array} \right) , \quad \mathbf { d } = \left( \begin{array} { l }

Edexcel AEA 2014 June Q6

6
1
1 \end{array} \right)$$

Find the vectors $\overrightarrow { A B } , \overrightarrow { A C } , \overrightarrow { A D } , \overrightarrow { B C } , \overrightarrow { B D }$ and $\overrightarrow { C D }$.
Find
1. the length of a side of the square base of $P$,
2. the cosine of the angle between one of the slanting edges of $P$ and its base,
3. the height of $P$,
4. the position vector of $E$. A second pyramid, identical to $P$, is attached by its square base to the base of $P$ to form an octahedron.
Find the position vector of the other vertex of this octahedron.
6. (i) A curve with equation $y = \mathrm { f } ( x )$ has $\mathrm { f } ( x ) \geqslant 0$ for $x \geqslant a$ and $$A = \int _ { a } ^ { b } \mathrm { f } ( x ) \mathrm { d } x \quad \text { and } \quad V = \pi \int _ { a } ^ { b } [ \mathrm { f } ( x ) ] ^ { 2 } \mathrm {~d} x$$ where $a$ and $b$ are constants with $b > a$. Use integration by substitution to show that for the positive constants $r$ and $h$ $$\pi \int _ { a + h } ^ { b + h } [ r + \mathrm { f } ( x - h ) ] ^ { 2 } \mathrm {~d} x = \pi r ^ { 2 } ( b - a ) + 2 \pi r A + V$$ (ii) \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{65b3fb2e-c603-48a0-b4ad-0f78603c203c-5_492_1038_799_504} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows part of the curve $C$ with equation $y = 4 + \frac { 2 } { \sqrt { 3 } \cos x + \sin x }$
This curve has asymptotes $x = m$ and $x = n$ and crosses the $y$-axis at $( 0 , p )$.

Edexcel AEA 2014 June Q8

8
- 6 \end{array} \right) , \quad \mathbf { c } = \left( \begin{array} { l } 2
5
3 \end{array} \right) , \quad \mathbf { d } = \left( \begin{array} { l } 6
1
1 \end{array} \right)$$

Find the vectors $\overrightarrow { A B } , \overrightarrow { A C } , \overrightarrow { A D } , \overrightarrow { B C } , \overrightarrow { B D }$ and $\overrightarrow { C D }$.
Find
1. the length of a side of the square base of $P$,
2. the cosine of the angle between one of the slanting edges of $P$ and its base,
3. the height of $P$,
4. the position vector of $E$. A second pyramid, identical to $P$, is attached by its square base to the base of $P$ to form an octahedron.
Find the position vector of the other vertex of this octahedron.
6. (i) A curve with equation $y = \mathrm { f } ( x )$ has $\mathrm { f } ( x ) \geqslant 0$ for $x \geqslant a$ and $$A = \int _ { a } ^ { b } \mathrm { f } ( x ) \mathrm { d } x \quad \text { and } \quad V = \pi \int _ { a } ^ { b } [ \mathrm { f } ( x ) ] ^ { 2 } \mathrm {~d} x$$ where $a$ and $b$ are constants with $b > a$. Use integration by substitution to show that for the positive constants $r$ and $h$ $$\pi \int _ { a + h } ^ { b + h } [ r + \mathrm { f } ( x - h ) ] ^ { 2 } \mathrm {~d} x = \pi r ^ { 2 } ( b - a ) + 2 \pi r A + V$$ (ii) \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{65b3fb2e-c603-48a0-b4ad-0f78603c203c-5_492_1038_799_504} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows part of the curve $C$ with equation $y = 4 + \frac { 2 } { \sqrt { 3 } \cos x + \sin x }$
This curve has asymptotes $x = m$ and $x = n$ and crosses the $y$-axis at $( 0 , p )$.
Find the value of $p$, the value of $m$ and the value of $n$.
Show that the equation of $C$ can be written in the form $y = r + \mathrm { f } ( x - h )$ and specify the function f and the constants $r$ and $h$. The region bounded by $C$, the $x$-axis and the lines $x = \frac { \pi } { 6 }$ and $x = \frac { \pi } { 3 }$ is rotated through $2 \pi$ radians about the $x$-axis.
Find the volume of the solid formed.
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{65b3fb2e-c603-48a0-b4ad-0f78603c203c-6_631_974_201_548} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A circular tower stands in a large horizontal field of grass．A goat is attached to one end of a string and the other end of the string is attached to the fixed point $O$ at the base of the tower．Taking the point $O$ as the origin（ 0,0 ），the centre of the base of the tower is at the point $T ( 0,1 )$ ．The radius of the base of the tower is 1 ．The string has length $\pi$ and you may ignore the size of the goat．The curve $C$ represents the edge of the region that the goat can reach as shown in Figure 2.
（a）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 2 ，the string 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 $A$ ．
（b）With the aid of a suitable diagram show that $$\begin{aligned} & x = \sin \theta + ( \pi - \theta ) \cos \theta
& y = 1 - \cos \theta + ( \pi - \theta ) \sin \theta \end{aligned}$$ （c）By considering $\int y \frac { \mathrm {~d} x } { \mathrm {~d} \theta } \mathrm {~d} \theta$ ，show that the area between $C$ ，the positive $x$－axis and the positive $y$－axis can be expressed in the form $$\int _ { 0 } ^ { \pi } u \sin u \mathrm {~d} u + \int _ { 0 } ^ { \pi } u ^ { 2 } \sin ^ { 2 } u \mathrm {~d} u + \int _ { 0 } ^ { \pi } u \sin u \cos u \mathrm {~d} u$$ （d）Show that $\int _ { 0 } ^ { \pi } u ^ { 2 } \sin ^ { 2 } u \mathrm {~d} u = \frac { \pi ^ { 3 } } { 6 } + \int _ { 0 } ^ { \pi } u \sin u \cos u \mathrm {~d} u$
（e）Hence find the area of grass that can be reached by the goat．

Edexcel AEA 2015 June Q1

1．（a）Sketch the graph of the curve with equation $$y = | \ln ( 2 x + 5 ) | \quad x > - \frac { 5 } { 2 }$$ On your sketch you should clearly state the equations of any asymptotes and mark the coordinates of points where the curve meets the coordinate axes．
（b）Solve the equation $| \ln ( 2 x + 5 ) | = \ln 9$

Edexcel AEA 2015 June Q2

2．（a）Show that $( x + 1 )$ is a factor of $2 x ^ { 3 } + 3 x ^ { 2 } - 1$
（b）Solve the equation
（b）Solve the equation $$\sqrt { x ^ { 2 } + 2 x + 5 } = x + \sqrt { 2 x + 3 }$$

Edexcel AEA 2015 June Q3

3．Solve for $0 < x < 360 ^ { \circ }$ $$\cot 2 x - \tan 78 ^ { \circ } = \frac { ( \sec x ) \left( \sec 78 ^ { \circ } \right) } { 2 }$$ where $x$ is not an integer multiple of $90 ^ { \circ }$

Edexcel AEA 2015 June Q4

4．（a）Find the binomial series expansion for $( 4 + y ) ^ { \frac { 1 } { 2 } }$ in ascending powers of $y$ up to and including the term in $y ^ { 3 }$ ．Simplify the coefficient of each term．
（3）
（b）Hence show that the binomial series expansion for $\left( 4 + 5 x + x ^ { 2 } \right) ^ { \frac { 1 } { 2 } }$ in ascending powers of $x$ up to and including the term in $x ^ { 3 }$ is $$2 + \frac { 5 x } { 4 } - \frac { 9 x ^ { 2 } } { 64 } + \frac { 45 x ^ { 3 } } { 512 }$$ （c）Show that the binomial series expansion of $\left( 4 + 5 x + x ^ { 2 } \right) ^ { \frac { 1 } { 2 } }$ will converge for $- \frac { 1 } { 2 } \leqslant x \leqslant \frac { 1 } { 2 }$
（d）Use the result in part（b）to estimate $$\int _ { - \frac { 1 } { 2 } } ^ { \frac { 1 } { 2 } } \sqrt { 4 + 5 x + x ^ { 2 } } d x$$ Give your answer as a single fraction．

Edexcel AEA 2015 June Q5

5.
\includegraphics[max width=\textwidth, alt={}, center]{3e18cb7c-1a67-4152-8628-76847e368882-4_639_1177_264_445} Figure 1 shows a sketch of the curve with equation $y = \mathrm { f } ( x )$ where $$f ( x ) = \frac { x ^ { 2 } + 16 } { 3 x } \quad x \neq 0$$ The curve has a maximum at the point $A$ with coordinates $( a , b )$.

Find the value of $a$ and the value of $b$. The function g is defined as $$\mathrm { g } : x \mapsto \frac { x ^ { 2 } + 16 } { 3 x } \quad a \leqslant x < 0$$ where $a$ is the value found in part (a).
Write down the range of g .
On the same axes sketch $y = \mathrm { g } ( x )$ and $y = \mathrm { g } ^ { - 1 } ( x )$.
Find an expression for $\mathrm { g } ^ { - 1 } ( x )$ and state the domain of $\mathrm { g } ^ { - 1 }$
Solve the equation $\mathrm { g } ( x ) = \mathrm { g } ^ { - 1 } ( x )$.

Edexcel AEA 2015 June Q6

6．The lines $L _ { 1 }$ and $L _ { 2 }$ have vector equations $$\begin{aligned} & L _ { 1 } : \mathbf { r } = \left( \begin{array} { r } 1
10
- 3 \end{array} \right) + \lambda \left( \begin{array} { r } 2
- 5
4 \end{array} \right)
& L _ { 2 } : \mathbf { r } = \left( \begin{array} { r } - 1
2
3 \end{array} \right) + \mu \left( \begin{array} { l } 1
2
2 \end{array} \right) \end{aligned}$$ （a）Show that $L _ { 1 }$ and $L _ { 2 }$ are perpendicular．
（b）Show that $L _ { 1 }$ and $L _ { 2 }$ are skew lines． The point $A$ with position vector $- \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k }$ lies on $L _ { 2 }$ and the point $X$ lies on $L _ { 1 }$ such that $\overrightarrow { A X }$ is perpendicular to $L _ { 1 }$
（c）Find the position vector of $X$ ．
（d）Find $| \overrightarrow { A X } |$ The point $B$（distinct from $A$ ）also lies on $L _ { 2 }$ and $| \overrightarrow { B X } | = | \overrightarrow { A X } |$
（e）Find the position vector of $B$ ．
（f）Find the cosine of angle $A X B$ ．

Edexcel AEA 2015 June Q10

10
- 3 \end{array} \right) + \lambda \left( \begin{array} { r } 2
- 5
4 \end{array} \right)
& L _ { 2 } : \mathbf { r } = \left( \begin{array} { r } - 1
2
3 \end{array} \right) + \mu \left( \begin{array} { l } 1
2
2 \end{array} \right) \end{aligned}$$ （a）Show that $L _ { 1 }$ and $L _ { 2 }$ are perpendicular．
（b）Show that $L _ { 1 }$ and $L _ { 2 }$ are skew lines． The point $A$ with position vector $- \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k }$ lies on $L _ { 2 }$ and the point $X$ lies on $L _ { 1 }$ such that $\overrightarrow { A X }$ is perpendicular to $L _ { 1 }$
（c）Find the position vector of $X$ ．
（d）Find $| \overrightarrow { A X } |$ The point $B$（distinct from $A$ ）also lies on $L _ { 2 }$ and $| \overrightarrow { B X } | = | \overrightarrow { A X } |$
（e）Find the position vector of $B$ ．
（f）Find the cosine of angle $A X B$ ． 7．（a）Use the substitution $x = \sec \theta$ to show that $$\int _ { \sqrt { 2 } } ^ { 2 } \frac { 1 } { \left( x ^ { 2 } - 1 \right) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x = \frac { \sqrt { 6 } - 2 } { \sqrt { 3 } }$$ （b）Use integration by parts to show that $$\int \operatorname { cosec } \theta \cot ^ { 2 } \theta \mathrm {~d} \theta = \frac { 1 } { 2 } [ \ln | \operatorname { cosec } \theta + \cot \theta | - \operatorname { cosec } \theta \cot \theta ] + c$$ （6） \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{3e18cb7c-1a67-4152-8628-76847e368882-6_592_1196_772_420} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a sketch of part of the curve with equation $y = \frac { 1 } { \left( x ^ { 2 } - 1 \right) ^ { \frac { 3 } { 2 } } }$ for $x > 1$
The region $R$ ，shown shaded in Figure 2，is bounded by the curve，the $x$－axis and the lines $x = \sqrt { 2 }$ and $x = 2$
The region $R$ is rotated through $2 \pi$ radians about the $x$－axis．
（c）Show that the volume of the solid formed is $$\pi \left[ \frac { 3 } { 8 } \ln \left( \frac { 1 + \sqrt { 2 } } { \sqrt { 3 } } \right) + \frac { 7 } { 36 } - \frac { \sqrt { 2 } } { 8 } \right]$$

Edexcel AEA 2016 June Q1

1．The function f is given by $$\mathrm { f } ( x ) = x ^ { 2 } - 4 x + 9 \quad x \in \mathbb { R } , x \geqslant 3$$ （a）Find the range of f ． The function g is given by $$\operatorname { g } ( x ) = \frac { 10 } { x + 1 } \quad x \in \mathbb { R } , x \geqslant 4$$ （b）Find an expression for $\operatorname { gf } ( x )$ ．
（c）Find the domain and range of gf．

Edexcel AEA 2016 June Q2

2．Find the value of $$\arccos \left( \frac { 1 } { \sqrt { 2 } } \right) + \arcsin \left( \frac { 1 } { 3 } \right) + 2 \arctan \left( \frac { 1 } { \sqrt { 2 } } \right)$$ Give your answer as a multiple of $\pi$ ． $$\text { (arccos } x \text { is an alternative notion for } \cos ^ { - 1 } x \text { etc.) }$$

Edexcel AEA 2016 June Q3

3．The points $A , B , C , D$ and $E$ are five of the vertices of a rectangular cuboid and $A E$ is a diagonal of the cuboid．With respect to a fixed origin $O$ ，the position vectors of $A , B , C$ and $D$ are $\mathbf { a , b , c }$ and d respectively，where $$\mathbf { a } = \left( \begin{array} { c } 1
2
- 1 \end{array} \right) , \quad \mathbf { b } = \left( \begin{array} { c } 0
- 3
- 8 \end{array} \right) , \quad \mathbf { c } = \left( \begin{array} { c }

Edexcel AEA 2016 June Q4

4
- 1
- 10 \end{array} \right) \text { and } \mathbf { d } = \left( \begin{array} { c } - 4
2
- 11 \end{array} \right)$$ （a）Find the position vector of $E$ ． The volume of a tetrahedron is given by the formula $$\text { volume } = \frac { 1 } { 3 } ( \text { area of base } ) \times ( \text { height } )$$ （b）Find the volume of the tetrahedron $A B C D$ ． 4．（a）Given that $x > 0 , y > 0 , x \neq 1$ and $n > 0$ ，show that $$\log _ { x } y = \log _ { x ^ { n } } y ^ { n }$$ （b）Solve the following，leaving your answers in the form $2 ^ { p }$ ，where $p$ is a rational number．
（i） $\log _ { 2 } u + \log _ { 4 } u ^ { 2 } + \log _ { 8 } u ^ { 3 } + \log _ { 16 } u ^ { 4 } = 5$
（ii） $\log _ { 2 } v + \log _ { 4 } v + \log _ { 8 } v + \log _ { 16 } v = 5$
（iii） $\log _ { 4 } w ^ { 2 } + \frac { 3 \log _ { 8 } 64 } { \log _ { 2 } w } = 5$

Edexcel AEA 2016 June Q5

5．（a）Show that $$\sum _ { r = 0 } ^ { n } x ^ { - r } = \frac { x } { x - 1 } - \frac { x ^ { - n } } { x - 1 } \quad \text { where } x \neq 0 \text { and } x \neq 1$$ （b）Hence find an expression in terms of $x$ and $n$ for $\sum _ { r = 0 } ^ { n } r x ^ { - ( r + 1 ) }$ for $x \neq 0$ and $x \neq 1$
Simplify your answer．
（c）Find $\sum _ { r = 0 } ^ { n } \left( \frac { 3 + 5 r } { 2 ^ { r } } \right)$ Give your answer in the form $a - \frac { b + c n } { 2 ^ { n } }$ ，where $a , b$ and $c$ are integers．

Edexcel AEA 2016 June Q6

6.
\includegraphics[max width=\textwidth, alt={}, center]{0214eebf-93f2-4338-9222-443000115225-4_346_1040_303_548} \section*{Figure 1} Figure 1 shows a sketch of the curve $C _ { 1 }$ with equation $$y = \cos ( \cos x ) \sin x \quad \text { for } \quad 0 \leqslant x \leqslant \pi$$ （a）Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$
（b）Hence verify that the turning point is at $x = \frac { \pi } { 2 }$ and find the $y$ coordinate of this point．
（c）Find the area of the region bounded by $C _ { 1 }$ and the positive $x$－axis between $x = 0$ and $x = \pi$ Figure 2 shows a sketch of the curve $C _ { 1 }$ and the curve $C _ { 2 }$ with equation $$y = \sin ( \cos x ) \sin x \quad \text { for } \quad 0 \leqslant x \leqslant \pi$$ \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{0214eebf-93f2-4338-9222-443000115225-4_519_1065_1631_484} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} The curves $C _ { 1 }$ and $C _ { 2 }$ intersect at the origin and the point $A ( a , b )$ ，where $a < \pi$
（d）Find $a$ and $b$ ，giving $b$ in a form not involving trigonometric functions．
（e）Find the area of the shaded region between $C _ { 1 }$ and $C _ { 2 }$

Edexcel AEA 2016 June Q7

7. (a) Find the set of values of $k$ for which the equation $$\frac { x ^ { 2 } + 3 x + 8 } { x ^ { 2 } + x - 2 } = k$$ has no real roots. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{0214eebf-93f2-4338-9222-443000115225-5_718_869_511_603} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Figure 3 shows a sketch of the curve $C _ { 1 }$ with equation $y = \mathrm { f } ( x )$ where $\mathrm { f } ( x ) = \frac { x ^ { 2 } + 3 x + 8 } { x ^ { 2 } + x - 2 }$ The curve has asymptotes $x = a , x = b$ and $y = c$, where $a , b$ and $c$ are integers.
(b) Find the value of $a$, the value of $b$ and the value of $c$.
(c) Find the coordinates of the points of intersection of $C _ { 1 }$ with the line $y = 2$
(d) Find all the integer pairs $( r , s )$ that satisfy $s = \frac { r ^ { 2 } + 3 r + 8 } { r ^ { 2 } + r - 2 }$ The curve $C _ { 2 }$ has equation $y = \mathrm { g } ( x )$ where $\mathrm { g } ( x ) = \frac { 2 x ^ { 2 } - 4 x + 6 } { x ^ { 2 } - 3 x }$
(e) Show that, for suitable integers $m$ and $n , \mathrm {~g} ( x )$ can be written in the form $\mathrm { f } ( x + m ) + n$.
(f) Sketch $C _ { 2 }$ showing any asymptotes and stating their equations.

Edexcel AEA 2017 June Q1

1．The function f is given by $$\mathrm { f } ( x ) = \sqrt { x + 2 } \quad \text { for } \quad x \in \mathbb { R } , x \geqslant 0$$ （a）Find $\mathrm { f } ^ { - 1 } ( x )$ and state the domain of $\mathrm { f } ^ { - 1 }$ The function g is given by $$\mathrm { g } ( x ) = x ^ { 2 } - 4 x + 5 \text { for } x \in \mathbb { R } , x \geqslant 0$$ （b）Find the range of g ．
（c）Solve the equation $\operatorname { fg } ( x ) = x$ ．

Edexcel AEA 2017 June Q2

2．（a）Show that the equation $$\tan x = \frac { \sqrt { 3 } } { 1 + 4 \cos x }$$ can be written in the form $$\sin 2 x = \sin \left( 60 ^ { \circ } - x \right)$$ （b）Solve，for $0 < x < 180 ^ { \circ }$ $$\tan x = \frac { \sqrt { 3 } } { 1 + 4 \cos x }$$

Questions — Edexcel AEA (165 questions)

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