Questions - Edexcel AEA - A-Level Maths

Edexcel AEA 2009 June Q6

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{dfb57dc0-5831-4bbb-b1e5-58c4798215cb-5_700_684_246_694} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a sketch of the curve $C$ with parametric equations $$x = 2 \sin t , \quad y = \ln ( \sec t ) , \quad 0 \leqslant t < \frac { \pi } { 2 }$$ The tangent to $C$ at the point $P$ ，where $t = \frac { \pi } { 3 }$ ，cuts the $x$－axis at $A$ ．
（a）Show that the $x$－coordinate of $A$ is $\frac { \sqrt { } 3 } { 3 } ( 3 - \ln 2 )$ ． The shaded region $R$ lies between $C$ ，the positive $x$－axis and the tangent $A P$ as shown in Figure 2 ．
（b）Show that the area of $R$ is $\sqrt { 3 } ( 1 + \ln 2 ) - 2 \ln ( 2 + \sqrt { 3 } ) - \frac { \sqrt { 3 } } { 6 } ( \ln 2 ) ^ { 2 }$ ．

Edexcel AEA 2009 June Q7

7．Relative to a fixed origin $O$ the points $A , B$ and $C$ have position vectors $$\mathbf { a } = - \mathbf { i } + \frac { 4 } { 3 } \mathbf { j } + 7 \mathbf { k } , \quad \mathbf { b } = 4 \mathbf { i } + \frac { 4 } { 3 } \mathbf { j } + 2 \mathbf { k } \text { and } \mathbf { c } = 6 \mathbf { i } + \frac { 16 } { 3 } \mathbf { j } + 2 \mathbf { k } \text { respectively. }$$ （a）Find the cosine of angle $A B C$ ． The quadrilateral $A B C D$ is a kite $K$ ．
（b）Find the area of $K$ ． A circle is drawn inside $K$ so that it touches each of the 4 sides of $K$ ．
（c）Find the radius of the circle，giving your answer in the form $p \sqrt { } ( q ) - q \sqrt { } ( p )$ ，where $p$ and $q$ are positive integers．
（d）Find the position vector of the point $D$ ．
（Total 18 marks）

Edexcel AEA 2010 June Q1

1．（a）Solve the equation $$\sqrt { } ( 3 x + 16 ) = 3 + \sqrt { } ( x + 1 )$$ （b）Solve the equation $$\log _ { 3 } ( x - 7 ) - \frac { 1 } { 2 } \log _ { 3 } x = 1 - \log _ { 3 } 2$$

Edexcel AEA 2010 June Q2

2．The sum of the first $p$ terms of an arithmetic series is $q$ and the sum of the first $q$ terms of the same arithmetic series is $p$ ，where $p$ and $q$ are positive integers and $p \neq q$ ． Giving simplified answers in terms of $p$ and $q$ ，find
（a）the common difference of the terms in this series，
（b）the first term of the series，
（c）the sum of the first $( p + q )$ terms of the series．

Edexcel AEA 2010 June Q3

3．The curve $C$ has equation $$x ^ { 2 } + y ^ { 2 } + f x y = g ^ { 2 }$$ where $f$ and $g$ are constants and $g \neq 0$ ．
（a）Find an expression in terms of $\alpha , \beta$ and $f$ for the gradient of $C$ at the point $( \alpha , \beta )$ ． Given that $f < 2$ and $f \neq - 2$ and that the gradient of $C$ at the point $( \alpha , \beta )$ is 1 ，
（b）show that $\alpha = - \beta = \frac { \pm g } { \sqrt { } ( 2 - f ) }$ ． Given that $f = - 2$ ，
（c）sketch $C$ ．

Edexcel AEA 2010 June Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{0396f61a-b844-40ed-98d1-82ee2d8a6807-3_643_332_246_870} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a cuboid $O A B C D E F G$, where $O$ is the origin, $A$ has position vector $5 \mathbf { i } , C$ has position vector $10 \mathbf { j }$ and $D$ has position vector $20 \mathbf { k }$.

Find the cosine of angle $C A F$. Given that the point $X$ lies on $A C$ and that $F X$ is perpendicular to $A C$,
find the position vector of point $X$ and the distance $F X$. The line $l _ { 1 }$ passes through $O$ and through the midpoint of the face $A B F E$. The line $l _ { 2 }$ passes through $A$ and through the midpoint of the edge $F G$.
Show that $l _ { 1 }$ and $l _ { 2 }$ intersect and find the coordinates of the point of intersection.

Edexcel AEA 2010 June Q5

5. $$I = \int \frac { 1 } { ( x - 1 ) \sqrt { } \left( x ^ { 2 } - 1 \right) } \mathrm { d } x , \quad x > 1$$ （a）Use the substitution $x = 1 + u ^ { - 1 }$ to show that $$I = - \left( \frac { x + 1 } { x - 1 } \right) ^ { \frac { 1 } { 2 } } + c$$ （b）Hence show that $$\int _ { \sec \alpha } ^ { \sec \beta } \frac { 1 } { ( x - 1 ) \sqrt { } \left( x ^ { 2 } - 1 \right) } \mathrm { d } x = \cot \left( \frac { \alpha } { 2 } \right) - \cot \left( \frac { \beta } { 2 } \right) , \quad 0 < \alpha < \beta < \frac { \pi } { 2 }$$

Edexcel AEA 2010 June Q6

6．（a）Given that $x ^ { 4 } + y ^ { 4 } = 1$ ，prove that $x ^ { 2 } + y ^ { 2 }$ is a maximum when $x = \pm y$ ，and find the maximum and minimum values of $x ^ { 2 } + y ^ { 2 }$ ．
（b）On the same diagram，sketch the curves $C _ { 1 }$ and $C _ { 2 }$ with equations $x ^ { 4 } + y ^ { 4 } = 1$ and $x ^ { 2 } + y ^ { 2 } = 1$ respectively．
（c）Write down the equation of the circle $C _ { 3 }$ ，centre the origin，which touches the curve $C _ { 1 }$ at the points where $x = \pm y$ ．

Edexcel AEA 2010 June Q7

7. $$\mathrm { f } ( x ) = \left[ 1 + \cos \left( x + \frac { \pi } { 4 } \right) \right] \left[ 1 + \sin \left( x + \frac { \pi } { 4 } \right) \right] , \quad 0 \leqslant x \leqslant 2 \pi$$

Show that $\mathrm { f } ( x )$ may be written in the form $$f ( x ) = \left( \frac { 1 } { \sqrt { 2 } } + \cos x \right) ^ { 2 } , \quad 0 \leqslant x \leqslant 2 \pi$$
Find the range of the function $\mathrm { f } ( x )$. The graph of $y = \mathrm { f } ( x )$ is shown in Figure 2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0396f61a-b844-40ed-98d1-82ee2d8a6807-5_426_938_849_591} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure}
Find the coordinates of all the maximum and minimum points on this curve. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0396f61a-b844-40ed-98d1-82ee2d8a6807-5_432_942_1535_589} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} The region $R$, bounded by $y = 2$ and $y = \mathrm { f } ( x )$, is shown shaded in Figure 3.
Find the area of $R$.

Edexcel AEA 2011 June Q1

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

Edexcel AEA 2011 June Q2

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 $b$ ．

Edexcel AEA 2011 June Q3

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 $\sum _ { r = 1 } ^ { \infty } 6 \times \left( \frac { 4 } { 3 } \right) ^ { \left[ \frac { r } { 2 } \right] } \times \left( \frac { 3 } { 5 } \right) ^ { \left[ \frac { r - 1 } { 2 } \right] }$

Edexcel AEA 2011 June Q4

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

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.
Show that the area of $R$ is $\sin \alpha \cos ^ { 3 } \alpha$
Find the maximum area of $R$, as $\alpha$ varies.

Edexcel AEA 2011 June Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{12d3f92f-8464-4ba1-93a2-c7b841e3d3de-4_739_1397_187_335} \captionsetup{labelformat=empty} \caption{Figure 2}

\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 $U$.

Write down the coordinates of the point $U$. The point $P$ with $x$-coordinate $a ( a \neq 0 )$ lies on $C$.
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 $C$.
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 $E$.

Edexcel AEA 2011 June Q6

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]

\includegraphics[alt={},max width=\textwidth]{12d3f92f-8464-4ba1-93a2-c7b841e3d3de-5_483_1367_1263_347} \captionsetup{labelformat=empty} \caption{Figure 3}

\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 $C$ ．

Edexcel AEA 2011 June Q13

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]

\includegraphics[alt={},max width=\textwidth]{12d3f92f-8464-4ba1-93a2-c7b841e3d3de-5_483_1367_1263_347} \captionsetup{labelformat=empty} \caption{Figure 3}

\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]

\includegraphics[alt={},max width=\textwidth]{12d3f92f-8464-4ba1-93a2-c7b841e3d3de-7_673_1338_591_367} \captionsetup{labelformat=empty} \caption{Figure 5}

\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 $D$.

Find the value of $p$ and the value of $q$.
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).
Find $\mathrm { m } ^ { - 1 }$
Write down the domain of $\mathrm { m } ^ { - 1 }$
Find the value of $t$ such that $\mathrm { m } ( t ) = \mathrm { m } ^ { - 1 } ( t )$

Edexcel AEA 2012 June Q1

1．The function f is given by $$\mathrm { f } ( x ) = x ^ { 2 } - 2 x + 6 , \quad x \geqslant 0$$ （a）Find the range of $f$ ． The function $g$ is given by $$\mathrm { g } ( x ) = 3 + \sqrt { } ( x + 4 ) , \quad x \geqslant 2$$ （b）Find $\operatorname { gf } ( x )$ ．
（c）Find the domain and range of gf．

Edexcel AEA 2012 June Q2

2．（a）Show that $$\sin 3 x = 3 \sin x - 4 \sin ^ { 3 } x$$ Hence find
（b） $\int \cos x ( 6 \sin x - 2 \sin 3 x ) ^ { \frac { 2 } { 3 } } \mathrm {~d} x$
（c） $\int ( 3 \sin 2 x - 2 \sin 3 x \cos x ) ^ { \frac { 1 } { 3 } } \mathrm {~d} x$

Edexcel AEA 2012 June Q3

3．The angle $\theta , 0 < \theta < \frac { \pi } { 2 }$ ，satisfies $$\tan \theta \tan 2 \theta = \sum _ { r = 0 } ^ { \infty } 2 \cos ^ { r } 2 \theta$$ （a）Show that $\tan \theta = 3 ^ { p }$ ，where $p$ is a rational number to be found．
（b）Hence show that $\frac { \pi } { 6 } < \theta < \frac { \pi } { 4 }$

Edexcel AEA 2012 June Q6

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{fc5d0d07-b750-4646-bdcb-419a290200c9-4_433_1011_221_529} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of the curve with equation $y = ( x + a ) ( x - b ) ^ { 2 }$, where $a$ and $b$ are positive constants. The curve cuts the $x$-axis at $P$ and has a maximum point at $S$ and a minimum point at $Q$.

Write down the coordinates of $P$ and $Q$ in terms of $a$ and $b$.
Show that $G$, the area of the shaded region between the curve $P S Q$ and the $x$-axis, is given by $G = \frac { ( a + b ) ^ { 4 } } { 12 }$. The rectangle $P Q R S T$ has $R S T$ parallel to $Q P$ and both $P T$ and $Q R$ are parallel to the $y$-axis.
Show that $\frac { G } { \text { Area of } P Q R S T } = k$, where $k$ is a constant independent of $a$ and $b$ and find the value of $k$.

Edexcel AEA 2013 June Q1

1．In the binomial expansion of $$\left( 1 + \frac { 12 n } { 5 } x \right) ^ { n }$$ the coefficients of $x ^ { 2 }$ and $x ^ { 3 }$ are equal and non－zero．
（a）Find the possible values of $n$ ．
（4）
（b）State，giving a reason，which value of $n$ gives a valid expansion when $x = \frac { 1 } { 2 }$
（2）

Edexcel AEA 2013 June Q2

2．（a）Use the formula for $\sin ( A - B )$ to show that $\sin \left( 90 ^ { \circ } - x \right) = \cos x$
（b）Solve for $0 < \theta < 360 ^ { \circ }$ $$2 \sin \left( \theta + 17 ^ { \circ } \right) = \frac { \cos \left( \theta + 8 ^ { \circ } \right) } { \cos \left( \theta + 17 ^ { \circ } \right) }$$

Edexcel AEA 2013 June Q3

3．The lines $L _ { 1 }$ and $L _ { 2 }$ have equations given by
$L _ { 1 } : \quad \mathbf { r } = \left( \begin{array} { c } - 7
7
1 \end{array} \right) + \lambda \left( \begin{array} { c } 2
0
- 3 \end{array} \right)$ and $L _ { 2 } : \quad \mathbf { r } = \left( \begin{array} { c } 7
p
- 6 \end{array} \right) + \mu \left( \begin{array} { c } 10
- 4
- 1 \end{array} \right)$
where $\lambda$ and $\mu$ are parameters and $p$ is a constant．
The two lines intersect at the point $C$ ．
（a）Find
（i）the value of $p$ ，
（ii）the position vector of $C$ ．
（b）Show that the point $B$ with position vector $\left( \begin{array} { c } - 13
11
- 4 \end{array} \right)$ lies on $L _ { 2 }$ ． The point $A$ with position vector $\left( \begin{array} { c } - 7
7
1 \end{array} \right)$ lies on $L _ { 1 }$ ．
（c）Find $\cos ( \angle A C B )$ ，giving your answer as an exact fraction． The line $L _ { 3 }$ bisects the angle $A C B$ ．
（d）Find a vector equation of $L _ { 3 }$ ．

Edexcel AEA 2013 June Q4

4．A sequence of positive integers $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$ has $r$ th term given by $$a _ { r } = 2 ^ { r } - 1$$ （a）Write down the first 6 terms of this sequence．
（b）Verify that $a _ { r + 1 } = 2 a _ { r } + 1$
（c）Find $\sum _ { r = 1 } ^ { n } a _ { r }$
（d）Show that $\frac { 1 } { a _ { r + 1 } } < \frac { 1 } { 2 } \times \frac { 1 } { a _ { r } }$
（e）Hence show that $1 + \frac { 1 } { 3 } + \frac { 1 } { 7 } + \frac { 1 } { 15 } + \frac { 1 } { 31 } + \ldots < 1 + \frac { 1 } { 3 } + \left( \frac { 1 } { 7 } + \frac { \frac { 1 } { 2 } } { 7 } + \frac { \frac { 1 } { 4 } } { 7 } + \ldots \right)$
（f）Show that $\frac { 31 } { 21 } < \sum _ { r = 1 } ^ { \infty } \frac { 1 } { a _ { r } } < \frac { 34 } { 21 }$

Edexcel AEA 2013 June Q5

5．In this question u and v are functions of $x$ ．Given that $\int \mathrm { u } \mathrm { d } x , \int \mathrm { v } \mathrm { d } x$ and $\int \mathrm { uv } \mathrm { d } x$ satisfy $$\int \text { uv } \mathrm { d } x = \left( \int \mathrm { u } \mathrm {~d} x \right) \times \left( \int \mathrm { v } \mathrm {~d} x \right) \quad \text { uv } \neq 0$$ （a）show that $1 = \frac { \int \mathrm { u } \mathrm { d } x } { \mathrm { u } } + \frac { \int \mathrm { v } \mathrm { d } x } { \mathrm { v } }$ Given also that $\frac { \int \mathrm { u } \mathrm { d } x } { \mathrm { u } } = \mathrm { sin } ^ { 2 } x$,
（b）use part（a）to write down an expression，in terms of $x$ ，for $\frac { \int \mathrm { v } \mathrm { d } x } { \mathrm { v } }$ ，
（c）show that $$\frac { 1 } { \mathrm { u } } \frac { \mathrm { du } } { \mathrm {~d} x } = \frac { 1 - 2 \sin x \cos x } { \sin ^ { 2 } x }$$ （d）hence use integration to show that $\mathrm { u } = A \mathrm { e } ^ { - \cot x } \operatorname { cosec } ^ { 2 } x$ ，where $A$ is an arbitrary constant．
（e）By differentiating $\mathrm { e } ^ { \tan x }$ find a similar expression for v ．

Questions — Edexcel AEA (165 questions)

Browse by module

Browse by board