Questions - Edexcel AEA - A-Level Maths

Edexcel AEA 2005 June Q7

(a) Use the substitution $x = \sec \theta$ to show that

$$\int \sqrt { } \left( x ^ { 2 } - 1 \right) d x$$ can be written as $$\int \sec \theta \tan ^ { 2 } \theta \mathrm {~d} \theta$$ (3)
(b) Use integration by parts to show that $$\int \sec \theta \tan ^ { 2 } \theta \mathrm {~d} \theta = \frac { 1 } { 2 } [ \sec \theta \tan \theta - \ln | \sec \theta + \tan \theta | ] + \text { constant. }$$ (c) Evaluate $\int _ { 0 } ^ { \frac { \pi } { 4 } } \sin x \sqrt { } ( \cos 2 x ) \mathrm { d } x$.

Edexcel AEA 2006 June Q1

1．（a）For $| y | < 1$ ，write down the binomial series expansion of $( 1 - y ) ^ { - 2 }$ in ascending powers of $y$ up to and including the term in $y ^ { 3 }$ ．
（b）Hence，or otherwise，show that $$1 + \frac { 2 x } { 1 + x } + \frac { 3 x ^ { 2 } } { ( 1 + x ) ^ { 2 } } + \ldots + \frac { r x ^ { r - 1 } } { ( 1 + x ) ^ { r - 1 } } + \ldots$$ can be written in the form $( a + x ) ^ { n }$ ．Write down the values of the integers $a$ and $n$ ．
（c）Find the set of values of $x$ for which the series in part（b）is convergent．

Edexcel AEA 2006 June Q2

2．Given that $( \sin \theta + \cos \theta ) \neq 0$ ，find all the solutions of $$\frac { 2 \cos 2 \theta ( \sin 2 \theta - \sqrt { } 3 \cos 2 \theta ) } { \sin \theta + \cos \theta } = \sqrt { } 6 ( \sin 2 \theta - \sqrt { } 3 \cos 2 \theta )$$ for $0 \leq \theta < 360 ^ { \circ }$ ．

Edexcel AEA 2006 June Q3

3．Given that $x > y > 0$ ，
（a）by writing $\log _ { y } x = z$ ，or otherwise，show that $\log _ { y } x = \frac { 1 } { \log _ { x } y }$ ．
（b）Given also that $\log _ { x } y = \log _ { y } x$ ，show that $y = \frac { 1 } { x }$ ．
（c）Solve the simultaneous equations $$\begin{gathered} \log _ { x } y = \log _ { y } x
\log _ { x } ( x - y ) = \log _ { y } ( x + y ) \end{gathered}$$

Edexcel AEA 2006 June Q4

4．The line with equation $y = m x$ is a tangent to the circle $C _ { 1 }$ with equation $$( x + 4 ) ^ { 2 } + ( y - 7 ) ^ { 2 } = 13$$ （a）Show that $m$ satisfies the equation $$3 m ^ { 2 } + 56 m + 36 = 0$$ The tangents from the origin $O$ to $C _ { 1 }$ touch $C _ { 1 }$ at the points $A$ and $B$ ．
（b）Find the coordinates of the points $A$ and $B$ ．
（8）
Another circle $C _ { 2 }$ has equation $x ^ { 2 } + y ^ { 2 } = 13$ ．The tangents from the point $( 4 , - 7 )$ to $C _ { 2 }$ touch it at the points $P$ and $Q$ ．
（c）Find the coordinates of either the point $P$ or the point $Q$ ．
（2）

Edexcel AEA 2006 June Q5

5．The lines $L _ { 1 }$ and $L _ { 2 }$ have vector equations
$L _ { 1 } : \quad \mathbf { r } = - 2 \mathbf { i } + 11.5 \mathbf { j } + \lambda ( 3 \mathbf { i } - 4 \mathbf { j } - \mathbf { k } )$,
$L _ { 2 } : \quad \mathbf { r } = 11.5 \mathbf { i } + 3 \mathbf { j } + 8.5 \mathbf { k } + \mu ( 7 \mathbf { i } + 8 \mathbf { j } - 11 \mathbf { k } )$,
where $\lambda$ and $\mu$ are parameters．
（a）Show that $L _ { 1 }$ and $L _ { 2 }$ do not intersect．
（b）Show that the vector $( 2 \mathbf { i } + \mathbf { j } + 2 \mathbf { k } )$ is perpendicular to both $L _ { 1 }$ and $L _ { 2 }$ ． The point $A$ lies on $L _ { 1 }$ ，the point $B$ lies on $L _ { 2 }$ and $A B$ is perpendicular to both $L _ { 1 }$ and $L _ { 2 }$ ．
（c）Find the position vector of the point $A$ and the position vector of the point $B$ ．
（8）
\includegraphics[max width=\textwidth, alt={}, center]{0df09d8a-7478-4679-b117-128ee226db6a-4_554_1017_404_571} Figure 1 shows a sketch of part of the curve $C$ with equation $$y = \sin ( \ln x ) , \quad x \geq 1 .$$ The point $Q$ ，on $C$ ，is a maximum．

Edexcel AEA 2006 June Q7

7.
\includegraphics[max width=\textwidth, alt={}, center]{0df09d8a-7478-4679-b117-128ee226db6a-5_648_1590_296_275} The circle $C _ { 1 }$ has centre $O$ and radius $R$. The tangents $A P$ and $B P$ to $C _ { 1 }$ meet at the point $P$ and angle $A P B = 2 \alpha , 0 < \alpha < \frac { \pi } { 2 }$. A sequence of circles $C _ { 1 } , C _ { 2 } , \ldots , C _ { n } , \ldots$ is drawn so that each new circle $C _ { n + 1 }$ touches each of $C _ { n } , A P$ and $B P$ for $n = 1,2,3 , \ldots$ as shown in Figure 2. The centre of each circle lies on the line $O P$.

Show that the radii of the circles form a geometric sequence with common ratio $$\frac { 1 - \sin \alpha } { 1 + \sin \alpha }$$
Find, in terms of $R$ and $\alpha$, the total area enclosed by all the circles, simplifying your answer. The area inside the quadrilateral $P A O B$, not enclosed by part of $C _ { 1 }$ or any of the other circles, is $S$.
Show that $$S = R ^ { 2 } \left( \alpha + \cot \alpha - \frac { \pi } { 4 } \operatorname { cosec } \alpha - \frac { \pi } { 4 } \sin \alpha \right) .$$
Show that, as $\alpha$ varies, $$\frac { \mathrm { d } S } { \mathrm {~d} \alpha } = R ^ { 2 } \cot ^ { 2 } \alpha \left( \frac { \pi } { 4 } \cos \alpha - 1 \right)$$
Find, in terms of $R$, the least value of $S$ for $\frac { \pi } { 6 } \leq \alpha \leq \frac { \pi } { 4 }$.

Edexcel AEA 2007 June Q1

1．（a）Write down the binomial expansion of $\frac { 1 } { ( 1 - y ) ^ { 2 } } , | y | < 1$ ，in ascending powers of $y$ up to and including the term in $y ^ { 3 }$ ．
（b）Hence，or otherwise，show that $$\frac { 1 } { 4 } \operatorname { cosec } ^ { 4 } \left( \frac { \theta } { 2 } \right) = 1 + 2 \cos \theta + 3 \cos ^ { 2 } \theta + 4 \cos ^ { 3 } \theta + \ldots + ( r + 1 ) \cos ^ { r } \theta + \ldots$$ and state the values of $\theta$ for which this result is not valid．
（4）
Find
（c） $$\begin{aligned} & 1 + \frac { 2 } { 2 } + \frac { 3 } { 2 ^ { 2 } } + \frac { 4 } { 2 ^ { 3 } } + \ldots + \frac { ( r + 1 ) } { 2 ^ { r } } + \ldots
& 1 - \frac { 2 } { 2 } + \frac { 3 } { 2 ^ { 2 } } - \frac { 4 } { 2 ^ { 3 } } + \ldots + ( - 1 ) ^ { r } \frac { ( r + 1 ) } { 2 ^ { r } } + \ldots \end{aligned}$$ （d）

Edexcel AEA 2007 June Q2

2．（a）On the same diagram，sketch $y = x$ and $y = \sqrt { } x$ ，for $x \geq 0$ ，and mark clearly the coordinates of the points of intersection of the two graphs．
（b）With reference to your sketch，explain why there exists a value $a$ of $x ( a > 1 )$ such that $$\int _ { 0 } ^ { a } x \mathrm {~d} x = \int _ { 0 } ^ { a } \sqrt { } x \mathrm {~d} x$$ （c）Find the exact value of $a$ ．
（d）Hence，or otherwise，find a non－constant function $\mathrm { f } ( x )$ and a constant $b ( b \neq 0 )$ such that $$\int _ { - b } ^ { b } \mathrm { f } ( x ) \mathrm { d } x = \int _ { - b } ^ { b } \sqrt { } [ \mathrm { f } ( x ) ] \mathrm { d } x$$

Edexcel AEA 2007 June Q3

3．（a）Solve，for $0 \leq x < 2 \pi$ ， $$\cos x + \cos 2 x = 0$$ （b）Find the exact value of $x , x \geq 0$ ，for which $$\arccos x + \arccos 2 x = \frac { \pi } { 2 }$$ ［ $\arccos x$ is an alternative notation for $\cos ^ { - 1 } x$ ．］

Edexcel AEA 2007 June Q4

4．The function $\mathrm { h } ( x )$ has domain $\mathbb { R }$ and range $\mathrm { h } ( x ) > 0$ ，and satisfies $$\sqrt { \int \mathrm { h } ( x ) \mathrm { d } x } = \int \sqrt { \mathrm { h } ( x ) } \mathrm { d } x$$ （a）By substituting $\mathrm { h } ( x ) = \left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) ^ { 2 }$ ，show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 ( y + c ) ,$$ where $c$ is constant．
（b）Hence find a general expression for $y$ in terms of $x$ ．
（c）Given that $\mathrm { h } ( 0 ) = 1$ ，find $\mathrm { h } ( x )$ ．

Edexcel AEA 2007 June Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f2290882-b9a4-43ec-a38f-c44d46477242-4_493_1324_279_367} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows part of a sequence $S _ { 1 } , S _ { 2 } , S _ { 3 } , \ldots$, of model snowflakes. The first term $S _ { 1 }$ consists of a single square of side $a$. To obtain $S _ { 2 }$, the middle third of each edge is replaced with a new square, of side $\frac { a } { 3 }$, as shown in Figure 1 . Subsequent terms are obtained by replacing the middle third of each external edge of a new square formed in the previous snowflake, by a square $\frac { 1 } { 3 }$ of the size, as illustrated by $S _ { 3 }$ in Figure 1.

Deduce that to form $S _ { 4 } , 36$ new squares of side $\frac { a } { 27 }$ must be added to $S _ { 3 }$.
Show that the perimeters of $S _ { 2 }$ and $S _ { 3 }$ are $\frac { 20 a } { 3 }$ and $\frac { 28 a } { 3 }$ respectively.
Find the perimeter of $S _ { n }$.
Describe what happens to the perimeter of $S _ { n }$ as $n$ increases.
Find the areas of $S _ { 1 } , S _ { 2 }$ and $S _ { 3 }$.
Find the smallest value of the constant $S$ such that the area of $S _ { n } < S$, for all values of $n$.
\includegraphics[max width=\textwidth, alt={}, center]{f2290882-b9a4-43ec-a38f-c44d46477242-5_590_1041_283_588} Figure 2 shows a sketch of the curve $C$ with equation $y = \tan \frac { t } { 2 } , \quad 0 \leq t \leq \frac { \pi } { 2 }$.
The point $P$ on $C$ has coordinates $\left( x , \tan \frac { x } { 2 } \right)$.
The vertices of rectangle $R$ are at $( x , 0 ) , \left( \frac { x } { 2 } , 0 \right) , \left( \frac { x } { 2 } , \tan \frac { x } { 2 } \right)$ and $\left( x , \tan \frac { x } { 2 } \right)$ as shown in Figure 2.

Edexcel AEA 2007 June Q7

7．The points $O , P$ and $Q$ lie on a circle $C$ with diameter $O Q$ ．The position vectors of $P$ and $Q$ ， relative to $O$ ，are $\mathbf { p }$ and $\mathbf { q }$ respectively．
（a）Prove that $\mathbf { p } . \mathbf { q } = | \mathbf { p } | ^ { 2 }$ ． \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f2290882-b9a4-43ec-a38f-c44d46477242-6_615_714_412_689} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} The point $R$ also lies on $C$ and $O P Q R$ is a kite $K$ as shown in Figure 3．The point $S$ has position vector，relative to $O$ ，of $\lambda \mathbf { q }$ ，where $\lambda$ is a constant．Given that $\mathbf { p } = \mathbf { i } + 2 \mathbf { j } - \mathbf { k } , \mathbf { q } = 2 \mathbf { i } + \mathbf { j } - 2 \mathbf { k }$ and that $O Q$ is perpendicular to $P S$ ，find
（b）the value of $\lambda$ ，
（c）the position vector of $R$ ，
（d）the area of $K$ ． Another circle $C _ { 1 }$ is drawn inside $K$ so that the 4 sides of the kite are each tangents to $C _ { 1 }$ ．
（e）Find the radius of $C _ { 1 }$ giving your answer in the form $( \sqrt { } 2 - 1 ) \sqrt { } n$ ，where $n$ is an integer． A second kite $K _ { 1 }$ is similar to $K$ and is drawn inside $C _ { 1 }$ ．
（f）Find that area of $K _ { 1 }$ ．

Edexcel AEA 2008 June Q1

1．The first and second terms of an arithmetic series are 200 and 197.5 respectively．
The sum to $n$ terms of the series is $S _ { n }$ ． Find the largest positive value of $S _ { n }$ ．

Edexcel AEA 2008 June Q2

2．The points $( x , y )$ on the curve $C$ satisfy $$( x + 1 ) ( x + 2 ) \frac { d y } { d x } = x y$$ The line with equation $y = 2 x + 5$ is the tangent to $C$ at a point $P$ ．
（a）Find the coordinates of $P$ ．
（b）Find the equation of $C$ ，giving your answer in the form $y = \mathrm { f } ( x )$ ．

Edexcel AEA 2008 June Q3

3．（a）Prove that $\tan 15 ^ { \circ } = 2 - \sqrt { 3 }$
（b）Solve，for $0 \leqslant \theta < 360 ^ { \circ }$ ， $$\sin \left( \theta + 60 ^ { \circ } \right) \sin \left( \theta - 60 ^ { \circ } \right) = ( 1 - \sqrt { } 3 ) \cos ^ { 2 } \theta$$

\includegraphics[max width=\textwidth, alt={}]{280c36e3-6bb5-44b7-8222-17ce25c1bdbe-3_1008_1343_369_465}

Figure 1 shows a sketch of the curve $C$ with equation $$y = \cos x \ln ( \sec x ) , \quad - \frac { \pi } { 2 } < x < \frac { \pi } { 2 } .$$ The points $A$ and $B$ are maximum points on $C$ ．
（a）Find the coordinates of $B$ in terms of e ． The finite region $R$ lies between $C$ and the line $A B$ ．
（b）Show that the area of $R$ is $$\frac { 2 } { \mathrm { e } } \arccos \left( \frac { 1 } { \mathrm { e } } \right) + 2 \ln \left( \mathrm { e } + \sqrt { } \left( \mathrm { e } ^ { 2 } - 1 \right) \right) - \frac { 4 } { \mathrm { e } } \sqrt { } \left( \mathrm { e } ^ { 2 } - 1 \right) .$$ $\left[ \arccos x \right.$ is an alternative notation for $\left. \cos ^ { - 1 } x \right]$

Edexcel AEA 2008 June Q5

5. (i) Anna, who is confused about the rules for logarithms, states that $$\left( \log _ { 3 } p \right) ^ { 2 } = \log _ { 3 } \left( p ^ { 2 } \right)$$ and $\quad \log _ { 3 } ( p + q ) = \log _ { 3 } p + \log _ { 3 } q$.
However, there is a value for $p$ and a value for $q$ for which both statements are correct.
Find the value of $p$ and the value of $q$.
(ii) Solve $$\frac { \log _ { 3 } \left( 3 x ^ { 3 } - 23 x ^ { 2 } + 40 x \right) } { \log _ { 3 } 9 } = 0.5 + \log _ { 3 } ( 3 x - 8 ) .$$

Edexcel AEA 2008 June Q6

6. $$\mathrm { f } ( x ) = \frac { a x + b } { x + 2 } ; \quad x \in \mathbb { R } , x \neq - 2$$ where $a$ and $b$ are constants and $b > 0$ ．
（a）Find $f ^ { - 1 } ( x )$ ．
（b）Hence，or otherwise，find the value of $a$ so that $\operatorname { ff } ( x ) = x$ ． The curve $C$ has equation $y = \mathrm { f } ( x )$ and $\mathrm { f } ( x )$ satisfies $\mathrm { ff } ( x ) = x$ ．
（c）On separate axes sketch
（i）$y = \mathrm { f } ( x )$ ，
（ii）$y = \mathrm { f } ( x - 2 ) + 2$ ． On each sketch you should indicate the equations of any asymptotes and the coordinates，in terms of $b$ ，of any intersections with the axes． The normal to $C$ at the point $P$ has equation $y = 4 x - 39$ ．The normal to $C$ at the point $Q$ has equation $y = 4 x + k$ ，where $k$ is a constant．
（d）By considering the images of the normals to $C$ on the curve with equation $y = \mathrm { f } ( x - 2 ) + 2$ ，or otherwise，find the value of $k$ ．

Edexcel AEA 2008 June Q7

7. Relative to a fixed origin $O$, the position vectors of the points $A , B$ and $C$ are $$\overrightarrow { O A } = - 3 \mathbf { i } + \mathbf { j } - 9 \mathbf { k } , \quad \overrightarrow { O B } = \mathbf { i } - \mathbf { k } , \quad \overrightarrow { O C } = 5 \mathbf { i } + 2 \mathbf { j } - 5 \mathbf { k } \text { respectively. }$$

Find the cosine of angle $A B C$. The line $L$ is the angle bisector of angle $A B C$.
Show that an equation of $L$ is $\mathbf { r } = \mathbf { i } - \mathbf { k } + t ( \mathbf { i } + 2 \mathbf { j } - 7 \mathbf { k } )$.
Show that $| \overrightarrow { A B } | = | \overrightarrow { A C } |$. The circle $S$ lies inside triangle $A B C$ and each side of the triangle is a tangent to $S$.
Find the position vector of the centre of $S$.
Find the radius of $S$.

Edexcel AEA 2009 June Q1

(a) On the same diagram, sketch

$$y = ( x + 1 ) ( 2 - x ) \quad \text { and } \quad y = x ^ { 2 } - 2 | x |$$ Mark clearly the coordinates of the points where these curves cross the coordinate axes.
(b) Find the $x$-coordinates of the points of intersection of these two curves.

Edexcel AEA 2009 June Q2

2. The curve $C$ has equation $y = x ^ { \sin x } , \quad x > 0$.

Find the equation of the tangent to $C$ at the point where $x = \frac { \pi } { 2 }$.
Prove that this tangent touches $C$ at infinitely many points.

Edexcel AEA 2009 June Q3

3. (a) Solve, for $0 \leqslant \theta < 2 \pi$, $$\sin \left( \frac { \pi } { 3 } - \theta \right) = \frac { 1 } { \sqrt { } 3 } \cos \theta$$ (b) Find the value of $x$ for which $$\begin{aligned} & \arcsin ( 1 - 2 x ) = \frac { \pi } { 3 } - \arcsin x , \quad 0 < x < 0.5
& { \left[ \arcsin x \text { is an alternative notation for } \sin ^ { - 1 } x \right] } \end{aligned}$$

Edexcel AEA 2009 June Q4

(a) The function $\mathrm { f } ( x )$ has $\mathrm { f } ^ { \prime } ( x ) = \frac { \mathrm { u } ( x ) } { \mathrm { v } ( x ) }$. Given that $\mathrm { f } ^ { \prime } ( k ) = 0$, show that $\mathrm { f } ^ { \prime \prime } ( k ) = \frac { \mathrm { u } ^ { \prime } ( k ) } { \mathrm { v } ( k ) }$.

\begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{dfb57dc0-5831-4bbb-b1e5-58c4798215cb-3_874_879_486_593} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} (b) The curve $C$ with equation $$y = \frac { 2 x ^ { 2 } + 3 } { x ^ { 2 } - 1 }$$ crosses the $y$-axis at the point $A$. Figure 1 shows a sketch of $C$ together with its 3 asymptotes.

Find the coordinates of the point $A$.
Find the equations of the asymptotes of $C$. The point $P ( a , b ) , a > 0$ and $b > 0$, lies on $C$. The point $Q$ also lies on $C$ with $P Q$ parallel to the $x$-axis and $A P = A Q$.
Show that the area of triangle $P A Q$ is given by $\frac { 5 a ^ { 3 } } { a ^ { 2 } - 1 }$.
Find, as $a$ varies, the minimum area of triangle $P A Q$, giving your answer in its simplest form.

Edexcel AEA 2009 June Q5

5．（a）The sides of the triangle $A B C$ have lengths $B C = a , A C = b$ and $A B = c$ ，where $a < b < c$ ．The sizes of the angles $A , B$ and $C$ form an arithmetic sequence．
（i）Show that the area of triangle $A B C$ is $a c \frac { \sqrt { 3 } } { 4 }$ ． Given that $a = 2$ and $\sin A = \frac { \sqrt { } 15 } { 5 }$ ，find
（ii）the value of $b$ ，
（iii）the value of $c$ ．
（b）The internal angles of an $n$－sided polygon form an arithmetic sequence with first term $143 ^ { \circ }$ and common difference $2 ^ { \circ }$ ． Given that all of the internal angles are less than $180 ^ { \circ }$ ，find the value of $n$ ．

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