Questions - A-Level Maths

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 $\mathbf { e }$ （a） $\overrightarrow { A B }$ （b） $\overrightarrow { O D }$ The point $R$ divides $A B$ internally in the ratio $1 : 2$ （c）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 $X$ ．
（d）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
- 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.
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
- $\tan \alpha = \frac { 5 } { 12 }$ 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 $T$
- determine, in terms of $m$ and $g$, an expression for $T$,
- draw a diagram showing the forces on block $A$, clearly labelling each of the forces acting on the block,
- determine the value of $\mu$, giving a justification for your answer.
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.
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. \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
- 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.

Edexcel AEA 2004 June Q1

9 marks Challenging +1.2

1．Solve the equation $\cos x + \sqrt { } \left( 1 - \frac { 1 } { 2 } \sin 2 x \right) = 0 , \quad$ in the interval $0 ^ { \circ } \leq x < 360 ^ { \circ }$ ．

Edexcel AEA 2004 June Q2

10 marks Challenging +1.2

2．（a）For the binomial expansion of $\frac { 1 } { ( 1 - x ) ^ { 2 } } , | x | < 1$ ，in ascending powers of $x$ ，
（i）find the first four terms，
（ii）write down the coefficient of $x ^ { n }$ ．
（b）Hence，show that，for $| x | < 1 , \sum _ { n = 1 } ^ { \infty } n x ^ { n } = \frac { x } { ( 1 - x ) ^ { 2 } }$ ．
（c）Prove that，for $| x | < 1 , \sum _ { n = 1 } ^ { \infty } ( a n + 1 ) x ^ { n } = \frac { ( a + 1 ) x - x ^ { 2 } } { ( 1 - x ) ^ { 2 } }$ ，where $a$ is a constant．
（d）Hence evaluate $\sum _ { n = 1 } ^ { \infty } \frac { 5 n + 1 } { 2 ^ { 3 n } }$ ．

Edexcel AEA 2004 June Q3

11 marks Challenging +1.8

3. $$\mathrm { f } ( x ) = x ^ { 3 } - ( k + 4 ) x + 2 k , \quad \text { where } k \text { is a constant. }$$ （a）Show that，for all values of $k$ ，the curve with equation $y = \mathrm { f } ( x )$ passes through the point $( 2,0 )$ ．
（b）Find the values of $k$ for which the equation $\mathrm { f } ( x ) = 0$ has exactly two distinct roots． Given that $k > 0$ ，that the $x$－axis is a tangent to the curve with equation $y = \mathrm { f } ( x )$ ，and that the line $y = p$ intersects the curve in three distinct points，
（c）find the set of values that $p$ can take． \includegraphics[max width=\textwidth, alt={}, center]{a243ceda-8175-4ae0-9bc7-b3048f468d10-3_573_899_343_704} The circle, with centre $C$ and radius $r$, touches the $y$-axis at $( 0,4 )$ and also touches the line with equation $4 y - 3 x = 0$, as shown in Fig. 1.

1. Find the value of $r$.
2. Show that $\arctan \left( \frac { 3 } { 4 } \right) + 2 \arctan \left( \frac { 1 } { 2 } \right) = \frac { 1 } { 2 } \pi$.
  (8) The line with equation $4 x + 3 y = q , q > 12$, is a tangent to the circle.
Find the value of $q$.
(4)

Edexcel AEA 2004 June Q5

15 marks

(a) Given that $y = \ln \left[ t + \sqrt { } \left( 1 + t ^ { 2 } \right) \right]$, show that $\frac { \mathrm { d } y } { \mathrm {~d} t } = \frac { 1 } { \sqrt { } \left( 1 + t ^ { 2 } \right) }$.

The curve $C$ has parametric equations $$x = \frac { 1 } { \sqrt { } \left( 1 + t ^ { 2 } \right) } , \quad y = \ln \left[ t + \sqrt { } \left( 1 + t ^ { 2 } \right) \right] , \quad t \in \mathbb { R }$$ A student was asked to prove that, for $t > 0$, the gradient of the tangent to $C$ is negative.
The attempted proof was as follows: $$\begin{aligned} y & = \ln \left( t + \frac { 1 } { x } \right) \\ & = \ln \left( \frac { t x + 1 } { x } \right) \\ & = \ln ( t x + 1 ) - \ln x \\ \therefore \frac { \mathrm {~d} y } { \mathrm {~d} x } & = \frac { t } { t x + 1 } - \frac { 1 } { x } \\ & = \frac { \frac { t } { x } } { t + \frac { 1 } { x } } - \frac { 1 } { x } \\ & = \frac { t \sqrt { } \left( 1 + t ^ { 2 } \right) } { t + \sqrt { } \left( 1 + t ^ { 2 } \right) } - \sqrt { } \left( 1 + t ^ { 2 } \right) \\ & = - \frac { \left( 1 + t ^ { 2 } \right) } { t + \sqrt { } \left( 1 + t ^ { 2 } \right) } \end{aligned}$$ As $\left( 1 + t ^ { 2 } \right) > 0$, and $t + \sqrt { } \left( 1 + t ^ { 2 } \right) > 0$ for $t > 0 , \frac { \mathrm {~d} y } { \mathrm {~d} x } < 0$ for $t > 0$.
(b) (i) Identify the error in this attempt.
(ii) Give a correct version of the proof.
(c) Prove that $\ln \left[ - t + \sqrt { } \left( 1 + t ^ { 2 } \right) \right] = - \ln \left[ t + \sqrt { } \left( 1 + t ^ { 2 } \right) \right]$.
(d) Deduce that $C$ is symmetric about the $x$-axis and sketch the graph of $C$.

Edexcel AEA 2004 June Q6

17 marks Challenging +1.8

6. $$\mathrm { f } ( x ) = x - [ x ] , \quad x \geq 0$$ where $[ x ]$ is the largest integer $\leq x$. For example, $f ( 3.7 ) = 3.7 - 3 = 0.7 ; f ( 3 ) = 3 - 3 = 0$.

Sketch the graph of $y = \mathrm { f } ( x )$ for $0 \leq x < 4$.
Find the value of $p$ for which $\int _ { 2 } ^ { p } \mathrm { f } ( x ) \mathrm { d } x = 0.18$. Given that $$\mathrm { g } ( x ) = \frac { 1 } { 1 + k x } , \quad x \geq 0 , \quad k > 0$$ and that $x _ { 0 } = \frac { 1 } { 2 }$ is a root of the equation $\mathrm { f } ( x ) = \mathrm { g } ( x )$,
find the value of $k$.
Add a sketch of the graph of $y = \mathrm { g } ( x )$ to your answer to part (a). The root of $\mathrm { f } ( x ) = \mathrm { g } ( x )$ in the interval $n < x < n + 1$ is $x _ { n }$, where $n$ is an integer.
Prove that $$2 x _ { n } ^ { 2 } - ( 2 n - 1 ) x _ { n } - ( n + 1 ) = 0$$
Find the smallest value of $n$ for which $x _ { n } - n < 0.05$.

Edexcel AEA 2004 June Q7

19 marks Challenging +1.8

7．Triangle $A B C$ ，with $B C = a , A C = b$ and $A B = c$ is inscribed in a circle．Given that $A B$ is a diameter of the circle and that $a ^ { 2 } , b ^ { 2 }$ and $c ^ { 2 }$ are three consecutive terms of an arithmetic progression（arithmetic series），
（a）express $b$ and $c$ in terms of $a$ ，
（b）verify that $\cot A , \cot B$ and $\cot C$ are consecutive terms of an arithmetic progression． In an acute－angled triangle $P Q R$ the sides $Q R , P R$ and $P Q$ have lengths $p , q$ and $r$ respectively．
（c）Prove that $$\frac { p } { \sin P } = \frac { q } { \sin Q } = \frac { r } { \sin R }$$ Given now that triangle $P Q R$ is such that $p ^ { 2 } , q ^ { 2 }$ and $r ^ { 2 }$ are three consecutive terms of an arithmetic progression，
（d）use the cosine rule to prove that $\frac { 2 \cos Q } { q } = \frac { \cos P } { p } + \frac { \cos R } { r }$ ．
（6）
（e）Using the results given in parts（c）and（d），prove that $\cot P , \cot Q$ and $\cot R$ are consecutive terms in an arithmetic progression． Marks for style，clarity and presentation： 7

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 $\cos \theta = \frac { \sqrt { 5 } } { 3 }$ （a）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．
（b）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．
（a）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 $y = \mathrm { g } ( x )$ ．
（b）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 $B$ ．
（c）Write down the value of $p$ and the value of $q$ ．
（d）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），
（e）find the function g ．
（f）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 $\left( \frac { 3 \pi } { 4 } , 0 \right)$ ．
（a）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.
（b）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 $r$ ．
（c）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 $( 0,1 )$ ，
（d）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．
（a）Sketch $S ( \pi )$ ，labelling the vertices clearly． The shape $S ( \phi )$ is a trapezium．
（b）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 $\beta > \pi$ ．
（c）Show that $\alpha = \arccos \left( \frac { 29 } { 40 } \right) \cdot \left[ \arccos ( x ) \right.$ is an alternative notation for $\left. \cos ^ { - 1 } ( x ) \right]$ （d）Find the value of $\beta$ ． The area，in $\mathrm { cm } ^ { 2 }$ ，of shape $S ( \theta )$ is $R ( \theta )$ ．
（e）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 $\alpha \leqslant \theta \leqslant \beta$ (f) show that $R ( \theta )$ has only one stationary point and that this occurs when $\theta = \frac { 2 \pi } { 3 }$ (g) find the maximum and minimum values of $R ( \theta )$. FOR STYLE, CLARITY AND PRESENTATION: 7 MARKS TOTAL FOR PAPER: 100 MARKS
END

OCR H240/01 Q1

4 marks Moderate -0.8

1 Solve the simultaneous equations. $$\begin{array} { r } x ^ { 2 } + 8 x + y ^ { 2 } = 84 \\ x - y = 10 \end{array}$$

OCR H240/01 Q2

5 marks Moderate -0.8

2 The points $A$, $B$ and $C$ have position vectors $3 \mathbf { i } - 4 \mathbf { j } + 2 \mathbf { k } , - \mathbf { i } + 6 \mathbf { k }$ and $7 \mathbf { i } - 4 \mathbf { j } - 2 \mathbf { k }$ respectively. M is the midpoint of BC .

Show that the magnitude of $\overrightarrow { O M }$ is equal to $\sqrt { 17 }$. Point D is such that $\overrightarrow { B C } = \overrightarrow { A D }$.
Show that position vector of the point D is $11 \mathbf { i } - 8 \mathbf { j } - 6 \mathbf { k }$.

OCR H240/01 Q3

3 marks Moderate -0.8

3 The diagram below shows the graph of $y = \mathrm { f } ( x )$. \includegraphics[max width=\textwidth, alt={}, center]{6c16d9e2-7698-48e4-a3ed-5aae3b6f041e-05_801_1483_413_251}

On the diagram in the Printed Answer Booklet, draw the graph of $y = \mathrm { f } \left( \frac { 1 } { 2 } x \right)$.
On the diagram in the Printed Answer Booklet, draw the graph of $y = \mathrm { f } ( x - 2 ) + 1$.

OCR H240/01 Q4

7 marks Moderate -0.3

4 The diagram shows a sector $A O B$ of a circle with centre $O$ and radius $r \mathrm {~cm}$. \includegraphics[max width=\textwidth, alt={}, center]{6c16d9e2-7698-48e4-a3ed-5aae3b6f041e-05_510_606_1745_274} The angle $A O B$ is $\theta$ radians. The arc length $A B$ is 15 cm and the area of the sector is $45 \mathrm {~cm} ^ { 2 }$.

Find the values of $r$ and $\theta$.
Find the area of the segment bounded by the arc $A B$ and the chord $A B$.

OCR H240/01 Q5

4 marks Moderate -0.8

5 In this question you must show detailed reasoning. Use logarithms to solve the equation $3 ^ { 2 x + 1 } = 4 ^ { 100 }$, giving your answer correct to 3 significant figures.

OCR H240/01 Q6

3 marks Moderate -0.5

6 Prove by contradiction that there is no greatest even positive integer.

OCR H240/01 Q7

10 marks Moderate -0.8

7 Business A made a $\pounds 5000$ profit during its first year.
In each subsequent year, the profit increased by $\pounds 1500$ so that the profit was $\pounds 6500$ during the second year, $\pounds 8000$ during the third year and so on. Business B made a $\pounds 5000$ profit during its first year.
In each subsequent year, the profit was 90\% of the previous year's profit.

Find an expression for the total profit made by business A during the first $n$ years. Give your answer in its simplest form.
Find an expression for the total profit made by business B during the first $n$ years. Give your answer in its simplest form.
Find how many years it will take for the total profit of business A to reach $\pounds 385000$.
Comment on the profits made by each business in the long term.

OCR H240/01 Q8

6 marks Standard +0.3

8

Show that $\frac { 2 \tan \theta } { 1 + \tan ^ { 2 } \theta } = \sin 2 \theta$.
In this question you must show detailed reasoning. Solve $\frac { 2 \tan \theta } { 1 + \tan ^ { 2 } \theta } = 3 \cos 2 \theta$ for $0 \leq \theta \leq \pi$.

OCR H240/01 Q9

9 marks Standard +0.3

9 The equation $x ^ { 3 } - x ^ { 2 } - 5 x + 10 = 0$ has exactly one real root $\alpha$.

Show that the Newton-Raphson iterative formula for finding this root can be written as $$x _ { n + 1 } = \frac { 2 x _ { n } ^ { 3 } - x _ { n } ^ { 2 } - 10 } { 3 x _ { n } ^ { 2 } - 2 x _ { n } - 5 }$$
Apply the iterative formula in part (a) with initial value $x _ { 1 } = - 3$ to find $x _ { 2 } , x _ { 3 } , x _ { 4 }$ correct to 4 significant figures.
Use a change of sign method to show that $\alpha = - 2.533$ is correct to 4 significant figures.
Explain why the Newton-Raphson method with initial value $x _ { 1 } = - 1$ would not converge to $\alpha$.

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