Questions AEA - A-Level Maths

Show that the length of $L _ { 2 }$ is $\frac { 23 } { 4 }$
Write down the number of
1. line segments in $L _ { n }$ that are parallel to $L _ { 0 }$
2. sloped line segments in $L _ { 2 }$ that are not in $L _ { 1 }$
3. new sloped line segments that are created by the ( $n + 1$ )th iteration.
Hence find the length of $L _ { n }$ as $n \rightarrow \infty$ The area enclosed between $L _ { 0 }$ and $L _ { n }$ is $A _ { n }$
Find the value of $A _ { 1 }$
Find, in terms of $n$, an expression for $A _ { n + 1 } - A _ { n }$
Hence find the value of $A _ { n }$ as $n \rightarrow \infty$ The same construction as described above is applied externally to the three sides of an equilateral triangle of side length $a$.
Given that the limit of the area of the resulting shape is $26 \sqrt { 3 }$
find the value of $a$.

Edexcel AEA 2022 June Q7

7．A circle $C$ has centre $X ( a , b )$ and radius $r$ ．
A line $l$ has equation $y = m x + c$
（a）Show that the $x$ coordinates of the points where $C$ and $l$ intersect satisfy $$\left( m ^ { 2 } + 1 \right) x ^ { 2 } - 2 ( a - m ( c - b ) ) x + a ^ { 2 } + ( c - b ) ^ { 2 } - r ^ { 2 } = 0$$ Given that $l$ is a tangent to $C$ ，
（b）show that $$c = b - m a \pm r \sqrt { m ^ { 2 } + 1 }$$ The circle $C _ { 1 }$ has equation $$x ^ { 2 } + y ^ { 2 } - 16 = 0$$ and the circle $C _ { 2 }$ has equation $$x ^ { 2 } + y ^ { 2 } - 20 x - 10 y + 89 = 0$$ （c）Find the equations of any lines that are normal to both $C _ { 1 }$ and $C _ { 2 }$ ，justifying your answer．
（d）Find the equations of all lines that are a tangent to both $C _ { 1 }$ and $C _ { 2 }$
［You may find the following Pythagorean triple helpful in this part： $7 ^ { 2 } + 24 ^ { 2 } = 25 ^ { 2 }$ ］

Edexcel AEA 2023 June Q1

1．（a）Write down the exact value of $\cos 405 ^ { \circ }$
（b）Hence，using a double angle identity for cosine，or otherwise，determine the exact value of $\cos 101.25 ^ { \circ }$ ，giving your answer in the form $$a \sqrt { b + c \sqrt { 2 + \sqrt { 2 } } }$$ where $a$ ，$b$ and $c$ are rational numbers．

Edexcel AEA 2023 June Q2

2．A student is attempting to prove that there are infinitely many prime numbers．
The student＇s attempt to prove this is in the box below． Assume there are only finitely many prime numbers，then there is a biggest prime number，$p$ ． Let $n = 2 p + 1$ ．Then $n$ is bigger than $p$ and since $2 p + 1$ is not divisible by $p$ ， $n$ is a prime number． Hence $n$ is a prime number bigger than $p$ ，contradicting the initial assumption． So we conclude there are infinitely many prime numbers．
（a）Use $p = 7$ to show that the following claim made in the student＇s proof is not true： since $2 p + 1$ is not divisible by $p , n$ is a prime number． The student changes their proof to use $n = 6 p + 1$ instead of $n = 2 p + 1$
（b）Show，by counter example，that this does not correct the student＇s proof．
（c）Write out a correct proof by contradiction to show that there are infinitely many prime numbers．

Edexcel AEA 2023 June Q3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{78ba3acc-4cca-4d15-8362-a27e425c5859-08_752_586_251_742} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows the curve $C$ given by the parametric equations $$x = \frac { 5 } { \sqrt { 3 } } \sin t \quad y = 5 ( 1 - \cos t ) \quad 0 \leqslant t \leqslant 2 \pi$$ The circle with centre at the origin $O$ and with radius $\frac { 5 \sqrt { 2 } } { 2 }$ meets the curve $C$ at the points $A$ and $B$ as shown in Figure 1.
（a）Determine the value of $t$ at the point $B$ ． The region $R$ ，shown shaded in Figure 1，is bounded by the curve $C$ and the circle．
（b）Determine the area of the region $R$ ．

Edexcel AEA 2023 June Q4

4．（a）Use the trapezium rule with 4 strips to find an approximate value for $$\int _ { 0 } ^ { 1 } 16 ^ { x } d x$$ （b）Use the trapezium rule with $n$ strips to write down an expression that would give an approximate value for $$\int _ { 0 } ^ { 1 } 16 ^ { x } d x$$ （c）Hence show that $$\int _ { 0 } ^ { 1 } 16 ^ { x } \mathrm {~d} x = \lim _ { n \rightarrow \infty } \left( \frac { 1 } { n } \left( 1 + 16 ^ { \frac { 1 } { n } } + \ldots + 16 ^ { \frac { n - 1 } { n } } \right) \right)$$ （d）Use integration to determine the exact value of $$\int _ { 0 } ^ { 1 } 16 ^ { x } d x$$ Given that the limit exists，
（e）use part（c）and the answer to part（d）to determine the exact value of $$\lim _ { x \rightarrow 0 } \frac { 16 ^ { x } - 1 } { x }$$

Edexcel AEA 2023 June Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{78ba3acc-4cca-4d15-8362-a27e425c5859-16_517_881_210_593} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a partially completed Venn diagram of sports that a year group of students enjoy，where $a , b , c , d$ and $e$ are non－negative integers． The diagram shows how many students enjoy a combination of football（ $F$ ），golf（ $G$ ） and hockey $( H )$ or none of these sports． There are $n$ students in the year group．
It is known that
－ $\mathrm { P } ( F ) = \frac { 3 } { 7 }$
－ $\mathrm { P } ( H \mid G ) = \frac { 1 } { 3 }$
－$F$ is independent of $H \cap G$
（a）Show that $\mathrm { P } ( F \cap H \cap G ) = \frac { 1 } { 7 } \mathrm { P } ( G )$
（b）Prove that if two events $X$ and $Y$ are independent，then $X ^ { \prime }$ and $Y$ are also independent．
（c）Hence find the value $k$ such that $\mathrm { P } \left( F ^ { \prime } \cap H \cap G \right) = k \mathrm { P } ( G )$
（d）Show that $c = \frac { 4 } { 3 } a$ Given further that $\mathrm { P } ( F \mid H ) = \frac { 1 } { 5 }$
（e）find an expression for $d$ in terms of $a$ ，and hence deduce the maximum possible value of $a$ ．
（f）Determine the possible values of $n$ ．

Edexcel AEA 2023 June Q6

\hspace{0pt} [In this question you may assume the following formulae for the volume and curved] surface area of a cone of base radius $r$ and height $h$ and of a sphere of radius $r$.

Cone: volume $V = \frac { 1 } { 3 } \pi r ^ { 2 } h$ and curved surface area $S = \pi r \sqrt { h ^ { 2 } + r ^ { 2 } }$
Sphere: volume $V = \frac { 4 \pi } { 3 } r ^ { 3 }$ and curved surface area $S = 4 \pi r ^ { 2 }$
\includegraphics[max width=\textwidth, alt={}, center]{78ba3acc-4cca-4d15-8362-a27e425c5859-22_782_755_637_657} Figure 3
Figure 3 shows the design for a garden ornament.
The ornament is made of a hemisphere on top of a truncated cone.
The truncated cone has base radius $2 r \mathrm {~cm}$, top radius $r \mathrm {~cm}$ and height $4 r \mathrm {~cm}$.
The hemisphere has radius $R \mathrm {~cm}$.
Given that the volume of the ornament is $2100 \pi \mathrm {~cm} ^ { 3 }$

show that $$R ^ { 3 } = 3150 - 14 r ^ { 3 }$$
Find an expression involving $\frac { \mathrm { d } R } { \mathrm {~d} r }$ in terms of $r$ and/or $R$. The base of the truncated cone of the ornament is fixed to the ground.
Show that the visible surface area of the ornament, $A \mathrm {~cm} ^ { 2 }$, is given by $$A = ( 3 \sqrt { 17 } - 1 ) \pi r ^ { 2 } + 3 \pi R ^ { 2 }$$
Hence show that $$\frac { \mathrm { d } A } { \mathrm {~d} r } = \gamma \pi r - \frac { \delta \pi r ^ { 2 } } { R }$$ where $\gamma$ and $\delta$ are real numbers to be determined. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{78ba3acc-4cca-4d15-8362-a27e425c5859-23_705_803_625_630} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows a sketch of $A$ against $r$, for $r \geqslant 0$
There is a local minimum at $r = 0$ and a local maximum at the point $M$. The overall minimum point is at the point $N$, where the gradient of the curve is undefined.
1. Determine the $r$ coordinate of the point $N$.
2. Explain why, for the ornament, $r$ must be less than this value.
Show that the $r$ coordinate of the point $M$ is $$\sqrt [ 3 ] { \frac { p ( 3 \sqrt { 17 } - 1 ) ^ { 3 } } { 3 q ^ { 2 } + ( 3 \sqrt { 17 } - 1 ) ^ { 3 } } }$$ where $p$ and $q$ are integers to be determined.

Edexcel AEA 2023 June Q7

A sequence of non-zero real numbers $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$ is defined by

$$a _ { n + 1 } = p + \frac { q } { a _ { n } } \quad n \in \mathbb { N }$$ where $p$ and $q$ are real numbers with $q \neq 0$
It is known that

one of the terms of this sequence is a
the sequence is periodic
1. Determine an equation for $q$, in terms of $p$ and $a$, such that the sequence is constant (of period/order one).
2. Determine the value of $p$ that is necessary for the sequence to be of period/order 2.
3. Give an example of a sequence that satisfies the condition in part (b), but is not of period/order 2.
4. Determine an equation for $q$, in terms of $p$ only, such that the sequence has period/order 4.

Edexcel AEA 2002 June Q1

1．Solve the following equation，for $0 \leq x \leq \pi$ ，giving your answers in terms of $\pi$ ． $$\sin 5 x - \cos 5 x = \cos x - \sin x$$

Edexcel AEA 2002 June Q2

2．In the binomial expansion of $$( 1 - 4 x ) ^ { p } , \quad | x | < \frac { 1 } { 4 }$$ the coefficient of $x ^ { 2 }$ is equal to the coefficient of $x ^ { 4 }$ and the coefficient of $x ^ { 3 }$ is positive．
Find the value of $p$ ．

Edexcel AEA 2002 June Q3

3．The curve $C$ has parametric equations $$x = 15 t - t ^ { 3 } , \quad y = 3 - 2 t ^ { 2 }$$ Find the values of $t$ at the points where the normal to $C$ at $( 14,1 )$ cuts $C$ again．

Edexcel AEA 2002 June Q4

4．Find the coordinates of the stationary points of the curve with equation $$x ^ { 3 } + y ^ { 3 } - 3 x y = 48$$ and determine their nature．
\includegraphics[max width=\textwidth, alt={}, center]{7f1bc552-3850-43c5-b435-abc87b264f0a-3_553_749_401_618} Figure 1 shows a sketch of part of the curve with equation $$y = \sin ( \cos x ) .$$ The curve cuts the $x$-axis at the points $A$ and $C$ and the $y$-axis at the point $B$.

Find the coordinates of the points $A , B$ and $C$.
Prove that $B$ is a stationary point. Given that the region $O C B$ is convex,
show that, for $0 \leq x \leq \frac { \pi } { 2 }$, $$\sin ( \cos x ) \leq \cos x$$ and $$\left( 1 - \frac { 2 } { \pi } x \right) \sin 1 \leq \sin ( \cos x )$$ and state in each case the value or values of $x$ for which equality is achieved.
Hence show that $$\frac { \pi } { 4 } \sin 1 < \int _ { 0 } ^ { \frac { \pi } { 2 } } \sin ( \cos x ) d x < 1$$
\includegraphics[max width=\textwidth, alt={}]{7f1bc552-3850-43c5-b435-abc87b264f0a-4_682_824_399_704}
Figure 2 shows a sketch of part of two curves $C _ { 1 }$ and $C _ { 2 }$ for $y \geq 0$.
The equation of $C _ { 1 }$ is $y = m _ { 1 } - x ^ { n _ { 1 } }$ and the equation of $C _ { 2 }$ is $y = m _ { 2 } - x ^ { n _ { 2 } }$, where $m _ { 1 }$, $m _ { 2 } , n _ { 1 }$ and $n _ { 2 }$ are positive integers with $m _ { 2 } > m _ { 1 }$. Both $C _ { 1 }$ and $C _ { 2 }$ are symmetric about the line $x = 0$ and they both pass through the points $( 3,0 )$ and $( - 3,0 )$. Given that $n _ { 1 } + n _ { 2 } = 12$, find