Questions - Edexcel - A-Level Maths

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Edexcel FD2 2022 June Q6

14 marks Challenging +1.2

Bernie makes garden sheds. He can build up to four sheds each month.

If he builds more than two sheds in any one month, he must hire an additional worker at a cost of $\pounds 250$ for that month. In any month in which sheds are made, the overhead costs are $\pounds 35$ for each shed made that month. A maximum of three sheds can be held in storage at the end of any one month, at a cost of $\pounds 80$ per shed per month. Sheds must be delivered at the end of the month.
The order schedule for sheds is

Month	January	February	March	April	May
Number ordered	1	3	3	5	2

There are no sheds in storage at the beginning of January and Bernie plans to have no sheds left in storage after the May delivery. Use dynamic programming to determine the production schedule that minimises the costs given above. Complete the working in the table provided in the answer book and state the minimum cost.

Edexcel FD2 2022 June Q7

17 marks Challenging +1.8

\multirow{2}{*}{}		Player B
		Option W	Option X	Option Y	Option Z
\multirow{3}{*}{Player A}	Option Q	4	3	-1	-2
	Option R	-3	5	-4	$k$
	Option S	-1	6	3	-3

A two person zero-sum game is represented by the pay-off matrix for player A shown above. It is given that $k$ is an integer.

Show that Q is the play-safe option for player A regardless of the value of $k$. Given that Z is the play-safe option for player B ,
determine the range of possible values of $k$. You must make your working clear.
Explain why player B should never play option X. You must make your reasoning clear. Player A intends to make a random choice between options $\mathrm { Q } , \mathrm { R }$ and S , choosing option Q with probability $p _ { 1 }$, option R with probability $p _ { 2 }$ and option S with probability $p _ { 3 }$ Player A wants to find the optimal values of $p _ { 1 } , p _ { 2 }$ and $p _ { 3 }$ using the Simplex algorithm.
Given that $k > - 4$, player A formulates the following objective function for the corresponding linear program. $$\text { Maximise } P = V \text {, where } V = \text { the value of the original game } + 4$$
1. Formulate the constraints of the linear programming problem for player A. You should write the constraints as equations.
2. Write down an initial Simplex tableau, making your variables clear. The Simplex algorithm is used to solve the linear programming problem. It is given that in the final Simplex tableau the optimal value of $p _ { 1 } = \frac { 7 } { 37 }$, the optimal value of $p _ { 2 } = \frac { 17 } { 37 }$ and all the slack variables are zero.
Determine the value of $k$, making your method clear.

Edexcel FD2 2022 June Q8

9 marks Standard +0.8

8. The owner of a new company models the number of customers that the company will have at the end of each month. The owner assumes that

a constant proportion, $p$ (where $0 < p < 1$ ), of the previous month's customers will be retained for the next month
a constant number of new customers, $k$, will be added each month.

Let $u _ { n }$ (where $n \geqslant 1$ ) represent the number of customers that the company will have at the end of $n$ months. The company has 5000 customers at the end of the first month.

By setting up a first order recurrence relation for $u _ { n + 1 }$ in terms of $u _ { n }$, determine an expression for $u _ { n }$ in terms of $n , p$ and $k$. The owner believes that $95 \%$ of the previous month's customers will be retained each month and that there will be 10000 new customers each month. According to the model, the company will first have at least 135000 customers by the end of the $m$ th month.
Using logarithms, determine the value of $m$. Please check the examination details below before entering your candidate information \section*{Further Mathematics} Advanced
PAPER 4D: Decision Mathematics 2 \section*{Answer Book} Do not return the question paper with the answer book.
1.
1 2 3 4
A 32 45 34 48
B 37 39 50 46
C 46 44 40 42
D 43 45 48 52
You may not need to use all of these tables.
1 2 3 4
A
B
C
D

1 2 3 4
A
B
C
D

1 2 3 4
A
B
C
D

1 2 3 4
A
B
C
D
VALV SIHI NI IIIIIM ION OC
VIAV SIUL NI JAIIM ION OC
VIAV SIHI NI III IM ION OC
1 2 3 4
A
B
C
D

1 2 3 4
$A$
$B$
$C$
$D$

1 2 3 4
A
B
C
D

1 2 3 4
A
B
C
D

1 2 3 4
A
B
C
D
2.
VAMV SIHI NI IIIHM ION OO VIAV SIHI NI JIIIM I ON OC VJYV SIHI NI JIIIM ION OC
3. 4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cea07472-f93b-4a7b-b362-89fb8c0af4a9-16_936_1317_255_376} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} 5.
$R$ $S$ $T$
$A$
$B$
$C$
$D$

$R$ $S$ $T$
$A$
$B$
$C$
$D$

$R$ $S$ $T$
$A$
$B$
$C$
$D$

$R$ $S$ $T$
$A$
$B$
$C$
$D$

$R$ $S$ $T$
$A$
$B$
$C$
$D$
6.
Stage State Action Destination Value
May 2 0 0 160
1 1 0 80 + 35
0 2 0 70

Stage State Action Destination Value

Month January February March April May
Number made
Minimum cost: $\_\_\_\_$ 7. Player A \begin{table}[h]
\captionsetup{labelformat=empty} \caption{Player B}
Option W Option X Option Y Option Z
Option Q 4 3 -1 -2
Option R -3 5 -4 $k$
Option S -1 6 3 -3
\end{table} The tableau for (d)(ii) can be found at the bottom of page 16
b.v. □ □ □ □ Value
\includegraphics[max width=\textwidth, alt={}]{cea07472-f93b-4a7b-b362-89fb8c0af4a9-24_56_77_2348_182} □ □
P
8.

Edexcel CP AS Specimen Q4

5 marks Standard +0.2

The cubic equation

$$x ^ { 3 } + 3 x ^ { 2 } - 8 x + 6 = 0$$ has roots $\alpha , \beta$ and $\gamma$.
Without solving the equation, find the cubic equation whose roots are $( \alpha - 1 ) , ( \beta - 1 )$ and $( \gamma - 1 )$, giving your answer in the form $w ^ { 3 } + p w ^ { 2 } + q w + r = 0$, where $p , q$ and $r$ are integers to be found.
(5)

Edexcel S2 Q1

Easy -1.8

It is estimated that $4 \%$ of people have green eyes. In a random sample of size $n$, the expected number of people with green eyes is 5 .
1. Calculate the value of $n$.

Edexcel S2 Q8

Standard +0.3

The continuous random variable $X$ has probability density function given by

Edexcel FP1 2023 June Q1

Moderate -0.8

Use Simpson's rule with 4 intervals to find an estimate for $$\int _ { 0 } ^ { 2 } \mathrm { e } ^ { \sin ^ { 2 } x } \mathrm {~d} x$$ Give your answer to 3 significant figures. Given that $\int _ { 0 } ^ { 2 } \mathrm { e } ^ { \mathrm { sin } ^ { 2 } x } \mathrm {~d} x = 3.855$ to 4 significant figures,
comment on the accuracy of your answer to part (a).

Edexcel FP1 2023 June Q2

Standard +0.8

The vertical height, $h \mathrm {~m}$, above horizontal ground, of a passenger on a fairground ride, $t$ seconds after the ride starts, where $t \leqslant 5$, is modelled by the differential equation

$$t ^ { 2 } \frac { \mathrm {~d} ^ { 2 } h } { \mathrm {~d} t ^ { 2 } } - 2 t \frac { \mathrm {~d} h } { \mathrm {~d} t } + 2 h = t ^ { 3 }$$

Given that $t = \mathrm { e } ^ { x }$, show that
1. $t \frac { \mathrm {~d} h } { \mathrm {~d} t } = \frac { \mathrm { d } h } { \mathrm {~d} x }$
2. $t ^ { 2 } \frac { \mathrm {~d} ^ { 2 } h } { \mathrm {~d} t ^ { 2 } } = \frac { \mathrm { d } ^ { 2 } h } { \mathrm {~d} x ^ { 2 } } - \frac { \mathrm { d } h } { \mathrm {~d} x }$
Hence show that the transformation $t = \mathrm { e } ^ { x }$ transforms equation (I) into the equation $$\frac { \mathrm { d } ^ { 2 } h } { \mathrm {~d} x ^ { 2 } } - 3 \frac { \mathrm {~d} h } { \mathrm {~d} x } + 2 h = \mathrm { e } ^ { 3 x }$$
Hence show that $$h = A t + B t ^ { 2 } + \frac { 1 } { 2 } t ^ { 3 }$$ where $A$ and $B$ are constants. Given that when $t = 1 , h = 2.5$ and when $t = 2 , \frac { \mathrm {~d} h } { \mathrm {~d} t } = - 1$
determine the height of the passenger above the ground 5 seconds after the start of the ride.

Edexcel FP1 2023 June Q3

Standard +0.8

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{c0ac1e1e-16bf-4a06-9eaa-8dcf01177722-08_748_814_392_621} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of the curve with equation $y = \frac { x ^ { 2 } - 2 x - 24 } { | x + 6 | }$ and the line with equation $y = 5 - 4 x$ Use algebra to determine the values of $x$ for which $$\frac { x ^ { 2 } - 2 x - 24 } { | x + 6 | } < 5 - 4 x$$

Edexcel FP1 2023 June Q4

Challenging +1.2

The ellipse $E$ has equation

$$\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 9 } = 1$$

Determine the exact value of the eccentricity of $E$ The points $P ( 4 \cos \theta , 3 \sin \theta )$ and $Q ( 4 \cos \theta , - 3 \sin \theta )$ lie on $E$ where $0 < \theta < \frac { \pi } { 2 }$ The line $l _ { 1 }$ is the normal to $E$ at the point $P$
Use calculus to show that $l _ { 1 }$ has equation $$4 x \sin \theta - 3 y \cos \theta = 7 \sin \theta \cos \theta$$ The line $l _ { 2 }$ passes through the origin and the point $Q$ The lines $l _ { 1 }$ and $l _ { 2 }$ intersect at the point $R$
Determine, in simplest form, the coordinates of $R$
Hence show that, as $\theta$ varies, $R$ lies on an ellipse which has the same eccentricity as ellipse $E$

Edexcel FP1 2023 June Q5

Challenging +1.2

Show that the substitution $t = \tan \left( \frac { x } { 2 } \right)$ transforms the integral $$\int \frac { 1 } { 2 \sin x - \cos x + 5 } d x$$ into the integral $$\int \frac { 1 } { 3 t ^ { 2 } + 2 t + 2 } \mathrm {~d} t$$
Hence determine $$\int \frac { 1 } { 2 \sin x - \cos x + 5 } d x$$

Edexcel FP1 2023 June Q6

Challenging +1.2

6. $$y = \ln \left( \mathrm { e } ^ { 2 x } \cos 3 x \right) \quad - \frac { 1 } { 2 } < x < \frac { 1 } { 2 }$$

Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 - 3 \tan 3 x$$
Determine $\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } }$
Hence determine the first 3 non-zero terms in ascending powers of $x$ of the Maclaurin series expansion of $\ln \left( \mathrm { e } ^ { 2 x } \cos 3 x \right)$, giving each coefficient in simplest form.
Use the Maclaurin series expansion for $\ln ( 1 + x )$ to write down the first 4 non-zero terms in ascending powers of $x$ of the Maclaurin series expansion of $\ln ( 1 + k x )$, where $k$ is a constant.
Hence determine the value of $k$ for which $$\lim _ { x \rightarrow 0 } \left( \frac { 1 } { x ^ { 2 } } \ln \frac { \mathrm { e } ^ { 2 x } \cos 3 x } { 1 + k x } \right)$$ exists.

Edexcel FP1 2023 June Q7

Challenging +1.8

With respect to a fixed origin $O$ the point $A$ has coordinates $( 3,6,5 )$ and the line $l$ has equation

$$( \mathbf { r } - ( 12 \mathbf { i } + 30 \mathbf { j } + 39 \mathbf { k } ) ) \times ( 7 \mathbf { i } + 13 \mathbf { j } + 24 \mathbf { k } ) = \mathbf { 0 }$$ The points $B$ and $C$ lie on $l$ such that $A B = A C = 15$ Given that $A$ does not lie on $l$ and that the $x$ coordinate of $B$ is negative,

determine the coordinates of $B$ and the coordinates of $C$
Hence determine a Cartesian equation of the plane containing the points $A , B$ and $C$ The point $D$ has coordinates $( - 2,1 , \alpha )$, where $\alpha$ is a constant.
Given that the volume of the tetrahedron $A B C D$ is 147
determine the possible values of $\alpha$ Given that $\alpha > 0$
determine the shortest distance between the line $l$ and the line passing through the points $A$ and $D$, giving your answer to 2 significant figures. \includegraphics[max width=\textwidth, alt={}, center]{c0ac1e1e-16bf-4a06-9eaa-8dcf01177722-24_2267_50_312_1980}

Edexcel PURE 2024 October Q1

Moderate -0.8

The line $l _ { 1 }$ passes through the point $A ( - 5,20 )$ and the point $B ( 3 , - 4 )$.

Find an equation for $l _ { 1 }$ giving your answer in the form $y = m x + c$, where $m$ and $c$ are constants. The line $l _ { 2 }$ is perpendicular to $l _ { 1 }$ and passes through the midpoint of $A B$
Find an equation for $l _ { 2 }$ giving your answer in the form $p x + q y + r = 0$, where $p , q$ and $r$ are integers.

Edexcel PURE 2024 October Q2

Easy -1.2

In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

Simplify fully $$\frac { 3 y ^ { 3 } \left( 2 x ^ { 4 } \right) ^ { 3 } } { 4 x ^ { 2 } y ^ { 4 } }$$
Find the exact value of $a$ such that $$\frac { 16 } { \sqrt { 3 } + 1 } = a \sqrt { 27 } + 4$$ Write your answer in the form $p \sqrt { 3 } + q$ where $p$ and $q$ are fully simplified rational constants.

Edexcel PURE 2024 October Q3

Standard +0.3

In this question you must show all stages of your working.

$$f ( x ) = \frac { ( x + 5 ) ^ { 2 } } { \sqrt { x } } \quad x > 0$$

Find $\int f ( x ) d x$
1. Show that when $\mathrm { f } ^ { \prime } ( x ) = 0$ $$3 x ^ { 2 } + 10 x - 25 = 0$$
2. Hence state the value of $x$ for which $$\mathrm { f } ^ { \prime } ( x ) = 0$$

Edexcel PURE 2024 October Q4

Moderate -0.3

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{c48e6503-9d26-4f55-bdca-feadfb1afb7c-10_812_853_255_607} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of part of the curves $C _ { 1 }$ and $C _ { 2 }$ Given that $C _ { 1 }$

has equation $y = \mathrm { f } ( x )$ where $\mathrm { f } ( x )$ is a quadratic function
cuts the $x$-axis at the origin and at $x = 4$
has a minimum turning point at ( $2 , - 4.8$ )
1. find $\mathrm { f } ( x )$

Given that $C _ { 2 }$

The curves $C _ { 1 }$ and $C _ { 2 }$ meet in the first quadrant at the point $P$, shown in Figure 1.

Use algebra to find the coordinates of $P$.

Edexcel PURE 2024 October Q5

Standard +0.8

A plot of land $O A B$ is in the shape of a sector of a circle with centre $O$.

Given

$O A = O B = 5 \mathrm {~km}$
angle $A O B = 1.2$ radians
1. find the perimeter of the plot of land.
  (2)

\begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{c48e6503-9d26-4f55-bdca-feadfb1afb7c-14_609_650_664_705} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A point $P$ lies on $O B$ such that the line $A P$ divides the plot of land into two regions $R _ { 1 }$ and $R _ { 2 }$ as shown in Figure 2. Given that $$\text { area of } R _ { 1 } = 3 \times \text { area of } R _ { 2 }$$

show that the area of $R _ { 2 } = 3.75 \mathrm {~km} ^ { 2 }$

Find the length of $A P$, giving your answer to the nearest 100 m .

Edexcel PURE 2024 October Q6

Standard +0.8

In this question you must show all stages of your working.

\section*{Solutions relying on calculator technology are not acceptable.}

Sketch the curve $C$ with equation $$y = \frac { 1 } { 2 - x } \quad x \neq 2$$ State on your sketch
- the equation of the vertical asymptote
- the coordinates of the intersection of $C$ with the $y$-axis
The straight line $l$ has equation $y = k x - 4$, where $k$ is a constant.
Given that $l$ cuts $C$ at least once,
1. show that $$k ^ { 2 } - 5 k + 4 \geqslant 0$$
2. find the range of possible values for $k$.

Edexcel PURE 2024 October Q7

Easy -1.2

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{c48e6503-9d26-4f55-bdca-feadfb1afb7c-22_841_999_251_534} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Figure 3 shows a plot of part of the curve $C _ { 1 }$ with equation $$y = - 4 \cos x$$ where $x$ is measured in radians.
Points $P$ and $Q$ lie on the curve and are shown in Figure 3.

State
1. the coordinates of $P$
2. the coordinates of $Q$ The curve $C _ { 2 }$ has equation $y = - 4 \cos x + k$ where $x$ is measured in radians and $k$ is a constant. Given that $C _ { 2 }$ has a maximum $y$ value of 11
1. state the value of $k$
2. state the coordinates of the minimum point on $C _ { 2 }$ with the smallest positive $x$ coordinate. On the opposite page there is a copy of Figure 3 labelled Diagram 1.
Using Diagram 1, state the number of solutions of the equation $$- 4 \cos x = 5 - \frac { 10 } { \pi } x$$ giving a reason for your answer. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c48e6503-9d26-4f55-bdca-feadfb1afb7c-23_860_1016_1676_529} \captionsetup{labelformat=empty} \caption{Diagram 1}
\end{figure}

Edexcel PURE 2024 October Q8

Standard +0.3

A curve $C$ has equation $y = \mathrm { f } ( x )$.

The point $P$ with $x$ coordinate 3 lies on $C$ \section*{Given}

$\mathrm { f } ^ { \prime } ( x ) = 4 x ^ { 2 } + k x + 3$ where $k$ is a constant
the normal to $C$ at $P$ has equation $y = - \frac { 1 } { 24 } x + 5$
1. show that $k = - 5$
2. Hence find $\mathrm { f } ( x )$.

Edexcel PURE 2024 October Q9

Moderate -0.3

9. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{c48e6503-9d26-4f55-bdca-feadfb1afb7c-26_732_730_251_669} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} Figure 4 shows a sketch of the curve $C$ with equation $y = \mathrm { f } ( x )$, where $$f ( x ) = ( x + 5 ) \left( 3 x ^ { 2 } - 4 x + 20 \right)$$

Deduce the range of values of $x$ for which $\mathrm { f } ( x ) \geqslant 0$
Find $\mathrm { f } ^ { \prime } ( x )$ giving your answer in simplest form. The point $R ( - 4,84 )$ lies on $C$.
Given that the tangent to $C$ at the point $P$ is parallel to the tangent to $C$ at the point $R$ (c) find the $x$ coordinate of $P$.
(d) Find the point to which $R$ is transformed when the curve with equation $y = \mathrm { f } ( x )$ is transformed to the curve with equation,
1. $y = \mathrm { f } ( x - 3 )$
2. $y = 4 \mathrm { f } ( x )$

Edexcel PURE 2024 October Q1

Moderate -0.8

A continuous curve has equation $y = \mathrm { f } ( x )$.

A table of values of $x$ and $y$ for $y = \mathrm { f } ( x )$ is shown below.

$x$	0.5	1.75	3	4.25	5.5
$y$	3.479	6.101	7.448	6.823	5.182

Using the trapezium rule with all the values of $y$ in the given table,

find an estimate for $$\int _ { 0.5 } ^ { 5.5 } \mathrm { f } ( x ) \mathrm { d } x$$ giving your answer to one decimal place.
Using your answer to part (a) and making your method clear, estimate $$\int _ { 0.5 } ^ { 5.5 } ( \mathrm { f } ( x ) + 4 x ) \mathrm { d } x$$

Edexcel PURE 2024 October Q2

Standard +0.8

A sequence of numbers $u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots$ is defined by

$$\begin{gathered} u _ { 1 } = 7 \\ u _ { n + 1 } = ( - 1 ) ^ { n } u _ { n } + k \end{gathered}$$ where $k$ is a constant.

Show that $u _ { 5 } = 7$ Given that $\sum _ { r = 1 } ^ { 4 } u _ { r } = 30$
find the value of $k$.
Hence find the value of $\sum _ { r = 1 } ^ { 150 } u _ { r }$

Edexcel PURE 2024 October Q3

Moderate -0.8

3. $$f ( x ) = 2 x ^ { 3 } - x ^ { 2 } + A x + B$$ where $A$ and $B$ are integers.
Given that when $\mathrm { f } ( x )$ is divided by $( x + 3 )$ the remainder is 55

show that $$3 A - B = - 118$$ Given also that $( 2 x - 5 )$ is a factor of $\mathrm { f } ( x )$,
find the value of $A$ and the value of $B$.
Hence find the quotient when $\mathrm { f } ( x )$ is divided by ( $x - 7$ )

	1	2	3	4
\(A\)
\(B\)
\(C\)
\(D\)

	\(R\)	\(S\)	\(T\)
\(A\)
\(B\)
\(C\)
\(D\)

	\(R\)	\(S\)	\(T\)
\(A\)
\(B\)
\(C\)
\(D\)

	\(R\)	\(S\)	\(T\)
\(A\)
\(B\)
\(C\)
\(D\)

	\(R\)	\(S\)	\(T\)
\(A\)
\(B\)
\(C\)
\(D\)

b.v.			□	□	□	□	Value
\includegraphics[max width=\textwidth, alt={}]{cea07472-f93b-4a7b-b362-89fb8c0af4a9-24_56_77_2348_182}	□	□



P

Stage	State	Action	Destination	Value
May	2	0	0	160
	1	1	0	80 + 35
	0	2	0	70

Stage	State	Action	Destination	Value

Questions — Edexcel (10514 questions)

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Edexcel FD2 2022 June Q6

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Edexcel CP AS Specimen Q4

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Edexcel S2 Q8

Edexcel FP1 2023 June Q1

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Edexcel PURE 2024 October Q1

Edexcel PURE 2024 October Q2

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Edexcel PURE 2024 October Q6

Edexcel PURE 2024 October Q7

Edexcel PURE 2024 October Q8

Edexcel PURE 2024 October Q9

Edexcel PURE 2024 October Q1

Edexcel PURE 2024 October Q2

Edexcel PURE 2024 October Q3