Questions - Edexcel - A-Level Maths

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Edexcel FD2 2024 June Q4

9 marks Standard +0.8

Four workers, A, B, C and D, are to be assigned to four tasks, P, Q, R and S.

Each task must be assigned to just one worker and each worker can do only one task.
Worker B cannot be assigned to task Q and worker D cannot be assigned to task R.
The amount, in pounds, that each worker would earn when assigned to each task is shown in the table below.

	P	Q	R	S
A	65	72	69	75
B	71	-	68	65
C	70	69	73	77
D	73	70	-	71

The Hungarian algorithm can be used to find the maximum total amount that would be earned by the four workers.

1. Explain how to modify the table so that the Hungarian algorithm could be applied.
2. Modify the table as described in (a)(i).
Formulate the above situation as a linear programming problem. You must define the decision variables and make the objective function and constraints clear.

Edexcel FD2 2024 June Q5

10 marks Standard +0.3

5. Sebastien needs to make a journey. He can choose between travelling by plane, by train or by coach. Sebastien knows the exact costs of all three travel options, but he also wants to account for his travel time, including any possible delays. The cost of Sebastien's time is $\pounds 50$ per hour.
The table below shows the costs, the journey times (without delays), and the corresponding probabilities of delays, for each travel option.

	Cost of travel option	Journey time (in hours) without delays	Probability of a 1-hour delay	Probability of a 2-hour delay	Probability of a 3-hour delay	Probability of a 24-hour delay
Plane	£200	3	0.09	0.05	0	0.03
Train	£130	5	0.07	0.03	0	0
Coach	£70	6	0.15	0.1	0.05	0

By drawing a decision tree, evaluate the EMV of the total cost of Sebastien's journey for each node of your tree.
Hence state the travel option that minimises the EMV of the total cost of Sebastien's journey.
A cube root utility function is applied to the total costs of each option. Determine the travel option with the best expected utility and state the corresponding value.

Edexcel FD2 2024 June Q6

10 marks Standard +0.3

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{931ccf1d-4b02-448c-b492-846b0f42c057-07_709_1507_214_280} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} The staged, directed network in Figure 2 represents the roads that connect 12 towns, S, A, B, C, D, E, F, G, H, I, J and T. The number on each arc shows the time, in hours, it takes to drive between these towns. Elena plans to drive from S to T . She must arrive at T by 9 pm .

By completing the table in the answer book, use dynamic programming to find the latest time that Elena can start her journey from S to arrive at T by 9 pm .
Hence write down the route that Elena should take.

Edexcel FD2 2024 June Q7

13 marks Challenging +1.2

\multirow{2}{*}{}		Player B
		Option X	Option Y	Option Z
\multirow{3}{*}{Player A}	Option R	3	2	-3
	Option S	4	-2	1
	Option T	-1	3	6

A two person zero-sum game is represented by the pay-off matrix for player A, shown above.

Verify that there is no stable solution to this game. Player A intends to make a random choice between options $\mathrm { R } , \mathrm { S }$ and T , choosing option R with probability $p _ { 1 }$, option S with probability $p _ { 2 }$ and option T with probability $p _ { 3 }$ Player A wants to find the optimal values of $p _ { 1 } , p _ { 2 }$ and $p _ { 3 }$ using the Simplex algorithm.
Player A formulates the following objective function for the corresponding linear programme. $$\text { Maximise } P = V \quad \text { where } V = \text { the value of the game } + 3$$

Determine an initial Simplex tableau, making your variables and working clear. After several iterations of the Simplex algorithm, a possible final tableau is

b.v.	$V$	$p _ { 1 }$	$p _ { 2 }$	$p _ { 3 }$	r	$s$	$t$	$u$	Value
$p _ { 3 }$	0	0	0	1	$\frac { 1 } { 10 }$	$- \frac { 3 } { 80 }$	$- \frac { 1 } { 16 }$	$\frac { 33 } { 80 }$	$\frac { 33 } { 80 }$
$p _ { 2 }$	0	0	1	0	$- \frac { 1 } { 10 }$	$\frac { 13 } { 80 }$	$- \frac { 1 } { 16 }$	$\frac { 17 } { 80 }$	$\frac { 17 } { 80 }$
V	1	0	0	0	$\frac { 1 } { 2 }$	$\frac { 5 } { 16 }$	$\frac { 3 } { 16 }$	$\frac { 73 } { 16 }$	$\frac { 73 } { 16 }$
$p _ { 1 }$	0	1	0	0	0	$- \frac { 1 } { 8 }$	$\frac { 1 } { 8 }$	$\frac { 3 } { 8 }$	$\frac { 3 } { 8 }$
$P$	0	0	0	0	$\frac { 1 } { 2 }$	$\frac { 5 } { 16 }$	$\frac { 3 } { 16 }$	$\frac { 73 } { 16 }$	$\frac { 73 } { 16 }$

1. State the best strategy for player A.
2. Calculate the value of the game for player B. Player B intends to make a random choice between options $\mathrm { X } , \mathrm { Y }$ and Z .
Determine the best strategy for player B, making your method and working clear.
(3)

Edexcel FD2 2024 June Q8

8 marks Challenging +1.2

8. A sequence $\left\{ u _ { n } \right\}$, where $n \geqslant 0$, satisfies the recurrence relation $$2 u _ { n + 2 } + 5 u _ { n + 1 } = 3 u _ { n } + 8 n + 2$$

Find the general solution of this recurrence relation.
(5) A particular solution of this recurrence relation has $u _ { 0 } = 1$ and $u _ { 1 } = k$, where $k$ is a positive constant. All terms of the sequence are positive.
Determine the value of $k$.
(3)

Edexcel FD2 Specimen Q1

6 marks Standard +0.8

Find the general solution of the recurrence relation $$u _ { n + 2 } = u _ { n + 1 } + u _ { n } , \quad n \geqslant 1$$ Given that $u _ { 1 } = 1$ and $u _ { 2 } = 1$
find the particular solution of the recurrence relation.

Edexcel FD2 Specimen Q2

12 marks Challenging +1.2

	D	E	F	Available
A	15	19	9	25
B	11	18	10	55
C	11	12	18	20
Required	38	24	38

A company has three factories, $\mathrm { A } , \mathrm { B }$ and C . It supplies mattresses to three shops, $\mathrm { D } , \mathrm { E }$ and F . The table shows the transportation cost, in pounds, of moving one mattress from each factory to each shop. It also shows the number of mattresses available at each factory and the number of mattresses required at each shop. A minimum cost solution is required.

Use the north-west corner method to obtain an initial solution.
Show how the transportation algorithm is used to solve this problem. You must state, at each appropriate step, the

Edexcel FD2 Specimen Q3

13 marks Moderate -0.5

Four workers, A, B, C and D, are to be assigned to four tasks, P, Q, R and S.

Each worker must be assigned to at most one task and each task must be done by just one worker. The amount, in pounds, that each worker would earn while assigned to each task is shown in the table below.

\cline { 2 - 5 } \multicolumn{1}{c\|}{}	P	Q	R	S
A	32	32	33	35
B	28	35	31	37
C	35	29	33	36
D	36	30	36	33

The Hungarian algorithm is to be used to find the maximum total amount which may be earned by the four workers.

Explain how the table should be modified.
Reducing rows first, use the Hungarian algorithm to obtain an allocation which maximises the total earnings, stating how each table was formed.
Formulate the problem as a linear programming problem. You must define your decision variables and make your objective function and constraints clear.

Edexcel FD2 Specimen Q4

8 marks Moderate -0.3

4. A game uses a standard pack of 52 playing cards. A player gives 5 tokens to play and then picks a card. If they pick a $2,3,4,5$ or 6 then they gain 15 tokens. If any other card is picked they lose. If they lose, the card is replaced and they can choose to pick again for another 5 tokens. This time if they pick either an ace or a king they gain 40 tokens. If any other card is picked they lose. Daniel is deciding whether to play this game.

Draw a decision tree to model Daniel's possible decisions and the possible outcomes.
Calculate Daniel's optimal EMV and state the optimal strategy indicated by the decision tree.

Edexcel FD2 Specimen Q5

12 marks Challenging +1.2

	B plays 1	B plays 2	B plays 3	B plays 4
A plays 1	4	-2	3	2
A plays 2	3	-1	2	0
A plays 3	-1	2	0	3

A two person zero-sum game is represented by the pay-off matrix for player A given above.

Explain, with justification, how this matrix may be reduced to a $3 \times 3$ matrix.
Find the play-safe strategy for each player and verify that there is no stable solution to this game. The game is formulated as a linear programming problem for player A .
The objective is to maximise $P = V$, where $V$ is the value of the game to player A.
One of the constraints is that $p _ { 1 } + p _ { 2 } + p _ { 3 } \leqslant 1$, where $p _ { 1 } , p _ { 2 } , p _ { 3 }$ are the probabilities that player A plays 1, 2, 3 respectively.
Formulate the remaining constraints for this problem. Write these constraints as inequalities. The Simplex algorithm is used to solve the linear programming problem.
The solution obtained is $p _ { 1 } = 0 , p _ { 2 } = \frac { 3 } { 7 } , p _ { 3 } = \frac { 4 } { 7 }$
Calculate the value of the game to player A.

Edexcel FD2 Specimen Q6

12 marks Challenging +1.8

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{a2bc4f5d-f7db-4ce7-860b-f53a743c7e2c-7_821_1433_205_317} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a capacitated, directed network. The number on each arc $( x , y )$ represents the lower $( x )$ capacity and upper $( y )$ capacity of that arc.

Calculate the value of the cut $C _ { 1 }$ and cut $C _ { 2 }$
Explain why the flow through the network must be at least 12 and at most 16
Explain why arcs DG, AG, EG and FG must all be at their lower capacities.
Determine a maximum flow pattern for this network and draw it on Diagram 1 in the answer book. You do not need to use the labelling procedure.
1. State the value of the maximum flow through the network.
2. Explain why the value of the maximum flow is equal to the value of the minimum flow through the network. Node E becomes blocked and no flow can pass through it. To maintain the maximum flow through the network the upper capacity of exactly one arc is increased.
Explain how it is possible to maintain the maximum flow found in (d).

Edexcel FD2 Specimen Q7

12 marks Challenging +1.2

7. A company assembles boats. They can assemble up to five boats in any one month, but if they assemble more than three they will have to hire additional space at a cost of $\pounds 800$ per month. The company can store up to two boats at a cost of $\pounds 350$ each per month.
The overhead costs are $\pounds 1500$ in any month in which work is done.
Boats are delivered at the end of each month. There are no boats in stock at the beginning of January and there must be none in stock at the end of May. The order book for boats is

Month	January	February	March	April	May
Number ordered	3	2	6	3	4

Use dynamic programming to determine the production schedule which minimises the costs to the company. Show your working in the table provided in the answer book and state the minimum production cost.

Edexcel AEA 2012 June Q4

11 marks Challenging +1.8

4. $$\mathbf { a } = \left( \begin{array} { r } - 3 \\ 1 \\ 4 \end{array} \right) , \quad \mathbf { b } = \left( \begin{array} { r } 5 \\ - 2 \\ 9 \end{array} \right) , \quad \mathbf { c } = \left( \begin{array} { r } 8 \\ - 4 \\ 3 \end{array} \right)$$ The points $A , B$ and $C$ with position vectors $\mathbf { a } , \mathbf { b }$ and $\mathbf { c }$ ,respectively,are 3 vertices of a cube.

Find the volume of the cube. The points $P , Q$ and $R$ are vertices of a second cube with $\overrightarrow { P Q } = \left( \begin{array} { l } 3 \\ 4 \\ \alpha \end{array} \right) , \overrightarrow { P R } = \left( \begin{array} { l } 7 \\ 1 \\ 0 \end{array} \right)$ and $\alpha$ a positive constant.
Given that angle $Q P R = 60 ^ { \circ }$ ,find the value of $\alpha$ .
Find the length of a diagonal of the second cube.

Edexcel AEA 2012 June Q5

14 marks Challenging +1.8

5.[In this question the values of $a , x$ ,and $n$ are such that $a$ and $x$ are positive real numbers,with $a > 1 , x \neq a , x \neq 1$ and $n$ is an integer with $n > 1$ ] Sam was confused about the rules of logarithms and thought that $$\log _ { a } x ^ { n } = \left( \log _ { a } x \right) ^ { n }$$

Given that $x$ satisfies statement(1)find $x$ in terms of $a$ and $n$ . Sam also thought that $$\log _ { a } x + \log _ { a } x ^ { 2 } + \ldots + \log _ { a } x ^ { n } = \log _ { a } x + \left( \log _ { a } x \right) ^ { 2 } + \ldots + \left( \log _ { a } x \right) ^ { n }$$
For $n = 3 , x _ { 1 }$ and $x _ { 2 } \left( x _ { 1 } > x _ { 2 } \right)$ are the two values of $x$ that satisfy statement(2).
1. Find,in terms of $a$ ,an expression for $x _ { 1 }$ and an expression for $x _ { 2 }$ .
2. Find the exact value of $\log _ { a } \left( \frac { x _ { 1 } } { x _ { 2 } } \right)$ .
Show that if $\log _ { a } x$ satisfies statement(2)then $$2 \left( \log _ { a } x \right) ^ { n } - n ( n + 1 ) \log _ { a } x + \left( n ^ { 2 } + n - 2 \right) = 0$$

Edexcel AEA 2012 June Q7

24 marks Hard +2.3

7. $\left[ \arccos x \right.$ and $\arctan x$ are alternative notation for $\cos ^ { - 1 } x$ and $\tan ^ { - 1 } x$ respectively $]$ \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{fc5d0d07-b750-4646-bdcb-419a290200c9-5_387_935_322_566} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a sketch of the curve $C _ { 1 }$ with equation $y = \cos ( \cos x ) , 0 \leqslant x < 2 \pi$ .
The curve has turning points at $( 0 , \cos 1 ) , P , Q$ and $R$ as shown in Figure 2.

Find the coordinates of the points $P , Q$ and $R$ . The curve $C _ { 2 }$ has equation $y = \sin ( \cos x ) , 0 \leqslant x < 2 \pi$ .The curves $C _ { 1 }$ and $C _ { 2 }$ intersect at the points $S$ and $T$ .
Copy Figure 2 and on this diagram sketch $C _ { 2 }$ stating the coordinates of the minimum point on $C _ { 2 }$ and the points where $C _ { 2 }$ meets or crosses the coordinate axes. The coordinates of $S$ are $( \alpha , d )$ where $0 < \alpha < \pi$ .
Show that $\alpha = \arccos \left( \frac { \pi } { 4 } \right)$ .
Find the value of $d$ in surd form and write down the coordinates of $T$ . The tangent to $C _ { 1 }$ at the point $S$ has gradient $\tan \beta$ .
Show that $\beta = \arctan \sqrt { } \left( \frac { 16 - \pi ^ { 2 } } { 32 } \right)$ .
Find,in terms of $\beta$ ,the obtuse angle between the tangent to $C _ { 1 }$ at $S$ and the tangent to $C _ { 2 }$ at $S$ .

Edexcel C12 2018 January Q8

6 marks Moderate -0.5

$y = \mathrm { f } ( - x )$
$y = \mathrm { f } ( 2 x )$ On each diagram, show clearly the coordinates of any points of intersection of the curve with the two coordinate axes and the coordinates of the stationary points.

Edexcel C1 2006 January Q6

9 marks Moderate -0.8

$y = \mathrm { f } ( x + 1 )$,
$y = 2 \mathrm { f } ( x )$,
$y = \mathrm { f } \left( \frac { 1 } { 2 } x \right)$. On each diagram show clearly the coordinates of all the points where the curve meets the axes.

Edexcel C2 2013 June Q7

9 marks Moderate -0.3

Find by calculation the $x$-coordinate of $A$ and the $x$-coordinate of $B$. The shaded region $R$ is bounded by the line with equation $y = 10$ and the curve as shown in Figure 1.
Use calculus to find the exact area of $R$.

Edexcel P3 2022 October Q8

9 marks Standard +0.3

Express $8 \sin x - 15 \cos x$ in the form $R \sin ( x - \alpha )$, where $R > 0$ and $0 < \alpha < \frac { \pi } { 2 }$ Give the exact value of $R$, and give the value of $\alpha$, in radians, to 4 significant figures. $$\mathrm { f } ( x ) = \frac { 15 } { 41 + 16 \sin x - 30 \cos x } \quad x > 0$$
Find
1. the minimum value of $\mathrm { f } ( x )$
2. the smallest value of $x$ at which this minimum value occurs.
State the $y$ coordinate of the minimum points on the curve with equation $$y = 2 \mathrm { f } ( x ) - 5 \quad x > 0$$
State the smallest value of $x$ at which a maximum point occurs for the curve with equation $$y = - \mathrm { f } ( 2 x ) \quad x > 0$$ \section*{8. In this question you must show all stages of your working.
In this question you must show all stages of your working.}

Edexcel P4 2021 October Q8

7 marks Standard +0.8

Find $\int x ^ { 2 } \ln x \mathrm {~d} x$ Figure 3 shows a sketch of part of the curve with equation $$y = x \ln x \quad x > 0$$ The region $R$, shown shaded in Figure 3, lies entirely above the $x$-axis and is bounded by the curve, the $x$-axis and the line with equation $x = \mathrm { e }$. This region is rotated through $2 \pi$ radians about the $x$-axis to form a solid of revolution.
Find the exact volume of the solid formed, giving your answer in simplest form. \section*{8. In this question you must show all stages of your working.
In this question you must show all stages of your working.}

Edexcel C4 2013 January Q5

15 marks Moderate -0.3

Show that $A$ has coordinates $( 0,3 )$.
Find the $x$ coordinate of the point $B$.
Find an equation of the normal to $C$ at the point $A$. The region $R$, as shown shaded in Figure 2, is bounded by the curve $C$, the line $x = - 1$ and the $x$-axis.
Use integration to find the exact area of $R$.

Edexcel M3 2003 January Q3

10 marks Challenging +1.2

Show that the distance $d$ of the centre of mass of the toy from its lowest point $O$ is given by $$d = \frac { h ^ { 2 } + 2 h r + 5 r ^ { 2 } } { 2 ( h + 4 r ) } .$$ When the toy is placed with any point of the curved surface of the hemisphere resting on the plane it will remain in equilibrium.
Find $h$ in terms of $r$.
(3)

Edexcel D1 2002 June Q4

10 marks Moderate -0.3

Use Dijkstra's algorithm to find the shortest route from $A$ to $I$. Show all necessary working in the boxes in the answer booklet and state your shortest route and its length.
(5) The park warden wishes to check each of the paths to check for frost damage. She has to cycle along each path at least once, starting and finishing at $A$.
1. Use an appropriate algorithm to find which paths will be covered twice and state these paths.
2. Find a route of minimum length.
3. Find the total length of this shortest route.
  (5)

Edexcel D1 2002 November Q4

7 marks Standard +0.3

Use the Route Inspection algorithm to find which paths, if any, need to be traversed twice. It is decided to start the inspection at node $C$. The inspection must still traverse each pipe at least once but may finish at any node.
Explaining your reasoning briefly, determine the node at which the inspection should finish if the route is to be minimised. State the length of your route.
(3)

Edexcel D1 2004 November Q5

10 marks Standard +0.3

find the route the driver should follow, starting and ending at $F$, to clear all the roads of snow. Give the length of this route. The local authority decides to build a road bridge over the river at $B$. The snowplough will be able to cross the road bridge.
Reapply the algorithm to find the minimum distance the snowplough will have to travel (ignore the length of the new bridge). \section*{6.} \section*{Figure 3}
\includegraphics[max width=\textwidth, alt={}]{4bbe6272-3900-42de-b287-599638ca75e4-07_1131_1118_347_502}
Peter wishes to minimise the time spent driving from his home $H$, to a campsite at $G$. Figure 3 shows a number of towns and the time, in minutes, taken to drive between them. The volume of traffic on the roads into $G$ is variable, and so the length of time taken to drive along these roads is expressed in terms of $x$, where $x \geq 0$.
1. On the diagram in the answer book, use Dijkstra's algorithm to find two routes from $H$ to $G$ (one via $A$ and one via $B$ ) that minimise the travelling time from $H$ to $G$. State the length of each route in terms of $x$.
2. Find the range of values of $x$ for which Peter should follow the route via $A$. \section*{7.} \begin{figure}[h]
  \captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{4bbe6272-3900-42de-b287-599638ca75e4-08_1495_1335_322_392}
  \end{figure} The company EXYCEL makes two types of battery, X and Y . Machinery, workforce and predicted sales determine the number of batteries EXYCEL make. The company decides to use a graphical method to find its optimal daily production of X and Y . The constraints are modelled in Figure 4 where $$\begin{aligned} & x = \text { the number (in thousands) of type } \mathrm { X } \text { batteries produced each day, } \\ & y = \text { the number (in thousands) of type } \mathrm { Y } \text { batteries produced each day. } \end{aligned}$$ The profit on each type X battery is 40 p and on each type Y battery is 20 p . The company wishes to maximise its daily profit.
3. Write this as a linear programming problem, in terms of $x$ and $y$, stating the objective function and all the constraints.
4. Find the optimal number of batteries to be made each day. Show your method clearly.
5. Find the daily profit, in $\pounds$, made by EXYCEL.

b.v.	\(V\)	\(p _ { 1 }\)	\(p _ { 2 }\)	\(p _ { 3 }\)	r	\(s\)	\(t\)	\(u\)	Value
\(p _ { 3 }\)	0	0	0	1	\(\frac { 1 } { 10 }\)	\(- \frac { 3 } { 80 }\)	\(- \frac { 1 } { 16 }\)	\(\frac { 33 } { 80 }\)	\(\frac { 33 } { 80 }\)
\(p _ { 2 }\)	0	0	1	0	\(- \frac { 1 } { 10 }\)	\(\frac { 13 } { 80 }\)	\(- \frac { 1 } { 16 }\)	\(\frac { 17 } { 80 }\)	\(\frac { 17 } { 80 }\)
V	1	0	0	0	\(\frac { 1 } { 2 }\)	\(\frac { 5 } { 16 }\)	\(\frac { 3 } { 16 }\)	\(\frac { 73 } { 16 }\)	\(\frac { 73 } { 16 }\)
\(p _ { 1 }\)	0	1	0	0	0	\(- \frac { 1 } { 8 }\)	\(\frac { 1 } { 8 }\)	\(\frac { 3 } { 8 }\)	\(\frac { 3 } { 8 }\)
\(P\)	0	0	0	0	\(\frac { 1 } { 2 }\)	\(\frac { 5 } { 16 }\)	\(\frac { 3 } { 16 }\)	\(\frac { 73 } { 16 }\)	\(\frac { 73 } { 16 }\)

Questions — Edexcel (10514 questions)

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