Questions - Edexcel - A-Level Maths

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

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．
（a）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．
（b）Given that angle $Q P R = 60 ^ { \circ }$ ，find the value of $\alpha$ ．
（c）Find the length of a diagonal of the second cube．

Edexcel AEA 2012 June Q5

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

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.
（a）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 $T$ ．
（b）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 $0 < \alpha < \pi$ ．
（c）Show that $\alpha = \arccos \left( \frac { \pi } { 4 } \right)$ ．
（d）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 $\tan \beta$ ．
（e）Show that $\beta = \arctan \sqrt { } \left( \frac { 16 - \pi ^ { 2 } } { 32 } \right)$ ．
（f）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

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

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 C1 Specimen Q2

Moderate -0.8

$y = \mathrm { f } ( x + 1 )$,
$y = \mathrm { f } ( 2 x )$. On each diagram, show clearly the coordinates of the maximum point, and of each point at which the curve crosses the coordinate axes.

Edexcel C2 2013 June Q7

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

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 2024 June Q6

show that \end{itemize} $$\frac { \mathrm { d } A } { \mathrm {~d} \theta } = K ( 1 - \cos \theta )$$ where $K$ is a constant to be found.
Find, in $\mathrm { cm } ^ { 2 } \mathrm {~s} ^ { - 1 }$, the rate of increase of the area of the segment when $\theta = \frac { \pi } { 3 }$ 5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e583bf92-d6a9-4f1a-b3c8-372afa6e0a0e-10_803_1086_248_493} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of the curve defined by the parametric equations $$x = t ^ { 2 } + 2 t \quad y = \frac { 2 } { t ( 3 - t ) } \quad a \leqslant t \leqslant b$$ where $a$ and $b$ are constants.
The ends of the curve lie on the line with equation $y = 1$
Find the value of $a$ and the value of $b$ The region $R$, shown shaded in Figure 2, is bounded by the curve and the line with equation $y = 1$
Show that the area of region $R$ is given by $$M - k \int _ { a } ^ { b } \frac { t + 1 } { t ( 3 - t ) } \mathrm { d } t$$ where $M$ and $k$ are constants to be found.
1. Write $\frac { t + 1 } { t ( 3 - t ) }$ in partial fractions.
2. Use algebraic integration to find the exact area of $R$, giving your answer in simplest form.

Edexcel P4 2021 October Q8

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

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 FP2 2008 June Q2

Challenging +1.2

Find, in terms of $a$, the polar coordinates of the points where the curve $C _ { 1 }$ meets the circle $C _ { 2 }$.(4) \end{enumerate} The regions enclosed by the curves $C _ { 1 }$ and $C _ { 2 }$ overlap and this common region $R$ is shaded in the figure.
Find, in terms of $a$, an exact expression for the area of the
\includegraphics[max width=\textwidth, alt={}, center]{863ef52d-ae75-450c-9eab-8102804868f5-1_523_707_1262_1255}
region $R$.(8)
In a single diagram, copy the two curves in the diagram above and also sketch the curve $C _ { 3 }$ with polar equation $r = 2 a \cos \theta , 0 \leq \theta < 2 \pi$ Show clearly the coordinates of the points of intersection of $\mathrm { C } _ { 1 } , \mathrm { C } _ { 2 }$ and $\mathrm { C } _ { 3 }$ with the initial line, $\theta = 0$.(3)(Total 15 marks)

Edexcel M1 2013 June Q2

Standard +0.3

the tension in the cable,
the magnitude of the force exerted on the woman by the floor of the lift.
\item \end{enumerate} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3c8dce6f-367a-42bb-be60-d03d0a23664f-04_616_780_118_584} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A box of mass 2 kg is held in equilibrium on a fixed rough inclined plane by a rope. The rope lies in a vertical plane containing a line of greatest slope of the inclined plane. The rope is inclined to the plane at an angle $\alpha$, where $\tan \alpha = \frac { 3 } { 4 }$, and the plane is at an angle of $30 ^ { \circ }$ to the horizontal, as shown in Figure 1. The coefficient of friction between the box and the inclined plane is $\frac { 1 } { 3 }$ and the box is on the point of slipping up the plane. By modelling the box as a particle and the rope as a light inextensible string, find the tension in the rope.

Edexcel M2 2006 January Q6

Standard +0.8

show that $\mu = 0.35$.
(6) A second load of weight $k W$ is now placed on the ladder at $A$. The load of weight $4 W$ is removed from $C$ and placed on the ladder at $B$. The ladder is modelled as a uniform rod which is in a vertical plane perpendicular to the wall. The loads are modelled as particles. Given that the ladder and the loads are in equilibrium,
find the range of possible values of $k$.
(7)

Edexcel M3 2003 January Q3

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 S2 2002 June Q2

Moderate -0.3

Explain what you understand by the statistic $Y$.
Give an example of a statistic.
Explain what you understand by the sampling distribution of $Y$. \item The continuous random variable $R$ is uniformly distributed on the interval $\alpha \leq R \leq \beta$. Given that $\mathrm { E } ( R ) = 3$ and $\operatorname { Var } ( R ) = \frac { 25 } { 3 }$, find
the value of $\alpha$ and the value of $\beta$,
$\mathrm { P } ( R < 6.6 )$. \item Past records show that $20 \%$ of customers who buy crisps from a large supermarket buy them in single packets. During a particular day a random sample of 25 customers who had bought crisps was taken and 2 of them had bought them in single packets.
Use these data to test, at the $5 \%$ level of significance, whether or not the percentage of customers who bought crisps in single packets that day was lower than usual. State your hypotheses clearly.
(6) \end{enumerate} At the same supermarket, the manager thinks that the probability of a customer buying a bumper pack of crisps is 0.03 . To test whether or not this hypothesis is true the manager decides to take a random sample of 300 customers.
Stating your hypotheses clearly, find the critical region to enable the manager to test whether or not there is evidence that the probability is different from 0.03 . The probability for each tail of the region should be as close as possible to $2.5 \%$.
Write down the significance level of this test.

Edexcel S3 2006 June Q2

Moderate -0.3

Write down the approximate distribution of the sample mean height. Give a reason for your answer.
Hence find the probability that the sample mean height is at least 91 cm . \item A biologist investigated whether or not the diet of chickens influenced the amount of cholesterol in their eggs. The cholesterol content of 70 eggs selected at random from chickens fed diet $A$ had a mean value of 198 mg and a standard deviation of 47 mg . A random sample of 90 eggs from chickens fed diet $B$ had a mean cholesterol content of 201 mg and a standard deviation of 23 mg .
Stating your hypotheses clearly and using a $5 \%$ level of significance, test whether or not there is a difference between the mean cholesterol content of eggs laid by chickens fed on these two diets.
State, in the context of this question, an assumption you have made in carrying out the test in part (a). \item The table below shows the price of an ice cream and the distance of the shop where it was purchased from a particular tourist attraction. \end{enumerate}
Shop Distance from tourist attraction (m) Price (£)
A 50 1.75
B 175 1.20
C 270 2.00
D 375 1.05
E 425 0.95
F 580 1.25
G 710 0.80
$H$ 790 0.75
I 890 1.00
J 980 0.85
Find, to 3 decimal places, the Spearman rank correlation coefficient between the distance of the shop from the tourist attraction and the price of an ice cream.
Stating your hypotheses clearly and using a $5 \%$ one-tailed test, interpret your rank correlation coefficient.

Edexcel D1 2002 January Q3

Easy -1.2

State the number of edges in a minimum spanning tree for $N$. A minimum spanning tree for a connected network has $n$ edges.
State the number of vertices in the network. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{2f4d2383-823a-4a6f-ad4b-69cf01566052-4_714_1129_380_476}
\end{figure} Figure 1 shows a network of roads. Erica wishes to travel from $A$ to $L$ as quickly as possible. The number on each edge gives the time, in minutes, to travel along that road.
Use Dijkstra's algorithm to find a quickest route from $A$ to $L$. Complete all the boxes on the answer sheet and explain clearly how you determined the quickest route from your labelling.
Show that there is another route which also takes the minimum time

Edexcel D1 2004 January Q2

Easy -1.2

State, giving your reason, whether this tableau represents the optimal solution.
State the values of every variable.
Calculate the profit made on each unit of $y$. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{a21a4565-800c-45c0-a513-bb813ef1086f-3_1193_1689_420_201}
\end{figure} Figure 1 shows a network of roads represented by arcs. The capacity of the road represented by that arc is shown on each arc. The numbers in circles represent a possible flow of 26 from $B$ to $L$. Three cuts $\mathrm { C } _ { 1 } , \mathrm { C } _ { 2 }$ and $\mathrm { C } _ { 3 }$ are shown on Fig. 1.
Find the capacity of each of the three cuts.
Verify that the flow of 26 is maximal. The government aims to maximise the possible flow from $B$ to $L$ by using one of two options.
Option 1: Build a new road from $E$ to $J$ with capacity 5 .
or
Option 2: Build a new road from $F$ to $H$ with capacity 3.
By considering both options, explain which one meets the government's aim
(3) \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{a21a4565-800c-45c0-a513-bb813ef1086f-4_780_1002_358_486}
\end{figure} An engineer needs to check the state of a number of roads to see whether they need resurfacing. The roads that need to be checked are represented by the arcs in Fig. 2. The number on each arc represents the length of that road in km. To check all the roads, he needs to travel along each road at least once. He wishes to minimise the total distance travelled. The engineer's office is at $G$, so he starts and ends his journey at $G$.
Use an appropriate algorithm to find a route for the engineer to follow. State your route and its length.
(6) The engineer lives at $D$. He believes he can reduce the distance travelled by starting from home and inspecting all the roads on the way to his office at $G$.
State whether the engineer is correct in his belief. If so, calculate how much shorter his new route is. If not, explain why not.
(3)
\includegraphics[max width=\textwidth, alt={}, center]{a21a4565-800c-45c0-a513-bb813ef1086f-5_1561_1568_347_178} Figure 3 describes an algorithm in the form of a flow chart, where $a$ is a positive integer. List $P$, which is referred to in the flow chart, comprises the prime numbers $2,3,5,7,11,13$, 17, ...
Starting with $a = 90$, implement this algorithm. Show your working in the table in the answer book.
Explain the significance of the output list.
Write down the final value of $c$ for any initial value of $a$.

Edexcel D1 2002 June Q4

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

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

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 $x \geq 0$.
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 $x$.
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.
Write this as a linear programming problem, in terms of $x$ and $y$, stating the objective function and all the constraints.
Find the optimal number of batteries to be made each day. Show your method clearly.
Find the daily profit, in $\pounds$, made by EXYCEL.

Edexcel C1 Q6

Moderate -0.5

Write down the gradient of $A B$ and hence the gradient of $B C$. The point $C$ has coordinates $( 8 , k )$, where $k$ is a positive constant.
Find the length of $B C$ in terms of $k$. Given that the length of $B C$ is 10 and using your answer to part (b),
find the value of $k$,
find the coordinates of $D$.

Edexcel C1 Q9

Standard +0.3

find the value of $x$,
find the expected value of sales in the eighth month,
show that the expected total of sales in pounds during the first $n$ months is given by $k n ( 51 - n )$, where $k$ is an integer to be found.
Explain why this model cannot be valid over a long period of time.

Shop	Distance from tourist attraction (m)	Price (£)
A	50	1.75
B	175	1.20
C	270	2.00
D	375	1.05
E	425	0.95
F	580	1.25
G	710	0.80
\(H\)	790	0.75
I	890	1.00
J	980	0.85

Questions — Edexcel (9670 questions)

Browse by module

Browse by board

Edexcel FD2 Specimen Q7

Edexcel AEA 2012 June Q4

Edexcel AEA 2012 June Q5

Edexcel AEA 2012 June Q7

Edexcel C12 2018 January Q8

Edexcel C1 2006 January Q6

Edexcel C1 Specimen Q2

Edexcel C2 2013 June Q7

Edexcel P3 2022 October Q8

Edexcel P4 2024 June Q6

Edexcel P4 2021 October Q8

Edexcel C4 2013 January Q5

Edexcel FP2 2008 June Q2

Edexcel M1 2013 June Q2

Edexcel M2 2006 January Q6

Edexcel M3 2003 January Q3

Edexcel S2 2002 June Q2

Edexcel S3 2006 June Q2

Edexcel D1 2002 January Q3

Edexcel D1 2004 January Q2

Edexcel D1 2002 June Q4

Edexcel D1 2002 November Q4

Edexcel D1 2004 November Q5

Edexcel C1 Q6

Edexcel C1 Q9