a level mathematics - pearson qualifications level/mathem… · mathematics mock papers – set 2...
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A levelMathematics
Version 1.0 • UKS • Mar 2020 • DCL1: Public
A guide to our question paper improvements from summer 2020 onwards
Please note: you’ll need your Edexcel Online username and password to access some of the resources. Your exams officer will be able to let you know your login details.
Refining our papers to improve the exam experience for all studentsFollowing the summer 2019 exam series for A level Mathematics, we have been continuing to gather feedback from teachers, parents and students. Building on this and our analysis of students’ performance in the exams, we’ve taken steps to refine our papers to improve the exam experience for all students from summer 2020 onwards.
To support you and your students, we’ve created this guide to show you what these improvements look like in practice and what you can expect in future exam papers. It reflects how these principles have guided the development of all our A level Mathematics Mock Papers – Set 2 from early drafts to the papers published in January 2020. This includes examples of how we moved, divided and reworded questions in our Pure Mathematics papers to enable students to better access and engage with papers and show what they know and can do.We’d like to take this opportunity to thank everyone who has offered feedback and been a vital part in helping us make these improvements. If you have any further questions or feedback on these improvements, please get in touch at [email protected].
The improvements focus on: • ensuring early questions are accessible to all and then steadily ramp in demand to encourage engagement and help
build students’ confidence through the papers pages 3–4
• dividing questions into parts so students are clear where marks can be achieved and can manage their focus and exam timings accordingly pages 5–8
• using clear, concise language to better enable all students to access the questions and understand the type of response expected pages 9–11.
2
Ensuring early questions are accessible to all (part 1)Our analysis of students’ performance has shown which types of questions contain the most accessible marks. That’s why we’ve reviewed and re-ordered questions in our exam papers to ensure that early questions are accessible to all, then steadily ramp in demand to encourage engagement and help build students’ confidence.
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1. Weed is completely covering the surface of a pond.
Fish are introduced into the pond in an effort to control the weed.
The surface area of the pond, A m ,2 covered by the weed, t days after the fish are introduced
is modelled by the equation
,t t∈ 0A 105= 12e0.08− t
According to the model,
(a) state the surface area of the pond covered by the weed at the start of the investigation
(1)
(b) find the time taken in days, to one decimal place, for the surface area of the pond coveredby the weed to fall to 40m2
(3)
Stuart wants to predict the surface area of the pond covered by the weed 30 days after the fish are introduced.
(c) Explain why he should not use this model.
(2)
Mark Scheme
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2. f ( )x 3= 4x 2+ 2x 12− +x 8
Given that =y f (x) has a single turning point at x =α ,
(a) show that α is a solution of the equation
3xx = 3 1−
(3)
The iterative formula
31 1 3
nn
xx
+= −
is used with 1x =1 to find an approximate value for α.
(b) Calculate the value of 2x and the value of 5x , giving each answer to 4 decimal places.
(3)
(c) Using a suitable interval and a suitable function that should be stated, show thatto 3 decimal places α is 0.889
(2)
Mark SchemeFinal Pure Mathematics Paper 1, Question 1 Final Pure Mathematics Paper 2, Question 2
Question 1 on the exponential model was placed at the beginning of the paper as it contains at least four marks that should be accessible to most students.
Question 2 on differentiation and iteration methods, was moved here from later in the paper. Analysis of past performance on similar A level Maths questions has shown that this is a very accessible topic.
3Questions here are taken from the A level Mathematics Mock Papers (Pure Mathematics) published in January 2020. They exemplify some of the improvements we’re making.
Ensuring early questions are accessible to all (Part 2)
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5. Vectors AB and BC are given by
4
2 p
AB q=
2
q
BC p −= 3
where p and q are constants
Given that AC
is parallel to 3− 4 3
find the value of p and the value of q.
(5)
Mark Scheme
Commentary
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11. The function f is defined by
f ( ) = +x 2 3xx
>x 0
(a) Find the value of a such that
f −1 a( ) = 53
(2)
(b) Explain why there are no values of x for which
f (x) < 32
(2)
(c) Show that f is a decreasing function.
(2)
Mark SchemeFinal Pure Mathematics Paper 2, Question 5 Final Pure Mathematics Paper 1, Question 11
Question 5 on vectors was originally Question 1, but it was felt an unsuitable start to the paper as it may affect students’ confidence.
Question 11 on functions was initially Question 4 but moved to later in the paper due to its relative difficulty.
4Questions here are taken from the A level Mathematics Mock Papers (Pure Mathematics) published in January 2020. They exemplify some of the improvements we’re making.
Dividing questions into parts (Part 1)Where possible, our exam papers will include questions divided into parts so students are clear where marks can be achieved and can manage their focus and exam timings accordingly.
10. (i) Find, using algebraic integration and making your method clear, the exact value of
5
1
4 9 d3x xx++∫
giving the answer in simplest form.
(5)
(ii) ( )
3 2
24 19 28 4h( ) 2
2x x xx x
x− + −
= >−
Given ( )2h( )
2Cx Ax B
x= + +
−where A, B and C are constants to be found, find
h( ) dx x∫(6)
(11 marks)
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(b) Hence find ∫ h( )x xd
(3)
4.( )
2
228 4 2
2x x3x −h( ) = 4 19x x
x+ −
>−
(a) Write h(x) in the form(x − 2)2
CAx + B + where A, B and C are constants to be found.
(3)
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Commentary
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12. Using algebraic integration and making your method clear, show that
5
1
4+ 3
9 dx=x a + ln bx
+∫where a and b are constants to be found.
Commentary II
Mark Scheme
Commentary I
Analysis of student responses has shown that some students will not attempt later parts to questions if they cannot do earlier parts. Question 4 and Question 12 were originally one large question on integration. The question has been divided into two separate questions so that students could attempt it in smaller bite-sized pieces. This makes Question 4 appear less daunting, and creates an accessible question on integration at a different point in the paper to Question 12, which requires more higher order thinking.
Draft questions Final Pure Mathematics Paper 1, Question 4 and Question 12
5Questions here are taken from the A level Mathematics Mock Papers (Pure Mathematics) published in January 2020. They exemplify some of the improvements we’re making.
Dividing questions into parts (Part 2)
Original Question 7
Red Squirrels were introduced into a large wood in Northumberland on 1st June 1996.
Scientists counted the number of red squirrels in the wood, P, on 1st June each year for t years after 1996.
The scientists found that over time the number of red squirrels can be modelled by the formula
tP ab=
where a and b are constants.
The line l, shown in Figure 1, illustrates the linear relationship between 10log P and t over a period of 20 years.
Using the information given on the graph and using the model,
(a) find the initial number of red squirrels that were introduced into the wood,
(2)
(b) find a complete equation for the model giving the value of b to 4 significant figures.
(4)
Original Question 3 Mock paper 2
Figure 1 shows the plan view of a design for a stage at a concert. The stage is modelled as a sector BCDF, of a circle centre F, joined to two congruent triangles ABF and EDF.
The straight line AFE has 10.7 mAF FE= = Given that 9.2mBF FD= = and angle 1.82BFD = radians, find
3
Originals For Preety
Hi Preety
This is the best I can do. It is simply a word version of the ‘’original’’ questions
Original 11 (Part (i) required for Appendix III as well)
(i) Find, using algebraic integration and making your method clear, the exact value of
5
1
4 9 d3x xx++∫
giving your answer in simplest form.
(5 )
(ii) ( )
3 2
24 19 28 4h( ) 2
2x x xx x
x− + −
= >−
Given ( )2h( )
2Cx Ax B
x= + +
− where A, B and C are constants to be found. find
h( ) dx x∫
(6)
(11 marks)
1
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12. Using algebraic integration and making your method clear, show that
5
1
4+ 3
9 dx=x a + ln bx
+∫where a and b are constants to be found.
Commentary II
Mark Scheme
Commentary I
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7.
Figure 1
Red Squirrels were introduced into a large wood in Northumberland on 1st June 1996.
Scientists counted the number of red squirrels in the wood, P, on 1st June each year for t years after 1996.
The scientists found that over time the number of red squirrels can be modelled by the formula
P = abt
where a and b are constants.
10log P and t over a The line l, shown in Figure 1, illustrates the linear relationship between period of 20 years.
Using the information given on the graph and using the model,
(2)
(2)
(a) find an equation for l,
(b) find the initial number of red squirrels that were introduced into the wood,
(c) find a complete equation for the model giving the value of b to 4 significant figures.
(2)
On 1st June 2019 there were found to be 198 red squirrels in the wood.
(d) (i) Use this information to show that the model is not valid on 1st June 2019.
(ii) Suggest a reason for the model not being valid at this time.
(3)
(0,2) (20,2.4)
10log P
O t
Mark Scheme
Commentary
Draft question
Draft question Final Pure Mathematics Paper 1, Question 12
Final Pure Mathematics Paper 1, Question 7
Additional scaffolding was added to Question 7 and Question 12 to aid accessibility. In Question 7, parts (a) and (c) were originally one part, and in Question 12, the form of the answer a+lnb, was added to aid students’ confidence.
6Questions here are taken from the A level Mathematics Mock Papers (Pure Mathematics) published in January 2020. They exemplify some of the improvements we’re making.
Dividing questions into parts (Part 3)Draft question Final Pure Mathematics Paper 2, Question 9
Additional scaffolding was added to Question 9. This was one complete 9-mark question which asked students to find the exact area of R. Adding the part (a) gives access to students who know how to differentiate but would not necessarily do so due to the complexity of the problem.
Mock 2 Original Question 9
9.
Figure 5
Figure 5 shows a sketch of part of the curve with equation
2e xy x −=
The point ( , )P a b is the turning point of the curve.
The finite region R, shown shaded in Figure 5, is bounded by the curve, the line with equation y b= and the y-axis. Find the exact area of R.
(9)
(9 marks)
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9.
Figure 3
In this question you must show all stages of your working.
Solutions relying on calculator technology are not acceptable.
(4)
Figure 3 shows a sketch of part of the curve with equation
y = x e− 2x
The point P(a , b ) is the turning point of the curve.
(a) Find the value of a and the exact value of b
The finite region R, shown shaded in Figure 3, is bounded by the curve, the line withequation y = b and the y-axis.
(b) Find the exact area of R.
22
(5)
Mark Scheme
Commentary
7Questions here are taken from the A level Mathematics Mock Papers (Pure Mathematics) published in January 2020. They exemplify some of the improvements we’re making.
Dividing questions into parts (Part 4)
Mock 2 Original Question 11
Figure 5 shows a sketch of the curve with parametric equations
3cos 2 , 2 tan 0 4x t y t t π= =
The region R, shown shaded in Figure 4, is bounded by the curve, the x-axis and the y-axis.
Find the exact area of R.
(Solutions relying entirely on calculator technology are not acceptable.)
(7 marks)
7
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11.
Figure 5
Figure 5 shows a sketch of the curve with parametric equations
0 4=x 3cos 2 ,t y = 2 tan t t π
The region R, shown shaded in Figure 5, is bounded by the curve, the x-axis and the y-axis.
(a) Show that the area of R is given by
4
0
24sin2 dt tπ
∫(4)
(b) Hence, using algebraic integration, find the exact area of R.
(3)
R
O
Mark Scheme
Commentary
Draft question Final Pure Mathematics Paper 2, Question 11
Additional scaffolding was added to Question 11. Adding the part (a) as a “show that” gives an opportunity for students to gain the final three marks without having gained the first four.
8Questions here are taken from the A level Mathematics Mock Papers (Pure Mathematics) published in January 2020. They exemplify some of the improvements we’re making.
Using clear, concise language (Part 1)We’ve refined the wording of our papers to better enable all students to access the questions and understand the type of response expected.
Draft question
Draft start to question 13
Final Pure Mathematics Paper 1, Question 5
Final Pure Mathematics Paper 1, start to Question 13
In Questions 5 and Question 13, the word count has been reduced, and bullet points have been added to improve readability and understanding.
Original question 5
Relative to a fixed origin O, the points A, B and C have position vectors 2 3− +i j , 3 p+i j and 7q +i j respectively.
Given that p is a constant and that 5 2AB =
(a) find the possible values of p.
(3)
Given that q is a constant and that the angle between AC
and the unit vector i is 3π radians,
(b) find the exact value of q.
(3)
(6 marks)
Original start to Question 13
Figure 4 shows a container in the shape of an inverted right circular cone which contains some water. The cone has an internal base radius of 2.5 m and a vertical height of 4 m.
At time t seconds, the height of the water is h m, the volume of the water is V m3 and water is
modelled as leaking from a hole at the bottom of the container at a rate of 3 1m s512
h − π
2
Original question 5
Relative to a fixed origin O, the points A, B and C have position vectors 2 3− +i j , 3 p+i j and 7q +i j respectively.
Given that p is a constant and that 5 2AB =
(a) find the possible values of p.
(3)
Given that q is a constant and that the angle between AC
and the unit vector i is 3π radians,
(b) find the exact value of q.
(3)
(6 marks)
Original start to Question 13
Figure 4 shows a container in the shape of an inverted right circular cone which contains some water. The cone has an internal base radius of 2.5 m and a vertical height of 4 m.
At time t seconds, the height of the water is h m, the volume of the water is V m3 and water is
modelled as leaking from a hole at the bottom of the container at a rate of 3 1m s512
h − π
2
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5. Relative to a fixed origin O,
• the point A has position vector − +2 3i j• the point B has position vector 3 +i jp , where p is constant•
Given that
the point C has position vector q +i j7 , where q is constant
AB = 5 2
(a) find the possible values of p.
(3)
Given that the angle between AC and the unit vector i is 3π
radians,
(b) find the exact value of q.
(3)
Mark Scheme
Commentary
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13.
Figure 4
[ The volume of a cone of base radius r and height h is 13π 2r h ]
Figure 4 shows a container in the shape of an inverted right circular cone which contains some water.
The cone has an internal base radius of 2.5 m and a vertical height of 4 m.
At time t seconds
• the height of the water is h m• the volume of the water is V m3
• the water is modelled as leaking from a hole at the bottom of the container at a rate of
512m s3 1− h
π
(a) Show that, while the water is leaking
32 d 1
d 200h h
t= −
(5)
(3)
Given that the container was initially full of water,
(b) find an equation, in terms of h and t, to model this situation.
It takes approximately 43 minutes for the container to empty.
(c) Use this information to comment on the suitability of this model.
(3)
Diagram not drawn to scale
Mark Scheme
Commentary
9Questions here are taken from the A level Mathematics Mock Papers (Pure Mathematics) published in January 2020. They exemplify some of the improvements we’re making.
Using clear, concise language (Part 2)
Original Question 7
Red Squirrels were introduced into a large wood in Northumberland on 1st June 1996.
Scientists counted the number of red squirrels in the wood, P, on 1st June each year for t years after 1996.
The scientists found that over time the number of red squirrels can be modelled by the formula
tP ab=
where a and b are constants.
The line l, shown in Figure 1, illustrates the linear relationship between 10log P and t over a period of 20 years.
Using the information given on the graph and using the model,
(a) find the initial number of red squirrels that were introduced into the wood,
(2)
(b) find a complete equation for the model giving the value of b to 4 significant figures.
(4)
Original Question 3 Mock paper 2
Figure 1 shows the plan view of a design for a stage at a concert. The stage is modelled as a sector BCDF, of a circle centre F, joined to two congruent triangles ABF and EDF.
The straight line AFE has 10.7 mAF FE= = Given that 9.2mBF FD= = and angle 1.82BFD = radians, find
3
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3.
Figure 1
Figure 1 shows the plan view of a design for a stage at a concert.
The stage is modelled as a sector BCDF, of a circle centre F, joined to two congruent triangles ABF and EDF.
Given that
AFE is a straight line
AF = FE =10.7 m
BF = FD = 9.2m
angle BFD =1.82 radians
find
(5)
(a) the perimeter of the stage, in metres, to one decimal place,
(b) the area of the stage, in m2, to one decimal place.
(4)
Mark Scheme
Commentary
Draft question Final Pure Mathematics Paper 2, Question 3
Significant changes were made to the look of Question 3. In the stem, spacing was used to aid readability and understanding.
10Questions here are taken from the A level Mathematics Mock Papers (Pure Mathematics) published in January 2020. They exemplify some of the improvements we’re making.
Using clear, concise language (Part 3)Draft question
Final Pure Mathematics Paper 2, Question 6
While Question 6 still has a large word count, particularly for a maths question, it was rewritten in order to describe the model more clearly and succinctly.
Mock 2 Original Question 6
A small company which makes batteries for electric cars has a 10-year plan for growth.
In year 1 the company will make 2 600 batteries
In year 10 the company aims to make 12 000 batteries
In order to calculate the number of batteries it will need to make each year from year 2 to year 9
the company considers two models.
Model A assumes that the number of batteries it will make each year will increase by the same
number each year.
(a) According to Model A, determine the number of batteries the company will make in year 2
(3)
Model B assumes that the number of batteries it will make each year will increase by the same
percentage each year.
(b) According to Model B, determine the number of batteries the company will make in year 2
Give your answer to the nearest 10 batteries.
(3)
(c) Calculate the difference between the total amounts of batteries that the company is predicted
to make from year 1 to year 10 inclusive, according to the two models. Give your answer to the
nearest 100 batteries.
(3)
(9 marks)
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6. A small company which makes batteries for electric cars has a 10-year plan for growth.
• In year 1 the company will make 2 600 batteries
• In year 10 the company aims to make 12 000 batteries
In order to calculate the number of batteries it will need to make each year, from year 2 to year 9,
the company considers two models, Model A and Model B.
In Model A the number of batteries made will increase by the same number each year.
(a) Using Model A, determine the number of batteries the company will make in year 2
(3)
In Model B the number of batteries made will increase by the same percentage each year.
(b) Using Model B, determine the number of batteries the company will make in year 2
Give your answer to the nearest 10 batteries.
(3)
Sam calculates the total number of batteries made from year 1 to year 10 inclusive using each of
the two models.
(c) Calculate the difference between the two totals, giving your answer to the nearest 100
batteries.
(3)
Mark Scheme
11Questions here are taken from the A level Mathematics Mock Papers (Pure Mathematics) published in January 2020. They exemplify some of the improvements we’re making.
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