Centre Number Paper Reference Surname Other Names Candidate Number Candidate Signature 1387 For Examiner’s Edexcel GCSE For Team Leader’s use Mathematics A Paper 5 HIGHER TIER Specimen Paper Time: 2 hours N0000 Materials required for the examination Items included with these question papers Ruler graduated in centimetres and millimetres, protractor, compasses, pen, HB pencil, eraser. Tracing paper may be used. Formulae sheets Instructions to Candidates In the boxes above, write your centre number, candidate number, the paper reference, your surname and other names and your signature. The paper reference is shown in the top left hand corner. Answer all questions in the spaces provided in this book. Supplementary answer sheets may be used Information for Candidates The total mark for this paper is 100. The marks for the various parts of questions are shown in round brackets: e.g. (2). Tracing paper may be used. Calculators must not be used. This question paper has 20 questions. There are no blank pages. Advice to Candidates Work steadily through the paper. Do not spend too long on one question. Show all stages in any calculations.
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Centre Number Paper Reference Surname Other Names
Candidate Number Candidate Signature
1387For Examiner’s use only
Edexcel GCSEFor Team Leader’s use only
Mathematics APaper 5
HIGHER TIER
Specimen PaperTime: 2 hours N0000
Materials required for the examination Items included with these question papersRuler graduated in centimetres and millimetres, protractor, compasses, pen, HB pencil, eraser.Tracing paper may be used.
Formulae sheets
Instructions to Candidates
In the boxes above, write your centre number, candidate number, the paper reference, your surname and other names and your signature. The paper reference is shown in the top left hand corner.Answer all questions in the spaces provided in this book. Supplementary answer sheets may be used
Information for Candidates
The total mark for this paper is 100. The marks for the various parts of questions are shown in round brackets: e.g. (2).Tracing paper may be used.Calculators must not be used.This question paper has 20 questions. There are no blank pages.
Advice to Candidates
Work steadily through the paper.Do not spend too long on one question.Show all stages in any calculations.If you cannot answer a question, leave it and attempt the next one.Return at the end to those you have left out.
In the same test the mean mark for the girls was m.
Work out an expression for the mean mark of the whole class of 27 students. HD4eGrade B
……………………………… (Total 3 marks)
11.
(a) Work out the length of AD. SSM2gGrade B
…………….. cm(2)
(b) Work out the length of BC. SSM2gGrade B
………… cm(2)
(Total 4 marks)
B
D
A C E
4.5 cm
6 cm 4 cm
5 cm
Diagram NOTaccurately drawn
12. A wax statue of a spaceman is on display in a museum..Wax models are to be sold in the museum shop. The statue and the wax models are similar.
The height of the statue is 1.8 mThe height of the model is 15 cm.The area of the flag in the model is 10 cm²
(a) Calculate the area of the statue’s flag. SSM3dGrade A
………………………. cm²(3)
The volume of the wax of the original statue is 172.8 litres.
(b) Calculate the volume of wax used to make the model. SSM3dGive your answer in ml. Grade A
………………………. ml(2)
(Total 5 marks)
13. K is an integer.
Find the value of K. NA4aGrade A
K = ………………..(Total 3 marks)
123
K3
14. The probability that a team will win a game is always 0.5.The team plays n games.The probability that the team will win all of the n games is less than 0.05.
Find the smallest value of n. HD4gGrade A
……………………(Total 3 marks)
15. The intensity of light L, measured in lumens, varies inversely as the square of the distance d m from the light.
When the distance is 2 m the light intensity is 250 lumens.
(a) Calculate the value of L when d is 2.5 m. NA5hGrade A
…………… lumens(3)
(b) Calculate the value of d when L is 90 lumens. NA5hGrade A
……………. m(2)
(Total 5 marks)
16.
ABCD is a parallelogram. Prove that triangles ABD and BCD are congruent. SSM2e
Grade A
(Total 3 marks)
Diagram NOT accurately drawn
A B
D C
17. The equation of a curve is
y = f(x)
where f(x) = x² – 10x + 32.
(a) Complete the square for f(x). NA5kGrade A*
………………………………..(2)
(b) Hence, sketch the graph of
(i) y = f(x)
(ii) y = f(x + 5) NA6gGrade A*
(4)
y = f(x) wheref(x) = sin x where 0 ≤ x ≤ 180
(c) By considering the function f(ax), sketch the graph of y = sin 2x. NA5kGrade A*
(2)(Total 8 marks)
18.
OA = a OB = b AC = 2a OD = 3b
Prove that AB is parallel to CD. SSM3fGrade A*
(Total 3 marks)
B
O
A
DC
ba
19. The diagram represents a right pyramid. The base is a square of side 2x.
The length of each of the slant edges is 8√3 cm.
The height of the pyramid is x cm.
Calculate the value of x.
SSM2fGrade A*
x
2x
8 3 2x
83
Diagram NOTaccurately drawn
………………….. cm(Total 6 marks)
20. Solve the equation
NA6e
Grade A*
x = ………………………….(Total 6 marks)
21. Leon recorded the lengths, in minutes, of the films shown on television in one week.His results are shown in the histogram.
20 films had length from 60 minutes up to, but not including, 80 minutes.
(a) Use the information in the histogram to complete the table. HD4aGrade A
Length in minutes (m) Frequency
60 ≤ m < 80 20
80 ≤ m < 90
90 ≤ m < 100
100 ≤ m ≤ 120(2)
Leon also recorded the lengths, in minutes, of all the films shown on television the following week. His results are given in the table below.
Length in minutes (m) Frequency Height
60 ≤ m < 100 72 36
100 ≤ m < 160 x
(b) Complete the table giving your answers in terms of x. HD4aGrade A*
(2)(Total 4 marks)
TOTAL FOR PAPER: 100 MARKS
60 80 100 120 140
Frequency density per unit interval
Time in minutes
Centre Number Paper Reference Surname Other Names
Candidate Number Candidate Signature
1387For Examiner’s use only
Edexcel GCSEFor Team Leader’s use only
Mathematics APaper 6
HIGHER TIER
Specimen PaperTime: 2 hours N0000
Materials required for the examination Items included with these question papersRuler graduated in centimetres and millimetres, protractor, compasses, pen, HB pencil, eraser, calculator.Tracing paper may be used.
Formulae sheets
Instructions to Candidates
In the boxes above, write your centre number, candidate number, the paper reference, your surname and other names and your signature. The paper reference is shown in the top left hand corner.Answer all questions in the spaces provided in this book. Supplementary answer sheets may be used
Information for Candidates
The total mark for this paper is 100. The marks for the various parts of questions are shown in round brackets: e.g. (2).Tracing paper may be used.Calculators may be used.This question paper has 20 questions. There are no blank pages.
Advice to Candidates
Work steadily through the paper.Do not spend too long on one question.Show all stages in any calculations.If you cannot answer a question, leave it and attempt the next one.Return at the end to those you have left out.
2. Tony carries out a survey about the words in a book.He chooses a page at random.He then counts the number of letters in each of the first hundred words on the page.The table shows Tony’s results.
Number of letters in a word 1 2 3 4 5 6 7 8
Frequency 6 9 31 24 16 9 4 1
The book has 25000 words.Estimate the number of 5 letter words in the book. HD4b
Grade C
……………………….(Total 3 marks)
25Turn over
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3.
The diagram shows a field which a farmer wants to fence.The field is in the shape of an isosceles trapezium in which AB is parallel to DC and AD = BC.
AB = 80 m and CD = 60 m.
The distance between the parallel sides is 30 m.
Calculate the length of fencing the farmer will need. SSM2fGrade C
……………………… m(Total 5 marks)
26
A
D C
B80 m
60 m
30 m
Diagram NOTaccurately drawn
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4. Wayne shares £360 between his children, Sharon and Liam, in the ratio of their ages. Sharon is 13 years old and Liam is 7 years old.
(a) Work out how much Sharon receives. NA3f Grade C
Sharon £ ………… (2)
(b) What percentage of the £360 does Sharon receive? NA3fGrade C
………………………. %(2)
(Total 4 marks)
27Turn over
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5. (a) Solve the equation Grade CNA5f
7p + 3 = 3(p –1)
p = …………….(2)
q is an integer such that 0 < 3q 16
(b) List all the possible values of q. Grade CNA5j
…………………………….(2)
(c) Solve the inequality Grade BNA5j
…………….……….(3)
(Total 7 marks)
28
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6. The Andromeda Galaxy is 21 900 000 000 000 000 000 km from the Earth.
(a) Write 21 900 000 000 000 000 000 in standard form. Grade BNA3h
………………………(1)
Light travels km in one year.
(b) Calculate the number of years that light takes to travel from the Andromeda Galaxy to Grade BEarth. NA3mGive your answer in standard form correct to 2 significant figures.
………………………(2)
(Total 3 marks)
29Turn over
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7.
A, B and C are points on the circumference of a circle, with centre O.
On the grid, triangle Q is the image of triangle P after a reflection.
(a) Rotate triangle Q through 90 clockwise about (2, 1). Grade CLabel this image R. SSM3b
(2)
(b) Describe fully the single transformation which maps triangle P onto triangle R. Grade B SSM3a
…………………………………….………………………………………………………...
…………………………………….………………………………………………………...
(2) (Total 4 marks)
32
-6 -4 -2 2 4 6 8
-4
-2
2
4
6
O x
y
PP Q
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10. The cumulative frequency graph gives information about the examination marks of a group of students.(a) How many students were in the group? Grade B
HD5d …………
(1)
(b) Use the graph to estimate the inter-quartile range. Grade BHD5d
………... (2)
The pass mark for the examination was 56.
(c) Use the graph to estimate the number of students who passed the examination. Grade BHD5d
………...(2)
(Total 5 marks)
11.
33Turn over
0 20 806040 100
40
60
30
20
10
50
Cumulative frequency
Mark
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The diagram shows a triangle ABC.The line CD is perpendicular to the line AB.AC = 7.3 cm, BD = 6.4 cm and angle BAC = 51.
Calculate the size of the angle marked x. Grade BGive your answer correct to 1 decimal place. SSM2g
……………….…
(Total 5 marks)
12. The population of the world is increasing at an annual rate of 1.6%.In 1990, it was 5250 million.Calculate an estimate for the population of the world in 2010. Grade A
NA3e
…………..……million (Total 3 marks)
13. Prove algebraically that the sum of the squares of any two consecutive integers is an odd Grade A
34
6.4 cm
D
51
7.3 cm
A B
C
Diagram NOT accurately drawn
x
Leaveblank
number. NA5b
(Total 3 marks)
35Turn over
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14.
The arc length of a sector of a circle is 15 cm.The radius of the circle is 12 cm.Work out size of the angle, θ°, of the sector. SSM4d
Grade A
…………………… (Total 3 marks)
36
θ
12 cm
15 cm
Diagram NOT accurately drawn
Leaveblank
15. The length of a rectangle is a centimetres.Correct to 2 decimal places, a = 6.37.
(a) For this value of a, write down Grade ANA3q
(i) the upper bound,
…………..……
(ii)the lower bound.
………..……… (1)
Correct to 1 significant figure, the area of the same rectangle is 20 cm².(b) Calculate the upper bound for the width of the rectangle. Grade A*
Write down all the figures on your calculator display. NA3q
………………………… (4)
A diagonal of the rectangle makes an angle of θ with one of the sides of length a centimetres.
(c) Calculate the upper bound for the value of tan θ. Grade A*Write down all the figures on your calculator display. NA3q
………………………… (2)
(Total 7 marks)
37Turn over
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16.
The diagram shows a rectangle with length x + 4 and width x 1.All measurements are given in centimetres.
The area of the rectangle is A square centimetres.
(a) Show that . Grade CNA5b
(2)
When A = 10, x satisfies the equation .
(b) Find the length of the rectangle. Grade AGive your answer correct to 2 decimal places. NA5k
………………cm (4)
(Total 6 marks)
38
x – 1
x + 4
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17.Year 10 Group Year 11 Group
Boys 75 50Girls 60 30
The table shows the number of boys and the number of girls in Year 10 and Year 11 of a school.The headteacher wants to find out what pupils think about a new Year 11 common room.A stratified sample of size 50 is to be taken from Year 10 and Year 11.
(a) Calculate the number of pupils to be sampled from Year 10. Grade AHD2d
………… (2)
Two pupils are to be chosen at random to speak to the headteacher.
One pupil is to be chosen from Year 10.One pupil is to be chosen from Year 11.
(b) Calculate the probability that both pupils will be boys. Grade AHD4g
…………(2)
However, the headteacher decides to choose one boy at random from all the boys in year 10 and 11 together and one girl at random from all the girls in year 10 and 11 together
(c) Calculate the probability that exactly one of the pupils will be from year 10. Grade A*HD4g
………… (3)
(Total 7 marks)
39Turn over
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18.
The diagram shows a quadrilateral ABCD.AB = 8.3 cm, BC = 7.8 cm, CD = 5.4 cm and AD = 6.1 cm.Angle BAD = 71.
(a) Calculate the area of triangle ABD. Grade AGive your answer correct to 3 significant figures. SSM2g
………….………… (3)
(b) Calculate the size of angle BCD. Grade A*Give your answer correct to 1 decimal place. SSM2g
……………..
(6)(Total 9 marks)
40
5.4 cm Diagram NOT accurately drawn
71
D
A B
C
8.3 cm
7.8 cm6.1 cm
A
C
B
D
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19. This is a sketch of the curve with equation y = f(x).
The maximum point of the curve is A (3, 8).Write down the co-ordinates of the maximum point for each of the curves having the following equations.
(i) y = f(x) + 2 Grade A*NA6g
(………, ………) (ii) y = f(x – 3)
(………, ………) (iii) y = f(–x)
(………, ………) (iv)y = f(3x)
(………, ………) (Total 4 marks)
41Turn over
y = f(x)
A (3, 8)
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20. Solve the simultaneous equations
Grade A*NA5l
………………………..
………………………..(Total 7 marks)
TOTAL FOR PAPER: 100
42
GCSE MATHEMATICS MARK SCHEME – PAPER 5
No NC Ref
Grade Working Answer Mark Notes
1 SSM2d C 360 30 = 12 12 sides 2 M1 for 360 30A1 cao
2 (a)
(b)
NA6c
NA6c
C
A
A and D selected
A and D selected
A and D
A and D
2
2
M1 for rearranging into y = mx + cA1 caoB1 for AB1 for D
3 (a) NA3h B 2 M1 for multiplying numbersA1 for
(b) NA2b B 4 1 B1 cao4 (a) NA3c C 3 M1 for using 12 as denominator
M1 for decomposing 2 wholesA1 cao
5 (a)(i) NA3a C 72 = 2 2 2 3 3 or
96 = 2 2 2 2 2 3 or
4 M1 for dividing through by 2 then 3A1 caoM1 for dividing through by 2 then 3A1 cao
(ii) NA2a C 2 2 2 3 = 24 2 M1 for selecting 2 and 3 as common prime factorsA1 cao
(b) NA3c A
Subtract
3 M1 for 0.454545 100M1 for 99x = 45A1 cao
6 HD4e C 92 1 = 9296 2 = 192
100 6 = 600104 1 = 104
98.8 4 M1 for f x; x within interval (accept both ends), at least 3 consistentM1 (dep) for use of all correct mid-interval values
43
GCSE MATHEMATICS MARK SCHEME – PAPER 5
No NC Ref
Grade Working Answer Mark Notes
988 10 = M1 (dep on 1st M1) for numerical (candidate’s numerical )A1 cao
7 (i) NA5d C 2 B1 cao(ii) NA5d C B1 cao
(b)(i) NA5b B 2x² – x – 6 2 B1 for 2x² – 6B1 for –x
(ii) NA5b B (3x – 2)(3x – 2)9x² – 12x + 4
2 B1 for 9x² + 4B1 for –12x
(c) NA5k B (x – 5)(x + 2) x = 5x = –2
3 M1 for factorisationA1 for correct factorsB1
(d) NA5f B – or 4 M1 for any multiple of 6M1 for multiplying out brackets correctlyM1 for 7h + 16 = 5A1 cao