gcse mathematics - st edmund campion catholic school past...gcse mathematics formulae: higher tier...
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NameFor Edexcel
GCSE MathematicsPaper 3D (Non-Calculator)
Higher TierTime: 1 hour and 45 minutes
Materials requiredRuler, protractor, compasses,pen, pencil, eraser.Tracing paper may be used.
Instructions and Information for CandidatesWrite your name in the box at the top of the page.Answer all the questions in the spaces provided in this question paper.The marks for each question and for each part of a question are shown in brackets.The total number of marks for this paper is 100. There are 23 questions in this paper.Calculators must not be used.
Advice to CandidatesShow all stages in any calculation.Work steadily through the paper. Do not spend too long on one question.If you cannot answer a question, leave it and attempt the next one.Return at the end to those you have left out.
Written by Shaun ArmstrongOnly to be copied for use in the purchaser's school or college
EH3D Page 1 © Churchill Maths Limited
GCSE Mathematics
Formulae: Higher Tier
Volume of a prism = area of cross section × length
Volume of sphere = 43 πr3 Volume of cone = 1
3 πr2hSurface area of sphere = 4πr2 Curved surface area of cone = πrl
In any triangle ABC The Quadratic EquationThe solutions of ax2 + bx + c = 0where a ≠ 0, are given by
x = −b±b2−4ac 2a
Sine Rule asin A =
bsin B =
csinC
Cosine Rule a2 = b2 + c2 – 2bc cos A
Area of triangle = 12 ab sin C
EH3D Page 2 © Churchill Maths Limited
sectioncross
length
rl h
r
c B
C
A
b a
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Q1
Answer ALL TWENTY THREE questions.
Write your answers in the spaces provided.
You must write down all the stages in your working.
You must NOT use a calculator.
1. Here is a list of ingredients for making 12 pancakes.
Work out how much of each ingredient is needed to make 18 pancakes.
…………………………. eggs
…………………… ml of milk
…………………… g of butter
………………. g of plain flour
(Total 3 marks)
EH3D Page 3 © Churchill Maths Limited
Ingredients2 eggs
250 ml of milk
50 g of butter
110 g of plain flour
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Q2
Number of pages
Time to read
(hours)
0
2
6
4
8
10
100 150 250200 300 350 400
2. Lisa records how long it takes her to read each book that she buys.
The scatter graph shows these times plotted against the number of pages in each book.
(a) Describe the correlation between the time it takes Lisa to read a book and the number of pages the book has.
…………………………(1)
(b) Draw a line of best fit on the diagram.(1)
Lisa buys a book with 260 pages.
(c) Use your line of best fit to estimate how long it will take Lisa to read it.
…………………… hours(1)
(Total 3 marks)
EH3D Page 4 © Churchill Maths Limited
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Q3
Q4
3. A is the point (–1, 7)B is the point (2, 5)
B is the midpoint of AC.
Find the coordinates of C.
( …… , …… )
(Total 2 marks)
4. Here are the first five terms of a number sequence.
5, 8, 11, 14, 17
(a) Write down an expression, in terms of n, for the nth term of this sequence.
…………………………(2)
(b) Explain why 81 will not be a term in this sequence.
………………………………………………………………………………………
………………………………………………………………………………………
………………………………………………………………………………………(2)
(Total 4 marks)
EH3D Page 5 © Churchill Maths Limited
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Q5
5.
Sox is a cat.
When he was five weeks old he weighed 400 grams.Over the following week his weight increased by 20%.
(a) Work out Sox's weight when he was six weeks old.
…………………… grams(2)
As an adult cat, Sox weighs 3.2 kg.
(b) Work out 400 grams as a percentage of 3.2 kg.
……………………… %(3)
(Total 5 marks)
EH3D Page 6 © Churchill Maths Limited
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Q6
Q7
0 10 3020 40 50 60Weight in grams
6. The box plot gives information about the distribution of the weights of fish in a pond.
(a) Write down the median weight of the fish.
…………………… g(1)
(b) Work out the interquartile range of the weights of the fish.
…………………… g(2)
(Total 3 marks)
7. v = 2t 2 – 7
t = 3
(a) Work out the value of v.
v = ……………………(2)
R = 4x + 3y
R = 9y = –5
(b) Work out the value of x.
x = ……………………(3)
(Total 5 marks)
EH3D Page 7 © Churchill Maths Limited
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Q8
O–4 –3 –1 1 2 3 x–6 –5 –2
–2
–1
1
2
3
4
y
5
A
C
B
8.
(a) The points A (–2, 0), B (–6, 1) and C (–5, 5) are plotted on the grid.
The quadrilateral ABCD is a square.
Write down the coordinates of the point D.
( …… , …… )(1)
(b) (i) Plot the points E (–1, –1), F (0, 1), G (2, 2) and H (3, –2) on the grid.
(ii) Write down the name of quadrilateral EFGH.
…………………………(3)
(Total 4 marks)
EH3D Page 8 © Churchill Maths Limited
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Q9
9. Dale makes his own pizzas.He always has a topping of ham, beef, chicken or pepperoni.The table shows the probability that he will have a ham, beef or chicken topping.
Topping Ham Beef Chicken Pepperoni
Probability 0.1 0.25 0.45
(a) Work out the probability that Dale has a pepperoni topping.
……………………(2)
In a year, Dale will make 60 pizzas.
(b) Work out an estimate for the number of times in a year that Dale will have a pizza with a beef topping.
………………………(2)
(Total 4 marks)
EH3D Page 9 © Churchill Maths Limited
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Q10
10.
(a) Reflect triangle P in the line y = 4.Label this image Q.
(2)
(b) Rotate triangle P 90° clockwise about O.Label this image R.
(2)
(c) Enlarge triangle P by scale factor 3, centre O.Label this image S.
(2)(Total 6 marks)
EH3D Page 10 © Churchill Maths Limited
1 2 43 5 x
y
–2 –1 6 87 9
–3
–2
–1O
7
9
8
1
2
4
3
5
6
P
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Q11
Q12
11.
The diagram shows a shaded semicircle of radius 4 cm, centre O.
Draw the locus of all points that are 2 cm from the edge of the semicircle.
(Total 3 marks)
12. Solve the equation
x − 34 +
x2 = 3
x = ……………………
(Total 4 marks)
EH3D Page 11 © Churchill Maths Limited
O
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Q13
13. (a) Factorise 6a – 9
…………………………(1)
(b) Simplify 3xy × 5x
…………………………(1)
(c) Expand and simplify r(r2 – 2) + 2r
…………………………(2)
(d) Find the value of p and the value of q such that for all values of x,
x2 + 4x – 1 = (x + p)2 + q
p = …………………
q = …………………(3)
(Total 7 marks)
EH3D Page 12 © Churchill Maths Limited
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Q14
a
c
D
A
C
BOE
14. Diagram NOTaccurately drawn
OABC is a rhombus.
OA = a.
OC = c.
D is the midpoint of AC.E is the midpoint of BD.
Express, in terms of a and c, the vectors
(a) OB
…………………………(1)
(b) OE
…………………………(2)
(Total 3 marks)
EH3D Page 13 © Churchill Maths Limited
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Q15
1 2 3 5 6 x–1–4 –3 –2
4
3
5
6
7
8
y
–1
–2
O 4
1
2
15. The shaded region on the grid below shows all the points which satisfy all three of these inequalities.
x ≤ a y ≥ b y ≤ cx + d
Write down the values of the integers a, b, c and d.
a = …………………
b = …………………
c = …………………
d = …………………
(Total 3 marks)
EH3D Page 14 © Churchill Maths Limited
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Q16
16. (a) Evaluate
(i) 2 –3
……………………
(ii) 6413
……………………
(iii) 49
32
……………………(4)
(b) (i) Express 12 in the form m3 , where m is an integer.
……………………
(ii) Rationalise the denominator of 1
12
Give your answer in the form 3n
, where n is an integer.
……………………(4)
(Total 8 marks)
EH3D Page 15 © Churchill Maths Limited
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Q17
6 cm x cmx cm
9 cm
17. Diagram NOTaccurately drawn
The diagram shows two candles which have the same volume.
One is a hemisphere of radius 6 cm.The other is a cuboid of height 9 cm which has a square base of side x cm.
Find the value of x in terms of π.Give your answer in its simplest form.
x = …………………………
(Total 4 marks)
EH3D Page 16 © Churchill Maths Limited
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Q18
10 cm12 cm
A B
18. Diagram NOTaccurately drawn
Two cylinders, A and B, are mathematically similar.
The height of cylinder A is 10 cm.The height of cylinder B is 12 cm.The total surface area of cylinder A is 200 cm2.
Work out the total surface area of cylinder B.
………………………… cm2
(Total 3 marks)
EH3D Page 17 © Churchill Maths Limited
y
x
1
–190
O180 360270
y
x
1
–190
O180 360270
y
x
1
–190
O180 360270
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19. Here is a sketch of the graph of y = cos x° for 0 ≤ x ≤ 360.
On each copy of this diagram, add a sketch of the graph indicated for 0 ≤ x ≤ 360.
(a) y = 2 cos x°
(1)
(b) y = cos (x + 60)°
(1)
EH3D Page 18 © Churchill Maths Limited
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Q19
y
x
1
–190
O180 360270
Q20
(c) y = 1 + cos ( 12 x)°
(2)(Total 4 marks)
20. (a) Change 512 to a decimal.
………………………(2)
(b) Convert the recurring decimal 0. 5̇ 4̇ to a fraction.Give your answer in its simplest form.
………………………(3)
(Total 5 marks)
EH3D Page 19 © Churchill Maths Limited
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Q21
21. A company has 400 employees.
The table gives some information about the employees.
The manager of the company wants to carry out a survey of employees' views about working hours.
She decides to take a sample of 40 employees, stratified by both gender and whether or not they have children.
Find the number of employees in each sub-group that should be in the sample.
Female, children …………………
Female, no children …………………
Male, children …………………
Male, no children …………………
(Total 4 marks)
EH3D Page 20 © Churchill Maths Limited
Children No children
Male
Female 156
129
47
68
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Q22
y
xO
A (–2, –2)
B (1, 7)
C
L1
L2
22. Diagram NOTaccurately drawn
The diagram shows the straight lines L1 and L2.
The line L1 passes through the points A (–2, –2) and B (1, 7) and crosses the y-axis at the point C.
The line L2 is perpendicular to L1 and also crosses the y-axis at the point C.
Find the equation of line L2.
…………………………
(Total 5 marks)
EH3D Page 21 © Churchill Maths Limited
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Q23
A C
E
B
D F
23
x + 2x2 – 7
x
23. Diagram NOTaccurately drawn
Triangles ABC and DEF are mathematically similar.
Angle ABC = angle DEF.Angle ACB = angle DFE.
All measurements are in centimetres.
(a) Show that 2x2 – 3x – 20 = 0
(3)
(b) Solve the equation 2x2 – 3x – 20 = 0
x = …………… or x = ……………(3)
(c) Find the length of AC.
………………… cm(2)
(Total 8 marks)
TOTAL FOR PAPER: 100 MARKS
END
EH3D Page 22 © Churchill Maths Limited
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