AQA GCSE MATHEMATICS Higher Tier FINAL REVISION

AQA GCSE MATHEMATICS Higher Tier
FINAL REVISION
KEY:
H = Higher level GCSE only, NC = No Calculator allowed
NC 1. For each of the following sequences, find an expression, in terms of n for the n th term.
(a) 4, 8, 12, 16, …… (b) 35, 30, 25, 20, ……
1 1
(c) , , , , ……
1
3
7
11
1
15
NC 2. A pressure washer is a section from a hemisphere of diameter D cm with a cylindrical
hole of diameter d cm in it. It is h cm high.
Which of these could be the formula for its curved surface area?
Give a reason for your answer.
(i) A = 0.5(D + d + h) (ii) A = 0.25h(D 2 + d 2 )
d
(iii) A = 0.25Ddh (iv) A = h(D + d).
h
H 3. (a) Find the two values of angle A, 0 A 360, such that sin A = 0.48
D
(b) You are given that cos 48° = 0.669
Find another angle between 0° and 360° that also has a cosine of 0.669.
NC 4. The diagram shows a right-angled triangle, ABC.
A
Angle ABC = 90°.
3
Not to scale
Tan x = 4
3 .
B
x
C
(a) Calculate sin x.
ABC and PQR are similar triangles.
P
PQ is the shortest side of triangle PQR.
15 cm
Not to scale
Angle PQR = 90° and PR = 15 cm.
(b) (i) What is the value of cos y?
Q
y
R
(ii)
What is the length of PQ?
5. ABCDEFGH is a cuboid with sides of 5 cm, 5 cm and l2 cm as shown.
A
Calculate angle DFH.
D
5 cm
B
Not to scale
C
E
5 cm
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H
G
12 cm
F

H 6. In triangle ABC, AB = 5 cm, BC = 8 cm and AC = 9 cm.
A
5 cm 9 cm
not drawn accurately
B
8 cm
C
Use the cosine rule to show that triangle ABC does not contain an obtuse angle.
H 7. A, B and C are three points which lie in a straight line on horizontal ground.
BT is a vertical tower.
T
A 21.5° 13.3°
C
B
1200 m
The angle of elevation of T from A is 21.5°. The angle of elevation of T from C is 13.3°.
AC = 1200 m.
Calculate the height of the tower.
H 8. A thin-walled glass paperweight consists of a hollow cylinder with a hollow cone on top
as shown.
The paperweight contains just enough sand to fill the cylinder.
2 cm
4 cm
6 cm
The paperweight is now turned upside down.
x
Calculate the depth of the sand, (marked x in the diagram).
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9. The diagram shows two similar triangles.
B
6 cm
A
36.9°
C
D
BC = 6 cm and angle BCA = 36.9°.
Not drawn accurately
E
(a)
Calculate the length of side AB.
(b) The ratio of BC : CE is 5 : 2.
Calculate the length of side DE.
H 10. AB is an arc of a circle, centre O,with radius 9.7cm.
Angle AOB = 110°.
O
110° 9.7 cm
not drawn accurately
Calculate the area of the sector OAB.
A
B
H 11. ABCD is a rhombus with side length 10 cm.
Angle BAD = 60°.
10 cm
B
ABD is a sector of a circle with centre A.
CBD is a sector of a circle with centre C.
A
60°
C
(a)
Calculate the area of triangle ABD.
(b)
H-NC 12. (a) Work out
Calculate the shaded area.
3 1
– 14
2 7
D
(b)
(c)
Find an approximate value of
289 × 4.13
0.19
You must show all your working.
1 × 9
Find the value of
3
1 2
× (2)
8
(d) Calculate
3
4
3
5
leaving your answer in the form 3 p
–
5
(e) Work out the value of 64
6
Leave your answer as a fraction.
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H-NC 13. (a) Work out 3.2 × 10 5 – 2.89 × 10 4
(b) Work out
1 2
3 – 1
4 5
(c)
2
3
Find the value of 8
(d)
3
7
Express 128 as a fraction.
(e) Express
8
3
4
3
in the form 3 p
14. The number of people waiting for an island ferry is 25% over the legal safety limit.
What percentage of the people waiting must the ferry leave behind so that it departs with
the maximum number of passengers allowed?
15. A shopkeeper normally sells his goods at 70% above cost price.
In a sale he reduces his prices by 40%.
What percentage profit does the shopkeeper make on goods sold in the sale?
NC 16. Brass is made from the metals copper, zinc and tin in the ratio 7 : 3 : 2
How much copper is needed to make 60 kg of brass?
17. Use trial and improvement to find a solution to the equation
x 3 + 2x = 60.
Give your answer correct to 1 decimal place.
12 .3(18.5 + 9.41)
18. (a) Find the greatest possible value of
15.8
All the numbers are given correct to three significant figures.
Write down your full calculator display.
(b)
A trailer can safely carry weights up to 5200 kg, correct to two significant figures.
It is loaded with boxes weighing 115 kg, correct to the nearest kilogram.
Calculate the greatest number of boxes that the trailer can carry safely.
You must show all your working.
NC 19. (a) Expand and simplify
(x + 4) 2
H (b) The diagram shows the circle x 2 + y 2 = 36 and the line y = x + 4.
The line and the circle intersect at the points A and B.
y
y
= x + 4
B
Not drawn accurately
A
O
x
x
+ y = 36
2 2
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Show that the x-coordinates of A and B are given by the solutions to the equation
x 2 + 4x – 10 = 0

H (c) Solve the equation x 2 + 4x – 10 = 0.
Give your answers to 2 decimal places.
You must show your working.
NC 20. (a) Factorise completely 12y 2 – 8y
(b)
n is an integer.
List the values of n such that –6 3n < 13
(c) Simplify (2xy 2 ) 3
H-NC 21.
The graph of y = 7 + 5x 2x 2 is sketched below.
y C
not drawn accurately
A
B
x
(a)
(b)
Find the coordinates of the points A and B, where the curve crosses the x-axis.
C is the point on the curve where the value of y is a maximum.
Use your answer to (a) to find the value of x at C.
22. Make r the subject of the formula
r – 3 = (t – 2r)
5 + x
23. You are given the formula y =
x
Rearrange the formula to give x in terms of y.
x 2
24. Solve the equation – = 1
x + 1 x – 1
H 25. Simplify
2
5x + 14x
– 3
x
2
– 9
H 26. You are given that x 2 – 6x + 13 = (x – a) 2 + b
(a) Find the values of a and b.
(b) Hence find the minimum value of x 2 – 6x + 13.
H 27. (a) Complete the table of values for y = x 2 – 4x – 2
x –2 –1 0 1 2 3 4 5 6
y 10 3 –2 –5 –5 –2 3 10
(b) On graph paper, draw the graph y = x 2 – 4x – 2 for values of x between –2 and 6.
(c) Use your graph to write down the solutions of the equation x 2 – 4x – 2 = 0
(d)
By drawing a straight line, the equation of which must be stated, find approximate
solutions of the equation x 2 – 5x –3 = 0
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28. In the diagram, the lines AC and BD intersect at E.
AB and DC are parallel and AB = DC.
A
B
E
D
C
Prove that triangles ABE and CDE are congruent.
H 29. (a) Factorise 2x 2 – 7x – 15
(b) The graph of y = 2x 2 – 7x – 15 is sketched below.
y
Not to scale
P
Q
x
Find the equation of the line of symmetry of this graph.
H 30. (a) The graph y = x 2 is transformed as shown.
y
y
y = x
2
Not drawn
accurately
O
x
(–3,0)
O
x
(b)
Write down the equation of the transformed graph.
The graph of y = 3x – 2 is sketched below.
y
On the same axes, sketch the graph of
y = 2 – 3x
O
x
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H 31. The diagram shows two sets of parallel lines.
Vector PQ = a and vector PS = b
PR =
3PQ
and PU = 2PS
U V W
S
b
P a Q R
(a) Write the vector PV in terms of a and b.
T
Not to scale
(b)
Write the vector RU in terms of a and b
(c) Find two vectors that can be written as 3a – b
H 32. PQRSTU is a regular hexagon and O is the centre of the hexagon.
OP = p and OQ = q
U
p
P
Express each of the following vectors in terms of p and q
T
O
q
Q
(a)
PQ
(b)
SP
S
R
(c)
SQ
H 33. The base of a triangle is 7 cm longer than its height.
The area of the triangle is 32 cm 2 .
(a)
(b)
Taking the height to be h cm, show that
h 2 + 7h 64 = 0
Solve this equation to find the height of the triangle.
Give your answer to 2 decimal places.
34. (a) Make x the subject of x 2 + k = 16.
(b) Make P the subject of
PRT
A = P +
100
35. (a) In the diagram, O is the centre of the circle.
Not drawn accurately
a
O
62°
Calculate the value of a.
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(b) In the diagram below, O is the centre of the circle and angle PSR = 100°.
Q
Not drawn accurately
P
O
b
100°
S
R
Calculate the value of b.
H (c) CD is a tangent to the circle at C.
A
Not drawn accurately
c
B
50°
44°
C
D
Calculate the value of c. Give reasons for your answer.
36. (a) O is the centre of the circle.
p
O
q
110°
not drawn accurately
(i)
(ii)
Calculate the value of angle p. Give a reason for your answer.
Calculate the value of angle q. Give a reason for your answer.
H (b) UPT is a tangent to the circle.
QRS is a straight line.
S
R
Q
not drawn accurately
U P T
Prove that angle PRS = angle QPT.
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37. The diagram shows part of a regular polygon.
Each interior angle is 162°.
Not drawn accurately
162°
Calculate the number of sides of the polygon.
H-NC 38.
Simplify fully
(a) 8 × 8 0 × 8 –1
(b)
1
5 –2 × (5 3 ) 3
(c) 15 2 1
× 3 2 1
× 5 2 1
H-NC 39. (a) By rationalising the denominator, simplify
2
=
(b) Show that ( 3 + 12 ) 27
15
5
H-NC 40. (a) Write 600 + 54 in the form p 6 where p is an integer.
(b)
600 + 54
Hence write
in the form q .
338
You may use 338 = 2 × 13 2
NC 41. Salt is sold in different sized blocks.
The weight of each block, B kilograms, is directly proportional to the cube of its
height, h metres.
A block of weight 54 kg has height 3m.
(a) Find an equation connecting h and B.
(b) Find the weight of a block with a height of 1m.
(c) Another block has a weight of 128 kg. Find its height.
H-NC 42. (a) Work out the exact value of ( 3 ) 4
(b) Write 32 in the form 2 p
(c) Find the value of (0.25) –1
3
(d) Find the value of 81 4 Leave your answer as a fraction.
H 43. Prove that the product of two odd numbers is always an odd number.
H-NC 44. (a) Write 0.1
8
as a fraction in its simplest form.
(b) Hence or otherwise express 0.51
8
as a fraction.
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H 45. The diagram shows the graph of y = 2x + 1.
y
6
5
4
3
2
1
–5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 x
–1
–2
–3
–4
–5
–6
A line passes through the point (2, 3) and is perpendicular to y = 2x + 1.
The equation of this line can be written in the from ax + by = c.
What are the values of a, b and c?
46. The line l passes through the points A (0, 3) and B (–4, 11).
(a) Calculate the gradient of the line l.
(b) Write down the equation of the line l.
H (c) Write down the equation of the line which also passes through the point (0, 3) but
is perpendicular to line l.
H 47. Gnomes ‘R’Us makes garden gnomes in two sizes.
The gnomes are similar in shape.
The smaller gnome is 28 cm high and the larger one is 35 cm high.
It takes 7936 cm 3 of plaster to make a small gnome.
How much plaster is needed to make a large gnome?
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48. A coffee machine dispenses 130 millilitres of black coffee into cups with a capacity of
175 millilitres.
These values are accurate to 3 significant figures.
Milk is supplied in small cartons which contain 21 millilitres, accurate to the nearest
millilitre.
Beryl likes milky coffee and always puts 2 cartons of milk in her coffee.
Will Beryl’s cup ever overflow?
You must show all your working.
49. In a village
5
3 of the pensioners have had a flu jab.
1
If a pensioner has had the flu jab the probability of catching flu is
30
7
If a pensioner has not had the flu jab the probability of catching flu is
10
(a)
(b)
Calculate the probability that a pensioner, picked at random, from this village
catches flu.
A statistician calculated that 120 pensioners from this village are expected to catch
flu.
Calculate how many pensioners live in the village.
50. Shaz has ten one pound coins.
Six have a thistle design and four have a leek design.
She chooses a one pound coin at random.
If the first coin has a thistle design she replaces it, and chooses again.
If the first coin has a leek design she does not replace it, but chooses again.
What is the probability that the second coin has a leek design?
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51. The cumulative frequency graphs represent the lengths of 40 programmes on Channel 1
and 40 programmes on Channel 3.
40
Channel 1
30
Cumulative
frequency
20
Channel 3
10
(a)
(b)
0
0 10 20 30 40 50
Length (minutes)
What is the difference between the median programme lengths for the two
channels?
How many programmes in total were more than 25 minutes long?
52. The scatter graph shows the number of petrol pumps and the number of cars queuing at
midday at six garages.
Number
of cars
queuing
6
5
4
3
2
1
×
×
×
× ×
×
0
0 1 2 3 4 5 6 7 8 9
(a)
(b)
(c)
Number of petrol pumps
State the type of correlation shown.
Use the scatter graph to estimate the number of cars queuing at a garage with
8 petrol pumps.
Explain why your answer in part (b) may be unreliable.
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H 53. The heights of 30 children are given in the table below.
Height in cm frequency
150 x < 155 2
155 x < 160 5
160 x < 165 8
165 x < 170 10
170 x < 175 5
(a)
(b)
Calculate an estimate of the mean height.
Calculate an estimate of the standard deviation of the heights.
H 54. The histogram shows information about how much time was spent in a supermarket by
100 shoppers.
2.0
1.5
Frequency
density
1.0
0.5
(a)
0
0 10 20 30 40 50 60 70 80
Time, t (minutes)
Complete this frequency table:
Time, t (minutes) 0 < t 5 5 < t 20 20 < t 30 30 < t 60 60 < t 80
Number of shoppers 6 15 25
(b)
20% of the shoppers are in the supermarket for more than T minutes.
Calculate an estimate of the value of T.
(2)
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