1
2
Determine if the number is rational (R) or irrational (I).
a) 61π
b) 42
c) √101
d) 67.714813...
e) √64
f) 98
4
1
(I).
c) (√5)
1
e) 3×3×3×7×7
f) 2 × 2 × 2 × 11
Evaluate:
1
1 −2
a) (11 )
9
a) (5 − √5)(5 + √5)
1 −2
b) (92 )
2
b) (√3 + 2)
c)
b) √2
d) 8×8×8
5
3
3
a) 6 × 6
16
Solve the following and write the following as rational (R) or irrational
Express in index form:
1
2√13
1
c) 814 ÷ 164
3√52 − 4√117
1
d)
d) √2 + 4√32 − 6√2
3
a) The population of China is estimated at 1,100,000,000. Write this
in standard form.
b) The mass of a hydrogen atom is 0.000000000000000000000
00167 grams. Write this mass in standard form.
c) The area of the surface of the Earth is about 510,000,000 km2.
Express this in standard form.
d) An atom is 0.000000000 25 cm in diameter. Write this in standard
form.
6
92
1
4− 2
Each shape below is regular. For each, find x and the length of each side.
7
For each shape below, find x.
10
a) Simplify:
𝑥 2 + 3𝑥
(𝑥 + 1)(𝑥 + 3)
b) Solve:
3𝑥 + 6
𝑥−2
÷ 2
𝑥 − 4 𝑥 − 4𝑥
c) Simplify:
7𝑥 − 14 8𝑥 + 6
+
7
2
11
8
Make the letter in [] the subject of the equation
Simplify the following:
2 3
a) 𝑥 𝑦 ÷ 𝑥𝑦
2
a)
9𝑧 − 5
= 3𝑥
2𝑥
[𝑥]
b) 9𝑓 4 × 48 × 2𝑔8 × 45
b) 12y + x = 3x − ey [𝑦]
c) (𝑐 2 𝑑3 )4
d) 45𝑥 9 𝑦10 𝑧 5 ÷ 5𝑥 12 𝑦 7
9
c)
3𝑚 + 10
= 2𝑛 + 1
2
d)
2 3
+ =5
𝑏 𝑐
Expand and simplify each expression:
a) (𝑎 − 1)(𝑎 + 4)
b) (𝑥 + 4)(𝑥 + 3)
c) (𝑥 − 3)(𝑥 − 2)
[𝑏]
[𝑚]
12
Express your answer using powers of 10 only.
13
Work out mentally:
a) 75.2 × 0.23
b) 12.4 × 0.7
c) 14.04 ÷ 6
d) 0.845 ÷ 5
14
a) Write 0.4 as percentage.
b) Write 28% as decimal.
c) Write 1.02 as percentage.
d) Write 8.5% as decimal.
15
a) Write 25% as fraction.
b) Write
3
as a percentage.
4
c) Write
13
as a percentage.
50
d) Write 68% as a fraction.
20
16
Sebastian leaves £3000 in the bank for two years.
It earns compound interest of 2% per year.
Calculate the total amount Sebastian has in the bank at the end of the
two years.
The weight of a bag of potatoes is 25 kg, correct to the nearest kg. (a)
Write down the smallest possible weight of the bag of potatoes.
............................... kg
(b) Write down the largest possible weight of the bag of potatoes.
............................... kg
17
Natalie invests £600 for 2 years at 10% per year compound interest.
How much interest does she earn?
18
A bag contains 9x green counters and 2x pink counters.
The number of green counters is decreased by 40%
The number of pink counters is increased by 10%
There are now 96 more green counter than pink counters in the bag.
Work out the value of x
21
A field is in the shape of a rectangle.
The length of the field is 340 m, to the nearest metre.
The width of the field is 117 m, to the nearest metre.
Calculate the upper bound for the perimeter of the field.
22
Zoe’s weight is 62 kg, correct to the nearest kilogram.
Leo’s weight is 85 kg, correct to the nearest kilogram.
a) Write down the upper bound for the weight of Zoe.
............................... kg
b) Write down the lower bound for the weight of Anu.
19
James bought a house.
In the first year the value of the house decreased by 10%.
In the second year the value of the house increased by 10%.
Is the house worth more, less, or the same as what James paid for it?
Explain your answer.
............................... kg
c) Work out the upper bound for the difference between Zoe’s weight
and Anu’s weight.
............................... kg
23
The volume of oil in a tank is 1000 litres, correct to the nearest 10 litres.
The oil is poured into tins of volume 2.5 litres, correct to one decimal
place.
Calculate the upper bound of the number of tins which will be required.
26
Lucy is three times as old as Alex.
Lucy is 7 years older than Megan.
The sum of their ages is 126.
Find the ratio of Alex’s age to Lucy’s age to Megan’s age.
24
The diagram shows a right-angled
triangle.
The area of the triangle is 294 cm2
Work out the value of x.
27
a) n is an integer such that –2 ≤ n < 3
Write down all the possible values of n.
b) Solve 2x – 5 > 8
c) On the number line, show the inequality -3 ≤ x + 2 < 2
25
ABCD is a parallelogram
All measurements are in
centimetres.
The perpendicular height of
the parallelogram is 5 cm.
Find the area of ABCD.
d) 1 ≤ 2y – 3 < 9 where y is an integer.
Write down all the possible values of y
28
a) Lee is y years old. Toby is 8 years younger than Lee. The sum of
their ages is less than 41.
i.
Write down in terms of y, an inequality to show this
information.
ii.
29
a) Solve the simultaneous equations.
5x + 3y = 41
2x + 3y = 20
Work out the oldest age that Lee can be. Give your answer as a
whole number of years.
b) David buys 2 scones and 2 coffees in a shop and the cost is £18.
Ellie buys 3 scones and 2 coffees in the same shop and they cost
£22.
Form two equations and solve to find the cost of each scone and
each coffee.
b) Annie, Beth and Carly go shopping.
Annie spends m pounds.
Beth spends twice as much as Annie.
Carly spends 5 pounds more than Annie.
The total amount of money spent, in pounds, is more than £60.
i.
Write down, in terms of m, an inequality to show this
information.
Each girl spends an whole number of pound.
ii.
Work out the least each girl could have spent.
Annie £...................... Beth £...................... Carly £......................
c) Shown
below
is
a
parallelogram. Each side is
measured in centimetres.
Work out the perimeter of the
parallelogram.
30
Complete these conversions:
a) 2580 grams to kilograms.
b) A circular field has a diameter of 32 metres. A farmer wants to build
a fence around the edge of the field. Each metre of fence will cost
£15.95 Work out the total cost of the fence.
b) 1.6 kilometres to metres.
c) 48 cm to mm.
d) 520 millilitres to litres.
e) 800 metres to kilometres.
31
Find the area of the compound shape.
a)
b)
c) An area is formed by a square, ABCD, and a semi circle. BD is the
diameter of the semi-circle. The radius of the semi-circle is 4m. The
area is going to be covered completely with lawn seed. A box of
lawn seed covers 25 m². How many boxes of lawn seed will be
needed? You must show your working.
32
a) A semi-circle has an area of 50 m2. Find the perimeter of the semicircle. Give your answer correct to one decimal place.
d) The diagram shows three-quarters of a circle with a radius of 12
metres. Find the perimeter of the shape.
33
a) The diagram shows a semi-circle inside a
sector of a circle, ABC. AB is the diameter of
the semi-circle.
Angle BAC = 90° AB = 12 cm
Find the area of the shaded region.
34
Write the following fractions as a recurring decimal.
52
4
a)
b)
45
3
c)
35
74
154
d)
6
9
a) Arrange the following in ascending order
0.5
b) Shape A is a semi-circle that has a radius of 12 cm.
Shape B is a circle.
The area of shape A is 8 times the area of shape B.
Show that the radius of shape B is 3 cm
0.562̇
0.5627
0.5̇62̇
Arrange the following in ascending order
61
330
36
c) A circle is enclosed by a square as shown in
the diagram.
Each side of the square measures 8cm.
Find the area of the shaded region.
Give your answer correct to 1 decimal place.
0.56̇2̇
0.17̇8̇
3-2
19
110
Write the following recurring decimals in its simplest fractions.
a) 0. 5̇
b) 0.47̇
2
3
3
2
c) 5 − 2 c) 3 × 2
3
4
4
3
c) 0. 5̇4̇
38
Work out:
1 1
3
1 3 3 2
√
a)
÷ × 2 b) ( ) + ×
16 2
4
2
4 9
d) 0. 1̇26̇
37
Work out:
1 1
1 5
a) 1 − ( + ) b) 12 ÷
2 6
2 8
1 15
2
1
1
c) (3 − ) ÷ d)
× √11
3 9
3
9
√64
39
Shape A is a regular triangle.
Shape B is a regular octagon.
Another regular polygon, P, is
shown on the diagram.
How many sides does polygon P
have?
You must show your work.
41
AB and CD are parallel lines.
Write down:
(a) the size of the angle x
(b) the size of the angle y
42
40
ABCDEF is a hexagon.
Angle BAF = Angle ABC = Angle
AFE = Angle BCD.
Angle DEF = Angle CDE = 130°
Work out the size of angle BAF.
You must show all your working
AB and CD are parallel lines.
Find the size of angle x
43
ABCD is a parallelogram.
CBE is a straight line.
Angle BAD = 128°
Angle AEB = 39° F
ind the size of angle BAE.
Give a reason for each stage of
your working.
45
AB and CD are parallel.
Angle HIK = 85°
Angle BFH = 32°
Find the size of angle FEG.
You must show how you got
your answer.
44
AB and CD are parallel lines.
EFG is an isosceles triangle
Angle AEG = 110°
Find the size of the angle FGD.
Give a reason for each stage of
your work.
46
AB and CD are parallel.
Find the size of angle x.
Give a reason for each stage of
your working.
47
48
The diagram shows three regular
pentagons meeting at a point.
Work out the size of the angle
marked x.
You must show all your working.
The diagram shows a regular pentagon,
ABCDE, and a square, EDFG.
The lines CD and DG are both sides of
another regular polgon, P.
How many sides does polygon P have?
You must show how you got your
answer.
49
Calculate the length of the AB.
Give your answer to 3 significant
figures.
50
A television has a diagonal length of 50 inches.
The ratio of the length of the television to the width of the television is
4:3
Calculate the length and the width of the television.
Give your answers correct to 1 decimal place.
51
ABCD is a square. The diagonal of the
square is 8 m. Calculate the perimeter of
the square. Give your answer correct to
one decimal place.
52
ABCD is a trapezium. AD is
parallel to BC. Angle A = angle
B = 90. AD = 2.1 m, AB = 1.9
m, CD = 3.2 m. Work out the
length of BC. Give your answer
correct to 3 significant figures.
53
Construct △ABC where AB = 5 cm, 𝐵𝐴̂𝐶 = 45o and 𝐴𝐵̂𝐶 = 60o.
Construct the perpendicular bisector of AB. If the perpendicular
bisector of AB meets AC at X, measure and write down the length of BX.
54
(a) Construct the quadrilateral PQRS in which PQ = 4 cm, SP = 3.6 cm,
QR =RS = 4.8 cm, and 𝑄𝑃̂𝑆 = 90o.
(b) Construct the perpendicular from S to QR and let it cut QR at X.
(c) Measure and write down the length of SX.
55
Construct the trapezium PQRS with PQ = 5 cm, 𝑄𝑃̂𝑆 = 120o, PS = 6 cm,
SR = 9.8 cm, and PQ parallel to SR. Measure and write down the length
of the QR.
56
The table shows the number of hours of sunshine in Eastbourne for each
of the first 11 months of 2015.
57
The graph shows the number of crimes reported in England and Wales for
the years 2006 to 2015.
a) Explain why there is no mode for this information.
b) Find the mean number of hours of sunshine per month.
Give your answer to the nearest whole number.
a) Write down the years that had between 9 million and 10 million
crimes reported.
c) Show that the range is 209
b) Write down one reason why this graph may be misleading.
In December 2015 there were 27 hours of sunshine in Eastbourne.
d) What effect will including this information have on
i.
the mean number of hours of sunshine per month?
c) Describe the trend in the number of crimes reported in England and
Wales between 2006 and 2015.
d) Between which two consecutive years did the biggest drop in the
number of crimes reported take place?
ii.
the range?
58
Supul is investigating how long pupils in Year 10 in his school spent on
homework.
He asked each pupil to record the time taken, to the nearest minute, to do
their homework one night.
a) Describe the type of data the pupils recorded.
59
A maths teacher asked 60 students to complete a puzzle. The time taken,
in minutes, for each student to complete the puzzle was recorded. The
histogram shows information about the results.
Supul collected each pupil’s recorded time.
b) Discuss how reliable the data might be
Supul selected a sample of 20 of the pupils.
Here are their recorded times.
55
53
35
31
21
47
50
32
58
51
40
45
c) Find the mean time.
64
63
53
33
23
41
37
60
a) Work out how many students completed the puzzle in 4 minutes or
less.
b) Work out how many students completed the puzzle more than 4
minutes.
c) Present the data in frequency table.
d) Find the median time.
e) The times were recorded to the nearest minute.
Find the maximum possible range for the times
60
The table shows information about the percentages of males and females
in different age groups in England who held a full driving licence each year
from 2010 to 2013
It also gives the total number of licence holders.
61
There are 40 scouts in a scout group.
The scout group leader needs to find out the activities the scouts want to
do at their summer camp.
He is going to give a questionnaire to all 40 scouts.
a) State the population.
b) Write down the statistical name for an investigation that gets
information from every member of the population.
c) Give one reason why using a sample of the scouts in the group is not
necessary.
a) Write down the percentage of females aged 21–29 who held a full
driving licence in 2012.
d) Give one possible problem with using a questionnaire with all 40
scouts.
b) Work out the percentage of males aged 30–39 who did not hold a full
driving licence in 2010.
In 2013 the percentage of females who held a full driving licence was
greater than the percentage of males in one age group.
c) Write down this age group.
d) Comment on the trend in the total number of licence holders between
2010 and 2013
The scout leader also wants to find out how long the scouts would like the
summer camp to be.
e) Design a suitable question for the questionnaire.