IGCSE

Math
Core
E n g . N a g y E l r a h e b
M r s . N e s r i n e A h m e d
Pre-IGCSE

0580 (Core)
 MATHEMATICS DEPARTMENT
Index

Topic Page Topic Page

1 – Numbers Simultaneous linear equations 50
Number systems 2 3- Geometry
Factors and Multiples 5 Lines, angles and shapes 56
HCF 6 Triangles 62
LCM 7 Quadrilaterals 66
Powers and roots 8 Circles 69
Directed numbers 10 Construction 70
Order of operations 11 Perimeter, area and volume 72
Rounding numbers 13 Pythagoras 77
Operations on fractions 14 Similarity 79
Percentages 16 Bearings 82
Standard form 19
Estimation 22
Working with ratio 22
Ratio and scale 24
Rates 26
Average speed 26
Upper and lower bounds 28
Indices 31
Sets 34
Set builder notation 37
Venn diagrams 38
Time 40
Managing money 41
Simple interest 42
Hire purchase 43
Compound interest 44
Exchange Rate 45
2-Algebra
Substitution 46
Working with brackets 46
Factorisation 47
Subject of a formula 48
Linear equations 49

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1-Numbers

Number systems

Different types of numbers:

(N)

(Z)

/ Rational
(Q)

Notes:

a. 1 is not prime because both factors are the same 1x1

b. 2 is the only even prime number, 2 is the smallest prime number

c. Cubic numbers: {1, 8, 27, 64, 125, ………}

A cubic number is any number that has an integer cubic root

E.g.: √27 = 3, √−27 = − 3

3 3

Think of two numbers that are square numbers and cube numbers at the same time

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Rational numbers (Q):

𝑎
any numbers that can be written in form of a fraction such that 𝑏 ≠ 0
𝑏

Notes:

a. Any whole number, natural number or integer is considered as a rational number.

b. Two types of decimals are rational.

i. Terminating: 0.1, 0.34, 1.335

ii. Recurring: 0. 1̇ = 0.111111…

0.23̇ = 0.233333333…

0. 1̇2̇ = 0.12121212…

Irrational number: Numbers that can’t be expressed as rational numbers.

E.g.: Surds: √2, √7 , √10, …………

3

Practice:

1. Look at the numbers 21, 35, 49, 31, 24. From this list write down,

a. a square number,

b. a prime number.

2. Find the value of

a. 50,

b. the square roots of 64,

c. the cube root of 64,

d. the integer closest in value to (1.8)3

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3. From the set of numbers below,

20 21 22 23 24 25 26 27 28 29 30

write down

a. a multiple of 8,

b. a square number

c. a cube number

d. two prime numbers,

e. a factor of 156,

f. the square root of 784,

g. two numbers whose product is 567.

5
4. From the following list 0. 3̇ 4 √8 √25 0.3333
2

Write down

a. the prime numbers

b. the irrational numbers

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Factors and Multiples (HCF and LCM)

Explaining example:

A. Factors, multiples and prime factors of 10:

10

Factors Prime Factors

1, 2, 5, 10 Multiples 2, 5

10, 20, 30, 40, 50, ..

Practice: ……

1. Write down

a. a common factor of 15 and 27, which is greater than 1,

b. a common multiple of 10 and 12.

c. two of the factors of 2007 are square numbers. One of these is 1.

Find the other square number.

2. Write down the factors of 48 which are between 10 and 40

Explaining example:

Write 12 as a product of its prime factors in an index form

Column method

2 12

2 6

3 3

12 = 22 × 3

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Practice:

1. Write down 40 as a product of i t s prime factors.

2. Write down 18 as a product of i t s prime factors.

3. Write down 100 as a product of prime factors.

HCF: Highest Common Factor:

Explaining example:

Find HCF of 12 and 24

2 12 24
Stop, when there
2 6 12 are no more
3 6 common factors
3
1 2

HCF = 2 × 2 × 3 = 12

Class work practice:

Find HCF of:

a. 24, 36

b. 12, 16, 24

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LCM: Lowest Common Multiple:

Find t h e LCM of 4 and 6

Key words: DON’T STOP untill the 1s

2 4 6
Stop, when you
2 2 3
reach 1s
3 1 3

1 1

LCM = 2 × 2 × 3 = 12

Ex. 1:

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Powers and Roots

Square numbers and square roots:

Cube numbers and cube roots:

We can write the integer as a product of primes and group the prime factors into pairs
or threes.

E.g. For square roots

For cube roots

Ex. 2:

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Directed numbers.

In mathematics, directed numbers are also known as integers. You can represent the
set of integers on a number line like this:

Ex. 3:

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Order of operations

Many people use the letters BODMAS to remember the order of operations. The
letters stand for:

Brackets pOwers Divide Multiply Add Subtract

Sometimes the word is BIDMAS replacing the O with an I for indices, which also means
powers.

E.g. Simplify:

(a) 7 × (3 + 4) = 7 × 7 = 49

(b) (10 − 4) × (4 + 9) = 6 × 13 = 78

(c) 3 + 82 = 3 + 64 = 67
6+28
(d) = (4 + 28) ÷ (17 - 9) = 32 ÷ 8 = 4
17−9
(e) √36 ÷ 4 + √100 − 36 = √9 + √64 = 3 + 8 = 11

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Ex. 4:

Ex. 5:

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Rounding numbers

E.g. Round 64.839906 to

(a) the nearest whole number = 65

(b) 1 decimal place = 64.8 (1dp)

(c) 3 decimal places = 64.840 (3dp) N.B The zero is important to show 3dp

E.g. Round 1.076 to 3 significant figures

= 1.08 (3sf)

E.g. Round 0.00736to 2 significant figures

= 0.0074 (2sf)

Ex. 6:

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Operations on fractions

Multiplying fractions:
3 2
E.g. (a) × =
4 7

(b)

3 1 3 9 27
(c) ×4 = × =
8 2 8 2 16

Ex. 7:

Dividing fractions:

𝑎 𝑐 𝑎 𝑑
÷ = ×
𝑏 𝑑 𝑏 𝑐

E.g. (a)

(b)

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(c)

Ex. 8:

Adding and subtracting fractions

E.g. (a)

(b)

(c)

Ex. 9:

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Percentages

A percentage is a fraction with a denominator of 100. Th e symbol used to represent

percentage is %.

E.g. Convert each of the following percentages to fractions in their simplest form

(a)

(b)

E.g. Convert each of the following fractions into percentages.

Some numbers can be

(a) changed to a 100 by simple
multiplication and they are
worth remembering.
(b)
20 × 5 = 100
25 × 4 = 100
(c) 50 × 2 = 100
8 × 12.5 = 100

(d) When those numbers are

not present just multiply
by 100, simplify then put a
Finding one number as a percentage of another % at the end.

E.g. Express 16 as a percentage of 48.

Increasing and decreasing by a given percentage

E.g. Increase 56 by 15%

Ans: If you consider the original to be 100% then 15% increase will lead to 115%

of the original.

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E.g. In a sale all items are reduced by 15%. If the normal selling price for a bicycle is

$120 calculate the sale price.

Ans: Note that reducing a number by 15% leaves you with 85% of the original

Percentage increases and decreases.

𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒/𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑒
% increase/decrease = × 100
𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙

E.g. The value of a house increases from $120 000 to $124 800 between August and

December. What percentage increase is this?

Ans:

Reverse percentages

Sometimes you are given the value or amount of an item aft er a percentage increase

or decrease has been applied to it and you need to know what the original value was.

E.g. A store is holding a sale in which every item is reduced by 10%. A jacket in this

sale is sold for $108.

How can you find the original price of the Jacket?

Ans: If an item is reduced by 10%, the new cost is 90% of the original (100–10). Use

x as the original value of the jacket

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Ex. 10:

Ex. 11:

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Standard form

When numbers are very small, like 0.0000362, or very large, like 358 000 000,

calculations can be time consuming, and it is easy to miss out some of the zeros.

Standard form is used to express very small and very large numbers in a compact and

efficient way. In standard form, numbers are written as a single non-zero digit

multiplied by 10 raised to a given power.

For large numbers:

E.g. Write 320 000 in standard form

Ans: Step 1: We leave one nonzero digit on the left. (Here the 3)

Step 2: Move the position of the decimal

Step3: You multiply by 10 raised to a power with the number of places that the

decimal has moved. So

E.g. Solve and give your answer in standard form.

(a)

(b) 16 is not in
standard form
so it needs
further work.

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(c)

(d)

Ex. 12:

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For small numbers

E.g. Write each of the following in standard form

(a) 0.004 = 4 × 10−3

(b) 0.00000034 = 3.4 × 10−7

(c)

Ex. 13:

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Estimation

To estimate, the numbers you are using need to be rounded before you do the
calculation.

Although you can use any accuracy, usually the numbers in the calculation are rounded to
one significant figure:

E.g. Estimate the value of:

(a)

(0.426 is the real calc.)

(b)

(6.09 is the real calc.)

Working with ratio

A ratio is a numerical comparison of two amounts. Th e order in which you write the
amounts is very important

E.g. For each of the following ratios find the missing value:

(a) 1 : 4 = x : 20

Ans:

(b) 4 : 9 = 24 : y

Ans:

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Dividing a quantity in a given ratio

E.g. Share $24 between Jess and Anne in the ratio 3:5.

(1) Unitary method

3+5=8

This is the value of 1 part 24 ÷ 8 = 3

Jess gets 3 parts:3 × 3 = $9

Anne gets 5 parts:5 × 3 = $15

(2) Ratio method

3+5=8

Ex. 14:

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Ratio and scale

All ratio scales must be expressed in the form of 1:n or n:1.

Expressing a ratio in the form of 1: n

E.g. Express 5:1000 in the form of 1: n E.g. Write 22 : 4 in the form of n : 1

E.g. Express 4 mm:50 cm as a ratio scale.

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E.g. The scale of a map is 1:25 000.

a. What is real distance between two points that are 5 cm apart on the diagram?

b. Express the real distance in kilometres.

Ans: a.

b.

E.g. A dam wall is 480 m long. How many centimetres long would it be on a map with a

scale of 1:1:2 000?

Ans:

Ex. 15:

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Rates

A rate is a comparison of two different quantities. In a rate, the quantity of one thing
is usually given in relation to one unit of the other thing. For example, 750 ml per bottle
or 60 km/h.

E.g. 492 people live in an area of 12 km2. Express this as a rate in its simplest terms.

Ans:

Average speed

E.g. A bus travels 210 km in three hours, what is its average speed in km/h?

Ans:

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E.g. I walk at 4.5 km/h. How far can I walk in 2 21 hours at the same speed?

Ans:

E.g. How long would it take to cover 200 km at a speed of 80 km/h?

Ans:

Ex. 16:

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Upper and lower bounds

E.g. Letting w represent the width of 47 cm to the nearest cm. Represent the possible
values of w on a number line then as an inequality.

Ans:

46.5 ≤ 𝑤 < 47.5

E.g. Find the upper and lower limits of L = 128000 expressed to the nearest 1000
1000
Ans: = 500 127500 ≤ 𝐿 < 128500
2

E.g. Find the upper and lower limits of h = 22.5 expressed to one decimal place

Ans: 22.5 = 22.50 22.45 ≤ ℎ < 22.55

Ex. 17:

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Problem solving:

E.g. If a = 3.6 (to 1dp) and b = 14 (to the nearest whole number), find the upper and
lower bounds for each of the following:
𝒂 𝒂+𝒃
(a) a + b (b) ab (c) b - a (d) (e)
𝒃 𝒂

Ans.: Firstly, find the upper and lower bounds of a and b:

3.55 ≤ a < 3.65 and 13.5 ≤ b < 14.5

(a) Upper bound for (a + b) = upper bound for a + upper bound for b

= 3.65 + 14.5 = 18.15

Lower bound for (a + b) = lower bound for a + lower bound for b

= 3.55 + 13.5 = 17.05

This can be written as: 17.05 ≤ (a + b) < 18.15

(b) Upper bound for ab = upper bound for a × upper bound for b

= 3.65 × 14.5 = 52.925

Lower bound for ab = lower bound for a × lower bound for b

= 3.55 × 13.5 = 47.925

This can be written as: 47.925 ≤ ab < 52.925

(c) Upper bound for (b – a) = upper bound for b - lower bound for a

= 14.5 – 3.55 = 10.95

Lower bound for (b – a) = lower bound for b – upper bound for a

= 13.5 – 3.65 = 9.85

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𝒂
(d) Upper bound for =
𝒃

𝒂
Lower bound for =
𝒃

𝒂+𝒃
(e) Upper bound for =
𝒂

𝒂+𝒃
Lower bound for =
𝒂

Ex. 18:

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Indices

E.g. Express these numbers as products of their prime factors in index form.

(a) 200 (b) 19 683

The laws of indices

Index/power
For numbers with the same base
𝑎𝑚
• 𝑚
𝑎 ×𝑎 =𝑎𝑛 𝑚+𝑛
E.g. 7 × 7 = 7
2 3 5

Base

• 𝑎𝑚 ÷ 𝑎𝑛 = 𝑎𝑚−𝑛 E.g. 75 ÷ 73 = 72

• (𝑎𝑚 )𝑛 = 𝑎𝑚𝑛 E.g. (52 )3 = 56

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1 1 2
• 𝑎−𝑚 = E.g. 3−2 = E.g. = 2 × 34
𝑎𝑚 32 3−4
−2 0
• 𝑎0 = 1 a ≠ 0 E.g. 50 = 1 E.g. ( 3 ) = 1

• (𝑎𝑏)𝑚 = 𝑎𝑚 𝑏 𝑚 E.g. (2𝑥 2 )3 = 23 𝑥 6 = 8𝑥 6

𝑎 𝑚 𝑎𝑚 2𝑥 2 4𝑥 2 3 −2 52
• (𝑏 ) = E.g. (3) = E.g. (5) =
𝑏𝑚 9 32

in
𝑚 6
𝑛 3
√𝑥 𝑚 =𝑥 𝑛 E.g. √𝑥 6 = 𝑥 = 𝑥2 3

Out

Ex. 19:

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Ex. 20:

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Ex. 21:
1

Sets

Some examples of sets are:

A = {2, 3, 5, 7} A is the set of prime numbers less than 10

B = {H, A, P, Y} B is the set of letters in the word ‘HAPPY’

• Two sets are equal if they contain exactly the same elements, even if the order is
different, so: {1, 2, 3, 4} = {4, 3, 2, 1} = {2, 4, 1, 3}

• A set that contains no elements is known as the empty set. The symbol ∅ is used
to represent the empty set.

E.g. {odd numbers that are multiples of two} = ∅

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• if x is a member (an element) of the set A then it is written: x ∈ A

If x is not a member of the set A, then it is written: x ∉ A

• Finite and infinite sets:

o If A = {letters of the alphabet}, then A has 26 members and is finite

o If B = {positive integers}, then B = {1, 2, 3, 4, 5, 6, . . .} and is

infinite.

• n(A) is the number of elements in set A

• The universal set : A universal set contains all possible elements that you would

consider for a set in a particular problem

• The complement of the set A is the set of all things that are in ℰ but NOT in the

set A. The symbol A′ is used to denote the complement of set A.

• Union and Intersection:

o A∪B: The union of two sets, A and B, is the set of all elements that

are members of A or members of B or members of both.

o A∩B: The intersection of two sets, A and B, is the set of all

elements that are members of both A and B.

E.g. if C = {4, 6, 8, 10} and D = {6, 10, 12, 14}, then:

C ∪D = {4, 6, 8, 10, 12, 14}

C ∩D = {6, 10}

• Subset: B ⊆ A: If every element of B is also a member of A and therefore, B is

completely contained within A. B is called a subset of A

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N.B: If B is not equal to A, then B is known as a proper subset.

If it is possible for B to be equal to A, then B is not a proper subset and you

write: B ⊂ A. If A is not a proper subset of B, we write A ⊄ B.

E.g. If W = {4, 8, 12, 16, 20, 24} and T = {5, 8, 20, 24, 28}.

1. List the sets:

a. W∪T b. W∩T

2. Is it true that T ⊂ W?

Ans.: 1. a. W ∪T = {4, 5, 8, 12, 16, 20, 24, 28}

b. W ∩T = {8, 20, 24}

2. 5 ∈ T but 5 ∉ W. So it is not true that every member of T is also a member

of W. So, T is not a subset of W.

Ex. 22:

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Set builder notation

E.g. List the members of the set C if: C = {x: x ∈ primes, 10 < x < 20}

Ans.: C = {11, 13, 17, 19}

E.g. Express the following set in set builder notation:

D = {right-angled triangles}

Ans.: D = {x : x is a triangle, x has a right-angle}

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Venn diagrams

if ℰ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {1, 2, 3, 4, 5, 6, 7} and B = {4, 5, 8}

then the Venn diagram looks like this:

Some important Venn diagrams:

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Ex. 23:

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Time

E.g. Sara and John left home at 2.15 p.m. Sara returned at 2.50 p.m. and John returned

at 3.05 p.m. How long was each person away from home?

Ans.:

E.g. A train leaves at 05.35 and arrives at 18.20. How long is the journey?

Ans.: 18.20 is equivalent to 17 hours and 80 minutes after 12 a.m. (20–35 is not

meaningful in the context of time, so carry one hour over to give 17 h 80 min.)

17h - 5h = 12h, 80 min - 35 min = 45 min

The journey took 12 hours and 45 minutes.

E.g. How much time passes from 19.35 on Monday to 03.55 on Tuesday?

Ans.: 19.35 to 24.00 is one part and 00.00 to 03.55 the next day is the other part.

Part one: 19.35 to 24.00. 24 h = 23 h 60 min (past 12 a.m.)

23 h 60 min – 19 h 35 min = 4 h 25 min

Part two: 0.00 to 03.55. 3h 55 min – 0 h 0 min = 3 h 55 min

4 h 25 min + 3 h 55 min = 7 h 80 min

80 min = 1 h 20 min

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7 h 0 min + 1 h 20 min = 8 h 20 min

so 8 hours and 20 min passes.

Reading timetables

Managing money

Earning money:

E.g. Emmanuel makes beaded necklaces for a curio stand. He is paid in South African

rand at a rate of R14.50 per completed necklace. He is able to supply 55 necklaces per

week. Calculate his weekly income.

Ans.:

E.g. Sanjay works as a sales representative for a company that sells mobile phones in

the United Arab Emirates. He is paid a retainer of 800 dirhams (Dhs) per week

plus a commission of 4.5% of all sales.

a. How much would he earn in a week if he made no sales?

b. How much would he earn if he sold four phones at Dhs3299 each in a week?

Ans.: a. Dhs800

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b.

E.g. Josh’s hourly rate of pay is $12.50. He is paid ‘time-and-a-half’ for work after

hours and on Saturdays and ‘double-time’ for Sundays and Public Holidays.

One week he worked 5.5 hours on Saturday and 3 hours on Sunday. How much

overtime pay would he earn?

Ans.:

Simple interest

E.g. $500 is invested at 10% per annum simple interest. How much interest is earned in

three years?

500×10×3
Ans.: 𝐼= = $150
100

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E.g. Sam invested $400 at 15% per annum for three years. How much money did he have

at the end of the period?

Ans.:

E.g. Calculate the rate of simple interest if a principal of $250 amounts to $400 in

three years.

Ans.:

Hire purchase

On HP you pay a part of the price as a deposit and the remainder in a certain number of
weekly or monthly instalments.

E.g. The cash price of a car was $20 000. The hire purchase price was $6000 deposit
and instalments of $700 per month for two years. How much more than the cash
price was the hire purchase price?

Ans.:

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Compound interest

E.g. $1500 is invested at 5% p.a. compound interest. What will the investment be worth

after 5 years? (p.a. means per annum)

Ans.:

E.g. A sum of money invested for 5 years at a rate of 5% interest, compounded yearly,

grows to $2500. What was the initial sum invested?

Ans.:

5 5 2500
2500 = 𝑃 (1 + 100) 2500 = 𝑃(1.05)5 P= = $ 1958.82
1.055

Ex. 24:

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Exchange Rate:

E.g. Convert £50 into Botswana pula, given that £1 = 9.83 pula

Ans.:

E.g. Convert 803 pesos into British pounds given that £1 = 146 pesos.

Ans.: 146 pesos = £1

1
803 pesos = £ × 803 = £5.50
146

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2-Algebra

Substitution

E.g. Given that a = 2 and b = 8, evaluate:

a. ab b. 3b - 2a c. 2a3 d. 2a(a + b)

Ans.: a. 𝑎𝑏 = 2 × 8 = 16

b. 3𝑏 − 2𝑎 = 3 × 8 − 2 × 2 = 24 − 4 = 20

c. 2𝑎3 = 2 × 23 = 2 × 8 = 16

d. 2𝑎(𝑎 + 𝑏) = 2 × 2 × (2 + 8) = 4 × 10 = 40
Working with brackets

E.g. Expand and simplify where possible.

a. 6(x + 3) + 4 b. 2(6x + 1) - 2x + 4 c. 2x(x + 3) + x(x - 4)

d. 8(p + 4) - 10(9p - 6)

Ans.: a. 6𝑥 + 18 + 4 = 6𝑥 + 22

b.12𝑥 + 2 − 2𝑥 + 4 = 10𝑥 + 6

c.2𝑥 2 + 6𝑥 + 𝑥 2 − 4𝑥 = 3𝑥 2 + 2𝑥

d.8𝑝 + 32 − 90𝑝 + 60 = −82𝑝 + 92

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Ex. 25:

Factorisation

1-H.C.F.

E.g. Factorise each of the following expressions as fully as possible.

Ans.: a.

b.

c.

d.

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2- Quadratic expressions

E.g. Factorise each of the following expressions:

a. 𝑥 2 + 3𝑥 − 10 b. 2𝑥 2 + 11𝑥 + 12 c. 𝑥 2 + 6𝑥 + 9

Ans.: a. 𝑥 2 + 3𝑥 − 10 = (𝑥 − 2)(𝑥 + 5)

b. 2𝑥 2 + 11𝑥 + 12 = (2𝑥 + 3)(𝑥 + 4)

c. 𝑥 2 + 6𝑥 + 9 = (𝑥 + 3)2

Subject of a formula

E.g. Make the variable shown in brackets the subject of the formula in each case

Ans.: a. y = c - x

b. √𝑥 = 𝑧 − 𝑦 𝑥 = (𝑧 − 𝑦)2

c. a - b = cd a - cd = b b = a - cd

Ex. 26:

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Linear equations

E.g. Solve the equation 5x - 2 = 3x + 6

8
Ans.: 5𝑥 − 3𝑥 = 6 + 2 2𝑥 = 8 𝑥=2=4

E.g. Solve the equation 5x + 12 = 20 - 11x

8 1
Ans.: 5𝑥 + 11𝑥 = 20 − 12 16𝑥 = 8 𝑥 = 16 = 2

E.g. Solve the equation 2(y - 4) + 4(y + 2) = 30

30
Ans.: 2𝑦 − 8 + 4𝑦 + 8 = 30 6𝑦 = 30 𝑦= =5
6

6
E.g. Solve the equation 7 𝑝 = 10
7 70 35
Ans.: 𝑝 = 10 × 6 = =
6 3

Ex. 27:

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Simultaneous linear equations

Graphical solution of simultaneous linear equations:

E.g. By drawing the graphs of each of the following equations on the same pair of axes,
find the simultaneous solutions to the equations.

𝑥 − 3𝑦 = 6 and 2𝑥 + 𝑦 = 5

Ans.:

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Ex. 28:

Algebraic solution of simultaneous linear equations

1-Solving by substitution

E.g. Solve simultaneously by substitution

3𝑥 − 2𝑦 = 29 (1)

4𝑥 + 𝑦 = 24 (2)

Ans.:

Substituting in equation (3) (or any equation)

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2-Solving by elimination

E.g. Solve the following pair of equations using elimination:

Ans.:

In equ. (2)

E.g.

Ans.:

The second equation is also satisfied by these values so x = -1 and y = 2.

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E.g.

Ans.:

Substituting in equation (1)

Equation (2) is also satisfied by these values, so x = -1 and y = 3.

E.g.

Ans.:

Subtracting (2)-(3)

Substituting in (1)

So the pair of values x = 2 and y = -4 satisfy the pair of simultaneous equations.

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E.g.

Ans.:

Check using equation (2).

So x = 3 and y = -3 satisfy the pair of simultaneous equations.

Ex. 29:

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Geometry
Lines, angles and shapes

Lines and angles

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Measuring and drawing angles

Exercise

For each angle listed: i) BAC ii) BAD iii) BAE

iv) CAD v) CAF vi) CAE

vii) DAB viii) DAE ix) DAF

a) state what type of angle it is (acute, right or obtuse)

b) estimate its size in degrees

c) use a protractor to measure the actual size of each

angle to the nearest degree.

d) What is the size of reflex angle DAB?

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Angle relationships

1) Complementary angles

Angles in a right angle add up to 90°.

When the sum of two angles is 90° those two angles are complementary angles.

2) Supplementary angles

Angles on a straight line add up to 180°.

When the sum of two angles is 180° those two angles are supplementary angles.

3) Angles round a point

Angles at a point make a complete revolution.

The sum of the angles at a point is 360°.

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4) Vertically opposite angles

When two lines intersect, two pairs of vertically opposite angles are formed.

Vertically opposite angles are equal in size

Exercise

In each diagram, find the value of the angles marked with a letter

Angles and parallel lines

When two parallel lines are cut by a third line (the transversal) eight angles are

formed. These angles form pairs which are related to each other in specific ways.

Corresponding angles (‘F’-shape)

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When two parallel lines are cut by a transversal four pairs of corresponding

angles are formed. Corresponding angles are equal to each other.

Alternate angles (‘Z’-shape)

When two parallel lines are cut by a transversal two pairs of alternate angles are

formed. Alternate angles are equal to each other.

Co-interior angles (‘C’-shape)

When two parallel lines are cut by a transversal two pairs of co-interior angles

are formed. Co-interior angles are supplementary (together they add up to 180°).

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Ex

Triangles

A triangle is a plane shape with three sides and three angles.

Triangles are classified according to the lengths of their sides and the sizes of

their angles(or both).

Scalene triangle

Scalene triangles have no sides of equal length

and no angles that are of equal sizes.

Isosceles triangle

Isosceles triangles have two sides of equal length.

The angles at the bases of the equal sides are equal in size.

Equilateral triangle

Equilateral triangles have three equal sides and

three equal angles (each being 60°).

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Other triangles

Acute-angled triangles

have three angles each < 90°.

Right-angled triangles

have one angle = 90°.

Obtuse-angled triangles

have one angle > 90°.

Angle properties of triangles

Look at the diagram below carefully to see two important angle properties of

triangles.

The diagram shows two things:

• The three interior angles of a triangle add up to 180°.

• Two interior angles of a triangle are equal to the opposite exterior angle.

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Angles in a triangle add up to 180°

To prove this you have to draw a line parallel to one side of the triangle.

x + a + y = 180° (angles on a line)

but:

b = x and c = y (alternate angles are equal)

so a + b + c = 180°

The exterior angle is equal to the sum of the opposite interior angles

c + x = 180° (angles on a line)

so, c = 180° − x

a + b + c = 180° (angle sum of triangle)

c = 180° − (a + b)

so, 180° − (a + b) = 180° − x

hence, a + b = x

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Ex

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Quadrilaterals

Quadrilaterals are plane shapes with four sides and four interior angles.

Quadrilaterals are given

special names according to their properties.

The angle sum of a quadrilateral

All quadrilaterals can be divided into two triangles by drawing one diagonal. You

already know that the angle sum of a triangle is 180°. Therefore, the angle sum of

a quadrilateral is 180° + 180° = 360°.

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Ex

Polygons

A polygon is a plane shape with three or more straight sides. Triangles are

polygons with three sides and quadrilaterals are polygons with four sides. Other

polygons can also be named according to the number of sides they have.

A polygon with all its sides and all its angles equal is called a regular polygon.

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Sum of interior angles = (n − 2) × 180°

EX

EX

The sum of exterior angles of a convex polygon

The sum of the exterior angles of a convex polygon is always 360°

Exercise:

1) Copy and complete this table.

2) Find the size of one interior angle for each of the following regular polygons.

a) pentagon b) hexagon c) octagon

3) A regular polygon has 15 sides. Find:

a) the sum of the interior angles

b) the sum of the exterior angles

c) the size of each interior angle

d) the size of each exterior angle.

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4) A regular polygon has n exterior angles of 15°. How many sides does it have?

5) Find the value of x in each of these irregular polygons.

Circles

In mathematics, a circle is defined as a set of points which are all the same

distance from a given fixed point. In other words, every point on the outside

curved line around a circle is the same distance from the centre of the circle.

Parts of a circle

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Exercise:

Circle 1 and circle 2 have the same centre (O). Use the correct terms or letters

to copy and complete each statement.

a) OB is a __ of circle 2.

b) DE is the __ of circle 1.

c) AC is a __ of circle 2.

d) __ is a radius of circle 1.

e) CAB is a __ of circle 2.

f)Angle FOD is the vertex of a __ of circle 1 and circle 2.

Construction

Using a ruler and a pair of compasses

Your ruler (sometimes called a straight-edge) and a pair of compasses are

probably your most useful construction tools. You use the ruler to draw straight

lines and the pair of compasses to measure and mark lengths, draw circles and

bisect angles and lines.

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Constructing triangles

You can draw a triangle if you know the length of three sides.

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Perimeter, area and volume

Perimeter

Perimeter is the measured or calculated length of the boundary of a shape.

The perimeter of a circle is its circumference.

You can add the lengths of sides or use a formula to calculate perimeter.

Area

The area of a region is the amount of space it occupies. Area is measured in

square units.

The surface area of a solid is the sum of the areas of its faces.

The area of basic shapes is calculated using a formula.

Volume

The volume of a solid is the amount of space it occupies.

Volume is measured in cubic units.

The volume of cuboids and prisms can be calculated using a formula.

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Perimeter and area in two dimensions

Polygons

A polygon is a flat (two-dimensional) shape with three or more straight sides. The

perimeter of a polygon is the sum of the lengths of its sides. The perimeter

measures the total distance around the outside of the polygon.

The area of a polygon measures how much space is contained inside it.

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Units of area

If the dimensions of your shape are given in cm, then the units of area are square

centimetres.

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Finding the circumference of a circle

Circumference is the word used to identify the perimeter of a circle.

Note that the diameter =2 × radius (2r).

Finding the area of a circle

Area = 𝛑 r2

Ex

Ex

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Surface areas and volumes of solids

Cuboids:

A cuboid has six rectangular faces,

12 edges and eight vertices.
Notice that the surface area is exactly
the same as the area of the cuboid’s net.
Surface area of cuboid = 2(ab + ac + bc)
Volume of cuboid = a × b × c

Prisms

A prism is a solid whose cross-section is the same all along its length.

(A cross-section is the surface formed when you

cut parallel to a face.)

Cylinders

A cylinder is another special case of a prism.

It is a prism with a circular cross-section.

Pyramids

A pyramid is a solid with a polygon-shaped base and triangular

faces that meet at a point called the apex.

If you find the area of the base

and the area of each of the triangles,

then you can add these up to find the total surface area

of the pyramid.

The volume can be found by using the following formula:

1
Volume = 3× base area ×perpendicular height

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Cones

A cone is a special pyramid with a circular base.

The length l is known as the slant height.

h is the perpendicular height.

Spheres

The diagram shows a sphere with radius r.

Pythagoras’ theorem and similar shapes

Pythagoras’ theorem

Centuries before the theorem of right-angled triangles was credited to Pythagoras, the

Egyptians knew that if they tied knots in a rope at regular intervals, as in the diagram

on the left , then they would produce a perfect right angle. In some situations you may

be given a right-angled triangle and then asked to calculate the length of an unknown

side. You can do this by using Pythagoras’ theorem if you know the lengths of the other

two sides.

Learning the rules

Pythagoras’ theorem describes the relationship between the sides of a right-angled

triangle. The longest side – the side that does not touch the right angle – is known as

the hypotenuse.

For this triangle, Pythagoras’ theorem states that:

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Ex:

Checking for right-angled triangles

You can also use the theorem to determine if a triangle is right-angled or not.

Substitute the values of a, b and c of the triangle into the formula and check to see if

it fits. If a2 + b2 does not equal c2 then it is not a right-angled triangle.

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Applications of Pythagoras’ theorem

This section looks at how Pythagoras’ theorem can be used to solve real-life problems.
In each case look carefully for right-angled triangles and draw them separately to make
the working clear.
Ex

Similar triangles

Two mathematically similar objects have exactly the same shape and proportions, but
may be different in size.

When one of the shapes is enlarged to produce the second shape, each part of the

original will correspond to a particular part of the new shape. For triangles,

corresponding sides join the same angles.

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All of the following are true for similar triangles:

If any of these things are true about two triangles, then all of them will be true for

both triangles.

Ex

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Understanding similar shapes

In the previous section you worked with similar triangles, but any shapes can be similar.
A shape is similar if the ratio of corresponding sides is equal and the corresponding
angles are equal.
Similar shapes are therefore identical in shape, but they differ in size.
You can use the ratio of corresponding sides to find unknown sides of similar shapes
just as you did with similar triangles.
Ex 1

Ex 2

Note:

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Bearings

Bearings are measured from north (0°)

in a clockwise direction to 360°

(which is the same bearing as north).

You measure bearings using your protractor and write them using three digits, so you

would write a bearing of 88 degrees as 088°.

You have now used scale drawings to find distances between objects and to measure

angles. When you want to move from one position to another, you not only need to know

how far you have to travel but you need to know the direction. One way of describing

directions is the bearing. This description is used around the world.

The angle 118°, shown in the diagram, is measured

clockwise from the north direction. Such an angle is called a bearing.

All bearings are measured clockwise from the north direction.

Here the bearing of P from O is 118°.

If the angle is less than 100° you still use three figures so that

it is clear that you mean to use a bearing.

Here the bearing of Q from O is 040°.

Since you always measure clockwise from north it is possible

for your bearing to be a reflex angle.

Here the bearing of R from O is 315°.

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Ex

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