178-IGCSE-E-Maths_5thPass_Batch2.pdf
Preface
This Extended Mathematics guidebook is as per
the latest syllabus of IGCSE. It is specially designed
for students who believe in self-study and intend to
do topical revision. Each chapter is carefully structured
in such a way that it begins with explanation of basic
concepts required to strengthen students’ mathematical
concepts.
Solved examples provide students with an opportunity
to understand how these concepts can be applied to
questions. Furthermore, the guidebook provides plenty
of practice questions crafted to cater to students’ needs
to prepare themselves for examinations. The practice
questions present increasing level of difficulty so that
the students can challenge themselves at an appropriate
pace hence building their level of confidence. Detailed
step-by-step worked solution of each practice question
is another highlight of this guidebook as the students
get an opportunity to check their answers and correct
themselves when in doubt.
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Contents
Chapter No.
Chapter Title
1
Numbers
2
Algebra
3
Sets
118
4
Mensuration
133
5
Geometry
184
6
Trigonometry
216
7
Vectors
257
8
Graphs
267
9
Matrices and Transformation
329
10
Statistics and Probability
376
Answers
419
Page No.
1
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Chapter
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1
Numbers
Type of Numbers
Type of Number
Definition
Natural Number
The set of numbers, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, ..., that
we see and use every day. The natural numbers are often referred to as the
counting numbers and the positive integers.
Whole Numbers
The natural numbers plus the zero.
Integers
Positive and negative counting numbers, as well as zero.
{..., −2, −1 , 0, 1, 2, ...}
Rational Number
Any number that is either an integer “a” or is expressible as the ratio of two
a
integers,
. The numerator, “a”, may be any whole number, and the
b
denominator, “b”, may be any positive whole number greater than zero. If
the denominator happens to be unity, b = 1, the ratio is an integer. If “b” is
a
other than 1,
is a fraction.
b
Irrational Numbers
Any number that cannot be expressed by an integer or the ratio of two
integers. Irrational numbers are expressible only as decimal fractions where
the digits continue forever with no repeating pattern. Some examples of
irrational numbers are 2 and 3 .
Real Numbers
The set of real numbers including all the rational and irrational numbers.
Real Numbers
bers
Num
l
a
n
tio
er
teg
Ra
e Numbers
In
hol
Irrational Numbers
W
Natural Numbers
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Square Number
Is a number which is the square of an integer.
For example: 1, 4, 9, 16…..
Cube Number
Is a number which is the cube of an integer.
For example: 1, 8, 27, 64……
Factor
Those numbers that divide into it exactly. This includes 1 and the number
itself.
For example, the factors of 6 are 1, 2, 3 and 6.
Prime Numbers
Natural number that only has two factors which are itself and one.
For example: 2, 3, 5, 7….
Composite Numbers
Natural number that has more than two factors.
For example: 4, 6, 8, 9….
Note
ÂÂ1 is neither prime nor composite as it has only one factor.
ÂÂAll even numbers are divisible by two and so all even numbers greater than two are
composite numbers.
ÂÂAll numbers that end in five are divisible by five. Therefore all numbers that end with
five and are greater than five are composite numbers.
ÂÂThe prime numbers between 2 and 100 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47,
53, 59, 61, 67, 71, 73, 79, 83, 89 and 97.
Highest Common Factor (HCF)
The factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24 and that the factors of 40 are 1, 2, 4, 5, 8, 10, 20 and 40.
The numbers 1, 2, 4 and 8 appear in both lists, so they are known as the common factors of 24 and 40.
The number 8 is the highest of them, and it is called the highest common factor (HCF).
Least Common Multiples (LCM)
The multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36, …
The multiples of 6 are 6, 12, 18, 24, 30, 36, 42, …
The numbers 12, 24 and 36 appear in both lists, so they are known as common multiples.
The number 12 is the least number to appear in both lists, and it is called the least common multiplex
(LCM).
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Exercise 1
1. For each of the numbers below, decide whether it is rational or irrational.
(a)5
(b)
5
8
(c)0.575
(d)5.7575....
(g)
48
3
(e) 7π
(h)
63
22
7
(i)
(f)
3
81
2. Arrange the following terms in order of size, starting from the smallest.
1
0.55, , 48%
2
3. Using the numbers given in the table, fill in the blanks:
243
3
75
6
75
71
2.3
(a) A rational number between
5 and
6 : __________________
(b) H.C.F of 30 and 54
: __________________
(c) A prime number
: __________________
(d) An irrational number
: __________________
(e) L.C.M of 15 and 25
: __________________
(f) A square number
: __________________
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4.0.52, 0.5 , 0.53
Write the numbers in order with the smallest first.
Answer: _________ < _________ < _________
5. Place brackets in the following calculation to make it a correct statement.
10 − 5 × 9 + 3 = 60
6. (a) Write down
(i) five numbers which are multiples of 7,
Answer: _________, _________ , _________ , _________ , _________
(ii) two common multiples of 4 and 7.
Answer: _________ and _________
(b)10 12 13 16 17 23 25 39
From the list above, write down
(i) a square number that is also an odd number,
(ii) a prime number that is one more than a square number.
(c) n is an integer and n3 is between 60 and 70.
Find the value of n.
(d) k and m are prime numbers.
k2 + m = 23
Find k and m.
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Estimation and Rounding Number
The rule is, if the next digit to the number we are rounding off is: 5 or more, we ‘round up’.
4 or less, it stays as it is.
33715 to the nearest thousand = 34000.
33715 to the nearest hundred = 33700.
Example:
Round 21724 to the nearest thousand.
The digit in the thousands place is 1. This means the number lies between 21000 and 22000. The digit
to the right of the 1 is 7. That means 21724 is closer to 22000 than 21000.
Solution:
Answer is 22000.
Exercise 2
1. Round off the following numbers:
Number
1 decimal
place
2 decimal
place
1
significant
figure
2
significant
figure
3
significant
figure
Scientific
Notation
987.87234
0.3456983
0.04567
35.098
0.00076543
0.70342
112.345
809.345
223.134
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2. Round of the following numbers and complete the table
Number
Hundred
Thousand
1 significant
figure
2 significant
figure
3 significant
figure
987123
567109876
345567
87943
1264344
231432
7865
24356
65476
3. The area of a small country is 78 133 square kilometres.
(a) Write this area correct to 1 significant figure.
(b) Write this area correct to 3 significant figure.
Upper Bound and Lower Bound
Example:
If a piece of ribbon is 22 cm long to the nearest cm, then what is the range of possible lengths it could be?
Solution:
It must be at least 21.5 cm long to round up to 22 cm. But it must be less than 22.5 cm, to round down
to 22 cm. If it was 22.5 cm or more we would round up to 23 cm.
Since 22.5 cm is the upper limit, it is called the ‘upper bound’.
21.5 cm is the lower bound and 22.5 cm is the upper bound for the length of the piece of ribbon.
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Exercise 3:
1. A rope is 22 m long to the nearest metre. What are the upper and lower bounds?
2. A block of wood is 7.5 cm wide measured to the nearest tenth of a cm. What are the upper and lower
bounds?
3. Write the appropriate lower and upper bound and complete the table:
Measurement
Lower Bound
Upper Bound
The mass of a dog is 17 kg to the nearest kg.
Temperature of the city is 34.4°C to the nearest 1 d.p.
Speed of the car is 3.05 km/hr to the nearest 2 d.p.
Capacity of a tank is 0.35 litre to the nearest litre.
Time taken to run 100 m race is 23.6 s to nearest 0.1 s.
Distance between school and home is 2 km to the
nearest km
4. If the sides of a rectangle are 3 cm and 5 cm to the nearest meter,
(a) Find the maximum possible perimeter.
(b) Find the lowest possible area of the perimeter.
5. The base of a parallelogram is 8 m correct to the nearest metre. If its area is 512 m2 correct to the
nearest m2, find the maximum possible height.
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Ratio
A “ratio” is a comparison between two different things. Suppose there are thirty-five people, fifteen of
whom are men. Then the ratio of men to women is 15 to 20. We can express it in
odds notation: 15 : 20
fractional notation:
15
20
Example:
Divide $100 in the ratio 2 : 3
Solution:
2
× 100 = $40
5
3
2nd Part = × 100 = $60
5
1st Part =
Proportion
When two ratios are equal then we say they are in proportion. Proportion is of two types:
Direct Proportion
Two quantities are in direct proportion when they increase or decrease in the same ratio.
If a and b are in direct proportion then we can write this as a ∝ b.
a
= k (constant)
b
a c
=
b d
ÂÂusing a colon, a : b = c : d
ÂÂtwo equal fractions,
When two ratios are equal, then the cross products of the ratios are equal.
That is, for the proportion, a : b = c : d, a × d = b × c (Product of extreme is equal to product of mean)
Inverse Proportion
Two quantities are in inverse proportion if one quantity increases then the other decreases in the same
ratio.
1
If a and b are in indirect proportion then a ∝ .
b
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If a and b are in inverse proportion their product is constant. (a × b = constant)
Example:
It takes 4 people 6 hours to paint the wall, how much time 8 people will take to do the same task.
Solution:
The time taken to paint a wall is indirectly proportional to the number of people doing the painting.
4 × 6 = 8 × No. of hours
4×6
Time taken by 8 people needed to paint the wall =
= 3 hrs
8
Exercise 4:
1. Divide £7400 among three people A, B and C in the ratio 3 : 5 : 12.
2. Roze lost her weight in the ratio 5 : 3. Her original weight was 80 kg. What is her new weight?
3. The ratio of the expenditure to the saving of a family is 6 : 3. Find the income if expenditure is $6300.
4. Adam can type 3600 words in one hour. How many words can he type in 15 minutes?
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