Preface - Fairfield Book Publisher · Preface This Extended ... The multiples of 4 are 4, 8, 12,...

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.

178-IGCSE-E-Maths_5thPass_Batch2.pdf 10/02/15

Chapter No. Chapter Title Page No.

1 Numbers 1

2 Algebra 37

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

Contents


Chapter

IGCSE E Maths Numbers1

Numbers1Type of Numbers

Type of Number Definition

Natural NumberThe 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.

IntegersPositive 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

integers, ab

. The numerator, “a”, may be any whole number, and the

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

other than 1, ab

is a fraction.

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.

Rational Numbers

Integer

Whole Numbers

Real Numbers

Irrational Numbers

Natural Numbers



Square NumberIs a number which is the square of an integer.For example: 1, 4, 9, 16…..

Cube NumberIs a number which is the cube of an integer.For example: 1, 8, 27, 64……

FactorThose 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 NumbersNatural number that only has two factors which are itself and one.For example: 2, 3, 5, 7….

Composite NumbersNatural number that has more than two factors.For example: 4, 6, 8, 9….

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).

ÂÂ 1 is neither prime nor composite as it has only one factor.

ÂÂ All even numbers are divisible by two and so all even numbers greater than two are composite numbers.

ÂÂ All numbers that end in five are divisible by five. Therefore all numbers that end with five and are greater than five are composite numbers.

ÂÂ The 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.

Note



Exercise 1

1. For each of the numbers below, decide whether it is rational or irrational.

(a) 5

(b) 58

(c) 0.575

(d) 5.7575....

(e) 7π

(f) 227

(g) 483

(h) 63

(i) 813

2. Arrange the following terms in order of size, starting from the smallest.

0.55, 12

, 48%

3. Using the numbers given in the table, fill in the blanks:

2433 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 : __________________



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 39From 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 = 23Find k and m.



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.

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

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.

Example:



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

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.

Example:



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.



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: 1520

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.

ab

k= (constant)

ÂÂ two equal fractions, ab

cd

=

ÂÂ using a colon, a : b = c : d

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.

If a and b are in indirect proportion then ab

∝ 1.

Divide $100 in the ratio 2 : 3

Solution:

1st Part = 25

100 40× = $

2nd Part = 35

100× = $60

Example:



If a and b are in inverse proportion their product is constant. (a × b = constant)

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?

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

Time taken by 8 people needed to paint the wall = 4 6

8×

= 3 hrs

Example:


Preface - Fairfield Book Publisher · Preface This Extended ... The multiples of 4 are 4, 8, 12,...

Documents

Transcript of Preface - Fairfield Book Publisher · Preface This Extended ... The multiples of 4 are 4, 8, 12,...