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Secondary Maths Resources
IGCSE Higher Tier Revision Guide
Update 1.1 – 03 May 2007
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Contents TYPES OF NUMBER, DECIMAL PLACES AND SIGNIFICANT FIGURES .......................................................................................... 5 TYPES OF NUMBERS: ........................................................................................................................................................................................ 5 DECIMAL PLACES ............................................................................................................................................................................................. 5 SIGNIFICANT FIGURES ...................................................................................................................................................................................... 5 STANDARD FORM .......................................................................................................................................................................................... 6 HIGHEST COMMON FACTOR AND LOWEST COMMON MULTIPLE ............................................................................................... 7 RATIO AND PROPORTION........................................................................................................................................................................... 8 SHARING IN A GIVEN RATIO .............................................................................................................................................................................. 8 PROPORTION .................................................................................................................................................................................................... 8 PERCENTAGES AND PROPORTIONAL CHANGE .................................................................................................................................. 9 PERCENTAGE INCREASE/DECREASE AND REVERSE PERCENTAGES ........................................................................................ 10 PERCENTAGE INCREASE ................................................................................................................................................................................. 10 REVERSE PERCENTAGES ................................................................................................................................................................................. 10 INDICES .......................................................................................................................................................................................................... 11 EQUATIONS INVOLVING INDICES ..................................................................................................................................................................... 11 SURDS .............................................................................................................................................................................................................. 12 EXPANDING BRACKETS .................................................................................................................................................................................. 12 FRACTIONS.................................................................................................................................................................................................... 13 CONVERTING DECIMALS TO FRACTIONS .......................................................................................................................................................... 13 ANGLES........................................................................................................................................................................................................... 14 POLYGONS ..................................................................................................................................................................................................... 16 TRANSFORMATIONS .................................................................................................................................................................................. 17 ENLARGEMENT .............................................................................................................................................................................................. 17 TRANSLATION................................................................................................................................................................................................ 18 ROTATION ..................................................................................................................................................................................................... 19 REFLECTION .................................................................................................................................................................................................. 20 MISCELLANEOUS QUESTIONS ......................................................................................................................................................................... 21 SECTORS AND SEGMENTS OF A CIRCLE ............................................................................................................................................. 22 CIRCLE THEOREMS .................................................................................................................................................................................... 23 THE INTERSECTING CHORD THEOREM ............................................................................................................................................................. 25 PYTHAGORAS’ THEOREM ........................................................................................................................................................................ 29 PROPORTION AND INVERSE PROPORTION ........................................................................................................................................ 30 INVERSE PROPORTION .................................................................................................................................................................................... 30 CONSTRUCTIONS ........................................................................................................................................................................................ 32 BEARINGS ...................................................................................................................................................................................................... 34 AREA AND VOLUME ................................................................................................................................................................................... 35 AREA AND VOLUME OF SIMILAR SHAPES .......................................................................................................................................... 38 AREA............................................................................................................................................................................................................. 38 VOLUME ........................................................................................................................................................................................................ 39 TRIGONOMETRY ......................................................................................................................................................................................... 40 SINE AND COSINE RULES .......................................................................................................................................................................... 42 THE SINE RULE .............................................................................................................................................................................................. 42 THE COSINE RULE ......................................................................................................................................................................................... 43 VECTORS ........................................................................................................................................................................................................ 45 GEOMETRIC PROBLEMS ........................................................................................................................................................................... 47 SOLVING LINEAR EQUATIONS ................................................................................................................................................................ 48 IGCSE Maths Revision Guide: Higher Tier
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NTH TERM ..................................................................................................................................................................................................... 50 SIMULTANEOUS EQUATIONS 1 ............................................................................................................................................................... 51 SLIGHTLY HARDER SIMULTANEOUS EQUATIONS .............................................................................................................................................. 52 REARRANGING FORMULAE..................................................................................................................................................................... 53 BASIC FACTORISATION ............................................................................................................................................................................ 54 FACTORISING EXPRESSIONS ............................................................................................................................................................................ 54 THE DIFFERENCE OF TWO SQUARES................................................................................................................................................................. 54 QUADRATIC EXPRESSIONS .............................................................................................................................................................................. 54 SOLVING QUADRATIC EQUATIONS ....................................................................................................................................................... 56 STRAIGHT LINES ......................................................................................................................................................................................... 57 INEQUALITIES .............................................................................................................................................................................................. 59 ERRORS IN MEASUREMENT .................................................................................................................................................................... 60 SHADING REGIONS ..................................................................................................................................................................................... 61 QUADRATIC INEQUALITIES..................................................................................................................................................................... 63 SIMPLIFYING A* ALGEBRAIC EXPRESSIONS AND SOLVING EQUATIONS................................................................................ 65 SIMULTANEOUS EQUATIONS 2 – ONE LINEAR AND ONE QUADRATIC ..................................................................................... 66 MEAN, MEDIAN, MODE, QUARTILES AND RANGE ............................................................................................................................ 68 QUARTILES .................................................................................................................................................................................................... 69 CUMULATIVE FREQUENCY ..................................................................................................................................................................... 71 PROBABILITY ............................................................................................................................................................................................... 73 SAMPLE SPACE DIAGRAM ............................................................................................................................................................................... 74 TREE DIAGRAMS ............................................................................................................................................................................................ 75 CONDITIONAL PROBABILITY ........................................................................................................................................................................... 76 HISTOGRAMS ................................................................................................................................................................................................ 77 SETS ................................................................................................................................................................................................................. 79 SUBSETS ........................................................................................................................................................................................................ 79 THE UNIVERSAL SET ....................................................................................................................................................................................... 79 INTERSECTION ............................................................................................................................................................................................... 80 DISJOINT SETS................................................................................................................................................................................................ 80 UNION ........................................................................................................................................................................................................... 80 ALTERNATIVE WAYS OF WRITING SETS ........................................................................................................................................................... 80 FUNCTIONS.................................................................................................................................................................................................... 82 RESTRICTING THE DOMAIN ............................................................................................................................................................................. 82 INVERSE FUNCTIONS. ..................................................................................................................................................................................... 83 COMPOSITE FUNCTIONS .................................................................................................................................................................................. 83 GRAPHS 1 ....................................................................................................................................................................................................... 85 TRIGONOMETRIC GRAPHS ...................................................................................................................................................................... 87 DIFFERENTIATION...................................................................................................................................................................................... 89 GRADIENT OF A CURVE .................................................................................................................................................................................. 89 NOTATION ..................................................................................................................................................................................................... 91 TURNING POINTS ............................................................................................................................................................................................ 92 MOTION IN A STRAIGHT LINE .......................................................................................................................................................................... 92 GRAPHS 2 ....................................................................................................................................................................................................... 94 TRAVEL GRAPHS ......................................................................................................................................................................................... 96 DISTANCE-TIME AND VELOCITY-TIME GRAPHS .............................................................................................................................. 97 DISTANCE-TIME GRAPHS ............................................................................................................................................................................... 97 VELOCITY-TIME GRAPHS ............................................................................................................................................................................... 97 ANSWERS ..................................................................................................................................................................................................... 100
IGCSE Maths Revision Guide: Higher Tier
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Introduction This revision guide is based on Edexcel’s International GCSE examination in Mathematics which was first examined in May 2004. It provides a comprehensive, easy to read and informative set of notes and is especially designed to be used regularly in the final months leading up to your exam. It is assumed that you have a good understanding of Grade D/C material. The guide outlines the topics you will be asked in the Higher Tier papers and gives simple examples to clarify and poses questions for you to try. The answers are in the back of this guide. When practising past papers, it is often very tempting to ask the teacher for help before you have really taken enough time to think about the problem. You will find that your knowledge of each topic will be reinforced if you research the things which trouble you before seeking help from the teacher. If you want to cement in place the topics which you find challenging then why not try teaching it to one of your friends. Having this guide by your side will prove to be an invaluable resource. I wish you every success in your forthcoming examination. P. D. Collins Head of Mathematics Portsmouth High School January 2006
Update 1.1 This update corrects several minor errors throughout the text. Corrected errors include: Page 11: 9x = 343 changed to 9x = 243 Page 14: 40° label added to diagram Page 53: More space given to example 2 Page 62: Marked co-ordinates stated Page 68:
75 75 changed to and 15 to 12.5. 5 6
In addition, this update contains major improvements to the formatting and layout.
This material has been endorsed by Edexcel and offers high quality support for the delivery of Edexcel qualifications. Edexcel endorsement does not mean that this material is essential to achieve any Edexcel qualification, nor does it mean that this is the only suitable material available to support any Edexcel qualification. No endorsed material will be used verbatim in setting any Edexcel examination and any resource lists produced by Edexcel shall include this and other appropriate texts. While this material has been through an Edexcel quality assurance process, all responsibility for the content remains with the publisher. Copies of official specifications for all Edexcel qualifications may be found on the Edexcel website www.edexcel.org.uk
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Types of number, decimal places and significant figures Types of numbers: Integers:
…–2, –1, 0, 1,2, 3,… etc
Square numbers:
1, 4, 9, 16,…etc
Cube numbers:
1, 8, 27, 64,…etc
Triangle numbers:
1, 3, 6, 10, 15,…etc
Prime numbers : these are numbers without any factors except 1 and the number itself. i.e. 2, 3, 5, 7, 11, 13, 17, 19,…etc
Decimal places 3.267 is 3.3 to 1 decimal place. The ‘6’ is greater than 5 and so the 2 is rounded up to a 3. 3.467 is 3.47 to 2 decimal places 3.424 is 3.42 to 2 decimal places
Significant figures Some examples: 8.9 to 1 sig fig is 9 56.8 to 1 sig fig is 60 56.8 to 2 sig figs is 57 0.5629 to 1 sig fig is 0.6 0.5629 to 2 sig figs is 0.56 0.0468 to 1 sig fig is 0.05
Note that the zeros are not significant.
0.0461 to 2 sig figs is 0.046
To try: a) b) c) d)
Write 67.452 to 2 decimal places Write 0.05037 to 3 significant figures Write 234.67 to 1 decimal place Write 24579 to 2 significant figures
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Standard Form A number is said to be written in standard form when it is expressed in the form a 10n where 1 a 10 and n is a whole number.
For example
45600 4.56 104 32.46 3.246 101 0.0672 6.72 102
Addition of numbers written in standard form For example
(8 105 ) (3 104 ) 800000 30000
830000
8.3 105 Multiplication of numbers written in standard form For example
(7 108 ) (5 104 )
7 5 1012
add powers
35 1012
3.5 101 1012 3.5 1013
To try: Give your answers in standard form a) (5 104 ) (3.6 102 ) b) (3.2 103 ) (3.5 104 ) c) (5 102 ) (6 103 ) d) The planet Mars is said to be 142 million miles from the Sun. Express this number in standard form.
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Highest Common Factor and Lowest Common Multiple First let us refresh our knowledge about factor trees. 30
Express 30 as a product of prime factors: 2
15
3
5
So 30 = 2 × 3 × 5 Find the highest common factor of 60 and 100. I will not draw the factor trees this time. 60 = 2 × 2 × 3 × 5 and 100 = 2 × 2 × 5 × 5 To find the highest common factor pick out the common factors. i.e. the two 2’s and 5 Therefore 2 × 2 × 5 is the highest common factor, i.e. 20 Find the lowest common multiple of 60 and 50 To do this select the most number of factors from each number. So select: the two 2’s and the 3 from 60 and the two 5’s from the 50. 2×2×3×5
2×5×5
We have 2 × 2 × 3 × 5 × 5 = 300 Alternatively: List the 50 and 60 times table and select the first common multiple seen 50 60 100 120 150 180 200 240 250 300 300
To try: a) Express 126 as a product of prime factors b) Find the HCF and LCM of i) 120 and 150 ii) 60 and 70
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Ratio and Proportion Sharing in a given ratio Jack and Susie share £85 in the ratio 5:12. How much do they each receive? There are 17 parts. (5 +12) Divide £85 by 17 to give £5. Jack will receive 5 £5 and Susie 12 £5. i.e
£25 and £60
To try: a) Rebecca, Andrew and Neha share $96 in the ratio of 2:3:7. How much will Andrew receive? Remember that the order is important. b) A jumper is knitted using the colours red and mauve in the ratio of 2:5. Twenty balls of mauve wool are needed. How many balls of red are required? c) Divide $560 in the ratio 1:6
Proportion An RAF jet uses 2500 litres of fuel on a 45 minute flight. (i) How long does 1 litre of fuel last in seconds? (ii) How many litres of fuel does it use in 1 minute? Answers: (i) Divide 45 by 2500 to find how long 1 litre of fuel lasts. Answer:
= 0.018 minutes = 1.08 seconds
(ii) Divide 2500 by 45. i.e 55.5 litres
To try: 3m of copper tubing costs £12 d) What is the cost of 1m? e) How many metres could I buy for £30?
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Percentages and Proportional Change Sam received £100 for his birthday. His Dad increased this amount by 5% when Sam scored a high mark in his music exam. How much did Sam eventually have? One way to tackle this is to find 5% of £100 and add it on to the £100. 5% of £100 is £5 and so Sam received £105. An alternative method is to simply multiply 100 by 1.05. So multiply by 1.05 for an increase of 5%. Listed below are some multipliers for various increases. Description Increase of 5% Increase of 8% Increase of 15% Increase of 17.5% (VAT)
Number by which we multiply 1.05 1.08 1.15 1.175
The price of a motorcycle in 2005 was £2000. There was a drop in price in 2006 by 20%. What was the price in 2006? One way to tackle this is to simply find 20% of £2000 and subtract it from the original price. Clearly this is £2000 – £400. So the price in 2006 is going to be £1600. An alternative method is to multiply £2000 by 0.80 So multiply by 0.80 (i.e 1 – 0.20) for a decrease of 20% Listed below are some multipliers for various decreases. Description Decrease of 20% Decrease of 8% Decrease of 15% Decrease of 22.5%
Number by which we multiply 0.80 0.92 0.85 0.775
To try: a) My house was worth £240 000 in 2005. House prices rose by 4% in 2006. How much was my house worth in 2006? b) EasyJet increased ticket prices by 4%. Calculate the new price of a £46 ticket. c) A virus reduced the population of wild cats on a remote island in 2005 by 24%. The original population was 2000. Find the new population.
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Percentage increase/decrease and reverse percentages Percentage increase In 2005 my house was worth £240 000 and in 2006 it was worth £249 600. What was the percentage increase?
Percentage increase =
Percentage increase
increase 100% original
Percentage decrease =
decrease 100% original
249600 240000 100% 240000
4%
Reverse percentages The price of a service for a car after VAT was added was £246.75. What was the actual cost of the service before VAT was added? Let £246.75
= 117.5 %
So 1%
=£
246.75 117.5
= £2.1 Therefore the actual cost (100%)
= 100 £2.1 = £210
To try: a) A rare stamp was valued at £80 in 2005. Its value rose to £100 in 2006. What was the percentage increase? b) The cost of an evening meal for four people was £141. This included a tip of 17.5%. What was the original cost of the bill before the tip was added? c) I bought a laptop from ebay for £400 and later sold it for £450. What was my percentage gain? d) Henry bought a car for £15 000 and sold it one year later for £11 000. What was his percentage loss? e) The price of a house is reduced by 10% to £180 000. What was the original asking price?
IGCSE Maths Revision Guide: Higher Tier
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Indices The three laws of indices:
am
an
a m n
am
an
a mn
a mn
(a m ) n
Remember also:
a0 1
an
1 an
1 2
1
1
an n a
m
a n n am
a n
m
Some examples: 2
27 3 (3 27 ) 2
25
32
25
1 2
1 5
9
Equations involving indices If 9 x 243 then write (32 ) x 35 Therefore 32 x 35 and thus 2 x 5 , i.e. x = 2
1 2
To try: a) Write in index notation: (i) 23 27 (ii) 28 23
(iii) (22)3
2 3
b) Write 64 as an integer c) Write 32 as a fraction
d) What is 81
3 4
as a fraction?
e) Solve the equation 4 x 128 f) If 9 2 x 3 , find x .
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Surds n look for the largest factor of n which is a perfect square.
To simplify
Some examples:
50
25 2
5 2
80
16 5 4 5
Sometimes the square root appears in the denominator. If this is the case, multiply the numerator and denominator by the denominator. For example:
5 2
5 2
2 2
5
2 2
notice that
2 2 2
Expanding brackets Expand and simplify
(2 3 1)(3 3 4) 18 8 3 3 3 4 14 5 3
To try: Simplify a)
150 and b)
3
3 Expand and simplify c) ( 3 1)(5 3 2) d) (5 3)( 3 4) e) ( 3 1) 2
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Fractions Some fractions can be expressed as a terminating decimal such as
1 which is 0.2 5 1 which is 0.125 and so on. 8
Other fractions do not terminate but recur such as
. 1 which is 0.3 3 . 2 which is 0.2 9
Converting decimals to fractions . Write 0.4 as a fraction.
Example 1.
Let x = 0.44444 (recurring)
(1)
10x
(2)
= 4.44444 (recurring)
Subtracting (1) from (2) gives 9x = 4 and so x =
4 . 9
.. Write 0.16 as a fraction
Example 2.
.. Let x
= 0.16
(1)
100x
.. = 16.16
(2)
Subtracting (1) from (2) gives 99x = 16 and so x =
16 99
. . Write 0.237 as a fraction
Example 3.
. . Let x
= 0.237
. . 1000x = 237.237
(1) (2)
Subtracting (1) from (2) gives 999x = 237 and so x
237 79 999 333
To try: a) Write
7 as a decimal 8
Write the following as fractions, simplifying when necessary
.
.
.
d) 0.08 1
b) 0.5
..
..
c) 0.15
e) 0.0 21
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Angles Angles on a line – angles on a straight line add up to 180.
a
40
25
Here a = 180 – 40 – 25 = 115 Angles in a triangle – the angles of a triangle add up to 180.
Here b = 180 – 30 – 36 = 114
36
30
b
Alternate angles – angles on opposite sides of the intersecting line are called alternate angles. Alternate angles are equal.
E F
Corresponding angles – Angles A and C are called corresponding angles. Corresponding angles are equal. Angles B and D are also corresponding.
C
A
IGCSE Maths Revision Guide: Higher Tier
D
B
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Interior angles – angles between parallel lines are called interior angles. Angles G and H are interior angles. Interior angles add up to 180.
G
H
To try: Find the angles marked with letters. Giving a reason for your answer.
a)
b)
30
n
m
d)
c)
75
110
p
s
100
r
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Polygons Regular polygons A regular polygon is simply a many sided shape with all of its edges the same length. There are special names for some of the regular polygons and a few are listed below. Number of sides 3 4 5 6 7 8 9 10
Name of polygon Equilateral triangle Square Pentagon Hexagon Heptagon Octagon Nonagon Decagon
Some important facts: The sum of the exterior angles = 360 An exterior angle =
360 where n represents the number of sides n
Exterior angle + interior angle = 180 Sum of interior angles = (n 2) 180
For example: The sum of the interior angles of an 8 sided polygon is 1080 because 8 – 2 = 6 and 6 180 = 1080
To try: a) Find an exterior angle of a regular octagon. b) Find an exterior angle of a regular 18 sided polygon. c) Find an interior angle of a regular octagon. d) Find the sum of the interior angles of an 18 sided polygon e) The sum of the interior angles of a regular polygon is 1800. How many sides does it have ?
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Transformations There are four transformations to learn: enlargement, translation, reflection and rotation.
Enlargement To specify an enlargement we need a scale factor and a centre of enlargement. In the diagram below the small triangle has been enlarged about (2, 0) with scale factor 2
To try: a) The smaller triangle shown above is enlarged about the point (3, 3) with scale factor 2. What are the co-ordinates of its new vertices? b) Using the diagram above, the large triangle is now ‘enlarged’ by a scale factor
1 about (4,0). 2
A fractional scale factor will generate a smaller image. What are the co-ordinates of its new vertices?
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Translation A vector is used to describe movement.
2
For example translates the object two units to the right and 3 units vertically up. 3
2 3
The example below shows a translation using the vector
The image
The object
To try: What vector translates the image to the object?
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Rotation When describing a rotation, always give the centre, angle of rotation and the sense of rotation. In the diagram below, the triangle has been rotated through 90 clockwise about the point (0, 4) image
object
When describing a rotation always give the centre of rotation as a co-ordinate, the angle in degrees and the sense of rotation.
To try: The object is rotated about (6, 5) through 90 in a clockwise sense. What are the co-ordinates of the vertices of the new image?
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Reflection You will be asked to reflect a shape in a specific mirror line or asked to give the equation of the mirror line. In the example below, the triangle is reflected in the y-axis (or x = 0)
In this example, the original shape called the object has, been reflected in the mirror line with equation y = x.
The object
The image
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Miscellaneous questions B'
6
D'
4
B''
C'
A'
A''
B
D
2
B’’’ B'''
C
D''
C''
D''''
B''''
D''' D’’’
C''' C’’’
A’’’ A'''
A''''
A
-4
-2
0
0
2
4
C''''
6
8
10
1
-2
Describe fully the transformation which maps: a) ABCD ABCD b) ABCD ABCD c) ABCD ABCD d) ABCD ABCD
Remember to ask for tracing paper in the examination.
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Sectors and segments of a circle
Minor sector
Minor arc
O r
x
Major arc
Major sector
x
r
B
A
Minor segment
x 2 r 360 x Area of minor sector = r2 360 Length of minor arc =
Area of minor segment = area of sector – area of triangle OAB =
x 1 r 2 r 2 sin x 360 2
To try: a) Find the radius of a circle given that the area of a sector is 50 cm2 and the angle is 70. b) Find the arc length of a sector of a circle radius 5cm and included angle 60. c) Find the perimeter of the sector in part (b). d) The area of a sector of a circle of radius 4cm is 8cm2. Find the angle of the sector to the nearest degree.
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Circle Theorems
Q
The angle at the centre of the circle is twice the angle at the circumference.
Angles in the same segment are equal
Opposite angles in a cyclic quadrilateral are supplementary. ˆ and EHG ˆ add to 180 as (i.e. angles EFG ˆ ) ˆ and HGF do angles HEF
m
o 2m P R
m
m o
H 85° G 110° o 70° E
IGCSE Maths Revision Guide: Higher Tier
95° F
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The angle in a semi circle is 90
90°
o
A tangent is perpendicular to the diameter.
R Q o 90° S
P
The alternate segment theorem: The angle between a tangent and a chord (BC) is equal to the angle in the alternate segment
B r C o
r A
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Tangents to a circle from a point outside are equal in length EF is equal in length to EG
G
E
o
F
The intersecting chord theorem
Case 1.
a b =c d
b c
a d
For example: 3 2 = 4 c and so c = 1.5
2 4
c
3
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Case 2.
a (a + b) = c (c + d)
b a d c
2 4
For example, (not to scale) Here we have 4 6 = 3(3 + x) and so 24 = 9 + 3x Therefore x = 5.
x
3
6 4 3
x
In this example we have 4 10 = x (x + 3) i.e. 40 = x2 + 3x 0 = x2 + 3x – 40 0 = (x + 8)(x – 5)
IGCSE Maths Revision Guide: Higher Tier
and so x = 5 (reject the other solution)
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To try:
A
Chords AB and DC intersect at R. Given that CR = 3.5cm, AR = 8cm and AB = 12cm, calculate the length of CD
C
R
B D H m
Find angles m and n. G
n o 93°
71° E
F
B
ˆ a) Find angle BAC b) Explain why BA is a diameter of the circle. 32°
o
90°
C
A
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B
ˆ a) Find angle AOD ˆ b) Find angle ABD
A o
47°
C
D ˆ is subtended either by the arc AB or Note – in the diagram below, we say that the angle AQB the chord AB. The term subtends may well be used in an examination question.
Q
o
A
B
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Pythagoras’ Theorem In a right angled triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides. i.e.
c
a
To try: b 2
a + b2 = c2
Example 1 – calculate the value of x to 3 significant figures
x
8cm
52 + 82
=
x2
25 + 64
=
x2
x2
=
89
x2
=
9.43 to 3s.f.
5cm
5cm
x
b) In a right angled triangle one of the shorter sides is 5cm and the longest side is 13cm. Find the length of the other shorter side. c) A ladder of length 4 m rests against a vertical wall. The foot of the ladder is 1.4m away from the wall. How far, to the nearest cm, is the top of the ladder from the foot of the wall? d) Calculate the distance between the points (2, 3) and (7, 15).
Example 2 – calculate the value of x as an exact value.
x2 + 22
a) In a right angled triangle the two shorter sides are 4cm and 6 cm exactly. Find the length of the hypotenuse giving your answer as an exact number.
= 52
e) Find the length of XZ and hence the length of the diagonal XT.
x2 = 25 – 4
T
x2 = 21
12cm x
=
21 cm Y
2cm
Notice that when an exact value is required we leave the answer as a surd.
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X
Z 3cm 4cm
W
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Proportion and inverse proportion If y is proportional to x then this means that if x doubles then y will also double. If x increases by a factor 5 then y will increase by a factor 5. We use the notation y x to represent the fact that y is proportional to x. Proportional, directly proportional, varies directly as… all mean the same thing. To solve problems involving proportionality simply replace y x with y kx where k is a constant which needs to be found. Example 1
i)
The tension (T) in an elastic string is proportional to the extension (e). Find the constant of proportionality given that T = 50 Newtons when e = 5cm ii) What is the formula connecting T and e? iii) If e increases to 6cm what will be the tension in the elastic string ? T = ke therefore 50 = 5k, k = 10 and so T = 10e. When e = 6, T = 60 Newtons. You will have to deal with questions such as: y is proportional to the square of x ( y x 2 ) j varies as the cube of k ( j k 3 ) t is directly proportional to the square root of h (t h ) Example 2
i)
The kinetic energy of a busy bee is proportional to the square of its speed. If the kinetic energy is 1 joule and its speed is 10m/s, calculate the constant of proportionality. ii) What is the formula connecting kinetic energy with speed? iii) If the bee increases its speed to 12m/s, find the kinetic energy of the bee.
E kv 2 therefore 1 k 10 2 so k
1 1 1 2 . i.e. E v . v 12, E 144 1.44 J 100 100 100
Inverse proportion If one variable increases the other decreases. If y is inversely proportional to x then y =
k x
If d is inversely proportional to the square of h then d
k h2
Example 3
i) P is inversely proportional to V. If P = 10 when V = 2 find k the constant of proportionality ii) Find P when V = 3 iii) Find V when P = 0.5
P
k 20 2 20 . When V 3, P 6 . When P 0.5, V 40 and k 10 2 20, so P 3 0.5 V V
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To try: a) M varies directly as the square of N. If M = 10 when N = 2, find k the constant of proportionality. If N = 5, find M. b) Z varies as the cube of W. If Z = 56 when W = 8 find the constant of proportionality. If W = 9, find Z. c) The volume V of a fixed mass of gas varies inversely as the pressure P. When V = 3m3, P = 500N/m2. Find the volume when the pressure is 300N/m2. Find the pressure when the volume is 10 m3 d) The force of attraction F between two magnets varies as the square of the distance d between them. When the magnets are 2cm apart, the force of attraction is 10N. How far apart are they if the attractive force is 4N? e) Given that d x d
1 x
complete the table of values shown below. 1
4 1
9 0.5
Draw a sketch of d against x.
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Constructions To construct an angle of 60 B
D
C
A
Draw the line AD. With compass point at A, draw an arc to intersect AD at C. (I have shown the complete circle in the diagram above). With compass point at C and using the same radius as before, draw an arc to intersect the first arc (circle) at B. Draw the line AB. DAB is now 60. Triangle ABC is of course equilateral.
To construct the perpendicular bisector of a line
Open the compass to over half the distance AB. With compass point at A draw an arc (I have drawn a circle) to intersect AB at C.
P
A
D
C
Keeping the same radius, draw another arc using centre B. I have drawn a circle once again. The arc will intersect AB at D.
B
Q
The two arcs should intersect at P and Q. Join P to Q. This line is the perpendicular bisector of line AB.
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To construct the angle bisector
X
A
Y
M
N B
Z
The two lines XY and YZ are given. Draw an arc from Y to intersect the lines at A and B. Draw an arc with centre A below the line XY (I have shown full circles) and keeping the same radius, draw another arc from B above the line YZ. These intersect at N and M. Join Y through N and M. This is the angle bisector. XYM = ZYM.
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Bearings A bearing is an angle. Bearings are measured clockwise starting from north. A bearing is given using 3 figures. If the angle is less than 100 put a zero in front of it. N
N A A B B
The bearing of B from A is the angle shown above (left). You start at A and face north. You then turn clockwise until you face B The bearing of A from B is also shown above (right).
To try: 1) In each of the following find the bearing of A from B. a)
N
b)
N
c)
N
N
d)
N A
A A B B B
B A
2. Captain Jones is trying to locate a boat in trouble off the coast. He measures the bearing of the boat from two points X and Y where Y is due East from X. The bearing of the boat from X is 036. The bearing of the boat from Y is 325. Points X and Y are 500m apart. (i) Draw a rough sketch to illustrate the information given above. (ii) Use the sine rule to find out how far the boat is from X, to the nearest metre.
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Area and volume The area of the triangle shown below is
1 bh 2
h
b
If two sides and an included angle are known we can use the formula
1 AB sin C 2
B
5
25° A
C 6
In the triangle above a = 5, b = 6 and C = 25. Using the formula, area =
1 5 6 sin 25° 2
= 6.34 (3 sig figs) The area of a parallelogram is b h
h
b
The area of a circle is r 2 The circumference is 2 r or d (where d is the diameter)
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To try: a) A right angled triangle has sides 5, 12 and 13cm. Find the area of the triangle. b) Find the area of a circle with a radius of 5cm giving your answer to 1 decimal place. c) Find the area of a semi-circle with a diameter 6cm giving your answer to 1 decimal place. d) The area of a circle is 25cm2, find its radius to 1 decimal place. e) The diameter of a semi-circle is 4cm. Calculate the perimeter of the semi-circle to 1 decimal place.
The area of a trapezium is
1 ( a b) h 2
a h b
To try: f) Find the area of a trapezium given that the two parallel sides are 6cm and 10cm respectively. The perpendicular height is 5cm g) In a trapezium, one of the parallel sides is xcm and the other is 6cm longer. The perpendicular height between the two parallel sides is 5cm. i) Find in terms of x the length of the longer parallel side. ii) Show that the area is (5x + 15)cm2
The surface area of a closed cylinder is 2 r 2 2 rh The volume of a cylinder is r 2 h
To try: h) Calculate the surface area of a cylinder, closed at one end, given that the radius is 5cm and the height 8cm. Leave your answer in terms of . i) A cylinder has a volume of 20cm3. Given that the height is 2cm, calculate the radius of the cylinder to 1 decimal place.
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The curved surface area of a cone is r l where l is the slant height. The volume of a cone =
1 2 r h 3
To try: j) A cone has a base radius of 4cm and a height of 3cm. Calculate the curved surface area of the cone leaving your answer in terms of . k) Calculate the volume of a cone with base radius 4cm, height 6cm giving your final answer correct to 3 significant figures. l) The volume of a cone of height 10cm is 20cm3. Show that the radius of the base of the cone is
The volume of a pyramid is
6
1 (base area) h 3
To try : m) The volume of a pyramid is 45cm3 . The height of the pyramid is 15cm and its base is a square. Find the dimensions of the base of the pyramid.
The volume of a sphere is
4 3 r and the surface area of a sphere is 4 r 2 3
To try: n) The moon has a radius approximately equal to 1700km. Find an estimate for the volume giving your answer in standard form. o) A table tennis ball has a diameter of 4cm. Find its surface area giving your answer in terms of . The volume of a prism is (area of cross section) (length)
To try: p) A triangular prism has a cross sectional area of 5cm 2. The length of the prism is 10cm. Find its volume. q) The volume of a prism is 100cm3. The cross sectional area is 20cm2. What is the length of the prism?
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Area and volume of similar shapes Area The triangles below are similar.
2cm
8cm
The enlargement scale factor is 4 The area scale factor is 42
If the small triangle has an area of 3cm2 then the area of the larger triangle is 42 3 i.e. 16 3 = 48cm2 An alternative method using the result:
If two shapes are similar then the ratio of their area is equal to the square of the ratio of their sides
For example, let the area of the larger triangle be A then
A 3 Therefore
A
8 2
2
=
42 3
=
48cm2
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Volume Two similar coca cola bottles have circular bases with radii 2cm and 3.5cm respectively. The volume of the smaller bottle is 330ml. What is the volume of the larger bottle? The enlargement scale factor is
3.5 = 1.75 2
The volume scale factor is 1.753 = 5.359375 Therefore the volume of the larger bottle is 330 5.359375 = 1768.59ml Or 1.77l (to 3 sig figs)
An alternative method using the result: If two shapes are similar then the ratio of their volumes is equal to the cube of the ratio of sides. V 3.5 For example, let the volume of the larger bottle be V, then 330 3
3
Therefore: V = 330 1.753 = 1768.59 ml
To try: a) Two shapes are similar. The height of the first shape is 5cm and that of the second is 7.5cm. The area of the smaller shape is 6cm2. Find the area of the larger shape. b) The two triangles shown are similar. The area of the larger triangle is 22cm2. Find the area of the smaller triangle. c) Two similar cylinders are shown below. Find the volume of the small cylinder given that the volume of the large cylinder is 56cm2.
8cm
IGCSE Maths Revision Guide: Higher Tier
3cm 6cm
4cm
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Trigonometry In a right angled triangle the hypotenuse is the longest side and is always opposite the right angle. The side opposite the angle is labelled opposite and the side which is not the hypotenuse is labelled adjacent.
sin
opposite hypotenuse
o h
cos
adjacent hypotenuse
a h
tan
opposite adjacent
hypotenuse opposite adjacent
S
H
C
o a
O
A
O
H
T
A Angle of depression
Angle of elevation
Example 1 - To find the length of a missing side
m 12 12 sin 25 m m 5.07cm sin 25
12cm m
25
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Example 2 - To find an angle given two sides
5 12 0.4166
cos
12cm
65.4 5cm .
5 b or shift cos 5a c12 12
Key presses on your calculator are: shift cos 0.41 6 or shift cos
To try: a) A 4m ladder rests against a vertical wall. It makes an angle of 64 with the ground. Find, to the nearest cm, the distance of the foot of the ladder from the wall. b) An isosceles triangle ABC with AB = BC, BC = 5 cm and BAC = 40, find the length of AC to 1 decimal place. c) A rectangle has one side 5cm and a pair of diagonals of length 13cm. Find the length of the other side of the rectangle and the angle which the diagonal makes with this side. d)
T 4cm Y
Z
(i) Find the length of XZ and hence the length of the diagonal XT in surd form. (ii) Find the angle that XT makes with the plane WXYZ (angle TXZ)
12cm X
W
5cm
e) The top of the school sixth form centre is observed from a position P, 42 metres away from the foot of the building. The height of the building is 15m. Find the angle of elevation x of the top of the building from P correct to the nearest degree.
15m x P
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42m
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Sine and Cosine Rules The Sine Rule a b c sin A sin B sin C
This rule is written
or
sin A sin B sin C a b c
Example 1
B
12 x sin 32 sin 64 12 sin 32 x sin 64 7.08cm
64 x 32 A
C
12cm
Example 2
Find ABC.
B
8cm 40 A
10cm
C
In this example make use of the second formula.
sin B
10 sin 40 8
Therefore:
sin 40° sin B 8 10
i.e.
sin B 0.8035 B 53.5
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The Cosine Rule Given the triangle ABC the rule is a 2 b 2 c 2 2bc cos A . This is the first rule which is used to calculate a missing side given two sides and an included angle. Here is an example which illustrates the use of this rule. Example 1
B
Here a = x, b =10, c = 8 and A = 22 x
x 2 102 82 2 10 8cos 22 100 64 160 cos 22 164 148.35 So 15.65 x 3.96cm (3 sig figs)
8cm C 10cm
22˚ A
The cosine rule can be rearranged into a different form which is useful if you need to find an angle.
cos A
b2 c2 a2 2bc
Here is an example to show how this rule can be used. Example 2
Find BAC B
So a = 5, b = 13 and c = 9 5cm
132 92 52 2 13 9 0.9615
So, cos A
9cm C
A 15.9 (3 sig figs) 13cm A Take care when entering the numbers into your calculator. If you are not confident in using your calculator, work out the numerator and denominator separately.
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Use the cosine rule if you know either
i) two sides and an included angle ii) three sides For all other cases use the sine rule.
To try: Give all answers to three significant figures. a) In triangle ABC, AB = 6cm, AC = 8cm and BAC = 50. Calculate the length of the side BC. b) In triangle QPR, PQ = 10cm, PR = 12cm and QR = 13cm. Find the angle QPR.
c) In the triangle ABC, AB = 5cm, AC = 6cm and BC = 10cm. 13 and hence A is obtuse. Show that cos A 20
d) In triangle RST, ST = 8cm, SRT = 70, RTS = 50. Find the length of RS. e) John was orienteering. He set off on a bearing of 030 and walked 3km. He then set a course on a bearing of 340 and walked until he was due north of his starting position. How far did John walk altogether?
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Vectors You should already be familiar with vectors when describing transformations.
Vector notation A vector quantity has both direction and magnitude (size). For example,
The arrow represents the direction, and the length of the line represents the magnitude. A scalar quantity only has magnitude, e.g. the numbers 2, 3, 4,…
2
This vector can be written as: AB or a, or . 2
The magnitude (or modulus) of this vector is
22 22 8 units.
In print, a is written in bold type. When using handwriting, the vector is indicated by putting a squiggle underneath the letter:
a
-a
3 4
Here the vector is either BA , a or
Equal vectors If two vectors have the same magnitude and direction, then they are equal.
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Adding vectors
a c a c b d b d Look at the graph on the right to see the movements between PQ, QR and PR.
2 4 6 5 3 2
Subtracting vectors
7 4 3 5 3 2
To try: 2
4
1
, find: Given x = , y = and z = 3 5 2 a) –y b) x y c) 2x + 3z
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Geometric problems In the triangle ABC, M and N are the midpoints of the sides AB and AC respectively.
AM = m and AN = n.
Find AB , AC , MN and BC in terms of m and n.
B
AB = 2m
M
AC = 2n
MN = –m + n
A
N C
BC = –2m + 2n
We can write the vector BC in a different form. For example
BC = –2m + 2n = 2(–m + n ) = 2 MN
Therefore BC and MN are parallel and BC is twice the
length of MN .
To try: a) In the diagram points C and D have position vectors c and d respectively referred to the point marked O. The point Q divides CD in the ratio 1:3. C i) Find DC in terms of c and d. ii) Find the position vector of Q in terms of c and d. Q
3 b) Given that a = , find the modulus of this vector. 4 3 c) Given that b = , find the angle which b makes with the 8
c
horizontal. D
d
O
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Solving linear equations
Let us start with something quite easy. For example,
3x 8 22 3 x 22 8
+8 to both sides
3 x 30 x 10
Now for x ' s on both sides 5 x 4 2 x 25 3 x 4 25
2x from both sides
3 x 21 x7
To try:
Now consider the equation c)
solve a) 6 x 9 57 and b) 6 x 4 x 21
18 x 9 2x 3
First multiply both sides by 3
18 x 27 6 x 18 27 7 x 9 7 x x
9 7
x 1
2 7
Notice the first step was multiply by 3. This single step reduced the equation to one like the second.
To try:
IGCSE Maths Revision Guide: Higher Tier
Solve c)
50 x 8 2x 4
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Finally, consider the equation : 2x 4 5
15(2 x 4) 5 3(2 x 4) 1
x2 3
5
15( x 2) 3
75
multiply each term by 15
5( x 2) 1
75
cancel
6 x 12 5 x 10 75 11x 2
75
11x
77
x
multiply out the brackets
7
To try:
IGCSE Maths Revision Guide: Higher Tier
d)
4x 5 2x 1 8 5 3
e)
2x 4 2x 3 5 4 3
f)
3x 1 2 x 1 1 2 3
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nth term In the simple sequence 5, 7, 9, 11,… a simple rule would be add 2. Therefore the first part of the algebraic rule will be 2n. Using n = 1 for the first term, 2n becomes 2 × 1 = 2 but the first term in the sequence is 5. Therefore we add 3. The rule is 2n + 3. Test it out on another term. The fourth term is 11 and 2 4 + 3 also equals 11 so our rule is correct.
To try: Find the nth term for each of the following sequences. a) 2, 4, 6, 8, 10,… b) 7, 12, 17, 22,… c) 10, 7, 4, 1,… Find the 5th term if the nth term is d) 6n + 1 e) 2n + 5
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Simultaneous Equations 1 When you are solving two equations at the same time you are solving simultaneous equations. When two straight lines are drawn on a graph then the intersection will be the solution. For example:
The diagram above shows x + y = 5 and x – y = 3. The lines intersect at (4, 1) and x = 4 and y = 1 are the solutions since 4 + 1 = 5 and 4 – 1 = 3. To solve this pair of equations algebraically proceed as follows: x+y=5
(A)
x–y=3
(B)
Add equation A to equation B 2x = 8 and therefore x = 4. Substituting x = 4 into equation A gives 4 + y = 5 so y = 1
We will now solve
3x + y = 23 x+y = 9
(A) (B)
In the previous example we added equation A to equation B. This time we will subtract. 2x = 14
therefore x = 7.
Substitute x = 7 into A gives 21 + y = 23 and so y = 2.
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The following simultaneous equations would only appear very early in the examination paper as they are really only grade C standard.
To try: a)
x y 10 x y 6
b)
x 3 y 14 x 3 y 10
2x y 7 x y 4
c)
Slightly harder simultaneous equations Example 1 4 x 3 y 10
(A)
2x y 0 (B) Notice that the y terms are different. If we multiply equation B by 3 we will then have:
6x 3y 0 4 x 3 y 10
now add so 10x = 10 and therefore x = 1
Substitute x = 1 into equation A.
4 3 y 10 3y 6 y2 It is a good idea to check your answer by using equation B. So 2(1) – 2 = 0 which is correct. Example 2 5 x 4 y 29 (A) 2 x 3 y 20 (B) Notice that the y terms are different again. This time we will multiply equation A by 3 and equation B by 4.
15 x 12 y 87 8 x 12 y 80
Now subtract. So 7x = 7 and x = 1. Substitute x = 1 in A.
5 4 y 29 y6 Check in B 2(1) + 3(6) = 20 which is correct.
To try: Solve the following simultaneous equations d)
3x 5 y 13 8x y 6
e)
5 x 2 y 17 3x 3 y 6
IGCSE Maths Revision Guide: Higher Tier
f)
4 x 3 y 28 5 x 5 y 35
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g)
4 x 3 y 14 8 x 6 y 28
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Rearranging Formulae To rearrange an equation or formula, do the same operation to both sides. Example 1 Make t the subject in
Example 2 Make x the subject in
4t 6 2m 4t 2m 6 2m 6 t 4
add 6 to both sides divide both sides by 4
a ( x b) p p xb a p x b a
or remove the brackets first:
ax ab p ax p ab x
Example 3 Make x the subject of the equation
p ab a
mx 2 n q mx 2 q n qn x2 m qn x m
To try: Make m the subject in each of the following formulae. a) 3m b c
e) d (m 2 t ) n
b) am b c
f)
IGCSE Maths Revision Guide: Higher Tier
c)
m 6 n w
j (a m 2 ) b c k
g)
d) d (m f ) n
m k r
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h) s ut
1 am 2 2
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Basic Factorisation Factorising expressions 5x + 15 = 5(x + 3) notice that 5 is a common factor of 5x and 15 ax + ay = a(x+ y) 3p2q 27p = 3p(pq – 9) here 3 and p are common factors of 3p2q and 27p 4fg + 8fh + 12fm = 4f(g + 2h + 3m)
The difference of two squares x 2 – y2 = (x y)(x + y)
9m2 – 25k2 = (3m – 5k)(3m + 5k)
Quadratic expressions 2
Example 1 x + 8x + 7 A quadratic expression containing three terms will need two sets of brackets. To be able to find what expressions are inserted into these brackets use the following method:
Find two numbers which multiply together to make +7 and at the same time, add to make + 8. 7 and 1 are the two numbers (both positive numbers)
The expression factorises into: ( x + 7)( x + 1) Example 2 x 2 + 4x – 12 Find two numbers which multiply together to make 12 and add to make +4. 6 and 2 are the numbers
The expression factorises into: ( x + 6)( x 2) Example 3 2x2 + 7x + 6 It seems sensible to try (2x
)(x
) as our first guess.
However, look at the + 6 and consider its factors. The factors of 6 are 6 and 1 or 3 and 2. Try one of these, e.g (2x + 3)(x + 2)
MULTIPLY OUT TO CHECK
2x2 + 4x + 3x + 6 = 2x2 + 7x + 6
IGCSE Maths Revision Guide: Higher Tier
which is correct.
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Finally: x2 8x has two terms and uses one bracket: x(x 8)
To try: Factorise the following expressions. a) 2 p 6q b) gh gk c) 3 g 2 d 9 g d) 16 p 2 25t 2 e) x 2 4 x 3 f) x 2 9 x 14 g) 3x 2 7 x 2 h) 2 x 2 11x 12 i) x 2 6 x j) 2m 2 4m
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Solving quadratic equations First factorise the left hand side if you can. For example: Solve x 2 + 4 x – 12 = 0 i.e
( x + 6)( x – 2) = 0
so
x + 6 = 0 or x – 2 = 0
thus
x = –6 or x = 2
If the question states: solve, giving your final answer to 1 or 2 decimal places then you know automatically that the left hand side of the equation will not factorise. In this case use the quadratic formula: For ax bx c 0, 2
x
b b2 4ac 2a
Take care if ‘b’ is negative! Example: solve 3 x 2 – 2 x – 7 = 0 Here a = 3, b = –2 and c = –7
(2) (2) 2 4(3)(7) x 6 2 4 84 6 2 88 6 ( x = 1.90 or –1.23 to 2 decimal places)
To try: Solve the equations below giving your final answer to 2 decimal places where appropriate. You do not need to use the formula for parts b) and c). a) x 2 + 5 x – 2 = 0
c) 2x2 14x + 8 = 12
b) x2 – 7x + 10 = 0
d) x 2 4 x 1 0
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Straight lines The equation of a straight line with a gradient m and y intercept c is: y = mx + c. Consider the graph of y = 3x + 4 below.
Equation: y = 3x + 4 The y-intercept is 4
Choose two points on the line and find the gradient. Here the two points are (‒1, 1) and (0, 4). The gradient is of course 3.
To sum up: In the equation y = mx + c, m represents the gradient of the line and c the y-intercept. Sometimes the equation of a line is given in a slightly different form. For example, 2x + y = 6. To draw this, either create a table of values and plot points, or rearrange the equation into the form y = mx + c. Rearranging: y = 6 – 2x i.e. y = 2x + 6
The line must slope ‘downhill’. Gradient = ‒2, y-intercept = 6 Therefore y = ‒2x+6
This is a straight line with a gradient of 2 and y-intercept +6. It is now possible to draw the line using these two pieces of information. The graph is shown below.
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You may be asked to find the equation of a line which is parallel to a given line. Example 1 Find the equation of a line which passes through (0, 5) and is parallel to y = 3x – 7.
Parallel lines must have the same gradient and therefore the gradient of the line must be 3. The line passes through (0, 5) and ‘c’ must therefore be 5. The equation of the line is: y = 3x + 5 Example 2 Find the equation of a line which passes through (2, 10) and is parallel to y = 3x – 7.
Once again the gradient is 3. However ‘c’ is not 10 as this point is NOT on the y-axis. The equation of the line so far is y = 3x + c. Substituting (2, 10) into the above equation gives: Solve for ‘c’. Therefore the equation of the line is :
10 = (3 2) + c c = 4 y = 3x + 4
To try: a) What is the gradient of the line y = 2x – 5 ? b) What is the y-intercept for the line above ? c) Find the equation of a line parallel to y = 2x – 5 which passes through (1, 2). d) Find the equation of a line which is parallel to y = –3x + 7 which also passes through (1, 3). e) A line passes through (2, 2) and (6, 3). Show that the equation of the line can be written in the form 4y – x = 6. f) Draw on a graph the line y = 2x + 1 and on the same axes draw the line y = x + 3. Write down the co-ordinate of the point of intersection of these lines.
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Inequalities Example 1 Solve 5x + 8 < 18
5x < 10 x <
2
Example 2 Solve 5x + 8 < 2x + 26
3x + 8 < 26 3x < 18
Subtract 2x from both sides Subtract 8 from both sides
x < 6
Example 3 Solve the inequality
7 2 x 3 3 and represent your solution on a number line. 4 2x 6 2 x 3
(Notice the filled in circle at x = 2) The solution to the problem is where the two lines overlap.
To try: Find the range of values of x which satisfy the given inequalities. a) 3x – 2 < 10 b) 3 – 2x > 7 c) 2 4x +1 < 9
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Errors in measurement The length of an envelope is 25cm correct to the nearest centimetre. The real length of the envelope can be anything from 24.5cm up to but not including 25.5cm. The lower limit (or bound) is 24.5cm. The upper limit (or bound) is 25.5cm. This can be shown on a number line. The circle representing the upper end point is not coloured in.
24.5
25
25.5
Patrick measured his pen. It is 12.4cm correct to 1 decimal place. What are the lower and upper limits? The lower limit is 12.35cm and the upper limit is 12.45cm.
To try: a) The capacity of a fridge is 5.3 cubic feet correct to 1 decimal place. What are the upper and lower limits for the capacity of the fridge? b) The length of the side of a square is 5.45cm correct to 3 significant figures. Find i) the lower limit for the length of the side ii) the lower limit for the perimeter of the square. c) The school car park is to be resurfaced. The length and width of the car park have been measured to the nearest metre. The length is 65m and the width 40m. Find the minimum and maximum area of the school car park.
IGCSE Maths Revision Guide: Higher Tier
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© ZigZag Education, 2006
Shading Regions Inequalities are represented on graphs using shading. For example, if y > 3x, the graph of y = 3x would be drawn. Then either all of the points greater than 3x would be shaded or all of the points less than or equal to 3x would be shaded. It is advisable to adopt the convention of shading out what is not required.
Example 1
Show graphically y < 3x + 1 As the inequality is strictly less than, the line is dotted. If in any doubt, consider a point not on the line such as (0, 0). y < 3x+1
Substitute x = 0 and y = 0 into the inequality y < 3x + 1 to see which side of the line is valid. 0 0
First draw a simple sketch of y = x2 5x + 4 It is relatively simple to sketch this as the right hand side factorises, So y = (x 4)(x 1). Now we can see that the roots are x = 1 and x = 4. The x2 term is a positive and this means that the curve will be shaped. y
4
2
x 2
4
6
–2
We now use the graph to solve the inequality y > 0 Consider the graph and find where the curve is above the x axis. Solution: x < 1 or x > 4
To try: Solve: a) x2 + 7x + 6 > 0 b) 3x2 – 9x + 6 < 0
IGCSE Maths Revision Guide: Higher Tier
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© ZigZag Education, 2006
Example 2 Solve the inequality
4x 2 < 25
4 x 2 25
Rearrange :
Factorise: (2 x 5)(2 x 5)
