I.G.C.S.E. Mathematics Extension Level Revision NotesJune 2008Primes, Factors, Powers and Standard Form..................................................................................7Prime numbers......................................................................................................................................................................7Factors..................................................................................................................................................................................7Prime Factors........................................................................................................................................................................7Common Factors and Highest Common Factor...................................................................................................................8Multiples...............................................................................................................................................................................8Common Multiples and Lowest Common Multiple.............................................................................................................8Square numbers....................................................................................................................................................................8Triangular numbers...............................................................................................................................................................9Square roots and prime factors.............................................................................................................................................9Indices(Powers)..................................................................................................................................................................9Standard Form....................................................................................................................................................................10Converting from standard form to an ordinary number...................................................................................................10Converting TO standard form.............................................................................................................................................10Combining numbers in standard form................................................................................................................................11Introduction to Algebra....................................................................................................................11Substituting in formulae.....................................................................................................................................................11Simplifying expressions and brackets................................................................................................................................11Solving equations...............................................................................................................................................................12Re-arranging formulae........................................................................................................................................................13English statements and equivalent mathematical expressions...........................................................................................14Forming equations..............................................................................................................................................................14Inequalities..........................................................................................................................................................................15Miscellaneous Number....................................................................................................................15Long Multiplication............................................................................................................................................................15Long Division.....................................................................................................................................................................16Fractions - proper, mixed, and improper fractions.........................................................................................................16Converting from mixed fractions to top heavy fractions..............................................................................................................................................16Converting from top heavy fractions to mixed fractions..............................................................................................................................................16Adding and subtracting fractions..................................................................................................................................................................................16Reciprocals....................................................................................................................................................................................................................17Multiplying and dividing fractions...............................................................................................................................................................................17Converting a decimal to a fraction................................................................................................................................................................................18Fractions and recurring decimals..................................................................................................................................................................................18Negative numbers...............................................................................................................................................................18Rounding off to the nearest ...............................................................................................................................................19Rough approximations........................................................................................................................................................19Sequences........................................................................................................................................19Sequences from rules..........................................................................................................................................................20Standard sequences.............................................................................................................................................................20Perimeters and Areas......................................................................................................................21Perimeter.............................................................................................................................................................................21Area....................................................................................................................................................................................21Volumes and Surface Areas............................................................................................................22Surface Areas......................................................................................................................................................................22Volumes..............................................................................................................................................................................22Prisms............................................................................................................................................................................................................................22Pyramids........................................................................................................................................................................................................................22Converting Units.................................................................................................................................................................23Lengths .........................................................................................................................................................................................................................23Areas.............................................................................................................................................................................................................................23Volumes........................................................................................................................................................................................................................23Statistics 1........................................................................................................................................24Mode, median and range....................................................................................................................................................24Median, quartiles and interquartile range...........................................................................................................................24Mean...................................................................................................................................................................................25Frequency tables and tally charts........................................................................................................................................2507/12/11IGCSE Extension Level RevisionSDB 2Bar chartsand line graphs...................................................................................................................................................26Histograms .........................................................................................................................................................................26Pie charts.............................................................................................................................................................................27Finding the mode and mean from a grouped frequency table............................................................................................28Surveys, questionnaires and sampling................................................................................................................................29Percentages......................................................................................................................................29Percentages as fractions and decimals................................................................................................................................29Finding a percentage of a given number............................................................................................................................29Percentage increases and decreases....................................................................................................................................30Finding the original number after a percentage change.....................................................................................................30Simple interest....................................................................................................................................................................30Compound increases or decreases......................................................................................................................................30Expressing one number as a percentage of another............................................................................................................31Geometry..........................................................................................................................................31Angle properties.................................................................................................................................................................31Acute, obtuse and reflex angles....................................................................................................................................................................................31Angles on a line, vertically opposite and round a point................................................................................................................................................32Parallel lines..................................................................................................................................................................................................................32Bearings..............................................................................................................................................................................32Angles of elevation and depression....................................................................................................................................33Triangles.............................................................................................................................................................................33Angles in a triangle add up to 180o..............................................................................................................................................................................33Exterior angle equals sum of the two interior opposite angles.....................................................................................................................................33Equilateral and isosceles triangles................................................................................................................................................................................34Congruence.........................................................................................................................................................................34Quadrilaterals......................................................................................................................................................................35Special quadrilaterals....................................................................................................................................................................................................35Polygons.............................................................................................................................................................................36Regular polygons................................................................................................................................................................36Sets...................................................................................................................................................36Set Notation........................................................................................................................................................................36Venn diagrams....................................................................................................................................................................37Functions..........................................................................................................................................37Notation..............................................................................................................................................................................37Composite functions...........................................................................................................................................................37Inverse functions.................................................................................................................................................................38Transformations...............................................................................................................................38Translations........................................................................................................................................................................38Combining translations.................................................................................................................................................................................................39Rotations.............................................................................................................................................................................39Finding the centre of rotation........................................................................................................................................................................................39.......................................................................................................................................................................................................................................39Reflections..........................................................................................................................................................................39Enlargements .....................................................................................................................................................................39Positive scale factor......................................................................................................................................................................................................39Fraction scale factor......................................................................................................................................................................................................40Negative scale factor ....................................................................................................................................................................................................40Pythagoras........................................................................................................................................40Ratio..................................................................................................................................................41Sharing in the ratio of two numbers...................................................................................................................................41Sharing in the ratio of three numbers.................................................................................................................................41Map Scales..........................................................................................................................................................................41Trigonometry 1.................................................................................................................................42SOH CAH TOA............................................................................................................................................................................................................42The awkward case,xon the denominator........................................................................................................................42Quadratic functions & factorising...................................................................................................43Multiplying out two brackets -F O I L...........................................................................................................................4307/12/11IGCSE Extension Level RevisionSDB 3Factorising..........................................................................................................................................................................43General examples of factorising.........................................................................................................................................44Algebraic fractions.............................................................................................................................................................44Quadratic equations.........................................................................................................................44Solution by factorising........................................................................................................................................................44Solution by completing the square.....................................................................................................................................45Quadratic equation formula................................................................................................................................................45Simultaneous equations..................................................................................................................45Solving by elimination............................................................................................................................................................................................45Graphs1...........................................................................................................................................46Distance between two points..............................................................................................................................................46Gradient..............................................................................................................................................................................47Mid-point............................................................................................................................................................................47Straight lines,y = mx + c...................................................................................................................................................47Parallel lines.......................................................................................................................................................................47Plotting curves....................................................................................................................................................................48Standard graphs.............................................................................................................................................................................................................49Probability.........................................................................................................................................50Possible outcomes...............................................................................................................................................................50Equally likely outcomes................................................................................................................................................................................................50Probability and relative frequency......................................................................................................................................50Two or three coins..............................................................................................................................................................50Two dice.............................................................................................................................................................................51Coin and spinner.................................................................................................................................................................51Tree diagrams.....................................................................................................................................................................51With replacement..........................................................................................................................................................................................................51Without replacement.....................................................................................................................................................................................................52Compound events...............................................................................................................................................................53Complements(not).......................................................................................................................................................................................................53Exclusive,OR............................................................................................................................................................................................................53Independent, AND......................................................................................................................................................................................................53Trigonometry 2.................................................................................................................................543-Dimensional problems.....................................................................................................................................................54Lengths and angles........................................................................................................................................................................................................54Sine & cosine for angles between 90o and 180o................................................................................................................55Graphs...........................................................................................................................................................................................................................55Solving equations .........................................................................................................................................................................................................55.......................................................................................................................................................................................................................................55Sine and Cosine rules,area of a triangle.............................................................................................................................56Sine rule........................................................................................................................................................................................................................56Cosine Rule...................................................................................................................................................................................................................56Area of a triangle................................................................................................................................................................57Proportion.........................................................................................................................................58Direct proportion...........................................................................................................................................................................................................58Inverse proportion.........................................................................................................................................................................................................58Similar figures..................................................................................................................................60Similarity............................................................................................................................................................................60Corresponding sides and enlargement ...............................................................................................................................61Corresponding sides factor............................................................................................................................................................................................61Enlargement factor........................................................................................................................................................................................................61Areas and volumes of similar figures.................................................................................................................................62Statistics 2........................................................................................................................................63Scatter graphs and lines of best fit......................................................................................................................................63Correlation..........................................................................................................................................................................63Cumulative frequency graphs ............................................................................................................................................64Finding median and quartiles from a cumulative frequency graph..............................................................................................................................65Percentiles.....................................................................................................................................................................................................................6507/12/11IGCSE Extension Level RevisionSDB 4Kinematics........................................................................................................................................66Distance, speed and time....................................................................................................................................................66Basic results..................................................................................................................................................................................................................66Gradient of a distance time curve..............................................................................................................................................................................66Gradient of a speed time curve..................................................................................................................................................................................66Area under a speed time curve...................................................................................................................................................................................66Converting units - speeds...................................................................................................................................................67 Accuracy..........................................................................................................................................68Decimal places and significant figures ..............................................................................................................................68To nearest 5 or nearest 10etc............................................................................................................................................68Limits of accuracy in calculations......................................................................................................................................68Graphical Inequalities......................................................................................................................69Inequalities in one variable.................................................................................................................................................69Lines and points............................................................................................................................................................................................................69Inequalities....................................................................................................................................................................................................................69Inequalities in two variables...............................................................................................................................................70Lines and points............................................................................................................................................................................................................70Inequalities....................................................................................................................................................................................................................70Practical situations..............................................................................................................................................................71Translating english statements into mathematics..........................................................................................................................................................71Types of number..............................................................................................................................73Natural numbers, integers and rational numbers................................................................................................................73Irrational numbers...............................................................................................................................................................73Real numbers......................................................................................................................................................................73Converting fraction to decimal.....................................................................................................................................................................................73Converting recurring decimal to fration........................................................................................................................................................................73Circle Geometry...............................................................................................................................74Segments and sectors..........................................................................................................................................................74Length of arc and area of sector.........................................................................................................................................74Tangents..............................................................................................................................................................................75Angle properties.................................................................................................................................................................75Constructions with compass and ruler..........................................................................................76Perpendicular bisector........................................................................................................................................................76Angle bisector.....................................................................................................................................................................77Dropping a perpendicular from a point to a line................................................................................................................77Constructing a 60o angle....................................................................................................................................................77Loci....................................................................................................................................................78Common loci......................................................................................................................................................................78Matrices.............................................................................................................................................79Size ....................................................................................................................................................................................79Adding matrices..................................................................................................................................................................79Multiplying matrices by a number......................................................................................................................................79Multiplying matrices...........................................................................................................................................................79Data matrices......................................................................................................................................................................802 2 matrices....................................................................................................................................................................81Identity matrix...............................................................................................................................................................................................................81Determinant...................................................................................................................................................................................................................81Inverse matrices..........................................................................................................................................................................................................81Transformations and 2 2 Matrices..............................................................................................83Matrices as transformations................................................................................................................................................83Base vectors........................................................................................................................................................................85Common transformations...................................................................................................................................................88Enlargements, stretches and shears....................................................................................................................................88Enlargements.................................................................................................................................................................................................................88Stretches........................................................................................................................................................................................................................89Shears............................................................................................................................................................................................................................90Vectors..............................................................................................................................................9107/12/11IGCSE Extension Level RevisionSDB 5Position vector....................................................................................................................................................................91Adding, multiplying............................................................................................................................................................91Magnitude...........................................................................................................................................................................92Adding vectors in a diagram.........................................................................................................................................................................................9207/12/11IGCSE Extension Level RevisionSDB 6Primes, Factors, Powers and Standard FormPrime numbersA prime numberis a number which has only two factors, 1 and the number itself:15 has factors 1, 3, 5 and 15 so 15 is not prime.7 can only has two factors, 1 and 7, so7is prime.The first few prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59 ...Factors2 divides into 12 so 2 is a factor of 12: other factors are 3, 4, and 6 - also 1 and 12: so the complete set of factors of 12 is1, 2, 3, 4, 6, 12.For large numbers write down the factors in pairs - be methodical about it:Factors of 80are 1802404 20516810and we can see easily that it is now time to stop!Prime FactorsFactors which are also prime numbers are called prime factors.21 is a factor of 42 but is not a prime so is not a prime factor.2 is a factor of 42 and is prime so is a prime factor; so also are 3 and 7 prime factors of 42 and we can write 42 as a product of its prime factors, 42= 2 3 7.50 has prime factors2and5(twice)and we write 50 as a product of its prime factors as50=2 5 5or 50=2 52.For larger numbers we find pairs of factors repeatedly to find the prime factors1260=10 126= 2 5 2 63= 2 5 297= 2 5 2 3 37= 2 2 3 3 5 7 writing factors in ascending orderso we write1260=22 32 5 7.07/12/11IGCSE Extension Level RevisionSDB 7Common Factors and Highest Common FactorThe complete set of factors of 12 is1, 2, 3, 4, 6, 12.The complete set of factors of 18 is1, 2, 3, 6, 9, 18.Notice that 1, 2, 3, 6are all factors of both 12 and 18 and are called common factors of 12 and 18.6 is the biggest common factor ofboth 12 and 18 and is called the Highest Common Factor,or H.C.F.of 12 and 18.For larger numbers it is usually helpful to write each one as a product of its prime factors: so to find the H.C.F. of120and 144 we first write120=2 2 2 3 5=2 2 2 3 5and 144=2 2 2 2 3 3 =2 2 2 2 3 3giving the H.C.F. as the product of all the common factors=2 2 2 3=24.MultiplesMultiples of 6 are 6, 12, 18, 24, ...since 6 divides into each of the numbers.Common Multiples and Lowest Common Multiple10, 20, 30, 40, 50, 60, 70, 80, 90, 100, ... are all multiples of10, and15, 30, 45, 6075, 90, 105, ...are all multiples of 15.Notice that 30, 60, 90, ...are multiples of both 10 and 15 and are called common multiples of 10 and 15.30 is the smallest common multiple of both 10 and 15and is called the Lowest Common Multiple, or L.C.M.of 10 and 15.For larger numbers it is usually helpful to write each one as a product of its prime factors: so to find the L.C.M. of48 and 140we first write48 =2 2 2 2 3and 140=2 2 5 7.To find the L.C.M. write down all the factors of one number140=2 2 5 7then include the factors from the other number until you just have all its factorsso L.C.M. =2 2 5 7 2 2 3 (including another2would be unnecessary)= 1680.Square numbers12 = 1, 22 = 4, 32 = 9, 42 = 16, 52 = 25, 62 = 36, etc. and so1, 4, 9, 16, 25, 36, 49, 64, 81, 100, ...are called square numbers.07/12/11IGCSE Extension Level RevisionSDB 8Triangular numbersThink of making triangles of balls on the snooker table1 ball 3 balls 6 balls 10 balls 15 ballsadd 2 add 3 add 4 add 5 add 6and these numbers are called triangular numbers: the first few are1 3 6 10 15 21 28 36 45 55...+2 +3 +4 +5 +6 +7 +8 +9 +10 +11Square roots and prime factors254016 can be expressed in prime factors as 26 34 72.We can use these prime factors to find254016 by halving each of the powers to give254016 = 23 32 7 = 8 9 7=72 7=504Indices(Powers)1.When multiplying, add the powers45 47 = 45 + 7 = 412but we cannot do anything with43 56because4and 5are different numbers.2. When dividing, subtract the powers79 74 = 79 - 4 = 75but we cannot do anything with 137 114 because 13 and 11 are different numbers.3. (43)2 = 43 2 = 46 ,since(43)2 = 43 43 = 43 + 3 = 43 2 = 46.4. 60 = 1,150 = 1.Any number to the power0is always equal to 1.5. 8181 111 1515337744 , , .A negative power meansone overorone divided by;5to the power of negative 4 means1divided by 5 to the power of positive 4.6.1 6 1 6 4 2 7 2 7 3 3 2 3 2 212133155 , , .A fractionone overmeans the square, cube, fourth etc. root of the number.07/12/11IGCSE Extension Level RevisionSDB 97. Rules 1, 2 and 3 still work for negative, fraction and zero powers.(i) 5 -354 = 5 -3 + 4 = 51 = 5.7 -47 -2 = 7 -4 + -2 = 7 -4 - 2 = 7-6 = 176.(ii) 35 3 -2 = 35 - -2 = 35 + 2 = 37.9 -4 96 = 9 -4 - 6 = 9-10 = 191011 -3 11 -5 = 11 -3 - -5 = 11 -3 + 5 = 112 = 121(iii) (6 -3)4 = 6 -3 4 = 6-12 = 1612.(iv) 64 64 4 16231322

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( )(v)1 2 511 2 52323since minus means turn upside down =152 =125,since 3 on bottom of fraction is cube root, 125 53 Standard Form6.42105is a number in standard form.The number bit, 6.42, must always be between 1 and 10 (can be 1 but not 10).56.3109 is not in standard form since the number bit, 56.3, is not between 1 and 10.Converting from standard form to an ordinary number.(i) 6.42105 =6.42 multiplied by 10five times = 642000.(ii) 7.34 10 -3=7.34 divided by 10 three times (negative power means divide)=0.00734.Converting TO standard form(i) 567.3 = 5.67 times 10 twice=5.67 102.(ii) 56.3104 =56.3 times 10 four times = 563000 =5.63 times 10 five times =5.63105.(iii) 0.000536 =5.36 divided by 10 four times(negative power means divide)=5.36 10 -4.07/12/11IGCSE Extension Level RevisionSDB 10Combining numbers in standard form1. Multiplying: multiply the number bits together, and then the powers of 10.4 1053.2 103 = (43.2) (105 103)=12.8108=1.28 101 108=1.28 109.2. Dividing: divide the number bits first and then the powers of 10.(2.7107) (3 102)=2 7 103 102 7310100 9 1072727 2. .. = 0.9 105 =9 10-1 105=9 104 .3. Adding and subtracting:you must be careful - you cannot treat this the same way as and .First convert to ordinary numbers, then add or subtract, then convert back to standard form.(i) 3.41 104+4.5 103 =34100 + 4500=34100450038600 + =3.86 104.(ii) 7.6310 -3 - 4.2 10-4= 0.00763-0.00042=0 007630 000420 00721... = 7.21 10 -3.Introduction to AlgebraSubstituting in formulaeWriting numbers in place of letters and working out:note that brackets mean do what is inside first.Find the value ofa - 3 (b - c) whena = 4,b = 7 and c = 2.a - 3 (b - c) =4 - 3 (7 - 2)=4 - 3 5=4 - 15 = -11.Simplifying expressions and bracketsSimplifying 3h - 4g + 5h - 7g equals8h - 11g.Before removing brackets multiple every term inside by the number outside (if there is one), but do not remove the brackets yet.When removing brackets with a minus sign in front, change the sign of every term inside the bracket, otherwise leave the signs unchanged.Examples: 07/12/11IGCSE Extension Level RevisionSDB 11(i) Simplify 3(2a + 5b)+4(7a - 2b)multiply by the number(s) outside= (6a + 15b) + (28a - 8b)remove brackets (no minus signs outside so leave signs unchanged)= 6a + 15b + 28a - 8b = 34a + 7b(ii) Simplify 4(a - 3b) - 5(2a - 3b)multiply by the number(s) outside= (4a - 12b) - (10a - 15b)remove brackets (minus sign outside second bracket so change signs)= 4a - 12b - 10a + 15b= -6a + 3b(iii) Simplify 5 - (3a + 7b)there is a minus separating the 5 and the bracket so the 5 does not multiply the bracket.We can remove brackets (minus outside means change signs inside)= 5 - 3a - 7band we can go no further.Solving equationsBalancing both sidesYou are allowed to add the same number to both sidessubtract the same number from both sidesmultiply both sides by the same numberdivide both sides by the same numberExamples:(i) 3k - 4 = 8add 4 to both sides 3k - 4 + 4 = 8 + 4Simplify each side 3k = 12divide both sides by 333123k k = 4(ii) 5c - 7 = 13 - 3cadd 3c to both sides 5c - 7 + 3c = 13 - 3c + 3cSimplify each side 8c - 7 = 13add 7 to both sides 8c - 7 + 7 = 13 + 7Simplify each side 8c = 20divide both sides by 807/12/11IGCSE Extension Level RevisionSDB 1288208c c=2.(iii)3 545x multiply both sides by 43 544 5 4x 3x - 5 = 20add 5 to both sides 3x - 5 + 5 = 20 + 5 3x = 25divide both sides by 333253x x =813.Re-arranging formulaeThis is just like solving equations but using letters instead of numbers.(i) Findkif3k - 4m = 8madd 4m to both sides 3k - 4m + 4m = 8m + 4mSimplify each side 3k = 12mdivide both sides by 333123k m k = 4m(ii) Find cif5c - 7d = 16b - 3cadd 3c to both sides 5c - 7d + 3c = 16b - 3c + 3cSimplify each side 8c - 7d = 16badd 7d to both sides 8c - 7d + 7d = 16b + 7dSimplify each side 8c = 16b + 7ddivide both sides by 8 c = 2b + 78d07/12/11IGCSE Extension Level RevisionSDB 13(iii) Findxifax - b = x + 3dadd b to both sides ax - b + b=x + 3d + b ax =x + 3d + bsubtractxfrom both sides ax - x=x + 3d + b - xfactorise L.H.S. and simplify R.H.S. x(a - 1)= 3d + bdivide both sides by (a - 1)x aad ba( ) +1131xd ba+31English statements and equivalent mathematical expressionsA farmer has some bulls but we dont know how many. He has twice as many cows. How can we write this using letters?Suppose that b is the number of bulls and that c is the number of cows and the above means that the number of cows is twice the number of bulls so c = 2b.The farmer has six fewer geese than bulls.Suppose that the number of geese is g then the number of geese is six fewer than the number of bulls so g = b - 6.The farmer has twice as many ducks as geese.Suppose that the number of ducks is d then the number of ducks is twice the number of geese sod = 2g.If 1 bar of chocolate costs35 pencethennbars cost35n pence.Forming equationsExample:John buys 4 oranges costing 15 pence each and 5 apples. (i) If the cost of each apple is x pence , write down an expression for the total cost of the apples and oranges together. (ii) In fact John spent 90 pence. Form an equation for x and solve it.Solution:(i) Each apple costsx pence so5 apples cost5xpence:07/12/11IGCSE Extension Level RevisionSDB 14the oranges cost 4 15 = 60penceso the total cost was (5x + 60)pence.(ii) We know that the total cost was (5x + 60)pence and also90penceso (5x + 60)= 905x + 60 - 60=90 - 605x=30 55305x x=6.Answer each apple cost6 p.InequalitiesSolving algebraic inequalities is just like solving equations, add, subtract, multiply or divide the same number to, from, etc. BOTH SIDESEXCEPT - if you multiply or divide both sides by a NEGATIVE number then you must TURN THE INEQUALITY SIGN ROUND.Example: Solve3 + 2x +4.Miscellaneous NumberLong MultiplicationExample: Multiply 324 by 162Solution: 324162648 2 32419440 60 32432400 100 3245248807/12/11IGCSE Extension Level RevisionSDB 15Long DivisionExample: Divide925by27and give the remainder.Solution: 3427 925 81 3 27 115 92 - 81 = 11and bring down the 5 108 4 27 7 115 - 108so925 divided by 27 equals 34 with remainder7.Fractions - proper, mixed, and improper fractionsExamples:381 34 778, ,are proper fractions since the numerator (top) is less than the denominator (bottom).4 1 7351 52 9, aremixed numbers -a whole number part and a fraction part.2 391 75, are improper, or top heavy, fractions since the numerator (top) is greater than the denominator (bottom).Converting from mixed fractions to top heavy fractions.Example: Convert 437to a top heavy fraction.Solution: 7 4 = 28then28 + 3 = 31so437 =317.Converting from top heavy fractions to mixed fractions.Example: Convert395to a mixed fraction.Solution: 5goes into397times with remainder 4, so 395=745.Adding and subtracting fractions1) Deal with the whole numbers first.2) Find the Lowest Common Denominator(L.C.D.)07/12/11IGCSE Extension Level RevisionSDB 163) Convert all fractions to have the same denominator(L.C.D.)4) Simplify and cancel down if possible.Example: Express2 5 3341659+ a simple mixed fractionSolution: 1) 2 5 3341659+ = 4 + 341659+ since2 + 5 3 =42) The L.C.D.of4,6and 9 is363) 4 + 341659+ = 4+2 73 663 62 03 6+ 4) = 4+ = 13364.ReciprocalsTo find the reciprocal of a fraction turn it upside downExamples: The reciprocal of 5995i s , the reciprocal of 1 73 23 21 7i s , etc.To find the reciprocal of a whole number like4,think of 4as41and turn it upside downExamples: The reciprocal of4is14, the reciprocalof23is123,etc.Multiplying and dividing fractions1) Mixed numbers must be converted into top-heavy fractions2) When dividing by a fraction, turn it upside down and multiply3) Cancel common factors by dividing top and bottom by the same number(s)4) Multiply out and simplify.Example: Simplify2 15857 .Solution: 1) 2 15857 = 2 181 27 make all top heavy3) =381 21dividing top and bottom by 7=3231 dividing top and bottom by 44) = 92124 Example: Simplify3 121 191 415 .Solution: 1) 3 121 191 415 =3 51 191 465 . make all top heavy07/12/11IGCSE Extension Level RevisionSDB 172) =3 51 191 456 turn 65 upside down and 3) =51 19256 dividing top and bottom by 7=51 13252 dividing top and bottom by 34) =75 944 441 Converting a decimal to a fraction(i) To convert0.345 to a fraction first notice that there are 3 decimal places so we write 0 3453451000. and then cancel 5 to give69200.(ii) 0.6428=64281000016072500.Fractions and recurring decimalsIf we calculate79as a decimal fraction we get79 = 0.7777777777777 . . . . .= 0 7 .which we read asnoughtpointsevenrecurringIf we calculate47as a decimal fraction we get47=0.571428571428571428571428571428 . . . . . .=057142 8 . which we read asnoughtpoint571428recurring.Some fractions do not recur when written as decimals: Example: 340 7 5 . which ends after 2 decimal places.All fractions either make recurring decimal fractions or decimal fractions which end.All recurring decimals can be written as fractions.Negative numbers-4 - 7means take away 4 then take away 7 so the result is take away 11 giving-4 - 7 = -11(-4) (-6) = +24 since minus times minus = plus(-3) 7 = -21 since minus times plus = minus(-12) (-3) = +4 since minus divided by minus = plus(-15) 5 = -3 since minus divided by plus = minus07/12/11IGCSE Extension Level RevisionSDB 1818 (-2) = -9 since plus divided by minus = minus.Rounding off to the nearest .....To round 43 to the nearest 10, first think of the tens around 43 which are20,30,40,50,60 and the one closest to 43 is40.Note that 45 is exactly half way between 40 and 50 and mathematicians always put 5s up so 45 rounded to the nearest 10 is 50.To round 77 to the nearest 5, first think of the fives around 77 which are65, 70, 75, 80, 85, 90 and the one closest to 77 is 75.Rough approximationsTo find a rough approximation first approximate every number to just 1 figure then do the calculation.67 3 4107 370 400710 40014000...SequencesA sequence is a list of numbers. Sometimes there is an easy pattern to spot which will let you find the next two terms of the sequence. Finding a pattern can be tricky but a good idea is to look at the differences between termsExamples:1) n 1 2 3 4 5 6 7term 4 7 10 13 16 ... ...add 3 add 3 add 3 add 3 add 3 add 3and so the 6th and 7th terms are19 (= 16 + 3)and22 (= 19 + 3)2) n 1 2 3 4 5 6 7term 3 6 12 24 48 ... ...times 2 times 2 times 2 times 2 times 2 times 2and so the 6th and 7th terms are96 (= 48 2)and192 (= 96 2)3) n 1 2 3 4 5 6 7term 5 7 10 14 19 ... ...add 2 add 3 add 4 add 5 add 6 add ??and so the 6th and 7th terms are25 (= 19 + 6)and32 (= 25 + 7)07/12/11IGCSE Extension Level RevisionSDB 19Sequences from rulesExamples:1) Rule2n + 3n 1 2 3 4 5 6 7term 5 7 ... ... ... ... ...when n is 3 the 3rd term is2 3 + 3 = 9,when n is 4 the 4th term is2 4 + 3 = 11,etc.2) Rulen2 - nn 1 2 3 4 5 6 7term 0 2 ... ... ... ... ...when n is 3 the 3rd term is32 - 3 = 6,when n is 4 the 4th term is42 - 4 = 12,when n is 5 the term is52 - 5 = 20, etc.Standard sequencesYou are expected to recognise some standard sequences formula1) Natural numbers 1 2 3 4 5 6 n2) Even numbers: 2 4 6 8 10 12 ... ... 2n3) Odd numbers 1 3 5 7 9 11 ... ... 2n + 14) Square numbers 1 4 9 16 25 36 ... ...= 122232425262... ... n25) Nearly square numbers(i) 3 6 11 18 27 38 notice that each number is two more than the square number above1+2 4+2 9+2 16+2 25+2 36+2 12+2 22+2 32+2 42+2 52+2 62+2 n2+2(ii) 3 12 27 48 75 108 notice that each number is three times the square number above3 1 3 4 3 9 3 16 3 25 3 36 3 123 223 323 423 523 62 3n26) Cube numbers 1 8 27 64 125 216 ... ... n307/12/11IGCSE Extension Level RevisionSDB 20= 132333435363... ...7) Triangular numbers 1 3 6 10 15 21 ... ...12( 1) n n +(snooker balls in triangle) +2 +3 +4 +5 +6 +7 +??Perimeters and AreasPerimeterThe perimeter is the distance round the outside of a figure so just add up all the lengths.The perimeter of a circle is also called the circumference and C = 2 r or C = d, where r is the radius and d is the diameter.AreaYou must learn the formulae for the following basic shapes.07/12/11IGCSE Extension Level RevisionSDB 21bhRectanglearea = b hbhParallelogramarea = b hhbarea = b hTrianglerCirclearea = r2bhaTrapeziumarea = (a + b) hVolumes and Surface AreasSurface AreasSurface area is the total of the areas of all the faces which are usually rectangles, triangles, circles etc.Plus the following shapes:(i) The area of the curved surface of a cylinder isarea = 2 r h =2r h(ii) The area of the curved surface of a cone is rlwherer is the radius of the base and lis the length of the slant height.A cone and its net are shown(iii) The surface area of a sphere is4 r2.VolumesYou must learn the following formulae for standard shapesVolume of a sphere=43 r3.PrismsA prism is any solid which has a constant cross-section - that is, if you cut it it is the same shape and size all the way up.Volume of a prism =area of the base height.A cylinder is a circular based prism with volume = r2h, since r2is area of base.PyramidsA pyramid has a base of any shape and comes straight up to a point.07/12/11IGCSE Extension Level RevisionSDB 22lwhCuboidVolume = l w hhrCylinderVolume = r2 hhrlrlVolume of a pyramid=13 area of the base height.A cone is a circular based pyramid with volume = 132 r h, since r2is area of base.Converting UnitsLengths Example:To convert5kmintocmwe multiply by 1000 to convert to metres and then by 100 to convert to cm 5km=5000m=500000cm.AreasExamples:(i) To convert4.3 km2intom2

we must first think of a 1 km1 kmsquareasa1000 m 1000 msquare and so1 km2=1000 1000 m2 = 1 000 000 m2 4.3km2=4.3 1 000 000 m2=4 300 000 m2.(ii) To convert634 mm2intocm2we must first think of a 1 cm1 cmsquareasa10 mm 10 mmsquare and so1 cm2=10 10 mm2 = 100 mm2 634mm2=634 100 cm2=6.34 cm2.VolumesExamples:(i) To convert25300litres intom3, we need to know that1000 litres= 1m3 25300litres=25300 1000 m3=25.3m3.(ii) To convert6.3m3 intocm3

we must first imagine a cube1 m 1 m 1 m =100 cm 100 cm 100 cm 1 m3=100 100 100cm3=1000000cm3 6.3m3=6.3 1000000cm3 =6300000cm3.(iii) To convert23400mm3intocm3we must first imagine a cube1 cm 1 cm 1 cm =10 mm 10 mm 10 mm 1 cm3=10 10 10mm3=1000mm307/12/11IGCSE Extension Level RevisionSDB 23 23400mm3=23400 1000cm3 =23.4cm3.Statistics 1Mode, median and range.The mode is the one which occurs most often.The range is found by subtracting the smallest value from the largest.To find the median you must first put in order of size then find the middle one.Note that if there are an even number of values there is no middle number so you find the middle pair and take the middle of this pair.Example:Find the mode, range and median of5,3,8,6,12,5,18,5,4,10,13,3.The mode is 5, since there are three 5s, more than any other number.The range is biggest - smallest = 18 - 3 = 15, so the range is 15.To find the median first put in order3,3,4,5,5,5,6,8,10,12,13,18.There are 12 numbers (even) so there is no middle number: the middle pair is5,6and the median is the middle of the middle pair which is5.Median, quartiles and interquartile rangeThe median is the middle number and divides all the numbers into two halves. The quartiles divide all the numbers into four quarters.Example: Find the median and quartiles of the following ages:5, 6, 8, 8, 9, 11, 12, 12, 13, 14, 15, 17, 17, 20, 21, 22, 25, 27, 27, 29, 31, 32, 35Solution: There are 23 ages and the quartiles will be the 6th, 12th and 18th ages. 07/12/11IGCSE Extension Level RevisionSDB 24middle pair(Note that6th = (n+1)th,12th = (n+1)th,12th = (n+1)th)and we can see the four quarters5, 6, 8, 8, 9, 11, 12, 12, 13, 14, 15, 17, 17, 20, 21, 22, 25, 27, 27, 29, 31, 32, 35Lower Quartile =11, Middle Quartile =17, which is the same as the median Upper Quartile =27,The interquartile range is the Upper Quartile minus the Lower QuartileI.Q.R.=UQ LQ= 27 11 = 16.MeanMean is 'normal average' - treat as a bag of apples - find the total weight of the bag and divide by the total number of apples.Example: The heights of 54 people are shown in the table. Find the mean.Solution: To find the mean we need to find the total height (as if all were lying in one long line) and divide by the total number of people.height (cm) frequencyx fxf140 3 420(420 is total height of the 3 people of height 140 cm)145 10 1450150 14 2100155 11 1705160 9 1440165 5 825170 2 340___ ____Totals 54 8280Thus the mean height=m total heighttotal number=828054 = 153.3 cm.Frequency tables and tally chartsExample: The marks of 200 pupils were recorded. Draw up a frequency table. Solution: First decide on suitable intervals(1 - 10,11 - 20,....,91 - 100marks) then record using 'five bar gates' as shown below:TALLYCHART FREQUENCYTABLE07/12/11IGCSE Extension Level RevisionSDB 25Interval Five bar gates Interval Frequency1 - 10 |||| || 1 - 10 711 - 20 |||| |||| || 11 - 20 1221 - 30 |||| |||| |||| ||| 21 - 30 1831 - 40 |||| |||| |||| |||| |||| |||| |||| | 31 - 40 3641 - 50 |||| |||| |||| |||| |||| |||| ||||41 - 50 3551 - 60 |||| |||| |||| |||| |||| |||| |||| |||| |||| 51 - 60 4461 - 70 |||| |||| |||| ||||61 - 70 2071 - 80 |||| |||| |||| | 71 - 80 1681 - 90 |||| || 81 - 90 791 - 100 ||||91 - 100 5Bar chartsand line graphsExample: Draw a bar chart and line graph for the above frequency table. Bar Chart Line GraphF r e q u e n c y1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 1 0 0M a r k1 02 03 04 0

Frequency10 20 30 40 50 60 70 80 90 100Mark10203040Note that(i) Frequency is plotted up(ii) Scale on horizontal axis is like an ordinary graph(iii) No gaps between bars.Note that the crosses are marked in the middle of the intervals, thus for a frequency of 7 in the interval 1 - 10, a cross is marked at(5, 7) etc.Histograms In a histogram the frequency is represented by the area of the bar.To draw a histogram we must first work out the width and height of each bar so that the area of the bar equals the frequency,height = frequencywidth.This is shown for the frequency table of ages below:07/12/11IGCSE Extension Level RevisionSDB 26age in years,x frequency width ofintervalheight of interval0 x < 515 5 35 x < 1023 5 4.610 x < 2046 10 4.620 x < 3052 10 5.230 x < 5082 20 4.150 x < 7073 20 3.6570 x < 11040 40 110 20 30 40 50 60 70 80 90 100 110123456xyPie chartsExample:The strength of the armed forces in 1956 is shown in the table below. Draw a pie chart to illustrate these figures.thousands of menRoyal Navy 112Army 380Royal Air Force 228Total 720Solution:07/12/11IGCSE Extension Level RevisionSDB 27frequency density720 men in total to share 360 so each man is equivalent to360720 = o, giving anglesRoyal Navy 1122 = 56Army 3802 = 190Royal Air Force 2282 = 114Total(to check) 360Finding the mode and mean from a grouped frequency tableFinding the mode is still easy - it is the most frequent class interval.In this case we use do not have exact information so we use the mid-interval value to calculate the mean.Example: Find the mode and an estimate for the mean weight of people from the frequency table below.Solution:The mode or modal class is the class which occurs most often and is 60weight < 70 kg.Note that we do not know the exact weights of the people in the 30 to 40 kg class so we take the mid-interval value, 35 kg, as the weight for all 12 people; similarly we take the mid-interval values for the other classes. Thus we cannot find an exact value for the and so we can only estimate the mean.Weightkgmid-intervalvalue xfrequency f xf30 weight < 4035 12 42040 weight < 5045 17 76550 weight < 6055 20 110060weight < 7065 22 143070 weight < 8075 15 112580 weight < 9085 10 850TOTALS 96 569007/12/11IGCSE Extension Level RevisionSDB 28The mean istotal weight of everybodytotal number of peoplekg 56909659 3 . ,to 1 D.P.Surveys, questionnaires and samplingQuestionnaires should ask for precise information requiring a yes/noor atick/cross answer:Do not ask what colour hair but give options -black, dark, brown, fair, ginger and ask to tick the colour which most nearly describes the colour.Do not ask how often a person goes to shop but ask0 -3, 4 - 6,7 - 9, more than 9 times a week etc.An observation sheet or data sheet is one on which you record the answers to your questions: it will probably take the form of a tally sheet.In selecting people to ask you must be careful to avoid bias.Choosing people from their e-mail addresses would give a biased sample as many people do not have an e-mail address, etc.Random samplesTo take a random sample of 22 boys from the school, list in any order and give each boy a unique number. Use the first 22 random numbers to select the sample.Stratified samplesIn the random sample above all the boys could be from the same house, or some houses might not be represented in the sample. To avoid this we use a stratified sample as follows.Divided the boys into houses and give each boy a unique number. Use random numbers to select 2 boys from each of the 11 houses. This will ensure that each house is equally represented.PercentagesPercentages as fractions and decimals45% means45 per 100 so45% = 4510092037% means37 per 100 so37% = 371000 37 .580 625 62 5% . .Finding a percentage of a given number.40% of 65 = 4010065 2607/12/11IGCSE Extension Level RevisionSDB 29Percentage increases and decreasesTo increase a price by 13% we are increasing from the original of 100% by 13% to give 100% + 13% = 113% which is 113100113 .times what we started with, so we multiply the original price by 1.13.Original price New price(increase by 13%) 75 75 1.13 multiply by 1.13 = 84.75.To decrease by 17% we would multiply by0.83,since 100% - 17% = 83% = 0.83Finding the original number after a percentage changeIf we know that the original price has been increased by 13% we know that it has been multiplied by 100% + 13% = 1.13. So if the new price is 135 then we must divide by 1.13 to find the original price.Original price New price(has been increased 13%) ? 135divide by 1.13so the original price was 135 1.13 = 119.47Simple interestIf money is invested and interest paid annually but not re-invested then this is called simple interest.To find the simple interest paid if 600 is invested at 4.5% each year (p.a.) for 7 years.The interest paid each yearis4.5% of 600=27, so over 7 years (not re-invested) the total interest paid is7 27=189.Compound increases or decreases.If money is invested and the interest is re-invested at the end of each year then this is called compound interestTo find the value of 600 invested at 4.5% p.a. after 7 years.Each year its value increases by 4.5% so we multiply the value at the beginning of each year by1.045 (104.5%).After 1 year the total value is600 1.04507/12/11IGCSE Extension Level RevisionSDB 30after 2 years the total value is (600 1.045) 1.045=600 1.0452after 3 years the total value is (600 1.0452) 1.045=600 1.0453and so on to give the value after 7 years as600 1.0457 = 816.52.With compound interest the interest paid over 7 years is 216.52which is 216.52- 189=27.52more than simple interest.Expressing one number as a percentage of anotherRule:percentage isfi rst numberond number sec 100Example: Find23 as a percentage of 81.Percentage is2 38 11 0 0 2 84 . so answer is28.4%Example:A coat is reduced from 135 to 110 in a sale. Find the percentage reduction.The actual reduction is135 - 110 = 25so the percentage reduction is25 as a percentage of 135(not 110)and is2 51 3 51 0 0 1 85 . so answer is 18.5%GeometryAngle propertiesAcute, obtuse and reflex angles07/12/11IGCSE Extension Level RevisionSDB 31acute angle, between 0o and 90oobtuse angle, between 90o and 180oreflex angle, between 180o and 360oAngles on a line, vertically opposite and round a pointParallel linesBearingsBearings are angles measured from North in a clockwise direction.To find the bearing of P from Q, imagine that you are standing at Q and measuring the angle from North round to P.The bearing of P from Q is 142oThe bearing of Y from X is305o07/12/11IGCSE Extension Level RevisionSDB 32angles on a straight line,a + b = 180oabbavertically opposite angles are equala = babdcangles round a point add up to 360oa + b + c + d = 360oabalternate angles are equala = bcdcorresponding angles are equalc = defallied anglese + f = 180oPQ142oNorthYX305oNorthAngles of elevation and depressionThe angle of elevation from your eye up to the top of a tree, say, is the angle made by the horizontal and the line from your eye up to the top of the tree.The angle of elevation from your eye to the top of the tree is 27o.The angle of depression from your eye down to the base of a building, say, is the angle made by the horizontal and the line from your eye down to the base of the building.The angle of depression from your eye to the base of the building is 19o.TrianglesAngles in a triangle add up to 180o.Proof:In any triangleABCdraw a line throughA and parallel to BC. ABC= DAB alternate ACB= EAC alternate DAB+ BAC+ EAC=180oangles on a straight line ABC+ BAC+ ACB=180oQ.E.D.Exterior angle equals sum of the two interior opposite anglesAngle sum of a triangle is180o a + b + c=180Angles on a line sum to 180o c + d=18007/12/11IGCSE Extension Level RevisionSDB 33BEDACabcdhorizontaltop of tree27oeyehorizontalbase of building19o eye c=a + bi.e.exterior angle = sum of interior opposite angles. Q.E.D.Equilateral and isosceles trianglesCongruenceTwo triangles are congruentexactly the same shape and size if(i) SSS all three sides are equal(ii) SAS Two sides and the included angle are equal i.e. the angle between the two equal sides.(iii) AAS Two angles and the corresponding side are equal.(iv) RHS A right angle, the hypotenuse and another side.Example: The triangleABCis isosceles withAB = AC.Dis the midpoint ofBC. Prove that the trianglesABDandACDare congruent and hence that ADis perpendicular toBC.Solution:In the trianglesADBandACDAB=AC (given)07/12/11IGCSE Extension Level RevisionSDB 34Equilateral triangles have all sides equal and all angles equal to 60o.Isosceles triangles have two equal sides and the base angles are equal.BAD CBD=DC (D is midpoint of BC)AD=AD (common) triangles are congruent, SSS. ADB= ADC (corresponding angles)but ADB+ ADC=180o(angles on a straight line) ADB= ADC=90o ADis perpendicular toBC. Q.E.D.QuadrilateralsQuadrilaterals are four sided figures.A trapezium is a quadrilateral with one pair of parallel sidesAparallelogram is a quadrilateral with two pairs of parallel sides (opposite sides and angles equal and diagonals bisect each other)A rhombus is a quadrilateral with all four sides equal a pushed over square (a parallelogram in which the diagonals bisect each other at right angles).Special quadrilaterals07/12/11IGCSE Extension Level RevisionSDB 35Rectangles have all angles equal to 90o.Squares have all sides equal and all angles equal to 90o.Parallelograms have two pairs of opposite sides parallel and equal: opposite angles are also equal and the diagonals bisect each other.A rhombus (diamond) has all four sides equal: opposite pairs are parallel and diagonals bisect each other at 90oA trapezium has one pair of opposite sides parallel.A kite is formed of two isosceles triangles with equal bases, joined at their bases.PolygonsA polygon is a figure consisting of any number of straight sides.The sum of all the exterior angles is 360o.The sum of all the interior anglesis(n 2) 1800,wherenis the number of sides of the polygon.Regular polygonsAregularpolygon is a polygon withequal sidesandequal angles.A regular pentagon has five sides, and the five equal exterior angles add up to 360o.Each exterior angle = 360 5 = 72o,and so each interior angle equals180 - 72 = 108o.In a regular n-sided polygon the exterior angle is360/nAnd the interior angle is180 360/nExample: A regular polygon has interior angles of 150o. How many sides does it have?Solution: Interior angle = 1500 exterior angle = 180 150 = 300and since360 30 = 12there must be 12 exterior angles and so 12 sides.SetsSet Notationn(A) the number of elements in the set Ax A x is an element of the set Ax A x is not an element of the set A07/12/11IGCSE Extension Level RevisionSDB 36exterior angle interior angleexterior angleinteriorangleA the complement of A all elements which are not in A the empty set the set with no elements the universal set the set of all elementsA B A is a subset of B all elements of A are also elements of B (A could be the same set as B)A B A is a proper subset of B all elements of A are also elements of B ( but A could not be the same set as B)A B A is not a proper subset of BA B A union B all elements in A or B or bothA B A intersection B all elements in both A and BVenn diagramsA B A BFunctionsA function is an expression (often in terms of x) which has only one value for each value of x.Notationy = x2 3x + 7,f (x) = x2 3x + 7andf : x x2 3x + 7are all ways of writing the same function.Composite functionsTo find the composite functionfgwe must dogfirst.Example: f : x 3x 2and g : x x2 + 1.Findfgandgf.Solution: Think offandgas rulesf is 07/12/11IGCSE Extension Level RevisionSDB 37A BA Btimes by3 subtract2square add1gis fgisgiving(x2 + 1) 3 2 =3x2 + 1 fg : x 3x2 + 1or fg(x)=3x2 + 1.gfisgiving(3x 2)2 + 1=9x2 12x + 5 gf : x 9x2 12x + 5orgf (x) = 9x2 12x + 5.Inverse functionsThe inverse offis the opposite off:thus the inverse of multiply by 3 is divide by 3and the inverse of squareis square root.The inverse offis written asf 1: note that this does not mean1 over f .Example:f : x 3x2 + 1f is To find the inverse we do the opposite of each box starting with the lastf 1 isgiving f 1: x 31 xTransformationsTranslationsWe usually describe a translation with a vector,

,_

75 means5 along and 7 up07/12/11IGCSE Extension Level RevisionSDB 38times by3 subtract2 square add1times by3 subtract2 square add1squaremultiply by 3 add1subtract 1divide by 3 square rootCombining translationsTo combine two translations we just add their vectors.Thus a translation through

,_

52followed by a translation through

,_

74is equivalent to a translation through

,_

52+

,_

74=

,_

122.RotationsTo describe a rotation you must always give the centre and the angle of rotationAngles of rotation are measured anticlockwise.A half turn is a rotation through180o,and a quarter turn is a rotation through 90o.Finding the centre of rotationWhen the diagram is drawn on squared paper you can usually spot the centre of rotation by inspection.Otherwise:to find the centre of rotation when triangleABChas been rotated onto A'B'C', draw the perpendicular bisectors ofAA' ,BB'andCC' ;the centre, O, is where these lines meet.ReflectionsWhen describing a reflection you must give the mirror line.To reflectAonto its image, A', go straight (at 90o)to the mirror and the same distance the other side.If A has been reflected onto A', you can find the mirror line by drawing the perpendicular bisector of AA'.Check by drawing the perp. bisector of BBas well.Enlargements Positive scale factorTo describe an enlargement you must give centre and scale factor.07/12/11IGCSE Extension Level RevisionSDB 39AC'COB'BA'5 105xyAA'BB'XAA'mirror lineBB'To find the image ofA under an enlargement, centre X, (2, 3), and factor 3.From X to A go along 3 and down 1, 31 _ ,, and for factor 3, just multiply all by 3to give from X to A go along 9 and down 3,

,_

39, to giveA'.Other points can be found similarly.Fraction scale factorTo enlarge a point A centre X and factor , measureXAand halve it to giveXA'.Similarly for all other points.Note that the image is smaller than the original.Negative scale factor Similar to positive factors except that you go to the other side of the centre.So to enlargeAcentreXfactor2, measureXA, double it to give the length ofXA', measured on the other side of the centre, X.Similarly for the other points.PythagorasPythogorass theorem states that in a right angled triangle,a2 + b2 = h2,whereh is the hypotenuse (the longest side)Examples:(i) Findx. (ii)Findx.07/12/11IGCSE Extension Level RevisionSDB 4068x1312xahbXA'AXA'A 62 + 82 = x2 36 + 64 = x2 100 = x2 x=10. Notice that x is not the hypotenuse so be careful!!x2 + 122 = 132 x2 + 144 = 169 x2=25 x=5.RatioSharing in the ratio of two numbersExample: Share 32 sweets between George, aged 5, and David, aged 3, in the ratio of their ages.Solution: Sharing in the ratio5 : 3, so for every5 + 3 = 8 sweets George has 5 and David has 3. Now 8 divides into 32four times so we multiply5 : 3by 4 to give20 : 12.George has 20 sweets and David 12.Sharing in the ratio of three numbersExample: Share 96 in the ratio7 : 3 : 2..Solution: 7 + 3 + 2 = 12 and 12divides into96eight times so we multiply everything by 8 to give56 : 24 : 16.Map ScalesMap scales are often given in as a ratio such as1 : 50 000. This means that 1 cm on the map represents 50 000cmon the ground; or that 1 mm on the map represents 50 000 mm on the ground.Examples:(i) The distance between two towns measured on a map is 13.5 cm. What is the actual distance between the towns in km?13.5 cmon the map 13.5 50000 = 675000 cmon the ground=675000 100 m=6750 m= 6750 1000 km=6.75km(ii) The actual distance between two houses is 850 m. Find the distance on the map between these houses in mm.We first need the actual distance in mm.850 m=850 1000 mm=850 000mmon the ground07/12/11IGCSE Extension Level RevisionSDB 41 850 000 50 000mm =17 mmon the map.(iii) The area of a lake on a map is given as 2.4 cm2. Find the actual area of the lake in m2.1 cm on the map = 50000 cm=50000 100 m= 500 mon the ground A square 1 cm by 1 cm= 1 cm2 on the mapis a square500 m by 500 m= 250 000 m2on the ground the area of the lake is2.4 250 000m2

= 600 000 m2.Trigonometry 1SOH CAH TOAsin A =opphyp, cos A =adjhyp, tan A =oppadjN.B.you must know two things before you can find a third.Ifsin A = 0.334 you must press shift sin 0.334' on your calculator to find A = 19.5Example:We know the angle = 57We know the hyp= 17 cmWe want the opp= x cmUsingSOH CAH TOAwe see that we need to use sin sin 57= opphypx x 17 170 83867 .x=0.83867 17 = 14.25739=14.3 cm to 3 S.F.The awkward case,xon the denominatorWe know the angle = 42We know the opp= 1207/12/11IGCSE Extension Level RevisionSDBAadjhypopp42xcmopp17 cmhyp57o42oopp12adj xWe want the adj= xUsingSOH CAH TOAwe see that we need to use tan 12 12tan 42 0.900404...x x 0. 900404 x=12 x = 120.900404...13.3 , to 3 S.F.Quadratic functions & factorisingMultiplying out two brackets -F O I LTo multiply out two brackets we useF O IL which stands for FirstOutsideInsideLastas shown below:(2x-3)(4x+5)(2x-3)(4x+5)giving8x2+(10x 12x)- 15=8x22x 15FactorisingTo factorise a quadratic use inspired guess work and check by multiplying out.A few tips: If the quadratic has only integers, only use integers when factorising. The first terms in each bracket must multiply to give the x2 term The last terms in each bracket must multiply to give the constant term. ax2 +bx + c=( ?x+? ) ( ?x+? ) ax2 t bx - c =( ?x+? ) ( ?x -? )07/12/11IGCSE Extension Level RevisionSDB 43+ & +means both brackets +- means one bracket +& one bracket -First = 2x 4x=8x2Outside= 2x 5=10xInside= -3 4x=12xLast=3 5=15General examples of factorising2ab + 6ac2 = 2a(b + 3c2)x2 - 5x + 6 = (x - 2)(x - 3)x2 - 6x=x(x - 6)6x2 - 11x - 10 = (3x + 2)(2x - 5)Learn the last threex2 - y2 = (x - y)(x + y)(x + y)2 = (x + y) (x + y)= x2 + 2xy + y2(x - y)2 = (x - y) (x - y)=x2 - 2xy + y2 Algebraic fractionsFactorise numerator and denominator fully and then cancel factors.Example: Simplify3 46 22+ ++x xxSolution:3 46 22+ ++x xx=12) 3 )( 1 () 3 ( 2++ ++x x xxExample: Simplify 10 722252222+ x xx xx xx.Solution:10 722252222+ x xx xx xx=) 2 )( 5 () 1 )( 2 () 2 () 5 )( 5 ( + + x xx xx xx x= ) 2 () 1 )( 5 (+ +x xx x.Quadratic equationsSolution by factorising(i) To solve 6x2 - 11x - 10 = 0first factorise to give (3x + 2)(2x - 5) = 0If two numbers multiplied together give 0 then one (or both) of the numbers must be 0(3x + 2)=0or (2x - 5) = 0 x x 2352122 o r(ii) Solve 3x2+7x=0 x(3x + 7)=0 x=0orx = 7307/12/11IGCSE Extension Level RevisionSDB 44Solution by completing the squareRule for completing the square:(i) Coefficient ofx2must be 1,(ii) Halve the coefficient ofx,square it and add this number to both sidesExample: Solve 3x2 18x + 8 = 0by completing the square.Solution: 3x2 18x + 8 = 0 3x2 18x=8 x2 6x=8/3dividing both sides by3to make coefficient of x2 equal 1 x2 6x+9=8/3+ 9=19/3halve coefficient of x (6/2), square it (9) add to both sides (x 3)2=19/3 x 3=319t x= 3319t =0.48or5.52.Quadratic equation formulaIf the expression will not factorise, use the formula:ax2 + bx + c = 0 xb b aca t 242To solve3x2-6x+2=0This will not factorise so use the formula, a = 3,b = -6,c = 2giving x t 6 6 4 3 22 32( ) x t t 6 36 2466 126 x + 6 1266 126or x=1.58 or 0.423 to 3 S.F.Simultaneous equationsSolving by eliminationIf we use algebra to solve the equations 3x + 5y = 17x 5y= 19 10x = 20 x=2.To find y we use the first equation and putx = 2 6 + 5y = 1 y = 1.Solutions are x = 2andy = 1.07/12/11IGCSE Extension Level RevisionSDB 45we notice that if we add the LHS of each equation the term in y disappears, so we add the two LHSs together and the two RHSsaddExamples:[1] 5x + 3y = 9[2] 5x +y= 13 2y = 4 y = 2in [1] 5x + 6 = 9 5x = 15 x = 3.Check in [2] LHS=15 2 = 13 = RHSSolutions are x = 3, y = 2[1] 3x + 4y = 7[2] 3x 2y = 196y= 12y = 2in [1] 3x + 8 = 7 3x = 15 x=5.Check in [2] LHS=15 4 = 19 = RHSSolutions arex = 5,y = 2.Sometimes the coefficients (numbers multiplying x and y) of neither of the variables are the same. In this case we can multiply one or both of the equations to give the same coefficients for one variable.Examples:[1] x y= 2[2]4x + 3y= 29Multiplying both sides of equation [1] by 3 will make the coefficients of y equal in size.3 [1] 3x 3y = 6[2] 4x + 3y= 29 7x=35 x = 5in [1] 5 y = 2 y = 3.Check in [2]LHS = 20 + 9 = 29 = RHS Solutions arex = 5,y = 3.[1] 2x + 5y = 3[2] 3x 2y = 14Multiplying both sides of equation [1] by 3 and both sides of equation [2] by 2will make the coefficients of x equal in size.3 [1] 6x + 15y = 92 [2] 6x4y = 2819y= 19 y = 1in [1] 2x 5 = 3 x = 4Check in [2]LHS = 12 2 = 14 = RHS Solutions arex = 4,y = 1.Graphs1Distance between two pointsThe distance between two points, X (a, b) and Y (c, d) isXY=2 2) ( ) ( b d a c + 07/12/11IGCSE Extension Level RevisionSDB 46addsubtractremember4 subtract 2=4 2=4 + 2 = 6subtractsubtractX (a, b)d bc aY (c, d)GradientGradientis increase inincrease inyx . (The 'increase' may be negative.)The gradient is the rate at which y is changing.In the diagram abovethe gradient of the line XYisa cb d.To find the gradient of a curve at a point draw the tangent at that point and find the gradient of the tangent by drawing a large triangle.Mid-pointThe mid-point, M, of the line XY in the above diagram isM is( (a + c), (b + d)) Straight lines,y = mx + cy = mx + c is a straight line with gradientmand intercept on the y-axis at (0, c).Example: Find the equation of the line through(1, 3)and(5, 15)Solution: The gradient of the line is26121 53 15 a cb d the equation is of the form y = 2x + c,sincem = 2.To findcwe use the fact that the line passes through(1, 3)which tells us thaty = 3 whenx = 1 3 = 2 (1) + c c = 5 y = 2x + 5is the equation of the line.Example: Find gradient and y-intercept of 3x + 2y = 8.Solution: First re-arrange to find y 2y=3x + 8 y = 1.5x + 4and so the gradient is1.5and the intercept on y-axis is(0, 4).Parallel linesParallel lines have the same gradient07/12/11IGCSE Extension Level RevisionSDB 47Example: Find the equation of the line which passes through(7, 3) and which is parallel toy = 2x 5.Solution: y = 2x 5has gradient2, and so any line parallel to y = 2x 5must also have gradient 2and so must have an equation of the formy = 2x + c.But the line we want passes through(7, 3)soy = 3whenx = 7 3 = 2 7 + c c = 11 the equation of the line is y = 2x 11Plotting curvesTo draw a graph form a table of values, plot the points with a cross and join up with a smooth curve.Example:a) Draw up a table of values and draw the graph ofy= x3 - 6x - 1. b) Using the graph solve the equations(i) x3 - 6x - 1 = 2;(ii) x3 - 6x = -2Solution:x -3 -2 -1 0 1 2 3x3-27 -8 -1 0 1 8 27-6x 18 12 6 0 -6 -12 -18-1 -1 -1 -1 -1 -1 -1 -1y -10 3 4 -1 -6 -5 8giving the graph below-3 -2 -1 1 2 3 4-10-8-6-4-22468xyb) (i)to solve x3 - 6x - 1= 2we need the values ofxto givey = 2.We see that there are three points with y-coordinate 2 for whichx = -2.1,-0.5and +2.7.07/12/11IGCSE Extension Level RevisionSDB 48(ii) to solve x3 - 6x = -2we must first re-arrange the equation so that the left hand side is x3 - 6x - 1 by subtracting 1 from each side to givex3 - 6x - 1 = -3and we now need the x values which give a y-coordinate of -3:these arex = -2.6, 0.3or2.3.Example: Draw the graphs ofy = x2 - 3x + 2and y = x + 1.Use your graphs to solve the equation x2 - 3x + 2=x + 1.Solution: The solution of this equation will be the x-coordinates of the points of intersection of the two graphs.1 2 3 4 5 6123456xyx= 0.7or4.3Standard graphs-4 -3 -2 -1 1 2 3 4 5-3-2-1123xy -4 -3 -2 -1 1 2 3 4 5-3-2-1123xy -4 -3 -2 -1 1 2 3 4 5-3-2-1123xy-4 -3 -2 -1 1 2 3 4 5-3-2-1123xy-4 -3 -2 -1 1 2 3 4 5-3-2-1123xy

-4 -3 -2 -1 1 2 3 4 5-3-2-1123xy07/12/11IGCSE Extension Level RevisionSDB 49y = xy = x2y = x3y = 1/xy = 1/x2y = -x-4 -3 -2 -1 1 2 3 4 5 6 7 8 9 101020xyy =3x2is likey = x2but steeper: similarly fory = 5x3likey = x3but steeper andy = 7/x likey = 1/xbut steeper,etc.ProbabilityPossible outcomesEqually likely outcomesIf all possible outcomes have the same chance of happening (are equally likely) then the probability is easily foundExample: In a normal pack of well shuffled cards, p(king) = 524since there are4 kingsand52 cards and all cards are equally likely.Example: A fair die has equal chances of any score from 1 to 6 sop(prime) = 63 since there are 3 primes (2, 3 and 5)and 6 possible scores.Probability and relative frequencyIf we spin a biased coin3000times and it lands Heads 2000 times thenthe frequency of Heads is2000but the relative frequency ofHeads is 3230002000and we would say that the probability of this coin landing Heads is2/3.Two or three coins1) For two coins we can consider HH,HT,TH,TTas a set of equally likely outcomes so thatp( exactly one Head)=2/4= 07/12/11IGCSE Extension Level RevisionSDB 50x + y = 3The exponential curve y =2x-4 -2 2 424xy2) For three coins we can considerHHH,HHT,HTH,THH,HTT,THT,TTH,TTTas equally likely outcomesso thatp( exactly one Head)=3/8 .Two diceExample: The total when two dice are thrown can be 2 ,3, 4, ...,11 or 12. BUT be careful since these are NOT equally likely.For two dice (red and green) the best approach is to make a table of 36 crosses to show all of the equally likely outcomes.To find the probability of a total of 10, we see that 10 can happen in only three ways of the 36 crosses shown at the points(4, 6), (5, 5), (6, 4)(marked by the larger c