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AS Level MathematicsA guide to help you prepare yourself for studying AS Level MathematicsMovie recommendationsThe Imitation Game is a 2014 American historical drama film directed by Morten Tyldum and written by Graham Moore, based on the biography Alan Turing: The Enigma by Andrew Hodges. It stars Benedict Cumberbatch as British cryptanalyst Alan Turing, who decrypted German intelligence messages for the British government during the Second World WarA Beautiful Mind is a 2001 American biographical drama film based on the life of the American mathematician John Nash, a Nobel Laureate in Economics and Abel Prize winner. The film was directed by Ron Howard, from a screenplay written by Akiva Goldsman. It was inspired by a bestselling, Pulitzer Prize-nominated 1998 book of the same name by Sylvia Nasar.Good Will Hunting is a 1997 American drama film directed by Gus Van Sant, and starring Robin Williams, Matt Damon, Ben Affleck, Minnie Driver, and Stellan Skarsg?rd. Written by Affleck and Damon, the film follows 20-year-old South Boston janitor Will Hunting, an unrecognized genius who, as part of a deferred prosecution agreement after assaulting a police officer, becomes a client of a therapist and studies advanced mathematics with a renowned professor.TED TalksIs maths discovered or inventedWould mathematics exist if people didn't? Did we create mathematical concepts to help us understand the world around us, or is math the native language of the universe itself? Jeff Dekofsky traces some famous arguments in this ancient and hotly debated question.LinkMaths is foreverWith humour and charm, mathematician Eduardo Sáenz de Cabezón answers a question that's wracked the brains of bored students the world over: What is math for? He shows the beauty of math as the backbone of science — and shows that theorems, not diamonds, are forever. In Spanish, with English subtitles.LinkFurther Mathematics Support Programme Wales - COVID-19: Opportunities for Yr11 (Summer 2020)Title/release date/ FMSP contact for more infoDescriptionWhat is this resource (W -Welsh, E-English or B-Bilingual)Does the resource require communication with learners?FMSPW Bridging Mathematics sessions for Y11, Date available from: 4th May 2020.Contact:Adrian Wells adw16@aber.ac.uk An introduction to AS Mathematics and Further Mathematics with an emphasis on fun and enjoyment of mathematics covering the topics on mathematical Investigations, Algebra, Introduction to applied mathematics, Mathematical proof, Differential calculus, Introduction to complex numbers, Problems and algorithms, Areas and volume: integration, Coordinate Geometry, Further Maths taster: Pureand Applied. A set of 10 pre-recorded tutorials followed by self-study tasks and materials with an end of study online assessment test (English and Welsh).Students will be required to subscribe to the programme through their school. Pre-recorded tutorials will be released weekly and available for limited time of 2 weeks, release of the solutions to self-study tasks will be delayed. Mathematical Worlds, Date available from: 18th May 2020Sofya Lyakhovas.lyakhova@swansea.ac.ukA set of 10 rich mathematical tasks prepared for Wales by mathematicians across the world (Germany, Australia, USA, Japan and UK). These are resources for slow contemplative mathematical thinking and problem solving for developing resilience and creativity in learners when approaching unseen mathematical problems.10 sets with an intro video from Dr Ian Roberts (Charles Darwin University), each set includes an introductory video (E), worksheet (B) and guidance/solutions (B).Year 11 students could pursue each world to the level they are comfortable with. Recommended for all SEREN students.Students will be required to subscribe to the programme through their school. Suitable for asynchronous learning. Learners could contact FMSPW by email if they need help or would like to share the results of their investigations.Careers in Mathematics talks, Date available from: 4th May 2020Elian RhindE.O.T.Rhind@Swansea.ac.ukPre-recorded video presentation with info about A-level and career choice. Session is 45 minutes.A series of 6 online live sessions recorded and uploaded to website. Weekly in June and July - held on Saturday mornings – each session 1.5 hours.Students will be required to subscribe to the programme. Learners can apply through their school or directly through FMSP Wales email fmspwales@swansea.ac.ukWebsites/Social MediaHegarty Maths There are numerous resources on his website and Colin Hegarty is also running free tutorials for Yr 11 pupils between now and September! Follow him on Twitter @hegartymaths. (Welsh Language) Numerous videos and resource, including for the Additional Mathematics bridging qualification from Yr11 to Yr12! Follow on Dr Gareth Evans on twitter @mathemateg.Reading ListIf you require more practice CGP Head Start to A-Level Mathematics is currently available for free as a Kindle EditionAs a student who is choosing to study Mathematics at A Level, it is logical to assume that you have an interest in the subject!With that said, the following books may be of interest to you.50 Mathematical Ideas You Really Need to Know (Tony Crilly)Alex’s Adventures in Numberland (Alex Bellos)Cabinet of Mathematical Curiosities (Ian Stewart)The Calculus Wars (Jason Socrates Bardi)The Code Book (Simon Singh)The Curious Incident of the Dog in the Night-time by Mark HaddonHow Many Socks Make a Pair?: Surprisingly Interesting Maths (Rob Eastway)Hello World: How to be Human in the Age of the Machine (Hannah Fry)Humble Pi: A Comedy of Maths Errors (Matt Parker)The Life-Changing Magic of Numbers (Bobby Seagull)The Num8er My5teries (Marcus du Sautoy)Mathematical connections (understanding not memorising!)Mathematics is a hierarchical subject and it would be extremely useful for you to ensure a solid understanding of basic algebra before commencing AS study. Good mathematicians make mathematical connections and do not look at topics in isolation. Consider the following example (don’t worry if some of the maths is unfamiliar!):Solve the equation 2x2+x-10=0I can answer in different ways (there are explanation videos underneath that might be useful):By factorising:2x2+x-10 =02x+5x-2=02x+5=0 neu x-2=0x= -52 neu x=2By applying the quadratic formula:2x2+x-10 =0a=2 b=1 c=-10x=-b±b2-4ac2ax=-1±12-(4×2 ×-10)2×2x=- 52 neu x=2By Completing the Square :2x2+x-10 =02(x2+12x-5 ) =02((x+14)2-116-5) =02((x+14)2-5116) =02(x+14)2-1018 =02(x+14)2 =1018 x+142 =5116 x+ 14 =±5116 x=- 52 neu x=2 Linc Cymraeg:English link:Linc Cymraeg:English link:Linc Cymraeg:English link:We can make further connections with graph work here e.g. if I plot: y=2x2+x-10 and y=0 The points of intersection will be x=- 52 or x=2, which are the solutions to the equation 2x2+x-10=0This is solving simultaneous equations graphically!Here’s a screenshot from Geogebra (which is freely available online – have a go!)The following worked examples, QR codes with link to explanations and exercises have been carefully selected to help you reinforce your understanding of key subject areas before you start AS Mathematics!BASIC ALGEBRA SKILLSHaving basic algebra skills is essential to start the A Level Mathematics course. Have a look over the notes and examples below before trying the ‘test yourself’ exercise. If you need further support and guidance use the video links.Expanding bracketsThis means multiplying the brackets out. Examples53435254387855305425419735501015057713494347576811279082595507275272510535248577513369648577512371053y-7 2) 3x(x2-2y) 3) (3x-1)(2x+3)2000256350We need to multiply everything in the bracket by 500We need to multiply everything in the bracket by 5 =3x3-6xy =6x2+9x-2x-3 =15y-35 left147320Video linksExpanding bracketsExpanding double brackets020000Video linksExpanding bracketsExpanding double brackets436245013335100965060960SimplifyingCollecting like termsExamples1171574411479001581150392430752475363855Simplify: 3x-2y+7x+5y = 10x+3yright33020Always look at what is in front of each term, if it helps circle the terms like this: 3x-2y+7x+5y so you can see you have 3x+7x=10x and-2y+5y=3y 00Always look at what is in front of each term, if it helps circle the terms like this: 3x-2y+7x+5y so you can see you have 3x+7x=10x and-2y+5y=3y 609600079375057143657048505342890800100509587571120058039064071500-3x+5y-2x-7y = -5x - 2y23x+2y-7x+y = 6x+4y-7x+y = -x+5y390525121920Expand the brackets first!00Expand the brackets first!35x-6y+3x+2y = 15x-18y+3x+6y = 18y-12x1028700217170left138430Video linksCollecting like terms020000Video linksCollecting like termsSolving linear equationsSolving a linear equation means working out the value of the unknown that’s represented by a letter. 409433254701137112624816125064773626803843693345582501552446601564639024621576414135463579+4x-9 ÷10020000+4x-9 ÷104612612465455+4x-9 ÷10020000+4x-9 ÷103773132330048-2x+1 ÷6020000-2x+1 ÷62162478329565-2x+1 ÷6020000-2x+1 ÷61227796243840 -1 ÷ 7020000 -1 ÷ 761405247782-1÷ 7020000-1÷ 7Examples1) 7x+1=36 2) 8x-1=2x+41 3) 32x+3= 25-4x 7x=35 6x-1=41 6x+9= 25-4x x=5 6x=42 10x+9= 25 x=7 10x=16 x=1.6 1248704376555 x 7 00 x 7 13602171603802265529137747434881225555551265822559245036023335261 x x ÷ 8 00 x x ÷ 8 3923295308307 x x ÷ 8 00 x x ÷ 8 2156346374536 x 7 00 x 7 4) x7=11 5) 56x=8 x=77 56=8x x=7 461010014732095250010922000045085Video linksSolving equationsEquations with letters on both sides020000Video linksSolving equationsEquations with letters on both sides-1905018605400BASIC ALGEBRA TEST YOURSELF EXERCISEExpand the following brackets:2a(4b-3a) b) 4y2(2y+3x) c) (x-3)(3x+4) d) (x-2y)2 Simplify:3x-2y+5x-7y b) 3x2+7y-5x2+yc) 8y-2(3y+5) d) 23a-4+3(4a+2) e) 5c-2d2-3c+7d2 f) 32x-5y-2(3x-4y) 3) Solve the following linear equations: a) 6x ? 2 ??2x ??8 b) 5x ? 7 ??2x ??25 c) x ? 11 ??3x ??3 d) 2(x ?3) ??5x ??6*Answers can be found at the back of this booklet*INDICESThe small number tells you how many times you multiply the number by itself. ie 23 means 2 x 2 x 2 = 8. It does NOT mean 2 x 3!Examples34 = 3 x 3 x 3 x 3 = 81 2) 23 x 52 = 2 x 2 x 2 x 5 x 5 3) 72 + 33 = 7 x 7 + 3 x 3 x 3 = 8 x 25 = 49 + 27 = 200 = 76Any number to the power of zero is equal to 1!Examples: 30 = 1 140 = 1 3430 = 1 x0 = 11695450202565Negative powers x-n=1xnExamplesright101602-3= 123 = 18 4-2= 142= 116 5-3= 153= 1125002-3= 123 = 18 4-2= 142= 116 5-3= 153= 1125 Fractional powers377190049529900539242050482400178117452387500228600523875Examples2512= 25=5 8112 = 81=9 813= 38=2 2713= 327=33686175205105A power of a third (13) means the same as working out the cube root of that number.00A power of a third (13) means the same as working out the cube root of that number.-76200186055A power of a half (12) means the same as working out the square root of that number.00A power of a half (12) means the same as working out the square root of that number.46672503543302486025363855276225349885Laws of indices xn×xm=xn+m xmxn=xm-n (xm)n=xmn Examples1) x3×x7=x3+7 2) y8÷y2=y8-2 3) 3x3y-5×2x4y3=(3×2)x3+4y-5+3 =x10 =y6 =6x7y-24) 24x6y56x3y-2=24÷6x6-3y5--2 5) (3x4)2=32x4×2 6) 823=(813)2 =4x3y7 =9x8 =22=4 5114925156845100012512827010477564770Video links Laws of indicesFractional indices020000Video links Laws of indicesFractional indicesCALCULATING WITH FRACTIONSSimplifying2447925634365÷9÷900÷9÷95553075640715÷7÷700÷7÷73914775634365÷15÷1500÷15÷151143000634365÷4÷400÷4÷4A fraction is in its simplest form if the numerator and denominator do not have a common factor other than 1. I.e. 46 is not in its simplest form as 2 is a common factor in the numerator and denominator. To simplify a fraction, divide the numerator and denominator by a common factor.Examples: 1216 = 34 2736 = 34 3045 = 23 6377 = 911Adding or Subtracting5667375139065x4x400x4x45372100151130x5x500x5x53409950174625x2x200x2x2857250165100x3x300x3x3To add or subtract fractions, the first thing you need to do is ensure the denominators are the sameExamples: 23-19=69-19 310+25=310+410 34-15=1520-420 =59 =710 =1120 MultiplyingMultiply the denominators together and the numerators together. Example: 711×34=7×311×4=2144 DividingUse the fact that dividing by ab will give you the same answer as multiplying by ba Examples: 710÷34=710×43 =2830 (=1415 if simplified) 45÷23=45×32=1210 (=65 if simplified)315277590170020002509017081915080645-2857533020Video linksFractions: AdditionFractions: MultiplicationFractions: Division00Video linksFractions: AdditionFractions: MultiplicationFractions: Division -7620034925000INDICES AND FRACTIONS TEST YOURSELF EXERCISE *You should attempt these questions without a calculator and show every step of your calculation* 1) Evaluate the following: a) 32×52 b) 53-72 c) 6412 d) 40012 e) 6413 f) 11-2 g) 9-3 h) 27-13 i) 3235 j) 125-23 2) Simplify: a) 3x2×4x5 b) 4x8y3×2x-3y4 c) 45a6b715a-2b d) 6(x-1)62(x-1)2 e) (3x4)2+6x62x-2 f) (4x-2y3)425612 3) Work out the following: a) 47+314 b) 4145-1115 c) 79+813 d) 913×612 e) 715÷811*Answers can be found at the back of this booklet*Rearranging formulaeFormulae are used in everyday life, from working out areas and volumes of shapes to converting units of measurement. Knowing how to use and rearrange formulae will be very useful in Pure and Applied Mathematicsleft6350Step-by-Step GuideStep 1: Firstly decide what needs to be on its own.Step 2: Secondly move all terms that contain that letter to one side. If the letter appears in more than one term then move all of these terms to one side. Step 3: Thirdly separate out the required letter on its own – this can be done by factorising. 020000Step-by-Step GuideStep 1: Firstly decide what needs to be on its own.Step 2: Secondly move all terms that contain that letter to one side. If the letter appears in more than one term then move all of these terms to one side. Step 3: Thirdly separate out the required letter on its own – this can be done by factorising. Examples:Rearrange the formulae to make a the subject: v = u + at v - u = at v - u = a t382270211455g = p + gf 0g = p + gf Rearrange the formulae to make g the subject:16764095885g - gf = p g - gf = p left209550g(1 – f) = p g(1 – f) = p 34163018415 g = p_ 1 - f g = p_ 1 - fRearrange the formulae to make n the subject:28575200660m(x? + n) = n(t? – s) m(x? + n) = n(t? – s) left146050mx? + mn = nt? - ns mx? + mn = nt? - ns 36004541275mx? = nt? - ns - mn mx? = nt? - ns - mn 37719013335mx? = n( t? – s - m) mx? = n( t? – s - m) 1047757620 mx?__ = n t? - s - m mx?__ = n t? - s - mleft114935Video links00Video links3457575222250Changing the subject (easier)Changing the subject (harder)Changing the subject (easier)Changing the subject (harder)15525751282703810032321500REARRANGING FORMULAE TEST YOURSELF EXERCISEMake a the subject of 14a + 6w = ac + 8wMake w the subject of the formula 4(g - w) = 5w – 3Make x the subject of y=x+3x-8Make v the subject of the formulae s= 12(u+v)t*Answers can be found at the back of this booklet*SIMULTANEOUS EQUATIONSConsider two linear equations in two variables (unknowns) such as 2x+3y=2 and 5x-2y=24. We have two equations and two unknowns. We call a pair of equations like these ‘simultaneous equations’ if the values of the unknowns are the same in both. That is, the values of the ‘x's would have to be the same in both equations and the values of the ‘y's would have to be the same. 264795090805Step 1: Get either the ‘y’s or the ‘x's the same. Here we’ve gone for getting the ‘y’s the same by multiplying the first equation with the coefficient of y from the second equation and multiplying the second equation with the coefficient of y from the first equation.Step 2: Either add or subtract both equations – which ever one will cancel the terms with ’y’ out. In this case if we subtract (6y- -6y = 12y) we would still have a term with y, whereas if we add (6y + -6y = 0) the terms with ‘y’ would cancel out. We therefore need to add in this example.Step 3: Solve the equation to find the value of x Step 4: Now choose any equation (we’ve gone for the first equation in this example as the numbers were smaller), sub the x value in and find the y value.020000Step 1: Get either the ‘y’s or the ‘x's the same. Here we’ve gone for getting the ‘y’s the same by multiplying the first equation with the coefficient of y from the second equation and multiplying the second equation with the coefficient of y from the first equation.Step 2: Either add or subtract both equations – which ever one will cancel the terms with ’y’ out. In this case if we subtract (6y- -6y = 12y) we would still have a term with y, whereas if we add (6y + -6y = 0) the terms with ‘y’ would cancel out. We therefore need to add in this example.Step 3: Solve the equation to find the value of x Step 4: Now choose any equation (we’ve gone for the first equation in this example as the numbers were smaller), sub the x value in and find the y value.1847850225425Example100012511493518478502197102x+3y=2 1 x 210572759969505x-2y=24 2 x 3 17430752178059525009779004x+6y=4 3114300012065001743075635015x-6y=72 417430752197101209675219710Adding equation 3 and 4 will give:19x=76 x=4 1390650225425Choosing equation 1 :2x+3y=2 As x=4, 2×4+3y=2 8+3y=2 3y= -6 y= -2 57150129540Video linksSimultaneous equations (elimination)020000Video linksSimultaneous equations (elimination)10337801333500right17526000SIMULTANEOUS EQUATIONS TEST YOURSELF EXERCISE Solve the following pairs of simultaneous equations using an algebraic (not graphical) method4x-3y=2 2) 3x+4y=7 6x-5y=1 2x-3y=16*Answers can be found at the back of this booklet*MAINLY QUADRATICS Factorising by extracting a common factorMeans putting one bracket in by extracting the highest common factor of all the terms in the expression.Examples1000125208915001) 21a + 14 = 7(3a + 2) 2) 12c - 18c2 = 6c(2 – 3c) 3) 6x2y + 12xy2 = 6xy(x + 2y)-19050146050You take the 7 out as it is a factor of 21 and 14, then ask yourself 7 times what is 21a and 7 times what is 1400You take the 7 out as it is a factor of 21 and 14, then ask yourself 7 times what is 21a and 7 times what is 141009650107950060325Video linksFactorising00Video linksFactorisingFactorising quadratic expressions221678535941000If the coefficient of x2 is 1 then there is a short cut. The examples below demonstrate the steps needed to factorise quadratic expressions.285751397000 2257425819150089535011049000062865Video linksFactorising quadratics (easier)Factorising quadratics (harder)00Video linksFactorising quadratics (easier)Factorising quadratics (harder)Difference between 2 squaresThis means when you have two terms which are square numbers and you’re subtracting one from the other. If this is the case then there’s an easy way to factorise such an expression.Examples4810125556260(8y-5)(8y+5) 020000(8y-5)(8y+5) 300990044196000160020047053500333375015621000495300015621064y2-25 0064y2-25 185737513716000914400111823400400050114681077152512623809 009 476251232535x2 00x2 4191001108709001238251127760-133350108585 102870025717576200216535Video linksFactorising (difference between 2 squares)00Video linksFactorising (difference between 2 squares)Solving quadratic equationsThere are three ways to solve quadratic equations algebraically, here we will focus on recapping two ways you should already be familiar with from the GCSE course. 30003752305050020002525527000Solving by factorising: 2) Solving by using the quadratic formula: 34480502362205524500207645You now have to learn this formula!020000You now have to learn this formula!5086350863604514850315595Solving by factorisingSolving with the formulaSolving graphicallySolving by factorisingSolving with the formulaSolving graphically323850028702000214439527749500981075248920left215265Video links00Video linksleft22479100MAINLY QUADRATICS TEST YOURSELF EXERCISE3190875108585e) 2x2-18x+28f) 3x2+7x-6 g) ??????y2-16h) ?????25x2-640e) 2x2-18x+28f) 3x2+7x-6 g) ??????y2-16h) ?????25x2-64 1) Factorise the following expressions: ??????a) 9x-6 ????? b) 12x2y+8xy2 ??????c) x2+7x+12 ??????d) x2-7x-18Solve the following quadratic equations by factorising:a)????x2+10x+24=0 b) 3x2+8x+5=0Solve the following equations, write your answers to 2 decimal places: a) 2x2+4x-3=0 b) 2x2-10x+7=0*Answers can be found at the back of this booklet*TRIGONOMETRYWe can use trigonometry to work out sides or angles in right angled triangles using the following ratios: Sinθ=opposide sidehypotenuse side Cosθ=adjacent sidehypotenuse side Tanθ=opposide sideadjacent sideWe can remember these ratios by learning ‘SOH CAH TOA’ with the following triangles: 8667756604000Examples of finding a missing side in a right angle triangle49530011366400364807544450003105150251460o020000o1657350105410h020000h-47625170180a 020000a 338137555880y 020000y 1438275194945x 020000x 47625139703.1m 26o0200003.1m 26o9620254000400285751733555495925192405005105400144780 50o020000 50o365760028067000495300170965-209550342265Label the relevant sides, that is, the one you know and the one you want to find out00Label the relevant sides, that is, the one you know and the one you want to find out496252510160a 020000a 45434251397012cm02000012cm x=3.1 ÷ cos(26) = 3.4m y=tan(50) x 12 = 14.3cm866775201295002495550210820-28575161290Video links00Video links388620062230IntroductionCalculating a sideIntroductionCalculating a side84772529654500Examples of finding a missing angle in a right angle triangle32385002063758cm 0200008cm 5000625120650h 020000h 447675014541512cm02000012cm364807640005004953009651900440055071120 yo020000 yo474345012827000365760021653500308610023495o020000o17430758255h 020000h left115570a 020000a 13906501498607cm 0200007cm 123825330205cm xo0200005cm xo16573501054100200004248150142875siny=oh siny=812 y=sin-1(812) y=41.8o 020000siny=oh siny=812 y=sin-1(812) y=41.8o 1619250207010cosx=ah cosx=57 x=cos-1(57) x=44.4o 020000cosx=ah cosx=57 x=cos-1(57) x=44.4o 73342514985900left46990495300170965-209550212725Label the relevant sides, that is, the two given sides 00Label the relevant sides, that is, the two given sides -19050889635Video linksCalculating an angle00Video linksCalculating an angle 895350232410We can also use trigonometry to find sides and angles of non-right angle triangles using the sine and cosine rules. These were given to you in a GCSE exam paper, for A Level Maths you will need to learn them as they will not be given!44742102349500The Sine Ruleleft38735In any triangle ABC: asinA=bsinB=ccosC Or this can be rearranged to be of the form: sinAa=sinBb=sinCc *The first format is most useful when finding a missing side and the second form is most useful when finding a missing angle*00In any triangle ABC: asinA=bsinB=ccosC Or this can be rearranged to be of the form: sinAa=sinBb=sinCc *The first format is most useful when finding a missing side and the second form is most useful when finding a missing angle*Finding a side with the Sine rule Finding an angle with the Sine rule25336501466852486025156210To use the SINE rule look out for a pair of opposites!020000To use the SINE rule look out for a pair of opposites!377190013335002000251333500The Cosine rule425767514414500-57150107950In any triangle ABC: a2=b2+c2-2bc?cosA Or this can be rearranged to be of the form: cosA=b2+c2-a22bc *The first format is most useful when finding a missing side and the second form is most useful when finding a missing angle*00In any triangle ABC: a2=b2+c2-2bc?cosA Or this can be rearranged to be of the form: cosA=b2+c2-a22bc *The first format is most useful when finding a missing side and the second form is most useful when finding a missing angle*Finding a side with the Cosine rule Finding an angle with the Cosine rule21717002025652181225164465It doesn’t matter which sides ‘b’ and ‘c’ are, as long as you label ‘a’ as the opposite side of the angle in question!020000It doesn’t matter which sides ‘b’ and ‘c’ are, as long as you label ‘a’ as the opposite side of the angle in question!left1206500410473712065004638675234951006191252254254895850352425Sine rule (ambiguous)Cosine rule (side)Cosine rule (angle)0Sine rule (ambiguous)Cosine rule (side)Cosine rule (angle)left275590Video links00Video links38195253435352276475343535885825334010-1905054038500412432534861500TRIGONOMETRY TEST YOURSELF EXERCISE*Note: None of the diagrams below are drawn to scale*Work out the length of the side BD in the following triangle:423100514160500Work out the length of the side AC in the following triangle:38271456223000Calculate the size of angle y in the following triangle:4324350283210b)020000b)53339924828400266700163195a)020000a)Calculate the size of angle x in the following diagrams:45720001333500 33210509398000331787583820d)020000d)28575036195c)020000c)5334001079500 *Answers can be found at the back of this booklet*left1725300EXTENSION TEST YOURSELF EXERCISEShowing all your working, simplify each of the following x-12x-34 ×x-12 b) y18 × y-24y16 c) (p13)4×(p4)13(3p)8 d) 28x17+ 7x277x17Evaluate the following (without a calculator!)1812 × 212(32)15 b) 2714 × 334 × 27 c) 913 × 27-123-16 × 3-23Expand and simplify the following x+4x-1x+1 b) x-12x+1x-3 c) x-5x+12 Solve the following equationsa) 12+7x-12x2=0 b) 40x2-17x-12=0 c) 3+x-63x=3x+12(x-3) Prove that 2x3-x-74+3x+15≡61x+117603800475825500A pyramid stands on a horizontal surface.The base of the pyramid is in the shape of a kite.The base of the pyramid is shown below.The apex (top vertex) of the pyramid is vertically above E.The vertical height of the pyramid is 17.3cm.The length of BD is 12.6cm and the angles are as shown on the diagram.Use the line EC to calculate the angle of elevation of the apex of the pyramid from the point C.01905000EXTENSION CHALLENGE!40957515875You’ve applied the quadratic formulae in order to solve equations at GCSE. Can you derive it? Your starting point is ax2+bx+c=0You need to rearrange until you get:x=-b±b2-4ac2a*HINT: you need to learn how to complete the square (see page 5 in this booklet)*00You’ve applied the quadratic formulae in order to solve equations at GCSE. Can you derive it? Your starting point is ax2+bx+c=0You need to rearrange until you get:x=-b±b2-4ac2a*HINT: you need to learn how to complete the square (see page 5 in this booklet)*TEST YOURSELF EXERCISE ANSWERSBASIC ALGEBRA a) 8ab-6a2 b) 8y3+12xy2 c) 3x2-5x-12 d) x2-4xy+4y2a) 8x-9y b) -2x2+8y c) 2y-10 d) 18a-2 e) 2c-3d2 f) -7y a) x=2.5 b) x=6 c) x=7 d) x=4 INDICES AND FRACTIONS a) 225 b) 76 c) 8 d) 20 e) 4 f) 1121 g) 1729 h) 13 i) 8 j) 125a) 12x7 b) 8x5y7 c) 3a8b6 d) 3(x-1)4 e) 9x8+3x8=12x8 f) 256x-2y1216=16x-2y12 a) 1114 b) 845 c) 163117 d) 54156=926 e) 77120REARRANGING FORMULAE SIMULTANEOUS EQUATIONS a=2w14-c 2) w=4g+39 3) x=3+8yy-1 4) v=2s-utt 1) x=3.5, y=4 2) x=5, y=-2MAINLY QUADRATICS a) 3x-2 b) 4xy3x+2y c) x+3x+4 d) (x-9)(x+25162550126365-(-10)±(-10)2-4(2)(7)2(2) 10±444x = 4.16 or x = 0.840-(-10)±(-10)2-4(2)(7)2(2) 10±444x = 4.16 or x = 0.843562351154940 -4±(4)2-4(2)(-3)2(2) -4±404 x = 0.58 or x = -2.580 -4±(4)2-4(2)(-3)2(2) -4±404 x = 0.58 or x = -2.581828800135890 3x+5x+1=0 3x+5=0?or?x+1=0x=-53????or?????x=-1 0 3x+5x+1=0 3x+5=0?or?x+1=0x=-53????or?????x=-1 -114300145415x+6x+4=0x+6=0?or?x+4=0x=-6?or?x=-40x+6x+4=0x+6=0?or?x+4=0x=-6?or?x=-4e) 2x2-9x+14=2x-2x-9 f) x+33x-2 g) y-4y+4 h) (5x-8)(5x+8)2) a) b) 3a) b) TRIGONOMETRY465772569853)0200003)220980088902)0200002)left88901)0200001)44005508890002181225889000left889000 48958502387600066675273685002305050508000 -16192516128900234315057154b)0200004b)left57154a)0200004a)23806151079500465772513589000441007595254d)0200004d)left88265000205740010839454c)0200004c)42456101130300y2=102+112-21011cos38 y2=47.6376…y=47.6376….? y=6.902…cm sinx6.902=sin359? sinx=sin359×6.902 sinx=0.439…. x=sin-1(0.439….) x=26.095… x=26.10o? (to 2 decimal places)00y2=102+112-21011cos38 y2=47.6376…y=47.6376….? y=6.902…cm sinx6.902=sin359? sinx=sin359×6.902 sinx=0.439…. x=sin-1(0.439….) x=26.095… x=26.10o? (to 2 decimal places)22098001282700sinx9=sin358 sinx=sin358×9 sinx=0.6452… x=sin-1(0.6452…) x=40.186183…? x=40.19o (to 2 decimal places)0sinx9=sin358 sinx=sin358×9 sinx=0.6452… x=sin-1(0.6452…) x=40.186183…? x=40.19o (to 2 decimal places)EXTENSION4649255767631d) 28x17+ 7x277x17=28x177x17 +7x277x17 =4x0+x17 =4+x17 0200001d) 28x17+ 7x277x17=28x177x17 +7x277x17 =4x0+x17 =4+x17 296676884101c) p134×p413(3p)8=p43×p43 ÷p83 =p83÷p83 =p0=1 0200001c) p134×p413(3p)8=p43×p43 ÷p83 =p83÷p83 =p0=1 161674681111b) y18 × y-24y16 =y18 × y-48y16 =y-38y16 =y-924y424 =y-1324=1y1324 001b) y18 × y-24y16 =y18 × y-48y16 =y-38y16 =y-924y424 =y-1324=1y1324 left4311a) x-12x-34 ×x-12=x-24x-34×(x-12)12 =x14×x-14 =x0=1 0200001a) x-12x-34 ×x-12=x-24x-34×(x-12)12 =x14×x-14 =x0=1 2a) 1812 × 212(32)15=18×2532=18×22=362=62=3 b) 2714 × 334 × 27 =2714×(33)14×2712=2714×2714×2712=271=27 c) 913 × 27-123-16 × 3-23=(32)13×(33)-123-16×3-46=323×3-323-56=346×3-963-56=3-563-56=13 a) x+4x-1x+1=x+4x2-1=x3+4x2-x-4 b) x-12x+1x-3=x-12x2-5x-3=2x3-7x2+2x+3162560280670-12x2+7x+12=0 -12x2-9x+16x+12=0 -3x4x+ 3+44x+3=0 4x+3-3x+4=0 x=-34 OR x=43020000-12x2+7x+12=0 -12x2-9x+16x+12=0 -3x4x+ 3+44x+3=0 4x+3-3x+4=0 x=-34 OR x=43 c) x-5x+12=x-5x2+2x+1=x3-3x2-9x-527863325943b) 40x2-17x-12=0 40x2+15x-32x-12=0 5x8x+3-48x+3=0 8x+35x-4=0 x=-38 OR x=45 00b) 40x2-17x-12=0 40x2+15x-32x-12=0 5x8x+3-48x+3=0 8x+35x-4=0 x=-38 OR x=45 4 c) 3+x-63x=3x+12(x-3) (×2(x-3)) 6x-3+2(x-3)(x-6)3x=3x+1 (×3x ) 18xx-3+2(x-3)(x-6)=3x(3x+1) 18x2-54x+2x2-18x+36=9x2+3x 18x2-54x+2x2-18x+36-9x2-3x=0 11x2-75x+36=0 a = 11 b = -75 c = 36x=-b±b2-4ac2a x=--75±(-75)2-4(11)(36)2(11) x=75+404122 or x=75-404122 x=6.30 or x=0.52 (to 2 dec places) 1155568279451500 5) 2x3-x-74+3x+15≡61x+11760 LHS: 2x3 -x-74 +3x+15 40x60-15x-760+12(3x+1)60 40x-15x-7+12(3x+1)60 40x-15x+105+36x+1260 61x+11760 =RHS 1050925-2849245x 20 x 15 x 12x 20 x 15 x 1200x 20 x 15 x 12x 20 x 15 x 12 6) Triangle ‘DEC’EC= 6.3tan14 EC=25.26791988cm Triangle ‘CEF’tanc=oppositeadjacent tanc=17.325.26791988 c=tan-1(17.325.26791988) c=34.3979…… =34.4o (to 1 dec place) Extension challenge!x=-b2a±-ca+b22a2x=-b2a±-ca+b24a2x=-b2a±-ca×4a4a+b24a2x=-b2a±-4ac4a2+b24a2x=-b2a±b2-4ac4a2x=-b2a±b2-4ac4a2x=-b2a±b2-4ac2a ................
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