STEP Specification 2021
STEP MATHEMATICS SPECIFICATIONSfor June 2021 Examinations UCLES 2020
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General IntroductionFrom the June 2021 sitting onwards, STEP Mathematics 1 will no longer be offered.Both STEP Mathematics 2 and STEP Mathematics 3 will continue to be offered.The specification, nature and style of both STEP Mathematics 2 and STEP Mathematics 3remain unchanged for 2021.STEP Mathematics 2 is based on the STEP Mathematics 2 specification set out in this document.Candidates should be aware that the STEP Mathematics 2 specification assumes all the content ofMathematics 1 set out in this document.STEP Mathematics 3 is based on the STEP Mathematics 3 specification set out in this document.Candidates should be aware that the STEP Mathematics 3 specification assumes all the content ofboth the Mathematics 1 and the STEP Mathematics 2 specifications set out in this document.The specifications have been written to follow, in the ways set out below, the content of theDepartment for Education’s A Level Mathematics1 and the Pure content of AS and A Level FurtherMathematics2 specifications. However, some topics have been removed and some additional topicshave been included. In the cases of STEP Mathematics 2 and STEP Mathematics 3, additionalsections have been included outlining which Probability, Statistics and Mechanics topics might betested. Whilst most questions will be set on areas mentioned in the respective specification, questionsmay also be set on areas that are not explicitly mentioned; when this is the case, appropriateguidance will be given in the question.Mathematics 1This sets outassumed knowledgefor both STEP 2 andSTEP 3STEP Mathematics 2STEP 2 is based onthis specificationAssumed knowledgefor STEP 3PureMechanicsPure content of ALevelMathematics withsomemodifications3and additions.Mechanics content of ALevel Mathematics withsome modifications3 andadditions.The prescribedPure content ofAS FurtherMathematics withsomemodifications3and additions.Assumed: Pure contentof Mathematics ent of A LevelMathematics withsome modifications3and additions.Assumed: Purecontent ofMathematics 1.Additional topics asoutlined.Additional topics asoutlined.Assumed: Pure andMechanics content ofMathematics 1;Pure content of STEPMathematics 2.Assumed: Pure andProbability/Statisticscontent ofMathematics 1;Pure content of STEPMathematics 2.Assumed: Purecontent ofMathematics hematicsNotesAdditions toDfE contentare indicatedin thespecificationby bolditalics.Additions toDfE contentare indicatedin thespecificationby bolditalics.A few topics have been removed and, occasionally, wording from the DfE document has been modified forclarity.33
STEP Mathematics 3STEP 3 is based onthis specificationThe prescribedPure content of ALevel FurtherMathematics withsomemodifications3and additions.Assumed:Pure content ofMathematics 1and STEPMathematics 2.Additional topics asoutlined.Additional topics asoutlined.Assumed:Mechanics content ofMathematics 1 andSTEP Mathematics 2;Pure content ofMathematics 1, STEPMathematics 2 andSTEP Mathematics 3.Assumed:Probability/Statisticscontent ofMathematics 1 andSTEP Mathematics 2;Pure content ofMathematics 1, STEPMathematics 2 andSTEP Mathematics 3.Additions toDfE contentare indicatedin thespecificationby bolditalics.Format of the papersSTEP Mathematics 2 and STEP Mathematics 3 will each be a 3-hour paper divided into threesections.Each paper will comprise 12 questions:Section A (Pure Mathematics)eight questionsSection B (Mechanics)two questionsSection C (Probability/Statistics)two questionsEach question will have the same maximum mark of 20. In each paper, candidates will be assessedon the six questions best answered; no restriction will be placed on the number of questions thatmay be attempted from any section.The marking scheme for each question will be designed to reward candidates who make goodprogress towards a complete solution. In some questions a method will be specified; otherwise, anycorrect and appropriately justified solution will receive full marks whatever the method used.Candidates’ solutions must be clear, logical and legible and their working fully set out. Standardnotational conventions should be followed, and final answers should be simplified. Marks may be lostif examiners are unable to follow a candidate’s working, even if a correct final answer appears.4
SpecificationsThese specifications are for the guidance of both examiners and candidates. The following pointsshould be noted:1.Whilst most questions will be set on topics mentioned in the relevant specification, questionsmay also be set on areas that are not explicitly mentioned, or in ways that extend topics thatare mentioned; when such questions are set, candidates will be given appropriate guidance inthe question.2.Individual questions will often require knowledge of several different specification topics.3.Questions may test a candidate’s ability to apply mathematical knowledge from thespecifications in unfamiliar ways.4.Questions may be set that require knowledge of topics from the higher tier GCSEMathematics.45.Solutions will frequently require insight, ingenuity, persistence, and the ability to work throughsubstantial sequences of algebraic manipulation.6.Examiners will aim to set questions on a wide range of topics, but it is not guaranteed thatevery topic will be examined every year.7.The Pure sections of each specification assume knowledge of the full Pure content of allpreceding specifications.8.The Mechanics and Probability/Statistics sections of each specification assume knowledge ofthe appropriate Pure Mathematics for that specification, and of the full Pure content of allpreceding specifications. In addition, each Mechanics section assumes knowledge of theMechanics sections of preceding specifications, and similarly for Probability/Statisticssections.9.Bold italics are used to indicate additional topics that do not fall under the compulsorycontent set out in the relevant government document. For STEP Mathematics 2 and STEPMathematics 3 this includes all additional topics in the Mechanics and Probability/Statisticssections.Formulae booklets and calculatorsCandidates will not be issued with a formulae book. Formulae that candidates are expected to knoware listed in the appendix to this document. Other formulae will be given in individual questions,should they be required.The required formulae for STEP extend beyond those required for the corresponding A levels.Calculators are not permitted or required.Bilingual dictionaries may be ernment/uploads/system/uploads/attachment data/file/254441/GCSE mathematics subject content and assessment objectives.pdf5
MATHEMATICS 1Section A: Pure MathematicsContentProofUnderstand and use the structure of mathematical proof,proceeding from given assumptions through a series of logicalsteps to a conclusion; use methods of proof, including proof bydeduction, proof by exhaustion, proof by induction.Understand and use the terms ‘necessary and sufficient’and ‘if and only if’.Disproof by counter-example.Proof by contradiction (including proof of the irrationality of 2 and the infinity of primes, and application to unfamiliar proofs).BAlgebra and functionsKnow, understand and use the laws of indices for all rationalexponents.Use and manipulate surds, including rationalising thedenominator.Work with quadratic functions and their graphs; the discriminantof a quadratic function, including the conditions for real andrepeated roots; completing the square; solution of quadraticequations including solving quadratic equations in a function ofthe unknown.Solve simultaneous equations in two (or more) variables byelimination and by substitution; including, for example, one linearand one quadratic equation.Solve linear and quadratic inequalities in a single variable andinterpret such inequalities graphically, including inequalities withbrackets and fractions.Express solutions through correct use of ‘and’ and ‘or’, or throughset notation.Represent linear and quadratic inequalities such as 𝑦 𝑥 1 and𝑦 𝑎𝑥 2 𝑏𝑥 𝑐 graphically.Solve inequalities and interpret them graphically; including,but not limited to, those involving rational algebraicexpressions (e.g.,𝟏𝒂 𝒙 𝒙𝒙 𝒃), trigonometric functions,exponential functions, and the modulus function.6
Manipulate polynomials algebraically, including expandingbrackets and collecting like terms, factorisation, and simplealgebraic division; use of the factor theorem and theremainder theorem; use of equating coefficients inidentities.Know, understand and use the relationship between the rootsand coefficients of quadratic equations.Simplify rational expressions including by factorising andcancelling, and algebraic division (by linear and higher degreeexpressions).Understand and use graphs of functions; sketch curves definedby simple equations including polynomials, the modulus of linear𝑎𝑎and other functions, 𝑦 and 𝑦 2 and other rationalfunctions such as 𝒚 𝑥𝒙(𝒙 𝒂)𝟐𝑥(including their vertical andhorizontal asymptotes); behaviour as 𝒙 ; interpret thealgebraic solution of equations graphically; use intersectionpoints of graphs to solve equations.Understand and use proportional relationships and their graphs.Understand and use the definition of a function; domain andrange of functions; composite functions; inverse functions andtheir graphs.Understand the effect of simple transformations on the graph of𝑦 f(𝑥) including sketching associated graphs:𝑦 𝑎f(𝑥), 𝑦 f(𝑥) 𝑎, 𝑦 f(𝑥 𝑎), 𝑦 f(𝑎𝑥), andcombinations of these transformations.Decompose rational functions into partial fractions (denominatorsnot more complicated than squared linear terms and with nomore than three terms, numerators constant or linear).Understand what is meant by the limit of a function 𝐟(𝒙) as 𝒙tends to a specific value at which the function is undefined,including the notation 𝒙 , and be able to find such limitsin simple cases.Use functions in modelling, including consideration of limitationsand refinements of the models.7
Coordinate geometryin the (𝒙, 𝒚) planeKnow, understand and use the equation of a straight line,including the forms 𝑦 𝑦1 𝑚( 𝑥 𝑥1 ) and 𝑎𝑥 𝑏𝑦 𝑐 0;gradient conditions for two straight lines to be parallel orperpendicular.Be able to use straight line models in a variety of contexts.Know, understand and use the coordinate geometry of the circleincluding using the equation of a circle in the form( 𝑥 𝑎 )2 ( 𝑦 𝑏 )2 𝑟2 ; completing the square to find thecentre and radius of a circle; know, understand and use basiccircle theorems: The angle subtended by an arc at the centre is twice theangle it subtends at the circumference.The angle on the circumference subtended by a diameter isa right angle.Two angles subtended by a chord in the same segmentare equal.A radius or diameter bisects a chord if and only if it isperpendicular to the chord.For a point P on the circumference, the radius or diameterthrough P is perpendicular to the tangent at P.For a point P on the circumference, the angle betweenthe tangent and a chord through P equals the anglesubtended by the chord in the alternate segment.Opposite angles of a cyclic quadrilateral aresupplementary.Understand and use the parametric equations of curves andconversion between Cartesian and parametric forms.Use parametric equations in modelling in a variety of contexts.8
Sequences and seriesKnow, understand and use the binomial expansion of(𝑎 𝑏𝑥)𝑛 for positive integer 𝑛; the notations 𝑛! and 𝑛𝐶𝑟 (and𝒏( ) and nCr) and their algebraic definitions; link to binomial𝒓probabilities.Extend the binomial expansion of (𝑎 𝑏𝑥)𝑛 to any rational 𝑛,including its use for approximation; be aware that the expansion is𝑏𝑥valid (converges) for 1 (proof not required).𝑎nUse 𝒏! and Cr in the context of permutations andcombinations.Work with sequences including those given by a formula for the 𝑛thterm and those generated by a simple relations of theform 𝑥𝑛 1 f(𝑥𝑛 ), or 𝒙𝒏 𝟏 𝐟(𝒙𝒏, 𝒙𝒏 𝟏 ); increasing sequences;decreasing sequences; periodic sequences.Understand and use sigma notation for sums of series.Understand and work with arithmetic sequences and series,including knowledge of the formulae for 𝑛th term and the sum to𝑛 terms.Understand and work with geometric sequences and seriesincluding knowledge of the formulae for the 𝑛th term and thesum of a finite geometric series; the sum to infinity of aconvergent geometric series, including the use of 𝑟 1 .Understand what is meant by the limit of a sequence,including the notation 𝒙𝒏 𝒂 as 𝒏 , and be able to findsuch a limit in simple cases.Use sequences and series in modelling.9
TrigonometryKnow, understand and use the definitions of sine, cosine, andtangent for all arguments; the sine and cosine rules; the area of1a triangle in the form 𝑎𝑏 sin 𝐶.2Work with radian measure, including use for arc length and area ofsector.Know, understand and use the standard small angleapproximations of sin 𝜃, cos 𝜃, and tan 𝜃 :sin 𝜃 𝜃, cos 𝜃 1 𝜃2,2tan 𝜃 𝜃 where 𝜃 is in radians.Understand and use the sine, cosine, and tangent functions; theirgraphs, symmetries, and periodicity.Know and use exact values of sin 𝜃 and cos 𝜃 for𝜃 0, , , , , and integer multiples.6 4 3 2Know and use exact values of tan 𝜃 for 𝜃 0, , , , and (appropriate) integer multiples.6 4 3Know, understand and use the definitions of sec, cosec, and cot-1-1and of sin-1, cos , and tan ; their relationships to sin, cos, andtan; understand their graphs, their ranges and domains.Know, understand and use tan 𝜃 sin 𝜃cos 𝜃.Know, understand and use sin2 𝜃 cos 2 𝜃 1,sec 2 𝜃 1 tan2 𝜃, and cosec 2 𝜃 1 cot 2 𝜃.Know, understand and use double angle formulae; use offormulae for sin(𝐴 𝐵), cos(𝐴 𝐵), and tan(𝐴 𝐵); understandgeometrical proofs of these formulae.Understand and use expressions for 𝑎 cos 𝜃 𝑏 sin 𝜃 in theequivalent forms of 𝑟 cos(𝜃 𝛼) or 𝑟 sin(𝜃 𝛼).Find general solutions to trigonometric equations, includingquadratic equations in sin, cos, or tan and equationsinvolving linear multiples of the unknown angle; for𝝅𝟏𝟓𝟐example, 𝐬𝐢𝐧(𝟑𝒙 ) .Construct proofs involving trigonometric functions and identities.Use trigonometric functions to solve problems in context,including problems involving vectors, kinematics and forces.10
Exponentialsand logarithmsKnow and use the function 𝑎 𝑥 and its graph, where 𝑎 is positive.Know and use the function e𝑥 and its graph.Know that the gradient of e𝑘𝑥 is equal to 𝑘e𝑘𝑥 , and henceunderstand why the exponential model is suitable in manyapplications.Know and use the definition of log 𝑎 𝑥 as the inverse of 𝑎 𝑥 , where 𝑎is positive (𝑎 1) and 𝑥 0.Know and use the function ln 𝑥 and its graph.Know and use ln 𝑥 as the inverse function of e𝑥 .Know, understand and use the laws of logarithms:log 𝑎 𝑥 log 𝑎 𝑦 log 𝑎 𝑥𝑦 ;𝑥log 𝑎 𝑥 log 𝑎 𝑦 log 𝑎 ;𝑦𝑘 log 𝑎 𝑥 log 𝑎 𝑥 𝑘1(including, for example, 𝑘 1 and 𝑘 ).2Understand and use the change of base formula forlogarithms:𝐥𝐨𝐠 𝒂 𝒙 𝐥𝐨𝐠 𝒃 𝒙𝐥𝐨𝐠 𝒃 𝒂Solve equations of the form 𝑎 𝑥 𝑏.Use logarithmic graphs to estimate parameters in relationships ofthe form 𝑦 𝑎𝑥 𝑛 and 𝑦 𝑘𝑏 𝑥 , given data for 𝑥 and 𝑦.Understand and use exponential growth and decay; use inmodelling (examples may include the use of e in continuouscompound interest, radioactive decay, drug concentration decay,or exponential growth as a model for population growth);consideration of limitations and refinements of exponentialmodels.11
DifferentiationCandidates should have an informal understanding ofcontinuity and differentiability.Understand and use the derivative of f(𝑥) as the gradient of thetangent to the graph of 𝑦 f(𝑥) at a general point (𝑥, 𝑦); thegradient of the tangent as a limit; interpretation as a rate ofchange; sketching the gradient function for a given curve; secondand higher derivatives; differentiation from first principles forsmall positive integer powers of 𝑥 , and for sin 𝑥 and cos 𝑥.Understand and use the second derivative as the rate of change ofgradient; connection to convex and concave sections of curvesand points of inflection.Differentiate 𝑥 𝑛 , for rational values of 𝑛, and related constantmultiples, sums and differences.Differentiate e𝑘𝑥 , 𝑎𝑘𝑥 , sin 𝑘𝑥, cos 𝑘𝑥, tan 𝑘𝑥 and othertrigonometric functions and related sums, differences andconstant multiples.Know, understand and use the derivative of ln 𝑥.Apply differentiation to find gradients, tangents and normals,maxima and minima and stationary points, points of inflection.Identify where functions are increasing or decreasing.Differentiate using the product rule, the quotient rule, and thechain rule, including problems involving connected rates ofchange and inverse functions.Differentiate simple functions and relations defined implicitly orparametrically, for first and higher derivatives.Apply the above to curve sketching.Construct simple differential equations in pure mathematics and incontext (contexts may include kinematics, population growth, andmodelling the relationship between price and demand).12
IntegrationCandidates should have an informal understanding ofintegrability.Know and use the Fundamental Theorem of Calculus, includingapplications to integration by inspection.Integrate 𝑥 𝑛 (including 𝒏 𝟏), and related sums, differencesand constant multiples.Integrate e𝑘𝑥 , sin 𝑘𝑥, and cos 𝑘𝑥, and related sums, differences,and constant multiples.Evaluate definite integrals; use a definite integral to find the areaunder a curve and the area between two curves.Understand and use integration as the limit of a sum.Carry out simple and more complex cases of integration bysubstitution and integration by parts; understand these methodsas the inverse processes of the chain and product rulesrespectively.(Integration by substitution includes finding a suitable substitutionand is not limited to cases where one substitution will lead to afunction which can be integrated; integration by parts includesmore than one application of the method but excludes reductionformulae.)Integrate using partial fractions that are linear and repeatedlinear in the denominator.Evaluate the analytical solution of simple first order differentialequations with separable variables, including finding particularsolutions.(Separation of variables may require factorisation involving acommon factor.)Interpret the solution of a differential equation in the context ofsolving a problem, including identifying limitations of the solution;includes links to kinematics.13
Numerical methodsLocate roots of f(𝑥 ) 0 by considering changes of sign of f(𝑥 ) inan interval of 𝑥 on which f(𝑥 ) is sufficiently well-behaved.Understand how change of sign methods can fail.Solve equations approximately using simple iterative methods; beable to draw associated cobweb and staircase diagrams.Solve equations using the Newton-Raphson method and otherrecurrence relations of the form 𝑥𝑛 1 g(𝑥𝑛 ). Understand how such methods can fail.Understand and use numerical integration of functions; includingthe use of the trapezium rule, and estimating the approximate areaunder a curve and limits that it must lie between.Use numerical methods to solve problems in context.VectorsUse vectors in two dimensions and in three dimensions.Calculate the magnitude and direction of a vector and conve
Pure content of Mathematics 1, STEP Mathematics 2 and STEP Mathematics 3. Additions to DfE content are indicated in the specification by . bold italics. Format of the papers . STEP Mathematics 2 and STEP Mathematics 3 will each be a 3-hour paper divided into three sections. Each paper will comprise 12 qu
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