STEP MATHEMATICS SPECIFICATIONS for June 2019 …

1 MATHEMATICS I, II and III (9465, 9470, and 9475)General IntroductionThere are two syllabuses, one for MATHEMATICS I and MATHEMATICS II, the other for MATHEMATICS III. The syllabusfor MATHEMATICS I and MATHEMATICS II is based on a single subject MATHEMATICS Advanced GCE. Questions onMathematics II are intended to be more challenging than questions on MATHEMATICS I. The syllabus for MathematicsIII is designing the syllabuses, the SPECIFICATIONS for all the UK Advanced GCE examinations, the Scottish AdvancedHigher and the International Baccalauriat were of MATHEMATICS I, II and III will be a 3-hour paper divided into three sections as follows:Section A ( pure MATHEMATICS )eight questionsSection B (Mechanics)three questionsSection C (Probability and Statistics)two questionsAll questions will carry the same weight.

2 Candidates will be assessed on the six questions best answered; norestriction will be placed on the number of questions that may be attempted from any section. Normally, a candidatewho answers at least four questions well will be awarded a grade marking scheme for each question will be designed to reward candidates who make good progress towards acomplete syllabuses given below are for the guidance of both examiners and candidates. The following points should Questions may test candidates ability to apply mathematical knowledge in novel and unfamiliar Solutions will frequently require insight, ingenuity, persistence and the ability to work through substantialsequences of algebraic Some questions may be largely independent of particular syllabus topics, being designed to test generalmathematical Questions will often require knowledge of several different syllabus Examiners will aim to set questions on a wide range of topics, but it is not guaranteed that every topic will beexamined every Questions may be set that require knowledge of elementary topics, such as basic geometry.

3 Mensuration (areasof triangles, volume of sphere, etc), sine and cosine formulae for triangles, rational and irrational booklets and calculatorsCandidates should use the STEP formula booklet which will be posted out with the question papers and can bedownloaded from the website ( ).Calculators are not permitted (or required).1 MATHEMATICS I (9465) and MATHEMATICS II (9470)Section A: pure MathematicsThis section comprises the Advanced GCE common core (typically, modules C1 C4 of an Advanced GCEspecification) broadly interpreted, together with a few additional items enclosed in asterisks .SpecificationNotesGeneralMathematical vocabulary and notationincluding: equivalent to; necessary and sufficient; if andonly if.

4 Methods of proofincluding proof by contradiction and disproof by counterex-ample; including, in simple cases, proof by induction .AlgebraIndices and surdsincluding rationalising proving positivity by completing a expansion for(a+b)nincluding knowledge of the general term;notation:(nr)=n!r! (n r)!.Algebraic operations on polynomials and factorisation, the factor theorem, the remaindertheorem;including understanding that, for example, ifx3+bx2+cx+d (x )(x )(x ),thend= .Partial fractionsincluding denominators with a repeated or quadratic and seriesincluding use of, for example,an+1= f(an)oran+1=f(an, an 1);including understanding of the terms convergent, divergentand periodic in simple cases;including use of nk=1kto obtain related binomial series for(1 +x)k, wherekis a ra-tional understanding of the condition|x|< and geometric seriesincluding sums to infinity and conditions for convergence,where appropriate.

5 Inequalities including solution of, eg,1a x>xx b;including simple inequalities involving the modulus func-tion;including the solution of simultaneous inequalities bygraphical , range, composition, inverseincluding use of functional notation such asy= f(ax+b),x= f 1(y)andz= f(g(x)).Increasing and decreasing functionsboth the common usages of the term increasing ( >y f(x)>f(y)orx > y f(x)>f(y)) will and logarithmsincludingx=ay y= logax,x= ey y= lnx; including the exponential seriesex= 1 +x+ +xn/n! + . The effect of simple transformationssuch asy=af(bx+c) + modulus of roots off(x) = 0by consideringchanges of sign off(x).Approximate solution of equations using simple it-erative sketchingGeneral curve sketchingincluding use of symmetry, tranformations, behaviour asx , points or regions where the function is unde-fined, turning points, asymptotes parallel to the measure, arc length of a circle, area of functionsincluding knowledge of standard values, such astan( /4),sin 30 ;including identities such assec2 tan2 = 1;including application to geometric problems in two andthree angle formulaeincluding their use in calculating, eg,tan( /8).

6 Formulae forsin(A B)andcos(A B)including their use in solving equations such asacos +bsin = trigonometric functionsdefinitions including domains and ranges;notation:arctan , etcCoordinate geometryStraight lines in two-dimensionsincluding the equation of a line through two given points, orthrough a given point and parallel to a given line or througha given point and perpendicular to a given line;including finding a point which divides a segment in a the general form(x a)2+ (y b)2=R2;including points of intersection of circles and and parametric equations of curves andconversion between the two of a derivative as a limit and as a rateof changeincluding knowledge of both notationsf (x) of standard functionsincluding algebraic expressions, trigonometric (but not in-verse trigonometric) functions, exponential and log of composite functions, productsand quotients and functions defined derivativesincluding knowledge of both notationsf (x)andd2ydx2.

7 Including knowledge of the of differentiation to gradients, tan-gents and normals, stationary points, increasingand decreasing functionsincluding finding maxima and minima which are not sta-tionary points;including classification of stationary points using the sec-ond as reverse of differentiationIntegral as area under a curveincluding area between two graphs;including approximation of integral by the rectangle andtrapezium within a surface of revolutionKnowledge and use of standard integralsincludingtheforms f (g(x))g (x)dxand f (x)/f(x)dx;including transformation of an integrand into standard (orsome given) form;including use of partial fractions;not including knowledge of integrals involving inversetrigonometric integrals including calculation, without justification, of simple im-proper integrals such as 0e xdxand 10x 12dx(if re-quired, information such as the behaviour ofxe xasx or ofxlnxasx 0will be given).

8 Integration by parts and by substitutionincluding understanding their relationship with differentia-tion of product and of a composite function;including application to and solution of differential equationsformulation of first order equations;solution in the case of a separable equation or by some othermethod given in the in two and three dimensionsincluding use of column vector andi,j, of a vectorincluding the idea of a unit addition and multiplication by scalarsincluding geometrical vectorsincluding application to geometrical distance between two equations of linesincluding the finding the intersection of two lines;understanding the notion of skew lines (knowledge of short-est distance between skew lines is not required).

9 The scalar productincluding its use for calculating the angle between two B: MechanicsThis section is roughly equivalent to two GCE modules M1 and M2; candidates who have only studied one mod-ule of mechanics are likely to lack both the knowledge and the experience of mechanics required to attempt thequestions. Questions may involve any of the material in the pure MATHEMATICS as a vectorincluding resultant of several forces acting at a point andthe triangle or polygon of forces;including equilibrium of a particle;forces include weight, reaction, tension and of massincluding obtaining the centre of mass of a system of par-ticles, of a simple uniform rigid body (possible composite)and, in simple cases, of non-uniform body by of a rigid body or several rigid bodiesin contactincluding use of moment of a force;for example, a ladder leaning against a wall or on a movablecylinder;including investigation of whether equilibrium is broken bysliding, toppling or rolling.

10 Including use of Newton s third law;excluding questions involving of a particle in a planeincluding the case when velocity or acceleration dependson time (but excluding use of acceleration=vdvdx) ;questions may involve the distance between two movingparticles, but detailed knowledge of relative velocity is (kinetic and potential), work and powerincluding application of the principle of conservation of of particlesincluding conservation of momentum, conservation of en-ergy (when appropriate);coefficient of restitution, Newton s experimental law;including simple cases of oblique impact (on a plane, forexample);including knowledge of the termsperfectly elastic(e= 1)andinelastic(e= 0);questions involving successive impacts may be s first and second laws of motionincluding motion of a particle in two and three dimensionsand motion of connected particles, such as trains, or parti-cles connected by means of of a projectile under gravityincluding manipulation of the equationy=xtan gx22V2cos2 ,viewed, possibly, as a quadratic intan ;not including projectiles on inclined C: Probability and StatisticsThe emphasis, in comparison with Advanced GCE and other comparable examinations, is towards probability andformal proofs, and away from data analysis and use of standard statistical tests.