Syllabus

The mathematical requirements for this book are based on the syllabus for STEP Mathematics I and II. The syllabus listed below serves as a rough guide. Some of the questions in the book require no knowledge of the syllabus; and some cover material that is not included in the syllabus, but is introduced in the question itself. You can ﬁnd more information about the STEP examinations from the web site http://www.stepmathematics.org.uk .

#### PURE MATHEMATICS

Speciﬁcation Notes
General
Mathematical vocabulary and notation including: equivalent to; necessary and sufficient; if and only if; $⇒\phantom{\rule{2.77695pt}{0ex}}$; $⇔\phantom{\rule{2.77695pt}{0ex}}$; $\equiv \phantom{\rule{2.77695pt}{0ex}}$.
Methods of proof including proof by contradiction and disproof by counterexample; proof by induction.
Algebra
Indices and surds including rationalising denominators.
Quadratics including proving positivity by completing a square.
The expansion for ${\left(a+b\right)}^{n}$ including knowledge of the general term;
notation: $\left(\right\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\begin{array}{c}\hfill n\hfill \\ \hfill r\hfill \end{array}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\left)\right=\frac{n!}{r!\phantom{\rule{0.3em}{0ex}}\left(n-r\right)!}\phantom{\rule{0.3em}{0ex}}$.
Algebraic operations on polynomials and rational functions including factorisation, the factor theorem, the remainder theorem;
including understanding that, for example, if ${x}^{3}+b{x}^{2}+cx+\mathrm{d}\equiv \left(x-\alpha \right)\left(x-\beta \right)\left(x-\gamma \right),$

then $\mathrm{d}=-\alpha \beta \gamma$.

Partial fractions including denominators with a repeated or quadratic factor.
Sequences and series including use of, for example, ${a}_{n+1}=\mathrm{f}\phantom{\rule{1.00006pt}{0ex}}\left({a}_{n}\right)$ or ${a}_{n+1}=\mathrm{f}\phantom{\rule{1.00006pt}{0ex}}\left({a}_{n},{a}_{n-1}\right)$;
including understanding of the terms convergent, divergent and periodic in simple cases;
including use of ${\sum }_{k=1}^{n}k$ to obtain related sums.

The binomial series for ${\left(1+x\right)}^{k}$, where $k$ is a rational number

including understanding of the condition $|x|<1\phantom{\rule{0.3em}{0ex}}$.

Arithmetic and geometric series

including sums to inﬁnity and conditions for convergence, where appropriate.

Inequalities

including solution of, eg, $\frac{1}{a-x}>\frac{x}{x-b}\phantom{\rule{0.3em}{0ex}}$;
including simple inequalities involving the modulus function;
including the solution of simultaneous inequalities by graphical means.
Functions

Domain, range, composition, inverse

including use of functional notation such as
$y=\mathrm{f}\phantom{\rule{1.00006pt}{0ex}}\left(ax+b\right)\phantom{\rule{2.77695pt}{0ex}}$, $x={\mathrm{f}\phantom{\rule{1.00006pt}{0ex}}}^{-1}\left(y\right)$ and $z=\mathrm{f}\phantom{\rule{1.00006pt}{0ex}}\left(\rightg\left(x\right)\left)\right$.

Increasing and decreasing functions

the precise deﬁnition of these terms will be included in the question.

Exponentials and logarithms

including $x={a}^{y}⇔y={log}_{a}x\phantom{\rule{2.77695pt}{0ex}}$,  $x={e}^{y}⇔y=lnx\phantom{\rule{0.3em}{0ex}}$;
including the exponential series

The effect of simple transformations

such as $y=a\mathrm{f}\phantom{\rule{1.00006pt}{0ex}}\left(bx+c\right)+\mathrm{d}$.

The modulus function.

Location of roots of $f\phantom{\rule{1.00006pt}{0ex}}\left(x\right)=0$ by considering changes of sign of $f\phantom{\rule{1.00006pt}{0ex}}\left(x\right)\phantom{\rule{0.3em}{0ex}}$

Approximate solution of equations using simple iterative methods

Curve sketching

General curve sketching

including use of symmetry, transformations, behaviour as $x\to ±\infty$, points or regions where the function is undeﬁned, turning points, asymptotes parallel to the axes.

Trigonometry

Radian measure, arc length of a circle, area of a segment

Trigonometric functions

including knowledge of standard values, such as $tan\left(\frac{1}{4}\pi \right)\phantom{\rule{0.3em}{0ex}}$, $sin3{0}^{\circ }\phantom{\rule{0.3em}{0ex}}$;
including identities such as ${sec}^{2}\varphi -{tan}^{2}\varphi =1\phantom{\rule{0.3em}{0ex}}$;
including application to geometric problems in two and three dimensions.

Double angle formulae

including their use in calculating, eg, $tan\left(\frac{1}{8}\pi \right)\phantom{\rule{0.3em}{0ex}}$.

Formulae for $sin\left(A±B\right)\phantom{\rule{2.77695pt}{0ex}}$ and $cos\left(A±B\right)$

including their use in solving equations such as
$acos𝜃+bsin𝜃=c.$

Inverse trigonometric functions

deﬁnitions including domains and ranges;
notation: either $arctan𝜃\phantom{\rule{2.77695pt}{0ex}}$ or ${tan}^{-1}𝜃$, etc.

Coordinate geometry

Straight lines in two-dimensions

including the equation of a line through two given points, or through a given point and parallel to a given line or through a given point and perpendicular to a given line;
including ﬁnding a point which divides a segment in a given ratio.

Circles

using the general form ${\left(x-a\right)}^{2}+{\left(y-b\right)}^{2}={R}^{2}\phantom{\rule{2.77695pt}{0ex}}$;
including points of intersection of circles and lines.

Cartesian and parametric equations of curves and conversion between the two forms.

Calculus

Interpretation of a derivative as a limit and as a rate of change

including knowledge of both notations ${\mathrm{f}\phantom{\rule{1.00006pt}{0ex}}}^{\prime }\left(x\right)$ and $\frac{\mathrm{d}y}{\mathrm{d}x}\phantom{\rule{0.3em}{0ex}}$.

differentiation of standard functions

including algebraic expressions, trigonometric and inverse trigonometric functions, exponential and log functions.

differentiation of composite functions, products and quotients and functions deﬁned implicitly.

Higher derivatives

including knowledge of both notations ${\mathrm{f}\phantom{\rule{1.00006pt}{0ex}}}^{″}\left(x\right)$ and $\frac{{d}^{2}y}{\mathrm{d}{x}^{2}}\phantom{\rule{2.77695pt}{0ex}}$;
including knowledge of the notation $\frac{{d}^{n}y}{\mathrm{d}{x}^{n}}\phantom{\rule{2.77695pt}{0ex}}$.

Applications of differentiation to gradients, tangents and normals, stationary points, increasing and decreasing functions

including ﬁnding maxima and minima which are not stationary points;
including classiﬁcation of stationary points using the second derivative.

Integration as reverse of differentiation

Integral as area under a curve

including area between two curves;
including approximation of integral by the rectangle and trapezium rules.

Volume within a surface of revolution

rotation about either $z$ or $y$ axes.

Knowledge and use of standard integrals

including the forms $\int {\mathrm{f}\phantom{\rule{1.00006pt}{0ex}}}^{\prime }\left(\mathrm{g}\left(x\right)\right){\mathrm{g}}^{\prime }\left(x\right)\phantom{\rule{0.3em}{0ex}}\mathrm{d}x\phantom{\rule{0.3em}{0ex}}$ and $\int {\mathrm{f}\phantom{\rule{1.00006pt}{0ex}}}^{\prime }\left(x\right)∕\mathrm{f}\phantom{\rule{1.00006pt}{0ex}}\left(x\right)\phantom{\rule{0.3em}{0ex}}\mathrm{d}x\phantom{\rule{0.3em}{0ex}}$;
including transformation of an integrand into standard (or some given) form;
including use of partial fractions;
not including knowledge of integrals involving inverse trigonometric functions.

Deﬁnite integrals

including calculation, without justiﬁcation, of simple improper integrals such as ${\int }_{0}^{\infty }{\mathrm{e}}^{-x}\phantom{\rule{0.3em}{0ex}}\mathrm{d}x$ and ${\int }_{0}^{1}{x}^{-\frac{1}{2}}\phantom{\rule{0.3em}{0ex}}\mathrm{d}x$ (if required, information such as the behaviour of $x{\mathrm{e}}^{-x}$ as $x\to \infty$ or of $xlnx$ as $x\to 0\phantom{\rule{2.77695pt}{0ex}}$ will be given).

Integration by parts and by substitution

including understanding their relationship with differentiation of product and of a composite function;
including application to (e.g.) $\int lnx\phantom{\rule{0.3em}{0ex}}\mathrm{d}x\phantom{\rule{0.3em}{0ex}}$.

Formulation and solution of differential equations

formulation of ﬁrst order equations;
solution in the case of a separable equation or by some other method given in the question.

Vectors

Vectors in two and three dimensions

including use of column vector and $\mathbf{i}$, $\mathbf{j}$, $\mathbf{k}$ notation.

Magnitude of a vector

including the idea of a unit vector.

Vector addition and multiplication by scalars

including geometrical interpretations.

Position vectors

including application to geometrical problems.

The distance between two points

Vector equations of lines

including the ﬁnding the intersection of two lines;
understanding the notion of skew lines (knowledge of shortest distance between skew lines is not required).

The scalar product

including its use for calculating the angle between two vectors.

#### MECHANICS

Questions on mechanics may involve any of the material in the Pure Mathematics syllabus.

Speciﬁcation Notes

Force as a vector

including resultant of several forces acting at a point and the triangle or polygon of forces;
including equilibrium of a particle;
forces include weight, reaction, tension and friction.

Centre of mass

including obtaining the centre of mass of a system of particles, of a simple uniform rigid body (possibly composite) and, in simple cases, of non-uniform body by integration.

Equilibrium of a rigid body or several rigid bodies in contact

including use of moment of a force;
for example, a ladder leaning against a wall or on a movable cylinder;
including investigation of whether equilibrium is broken by sliding, toppling or rolling;
including use of Newton’s third law;
excluding questions involving frameworks.

Kinematics of a particle in a plane

including the case when velocity or acceleration depends on time (but excluding knowledge of acceleration in the form $v\frac{\mathrm{d}v}{\mathrm{d}x}$);
questions may involve the distance between two moving particles, but detailed knowledge of relative velocity is not required.

Energy (kinetic and potential), work and power

including application of the principle of conservation of energy.

Collisions of particles

including conservation of momentum, conservation of energy (when appropriate);
coefficient of restitution, Newton’s experimental law;
including simple cases of oblique impact (on a plane, for example);
including knowledge of the terms perfectly elastic ($e=1$) and inelastic ($e=0$);
questions involving successive impacts may be set.

Newton’s ﬁrst and second laws of motion

including motion of a particle in two and three dimensions and motion of connected particles, such as trains, or particles connected by means of pulleys.

Motion of a projectile under gravity

including manipulation of the equation $y=xtan\alpha -\frac{g{x}^{2}}{2{V}^{2}{cos}^{2}\alpha },$

viewed, possibly, as a quadratic in $tan\alpha \phantom{\rule{0.3em}{0ex}}$;
not including projectiles on inclined planes.

#### PROBABILITY AND STATISTICS

The emphasis is towards probability and formal proofs, and away from data analysis and use of standard statistical tests. Questions may involve use of any of the material in the Pure Mathematics syllabus.

Speciﬁcation Notes
Probability

Permutations, combinations and arrangements

including sampling with and without replacement.

Exclusive and complementary events

including understanding of $\mathrm{P}\left(A\cup B\right)=\mathrm{P}\left(A\right)+\mathrm{P}\left(B\right)-\mathrm{P}\left(A\cap B\right)\phantom{\rule{0.3em}{0ex}}$, though not necessarily in this form.

Conditional probability

informal applications, such as tree diagrams.

Distributions

Discrete and continuous probability density functions and cumulative distribution functions

including calculation of mean, variance, median, mode and expectations by explicit summation or integration for a given (possibly unfamiliar) distribution (eg exponential or geometric or something similarly straightforward);
notation: $\mathrm{f}\phantom{\rule{1.00006pt}{0ex}}\left(x\right)={\mathrm{F}}^{\prime }\left(x\right)$.

Binomial distribution

including explicit calculation of mean.

Poisson distribution

including explicit calculation of mean;
including use as approximation to binomial distribution where appropriate.

Normal distribution

including conversion to the standard normal distribution by translation and scaling;
including use as approximation to the binomial or Poisson distributions where appropriate;
notation: $X\sim \mathbf{N}\left(\mu ,{\sigma }^{2}\right)\phantom{\rule{0.3em}{0ex}}$.

Hypothesis testing

Basic concepts in the case of a simple null hypothesis and simple or compound alternative

including knowledge of the terminology null hypothesis and alternative hypothesis, one and two tailed tests.