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; $\Rightarrow \phantom{\rule{2.77695pt}{0ex}}$; $\iff \phantom{\rule{2.77695pt}{0ex}}$; $\equiv \phantom{\rule{2.77695pt}{0ex}}$. 
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: $\left(\right.\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\begin{array}{c}\hfill n\hfill \\ \hfill r\hfill \end{array}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\left)\right.=\frac{n!}{r!\phantom{\rule{0.3em}{0ex}}\left(nr\right)!}\phantom{\rule{0.3em}{0ex}}$. 
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 ${a}_{n+1}=\mathrm{f}\phantom{\rule{1.00006pt}{0ex}}\left({a}_{n},{a}_{n1}\right)$; 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 $\leftx\right<1\phantom{\rule{0.3em}{0ex}}$. 
Arithmetic and geometric series 
including sums to inﬁnity and conditions for convergence, where appropriate. 
Inequalities 
including solution of, eg, $\frac{1}{ax}>\frac{x}{xb}\phantom{\rule{0.3em}{0ex}}$; 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 $z=\mathrm{f}\phantom{\rule{1.00006pt}{0ex}}\left(\right.g\left(x\right)\left)\right.$. 
Increasing and decreasing functions 
the precise deﬁnition of these terms will be included in the question. 
Exponentials and logarithms 
including $x={a}^{y}\iff y={log}_{a}x\phantom{\rule{2.77695pt}{0ex}}$, $x={e}^{y}\iff y=lnx\phantom{\rule{0.3em}{0ex}}$; 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 

General curve sketching 
including use of symmetry, transformations, behaviour as $x\to \pm \infty $, points or regions where the function is undeﬁned, turning points, asymptotes parallel to the axes. 
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 $sin\left(A\pm B\right)\phantom{\rule{2.77695pt}{0ex}}$ and $cos\left(A\pm B\right)$ 
including their use in solving equations such as $acos\mathit{\theta}+bsin\mathit{\theta}=c.$ 
Inverse trigonometric functions 
deﬁnitions including domains and ranges; notation: either $arctan\mathit{\theta}\phantom{\rule{2.77695pt}{0ex}}$ or ${tan}^{1}\mathit{\theta}$, etc. 
Straight lines in twodimensions 
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 ${\left(xa\right)}^{2}+{\left(yb\right)}^{2}={R}^{2}\phantom{\rule{2.77695pt}{0ex}}$; including points of intersection of circles and lines. 
Cartesian and parametric equations of curves and conversion between the two forms. 

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 ${\mathrm{f}\phantom{\rule{1.00006pt}{0ex}}}^{\u2033}\left(x\right)$ 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 $\int {\mathrm{f}\phantom{\rule{1.00006pt}{0ex}}}^{\prime}\left(x\right)\u2215\mathrm{f}\phantom{\rule{1.00006pt}{0ex}}\left(x\right)\phantom{\rule{0.3em}{0ex}}\mathrm{d}x\phantom{\rule{0.3em}{0ex}}$; 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 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 nonuniform 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 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 $tan\alpha \phantom{\rule{0.3em}{0ex}}$; 
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. 
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: $X\sim \mathbf{N}\left(\mu ,{\sigma}^{2}\right)\phantom{\rule{0.3em}{0ex}}$. 
Basic concepts in the case of a simple null hypothesis and simple or compound alternative 
including knowledge of the terminology 