### Paper 1

__SL__

- Time: 45 minutes (30 marks)
- 30 multiple – choice questions (core)
- No marks deducted from incorrect answers
- NO CALCULATOR ALLOWED
- Data booklet provided
- 20% weight

__HL__

- Time: 60 minutes (40 marks)
- 40 multiple – choice questions (core & AHL)
- No marks deducted from incorrect answers
- NO CALCULATOR ALLOWED
- Data booklet provided
- 20% weight

**Topic 1 – Core: Algebra**

- Topic 1.1 :
- Arithmetic sequences and series; sum of finite arithmetic series; geometric sequences and series; sum of finite and infinite geometric series,
- Sigma notation,
- Applications.

- Topic 1.2 :
- Exponents and logarithms,
- Laws of exponents; laws of logarithms,
- Change of base

- Topic 1.3 :
- Counting principles, including permutations and combinations
- The binomial theorem: expansion of \({\left( {a + b} \right)^n}\), \(n \in N\)

- Topic 1.4 :
- Proof by mathematical induction.

- Topic 1.5 :
- Complex numbers: the number \({\text{i}} = \sqrt { – 1} \) ; the terms real part, imaginary part, conjugate, modulus and argument.
- Cartesian form \(z = a + {\text{i}}b\) .
- Sums, products and quotients of complex numbers.

- Topic 1.6 :
- Modulus–argument (polar) form \(z = r\left( {\cos \theta + {\text{i}}\sin \theta } \right) = r{\text{cis}}\theta = r{e^{{\text{i}}\theta }}\).
- The complex plane.
- Sums, products and quotients of complex numbers.

- Topic 1.7 :
- Powers of complex numbers: de Moivre’s theorem.
- \(n\)th roots of a complex number.

- Topic 1.8 :
- Conjugate roots of polynomial equations with real coefficients.

- Topic 1.9 :
- Solutions of systems of linear equations (a maximum of three equations in three unknowns), including cases where there is a unique solution, an infinity of solutions or no solution.

**Topic 2 – Core: Functions and equations**

- Topic 2.1
- Concept of function \(f:x \mapsto f\left( x \right)\) : domain, range; image (value)
- Odd and even functions.
- Composite functions \(f \circ g\)
- Identity function.
- One-to-one and many-to-one functions.
- Inverse function \({f^{ – 1}}\), including domain restriction. Self-inverse functions.

- Topic 2.2
- The graph of a function; its equation \(y = f\left( x \right)\) .
- Investigation of key features of graphs, such as maximum and minimum values, intercepts, horizontal and vertical asymptotes and symmetry, and consideration of domain and range.
- The graphs of the functions \(y = \left| {f\left( x \right)} \right|\) and \(y = f\left( {\left| x \right|} \right)\)
- The graph of \(\frac{1}{{f\left( x \right)}}\) given the graph of \(y = f(x)\) .

- Topic 2.3
- Transformations of graphs: translations; stretches; reflections in the axes.
- The graph of the inverse function as a reflection in \(y = x\).

- Topic 2.4
- The rational function \(x \mapsto \frac{{ax + b}}{{cx + d}}\) and its graph.
- The function \(x \mapsto {a^x}\) , \(a > 0\) , and its graph.
- The function \(x \mapsto {\log _a}x\) , \(x > 0\) , and its graph

- Topic 2.5
- Polynomial functions and their graphs.
- The factor and remainder theorems.
- The fundamental theorem of algebra.

- Topic 2.6
- Solving quadratic equations using the quadratic formula.
- Use of the discriminant \(\Delta = {b^2} – 4ac\) to determine the nature of the roots.
- Solving polynomial equations both graphically and algebraically.
- Sum and product of the roots of polynomial equations.
- Solution of \({a^x} = b\) using logarithms.
- Use of technology to solve a variety of equations, including those where there is no appropriate analytic approach.

- Topic 2.7
- Solutions of \(g\left( x \right) \geqslant f\left( x \right)\) .
- Graphical or algebraic methods, for simple polynomials up to degree 3.
- Use of technology for these and other functions.

**Topic 3 – Core: Circular functions and trigonometry**

- Topic 3.1 :
- The circle: radian measure of angles.
- Length of an arc; area of a sector.

- Topic 3.2 :
- Definition of \(\cos \theta \) , \(\sin \theta \) and \(\tan \theta \) in terms of the unit circle.
- Exact values of \(\sin\), \(\cos\) and \(\tan\) of \(0\), \(\frac{\pi }{6}\), \(\frac{\pi }{4}\), \(\frac{\pi }{3}\), \(\frac{\pi }{2}\) and their multiples.
- Definition of the reciprocal trigonometric ratios \(\sec \theta \) , \(\csc \theta \) and \(\cot \theta \) .
- Pythagorean identities: \({\cos ^2}\theta + {\sin ^2}\theta = 1\) ; \(1 + {\tan ^2}\theta = {\sec ^2}\theta \) ; \(1 + {\cot ^2}\theta = {\csc ^2}\theta \) .

- Topic 3.3 :
- Compound angle identities.
- Double angle identities

- Topic 3.4 :
- Composite functions of the form \(f(x) = a\sin (b(x + c)) + d\) .
- Applications.

- Topic 3.5 :
- The inverse functions \(x \mapsto \arcsin x\) , \(x \mapsto \arccos x\) , \(x \mapsto \arctan x\) ; their domains and ranges; their graphs.

- Topic 3.6 :
- Algebraic and graphical methods of solving trigonometric equations in a finite interval, including the use of trigonometric identities and factorization.

- Topic 3.7 :
- The cosine rule.
- The sine rule including the ambiguous case.
- Area of a triangle as \(\frac{1}{2}ab\sin C\) .
- Applications.

**Topic 4 – Core: Vectors**

- Topic 4.1
- Concept of a vector.
- Representation of vectors using directed line segments.
- Unit vectors; base vectors \(i\), \(j\), \(k\).
- Components of a vector: \(v = \left( {\begin{array}{*{20}{c}} {{v_1}} \\ {{v_2}} \\ {{v_3}} \end{array}} \right) = {v_1}i + {v_2}j + {v_3}k\) .
- Algebraic and geometric approaches to the sum and difference of two vectors.
- Algebraic and geometric approaches to the zero vector \(0\), the vector \( – v\) .
- Algebraic and geometric approaches to multiplication by a scalar, \(kv\) .
- Algebraic and geometric approaches to magnitude of a vector, \(\left| v \right|\) .
- Algebraic and geometric approaches to position vectors \(\overrightarrow {OA} = a\) .
- \(\overrightarrow {AB} = b – a\) .

- Topic 4.2
- The definition of the scalar product of two vectors.
- Properties of the scalar product: \({\boldsymbol{v}} \cdot {\boldsymbol{w}} = {\boldsymbol{w}} \cdot {\boldsymbol{v}}\) ; \({\boldsymbol{u}} \cdot \left( {{\mathbf{v}} + {\boldsymbol{w}}} \right) = {\boldsymbol{u}} \cdot {\boldsymbol{v}} + {\boldsymbol{u}} \cdot {\boldsymbol{w}}\) ; \(\left( {k{\boldsymbol{v}}} \right) \cdot {\boldsymbol{w}} = k\left( {{\boldsymbol{v}} \cdot {\boldsymbol{w}}} \right)\) ; \({\boldsymbol{v}} \cdot {\boldsymbol{v}} = {\left| {\boldsymbol{v}} \right|^2}\) .
- The angle between two vectors.
- Perpendicular vectors; parallel vectors.

- Topic 4.3
- Vector equation of a line in two and three dimensions: \(r = a + \lambda b\) .
- Simple applications to kinematics.
- The angle between two lines.

- Topic 4.4
- Coincident, parallel, intersecting and skew lines; distinguishing between these cases.
- Points of intersection.

- Topic 4.5
- The definition of the vector product of two vectors.
- Properties of the vector product: \({\text{v}} \times {\text{w}} = – {\text{w}} \times {\text{v}}\) ; \({\text{u}} \times ({\text{v}} + {\text{w}}) = {\text{u}} \times {\text{v}} + {\text{u}} \times {\text{w}}\) ; \((k{\text{v}}) \times {\text{w}} = k({\text{v}} + {\text{w}})\) ; \({\text{v}} \times {\text{v}} = 0\) .
- Geometric interpretation of \({\text{v}} \times {\text{w}}\) .

- Topic 4.6
- Vector equation of a plane \(r = a + \lambda b + \mu c\) .
- Use of normal vector to obtain the form \(r \cdot n = a \cdot n\) .
- Cartesian equation of a plane \(ax + by + cz = d\) .

- Topic 4.7
- Intersections of: a line with a plane; two planes; three planes.
- Angle between: a line and a plane; two planes.

**Topic 5 – Core: Statistics and probability**

- Topic 5.1 :
- Concepts of population, sample, random sample and frequency distribution of discrete and continuous data.
- Grouped data: mid-interval values, interval width, upper and lower interval boundaries.
- Mean, variance, standard deviation.

- Topic 5.2 :
- Concepts of trial, outcome, equally likely outcomes, sample space (U) and event.
- The probability of an event \(A\) is \(P\left( A \right) = \frac{{n\left( A \right)}}{{n\left( U \right)}}\)
- The complementary events \(A\) and \({A’}\) (not \(A\)).
- Use of Venn diagrams, tree diagrams, counting principles and tables of outcomes to solve problems.

- Topic 5.3 :
- Combined events; the formula for \(P\left( {A \cup B} \right)\) .
- Mutually exclusive events.

- Topic 5.4 :
- Conditional probability; the definition \(P\left( {\left. A \right|P} \right) = \frac{{P\left( {A\mathop \cap \nolimits B} \right)}}{{P\left( B \right)}}\) .
- Independent events; the definition \(P\left( {\left. A \right|B} \right) = P\left( A \right) = P\left( {\left. A \right|B’} \right)\) .
- Use of Bayes’ theorem for a maximum of three events.

- Topic 5.5 :
- Concept of discrete and continuous random variables and their probability distributions.
- Definition and use of probability density functions.
- Expected value (mean), mode, median, variance and standard deviation.
- Applications.

- Topic 5.6 :
- Binomial distribution, its mean and variance.
- Poisson distribution, its mean and variance.

- Topic 5.7 :
- Normal distribution.
- Properties of the normal distribution.
- Standardization of normal variables.

**Topic 6 – Core: Calculus**

- Topic 6.1
- Informal ideas of limit, continuity and convergence.
- Definition of derivative from first principles as \(f’\left( x \right) = \mathop {\lim }\limits_{h \to 0} {\frac{{f\left( {x + h} \right) – f\left( x \right)}}{h}} \).
- The derivative interpreted as a gradient function and as a rate of change.
- Finding equations of tangents and normals.
- Identifying increasing and decreasing functions.
- The second derivative.
- Higher derivatives.

- Topic 6.2
- Derivatives of \({x^n}\) , \(\sin x\) , \(\cos x\) , \(\tan x\) , \({{\text{e}}^x}\) and \\(\ln x\) .
- Differentiation of sums and multiples of functions.
- The product and quotient rules.
- The chain rule for composite functions.
- Related rates of change.
- Implicit differentiation.
- Derivatives of \(\sec x\) , \(\csc x\) , \(\cot x\) , \({a^x}\) , \({\log _a}x\) , \(\arcsin x\) , \(\arccos x\) and \(\arctan x\) .

- Topic 6.3
- Local maximum and minimum values.
- Optimization problems.
- Points of inflexion with zero and non-zero gradients.
- Graphical behaviour of functions, including the relationship between the graphs of \(f\) , \({f’}\) and \({f”}\) .

- Topic 6.4
- Indefinite integration as anti-differentiation.
- Indefinite integral of \({x^n}\) , \(\sin x\) , \(\cos x\) and \({{\text{e}}^x}\) .
- Other indefinite integrals using the results from 6.2.
- The composites of any of these with a linear function.

- Topic 6.5
- Anti-differentiation with a boundary condition to determine the constant of integration.
- Definite integrals.
- Area of the region enclosed by a curve and the \(x\)-axis or \(y\)-axis in a given interval; areas of regions enclosed by curves.
- Volumes of revolution about the \(x\)-axis or \(y\)-axis.

- Topic 6.6
- Kinematic problems involving displacement \(s\), velocity \(v\) and acceleration \(a\).
- Total distance travelled.

- Topic 6.7
- Integration by substitution.
- Integration by parts.

**Topic 7 – Option: Statistics and probability**

- Topic 7.1
- Cumulative distribution functions for both discrete and continuous distributions.
- Geometric distribution.
- Negative binomial distribution.
- Probability generating functions for discrete random variables.
- Using probability generating functions to find mean, variance and the distribution of the sum of \(n\) independent random variables.

- Topic 7.2
- Linear transformation of a single random variable.
- Mean of linear combinations of \(n\) random variables.
- Variance of linear combinations of \(n\) independent random variables.
- Expectation of the product of independent random variables.

- Topic 7.3
- Unbiased estimators and estimates.
- Comparison of unbiased estimators based on variances.
- \({\bar X}\) as an unbiased estimator for \(\mu \) .
- \({S^2}\) as an unbiased estimator for \({\sigma ^2}\) .

- Topic 7.4
- A linear combination of independent normal random variables is normally distributed. In particular, \(X{\text{ ~ }}N\left( {\mu ,{\sigma ^2}} \right) \Rightarrow \bar X{\text{ ~ }}N\left( {\mu ,\frac{{{\sigma ^2}}}{n}} \right)\) .
- The central limit theorem.

- Topic 7.5
- Confidence intervals for the mean of a normal population.

- Topic 7.6
- Null and alternative hypotheses, \({H_0}\) and \({H_1}\) .
- Significance level.
- Critical regions, critical values, \(p\)-values, one-tailed and two-tailed tests.
- Type I and II errors, including calculations of their probabilities.
- Testing hypotheses for the mean of a normal population.

- Topic 7.7
- Introduction to bivariate distributions.
- Covariance and (population) product moment correlation coefficient \(\rho \).
- Proof that \(\rho = 0\) in the case of independence and \( \pm 1\) in the case of a linear relationship between \(X\) and \(Y\).
- Definition of the (sample) product moment correlation coefficient \(R\) in terms of n paired observations on \(X\) and \(Y\).
- Its application to the estimation of \(\rho \).Informal interpretation of \(r\), the observed value of \(R\). Scatter diagrams.
- Topics based on the assumption of bivariate normality: use of the \(t\)-statistic to test the null hypothesis \(\rho = 0\) .
- Topics based on the assumption of bivariate normality: knowledge of the facts that the regression of \(X\) on \(Y\) (\({E\left. {\left( X \right)} \right|Y = y}\)) and \(Y\) on \(X\) (\({E\left. {\left( Y \right)} \right|X = x}\)) are linear.
- Topics based on the assumption of bivariate normality: least-squares estimates of these regression lines (proof not required).
- Topics based on the assumption of bivariate normality: the use of these regression lines to predict the value of one of the variables given the value of the other.

**Topic 8 – Option: Sets, relations and groups**

- Topic 8.1
- Finite and infinite sets.
- Subsets.
- Operations on sets: union; intersection; complement; set difference; symmetric difference.
- De Morgan’s laws: distributive, associative and commutative laws (for union and intersection).

- Topic 8.2
- Ordered pairs: the Cartesian product of two sets.
- Relations: equivalence relations; equivalence classes.

- Topic 8.3
- Functions: injections; surjections; bijections.
- Composition of functions and inverse functions.

- Topic 8.4
- Binary operations.
- Operation tables (Cayley tables).

- Topic 8.5
- Binary operations: associative, distributive and commutative properties.

- Topic 8.6
- The identity element \(e\).
- The inverse \({a^{ – 1}}\) of an element \(a\).Proof that left-cancellation and right-cancellation by an element a hold, provided that a has an inverse.
- Proofs of the uniqueness of the identity and inverse elements.

- Topic 8.7
- The definition of a group \(\left\{ {G, * } \right\}\) .
- The operation table of a group is a Latin square, but the converse is false.
- Abelian groups.

- Topic 8.8
- Example of groups: \(\mathbb{R}\), \(\mathbb{Q}\), \(\mathbb{Z}\) and \(\mathbb{C}\) under addition.
- Example of groups: integers under addition modulo \(n\).
- Example of groups: non-zero integers under multiplication, modulo \(p\), where \(p\) is prime.
- Symmetries of plane figures, including equilateral triangles and rectangles.
- Invertible functions under composition of functions.

- Topic 8.9
- The order of a group.
- The order of a group element.
- Cyclic groups.
- Generators.
- Proof that all cyclic groups are Abelian.

- Topic 8.10
- Permutations under composition of permutations.
- Cycle notation for permutations.
- Result that every permutation can be written as a composition of disjoint cycles.
- The order of a combination of cycles.

- Topic 8.11
- Subgroups, proper subgroups.
- Use and proof of subgroup tests.
- Definition and examples of left and right cosets of a subgroup of a group.
- Lagrange’s theorem.
- Use and proof of the result that the order of a finite group is divisible by the order of any element. (Corollary to Lagrange’s theorem.)

- Topic 8.12
- Definition of a group homomorphism.
- Definition of the kernel of a homomorphism.
- Proof that the kernel and range of a homomorphism are subgroups.
- Proof of homomorphism properties for identities and inverses.
- Isomorphism of groups.
- The order of an element is unchanged by an isomorphism.

**Topic 9 – Option: Calculus**

- Topic 9.1
- Infinite sequences of real numbers and their convergence or divergence.

- Topic 9.2
- Convergence of infinite series.
- Tests for convergence: comparison test; limit comparison test; ratio test; integral test.
- The \(p\)-series, \(\mathop \sum \nolimits \frac{1}{{{n^p}}}\) .Series that converge absolutely.
- Series that converge conditionally.
- Alternating series.
- Power series: radius of convergence and interval of convergence. Determination of the radius of convergence by the ratio test.

- Topic 9.3
- Continuity and differentiability of a function at a point.
- Continuous functions and differentiable functions.

- Topic 9.4
- The integral as a limit of a sum; lower and upper Riemann sums.
- Fundamental theorem of calculus.
- Improper integrals of the type \(\int\limits_a^\infty {f\left( x \right){\text{d}}} x\) .

- Topic 9.5
- First-order differential equations.
- Geometric interpretation using slope fields, including identification of isoclines.
- Numerical solution of \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = f\left( {x,y} \right)\) using Euler’s method.
- Variables separable.
- Homogeneous differential equation \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = f\left( {\frac{y}{x}} \right)\) using the substitution \(y = vx\) .Solution of \(y’ + P\left( x \right)y = Q\left( x \right)\), using the integrating factor.

- Topic 9.6
- Rolle’s theorem.
- Mean value theorem.
- Taylor polynomials; the Lagrange form of the error term.
- Maclaurin series for \({{\text{e}}^x}\) , \(\\sin x\) , \(\cos x\) , \(\ln (1 + x)\) , \({(1 + x)^p}\) , \(P \in \mathbb{Q}\) .
- Use of substitution, products, integration and differentiation to obtain other series.
- Taylor series developed from differential equations.

- Topic 9.7
- The evaluation of limits of the form \(\mathop {\lim }\limits_{x \to a} \frac{{f\left( x \right)}}{{g\left( x \right)}}\) and \(\mathop {\lim }\limits_{x \to \infty } \frac{{f\left( x \right)}}{{g\left( x \right)}}\) .
- Using l’Hôpital’s rule or the Taylor series.

**Topic 10 – Option: Discrete mathematics**

- Topic 10.1
- Strong induction.
- Pigeon-hole principle.

- Topic 10.2
- \(\left. a \right|b \Rightarrow b = na\) for some \(n \in \mathbb{Z}\) .
- The theorem \(\left. a \right|b\) and \(a\left| {c \Rightarrow a} \right|\left( {bx \pm cy} \right)\) where \(x,y \in \mathbb{Z}\) .
- Division and Euclidean algorithms.
- The greatest common divisor, gcd(\(a\),\(b\)), and the least common multiple, lcm(\(a\),\(b\)), of integers \(a\) and \(b\).
- Prime numbers; relatively prime numbers and the fundamental theorem of arithmetic.

- Topic 10.3
- Linear Diophantine equations \(ax + by = c\) .

- Topic 10.4
- Modular arithmetic.
- The solution of linear congruences.
- Solution of simultaneous linear congruences (Chinese remainder theorem).

- Topic 10.5
- Representation of integers in different bases.

- Topic 10.6
- Fermat’s little theorem.

- Topic 10.7
- Graphs, vertices, edges, faces.
- Adjacent vertices, adjacent edges.
- Degree of a vertex, degree sequence.
- Handshaking lemma.
- Simple graphs; connected graphs; complete graphs; bipartite graphs; planar graphs; trees; weighted graphs, including tabular representation.
- Subgraphs; complements of graphs
- .Euler’s relation: \(v – e + f = 2\) ; theorems for planar graphs including \(e \leqslant 3v – 6\) , \(e \leqslant 2v – 4\) , leading to the results that \({\kappa _5}\) and \({\kappa _{3,3}}\) are not planar.

- Topic 10.8
- Walks, trails, paths, circuits, cycles.
- Eulerian trails and circuits.
- Hamiltonian paths and cycles.

- Topic 10.9
- Graph algorithms: Kruskal’s; Dijkstra’s.

- Topic 10.10
- Chinese postman problem.
- Travelling salesman problem.
- Nearest-neighbour algorithm for determining an upper bound.
- Deleted vertex algorithm for determining a lower bound.

- Topic 10.11
- Recurrence relations.
- Initial conditions, recursive definition of a sequence.
- Solution of first- and second-degree linear homogeneous recurrence relations with constant coefficients.
- The first-degree linear recurrence relation \({u_n} = a{u_{n – 1}} + b\) .
- Modelling with recurrence relations.