Paper 2
HL
- Time: 150 minutes (150 marks)
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Topic 1 – Linear Algebra
- Topic 1.1
- Definition of a matrix: the terms element, row, column and order for \(m \times n\) matrices.
- Algebra of matrices: equality; addition; subtraction; multiplication by a scalar for \(m \times n\) matrices.
- Multiplication of matrices.
- Properties of matrix multiplication: associativity, distributivity.
- Identity and zero matrices.
- Transpose of a matrix including \({A^{\text{T}}}\) notation: \({(AB)^{\text{T}}} = {B^{\text{T}}}{A^{\text{T}}}\) .
- Topic 1.2
- Definition and properties of the inverse of a square matrix: \({\left( {AB} \right)^{ – 1}} = {B^{ – 1}}{A^{ – 1}}\) , \({\left( {{A^{\text{T}}}} \right)^{ – 1}} = {\left( {{A^{ – 1}}} \right)^{\text{T}}}\) , \({\left( {{A^n}} \right)^{ – 1}} = {\left( {{A^{ – 1}}} \right)^n}\) .
- Calculation of \({A^{ – 1}}\) .
- Use of elementary row operations to find \({A^{ – 1}}\) .
- Formulae for the inverse and determinant of a \(2 \times 2\) matrix and the determinant of a \(3 \times 3\) matrix.
- Topic1.3
- Elementary row and column operations for matrices.
- Scaling, swapping and pivoting.
- Corresponding elementary matrices.
- Row reduced echelon form.
- Row space, column space and null space.
- Row rank and column rank and their equality.
- Topic 1.4
- Solutions of \(m\) linear equations in \(n\) unknowns: both augmented matrix method, leading to reduced row echelon form method, and inverse matrix method, when applicable.
- Topic 1.5
- The vector space \({\mathbb{R}^n}\) .
- Linear combinations of vectors.
- Spanning set.
- Linear independence of vectors.
- Basis and dimension for a vector space.
- Subspaces.
- Topic 1.6
- Linear transformations: \(T(u + v) = T(u) + T(v)\) , \(T(ku) = kT(u)\) .
- Composition of linear transformations.
- Domain, range, codomain and kernel.
- Result and proof that the kernel is a subspace of the domain.
- Result and proof that the range is a subspace of the codomain.
- Rank-nullity theorem (proof not required).
- Topic 1.7
- Result that any linear transformation can be represented by a matrix, and the converse of this result.
- Result that the numbers of linearly independent rows and columns are equal, and this is the dimension of the range of the transformation (proof not required).
- Application of linear transformations to solutions of system of equations.
- Solution of \(A{\text{x}} = {\text{b}}\) .
- Topic 1.8
- Geometric transformations represented by \(2 \times 2\) matrices include general rotation, general reflection in \(y = (\tan \alpha )x\) , stretches parallel to axes, shears parallel to axes, and projection onto \(y = (\tan \alpha )x\).
- Compositions of the above transformations.
- Geometric interpretation of determinant.
- Topic 1.9
- Eigenvalues and eigenvectors of \(2 \times 2\) matrices.
- Characteristic polynomial of \(2 \times 2\) matrices.
- Diagonalization of \(2 \times 2\) matrices (restricted to the case where there are distinct real eigenvalues).
- Applications to powers of \(2 \times 2\) matrices.
Topic 2 – Geometry
- Topic 2.1
- Similar and congruent triangles.
- Euclid’s theorem for proportional segments in a right-angled triangle.
- Topic 2.2
- Centres of a triangle: orthocentre, incentre, circumcentre and centroid.
- Topic 2.3
- Circle geometry.
- Tangents; arcs, chords and secants.
- In a cyclic quadrilateral, opposite angles are supplementary, and the converse.
- Topic 2.4
- Angle bisector theorem; Apollonius’ circle theorem, Menelaus’ theorem; Ceva’s theorem; Ptolemy’s theorem for cyclic quadrilaterals.
- Proofs of these theorems and converses.
- Topic 2.5
- Finding equations of loci.
- Coordinate geometry of the circle.
- Tangents to a circle.
- Topic 2.6
- Conic sections.
- The parabola, ellipse and hyperbola, including rectangular hyperbola.
- Focus–directrix definitions.
- Tangents and normals.
- Topic 2.7
- Parametric equations.
- Parametric differentiation.
- Tangents and normals.
- The standard parametric equations of the circle, parabola, ellipse, rectangular hyperbola, hyperbola.
- Topic 2.8
- The general conic \(a{x^2} + 2bxy + c{y^2} + dx + ey + f = 0\) and the quadratic form \({x^{\text{T}}}Ax = a{x^2} + 2bxy + c{y^2}\) .
- Diagonalizing the matrix \(A\) with the rotation matrix \(P\) and reducing the general conic to standard form.
Topic 3 – Statistics and probability
- Topic 3.1
- 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 3.2
- 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 3.3
- Unbiased estimators and estimates.
- Comparison of unbiased estimators based on variances.\(\overline X \) as an unbiased estimator for \(\mu \) .\({S^2}\) as an unbiased estimator for \({\sigma ^2}\) .
- Topic 3.4
- 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 3.5
- Confidence intervals for the mean of a normal population.
- Topic 3.6
- 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 3.7
- 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 ρ.
- Informal interpretation of \(r\), the observed value of \(R\).
- Scatter diagrams.
- The following topics are based on the assumption of bivariate normality.
- Use of the \(t\)-statistic to test the null hypothesis \(\rho = 0\) .Knowledge of the facts that the regression of \(X\) on \(Y\) (\({\text{E}}(X)|Y = y\)) and \(Y\) on \(X\) (\({\text{E}}(Y)|X = x\)) are linear.
- Least-squares estimates of these regression lines (proof not required).The use of these regression lines to predict the value of one of the variables given the value of the other.
Topic 4 – Sets, relations and groups
- Topic 4.1
- 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 4.2
- Ordered pairs: the Cartesian product of two sets.
- Relations: equivalence relations; equivalence classes.4.3Functions: injections; surjections; bijections.
- Composition of functions and inverse functions.4.4Binary operations.
- Operation tables (Cayley tables).
- Topic 4.5
- Binary operations: associative, distributive and commutative properties.
- Topic 4.6
- 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 4.7
- 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 4.8
- 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.
- Example of groups: symmetries of plane figures, including equilateral triangles and rectangles.
- Example of groups: invertible functions under composition of functions.
- Topic 4.9
- The order of a group.
- The order of a group element.
- Cyclic groups.
- Generators.
- Proof that all cyclic groups are Abelian.
- Topic 4.10
- 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 4.11
- 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 4.12
- 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 5 – Calculus
- Topic 5.1
- Infinite sequences of real numbers and their convergence or divergence.
- Topic 5.2
- 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 5.3
- Continuity and differentiability of a function at a point.
- Continuous functions and differentiable functions.
- Topic 5.4
- 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 5.5
- 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 5.6
- 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 5.7
- The evaluation of limits of the form \(\mathop {\lim }\limits_{x \to a} \frac{{f(x)}}{{g(x)}}\) and \(\mathop {\lim }\limits_{x \to \infty } \frac{{f(x)}}{{g(x)}}\) .
- Using l’Hôpital’s rule or the Taylor series.
Topic 6 – Discrete mathematics
- Topic 6.1
- Strong induction.
- Pigeon-hole principle.
- Topic 6.2
- \(\left. a \right|b \Rightarrow b = na\) for some \(n \in \mathbb{Z}\) .
- The theorem \(\left. a \right|b\) and \(\left. a \right|c \Rightarrow \left. a \right|(bx \pm cy)\) 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 6.3
- Linear Diophantine equations \(ax + by = c\) .
- Topic 6.4
- Modular arithmetic.
- The solution of linear congruences.
- Solution of simultaneous linear congruences (Chinese remainder theorem).
- Topic 6.5
- Representation of integers in different bases.
- Topic 6.6
- Fermat’s little theorem.
- Topic 6.7
- 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 6.8
- Walks, trails, paths, circuits, cycles.Eulerian trails and circuits.
- Hamiltonian paths and cycles.
- Topic 6.9
- Graph algorithms: Kruskal’s; Dijkstra’s.
- Topic 6.10
- Chinese postman problem.
- Travelling salesman problem.
- Nearest neighbour algorithm for determining an upper bound.
- Deleted vertex algorithm for determining a lower bound.
- Topic 6.11
- 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.