### Paper 3

__HL__

- Time: 60 minutes [55 Maximum marks].
- 2 questions only
- No marks deducted from incorrect answers
- Answer all the questions
- A graphic display calculator is required for this paper
- Only HL syllabus for Maths AA

**Topic 1: Number and algebra****– AH**L content

- Topic : AHL 1.10
- Counting principles, including permutations and combinations.
- Extension of the binomial theorem to fractional and negative indices, ie (a + b)
^{n }, n ∈ ℚ.

- Topic : AHL 1.11
- Partial fractions

- Topic : AHL 1.12
- Complex numbers: the number i, where i
^{2}= − 1. - Cartesian form z = a + bi; the terms real part, imaginary part, conjugate, modulus and argument.
- The complex plane.

- Complex numbers: the number i, where i
- Topic : AHL 1.13
- Modulus–argument (polar) form \(z = r\left( {\cos \theta + {\text{i}}\sin \theta } \right) = r{\text{cis}}\theta = r{e^{{\text{i}}\theta }}\).
- Euler form: z = re
^{iθ}. - Sums, products and quotients in Cartesian, polar or Euler forms and their geometric interpretation.

- Topic : AHL 1.14
- Complex conjugate roots of quadratic and polynomial equations with real coefficients.
- De Moivre’s theorem and its extension to rational exponents.
- Powers and roots of complex numbers.

- Topic : AHL 1.15
- Proof by mathematical induction.
- Proof by contradiction.
- Use of a counterexample to show that a statement is not always true.

- Topic : AHL 1.16
- 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: Functions**– **AHL content

- Topic: AHL 2.12
- Topic: AHL 2.13
- Rational functions of the form
- \(f(x)=(\frac{ax + b}{cx^2 + dx + e}),and \; f(x)=\frac{ax^2 + bx + c}{dx + e}\)

- Rational functions of the form
- Topic: AHL 2.14
- Odd and even functions.
- Finding the inverse function, \({f^{ – 1}}\), including domain restriction. Self-inverse functions.

- Topic: AHL 2.15
- Topic: AHL 2.16
- The graphs of the functions, \(y = \left| {f\left( x \right)} \right|\) and \(y = f\left( {\left| x \right|} \right)\),
- \(\frac{1}{{f\left( x \right)}}\) , y = f(ax + b), y = [f(x)]
^{2}

### Topic 3: Geometry and trigonometry-AHL content

- Topic : AHL 3.9
- Definition of the reciprocal trigonometric ratios secθ, cosecθ and cotθ.
- Pythagorean identities:
- 1 + tan
^{2}θ = sec^{2}θ - 1 + cot2θ = cosec
^{2}θ

- 1 + tan
- The inverse functions f(x) = arcsinx, f(x) = arccosx, f(x) = arctanx; their domains and ranges; their graphs.

- Topic : AHL 3.10
- Compound angle identities.
- Double angle identity for tan.

- Topic : AHL 3.11
- Relationships between trigonometric functions and the symmetry properties of their graphs.

- Topic : AHL 3.12
- Concept of a vector; position vectors; displacement vectors.
- Representation of vectors using directed line segments.
- 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 following:
- sum and difference of two vectors.
- the zero vector \(0\), the vector \( – v\) .
- multiplication by a scalar, \(kv\) , parallel vectors
- magnitude of a vector, \(\left| v \right|\) .unit vectors=\(\frac{\vec{v}}{\left | \vec{v} \right |}\)
- position vectors \(\overrightarrow {OA} = a\) .
- displacement vector \(\overrightarrow {AB} = b – a\) .

- Proofs of geometrical properties using vectors.

- Topic : AHL 3.13
- 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 : AHL 3.14
- Vector equation of a line in two and three dimensions: \(r = a + \lambda b\) .
- The angle between two lines.
- Simple applications to kinematics.

- Topic : AHL 3.15
- Coincident, parallel, intersecting and skew lines; distinguishing between these cases.
- Points of intersection.

- Topic : AHL 3.16
- 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 : AHL 3.17
- 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 : AHL 3.18
- Intersections of: a line with a plane; two planes; three planes.
- Angle between: a line and a plane; two planes.

**Topic 4 : Statistics and probability-AHL content**

- Topic: AHL 4.13
- Topic: AHL 4.14

### Topic 5: Calculus-AHL content

- Topic: AHL 5.12
- Topic: AHL 5.13
- Topic: AHL 5.14
- Implicit differentiation.
- Related rates of change. Optimization problems.

- Topic: AHL 5.15
- Derivatives of \(\tan x\), \(\sec x\) , cosec x , \(\cot x\) , \({a^x}\) , \({\log _a}x\) , \(\arcsin x\) , \(\arccos x\) and \(\arctan x\) .
- Indefinite integrals of the derivatives of any of the above functions. The composites of any of these with a linear function.
- Use of partial fractions to rearrange the integrand.

- Topic: AHL 5.16
- Integration by substitution.
- Integration by parts.
- Repeated integration by parts.

- Topic: AHL 5.17
- Area of the region enclosed by a curve and the y-axis in a given interval.
- Volumes of revolution about the \(x\)-axis or \(y\)-axis.

- Topic: AHL 5.18
- Topic: AHL 5.19

### IBDP Mathematics – Paper 3 – Old Syllabus

**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
- Topic 7.3
- Topic 7.4
- Topic 7.5
- Topic 7.6
- 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
- Topic 8.2
- Topic 8.3
- Topic 8.4
- Topic 8.5
- Topic 8.6
- Topic 8.7
- 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
- Topic 8.10
- Topic 8.11
- 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
- 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
- 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
- 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
- Topic 10.4
- Topic 10.5
- Topic 10.6
- 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