1. Number patterns and sigma notation
2. Arithmetic and geometric sequences
3. Arithmetic and geometric series
4. Applications of arithmetic and geometric patterns
5. The binomial theorem
6. Proof

1. What is function?
2. Functional notation
3. Drawing graphs of functions
4. The domain and range of a function
5. Composite functions
6. Inverse functions

1. Gradient of a linear function
2. Linear functions
3. Transformations of functions
4. Graphing quadratic functions
5. Solving quadratic equations by factorization and completing the square

1. The reciprocal function
2. Transforming the reciprocal function
3. Rational functions of the form f(x) = (ax+b)/(cx+d)
4. Chapter review Modelling and investigation activity

1. Limits and convergence
2. The derivative function
3. Differentiation rules
4. Graphical interpretation of first and second derivatives
5. Application of differential calculus:

1. Sampling
2. Presentation of data
3. Measures of central tendency
4. Measures of dispersion

1. Scatter diagrams
2. Measuring correlation
3. The line of best fit
4. Least squares regression

1. Theoretical and experimental probability
2. Representing probabilities: Venn diagrams and sample spaces
3. Independent and dependent events and conditional probability
4. Probability tree diagrams

1. Exponents
2. Logarithms
3. Derivatives of exponential functions and the natural logarithmic function

1. Antiderivatives and the indefinite integral
2. More on indefinite integrals
3. 3 Area and definite integrals
4. Functional theorem of calculus
5. Area between two curves

1. The geometry of 3D shapes
2. Right-angled triangle trigonometry
3. The sine rule
4. The cosine rule
5. Applications of right and non-right-angled trigonometry

1. Radian measure, arcs, sectors and segments
2. Trigonometric ratios in the unit circle
3. Trigonometric identities and equations
4. Trigonometric functions

1. Derivatives and with sine and cosine
2. Applications of derivatives
3. Integration with sine, cosine and substitution
4. Kinematics and accumulating change

1. Random variables
2. The binomial distribution
3. The normal distribution

Past Paper SL

Past Paper by Topics

TOPIC 1 - Algebra

1.1

• Arithmetic sequences and series;
• sum of finite arithmetic series;
• geometric sequences and series;
• sum of finite and infinite geometric series.
• Sigma notation.

1.2

• Elementary treatment of exponents and logarithms.
• Laws of exponents; laws of logarithms.
• Change of base.

1.3

• The binomial theorem: expansion of (a+b)^n, n∈
• Calculation of binomial coefficients using Pascal’s triangle and (n/r)

Topic 2 - Functions and equations

2.1

• Concept of function f : x ↦ f (x) f : x ↦ f(x)
• Domain, range; image (value).
• Composite functions.
• Identity function. Inverse function f^-1

2.2

• The graph of a function; its equation :

• Function graphing skills.
• Investigation of key features of graphs, such as maximum and minimum values, intercepts, horizontal and vertical asymptotes, symmetry, and consideration of domain and range.
• Use of technology to graph a variety of functions, including ones not specifically mentioned.
• The graph of y = f^-1(x) as the reflection in the line y=x of the graph of y= f(x)

2.3

• Transformations of graphs.
• Translations: y = f(x) + b; y = f(x-1)
• Reflections (in both axes): y = - f(x); y=f(-x)
• Vertical stretch with scale factor p; y=pf(x)
• Stretch in the x-direction with scale factor 1/q; y-f(qx)
• Composite transformations.

2.4

• The quadratic function x ↦ ax^2 +bx +c; its graph, y-intercept (0,c). Axis of symmetry.
• The form x ↦ a(x-p)(x-q), x-intercepts

• The form x ↦ a(x-h)^2 +k ,vertex (h , k)

2.5

• The reciprocal function x ↦ 1x , x ≠ 0: its graph and self-inverse nature.
• The rational function x->(ax+b)/(cx+d) and its graph.

• Vertical and horizontal asymptotes.

Table of Content

• Topic 1: Measuring space: accuracy and 2D geometry

1. Measurements and estimates
2. Recording measurements, significant digits and rounding
3. Measurements: exact or approximate?
4. Speaking scientifically
5. Trigonometry of right-anged trianges and indirect measurements
6. Angeles of elevation and depression

• Topic 2: Representing space: Non right angled trigonometry and volumes

1. Trigonometry of non-right triangles
2. Area of a triangle formula: applications of right and non-right angled trigonometry
3. geometry: solids, surface area and volume

• Topic 3: Representing and describing data-descriptive statistics

1. Collecting and organizing univariate data
2. Sampling tecuiques
3. Presentation of data
4. Vivariate data

• Topic 4: Dividing up space-coordinate geometry, lines, Voronoi diagramsTopic

1. Coordinates, distance and the midpoint formula in 2D and 3D
2. Gradient of a line and its applications
3. Equations of straight lines: different forms of equations
4. Parallel and perpendicular lines
5. Voronoi diagrams and the toxic waste dump problem

• 5: Modelling constant rates of change-linear functionTopic

1. Functions
2. Linear models
3. Arithmetic sequences
4. Modelling

• 6: Modelling relationships-linear correlation geometry

1. Measuring correlation
2. The line of best fit
3. Interpreting the regression line

• Topic 7: Quantifying uncertainty-probability, binomial and normal distributionsTopic

1. Theoretical and experimental probability
2. Representing combined probabilities with diagrams
3. Representing combined probabilities with diagrams and formulae
4. Complete, concise and consistent representations
5. Modelling random behaviour, random variables and probability distributions
6. Modelling the number of successes in a fixed number of trials
7. Modelling measurements that we distributed randomly

• 8: Testing for validity-Spearman's hypothesis testing and X^2 test for independence

1. Spearman's rank correlation coefficient
2. X^2 test for independence
3. X^2 goodness of fit test
4. The T-test

• Topic 9: Modelling relationships with Functions-power functions

1. Quadratic models
2. Problems involving quadratics
3. Cubic models, power functions and direct and inverse variation
4. Optimization

• Topic 10: Modelling rates of change-exponential and logarithmic functions

1. Geometric sequences and series
2. Compound interest, annuities amortization
3. Exponential models
4. Exponential equations and logarithms

• Topic 11: Modelling periodic phenomena-trigonometric functions

1. An intro to periodic functions
2. An infinity of sinusoidal functions
3. A world of sinusoidal models

• Topic 12 Analyzing rates of change-differential calculus

1. Limits and erivates
2. Equations of tangent and normal
3. Maximum and minimum points and optimization

• Topic 13: Approximating irregular spaces INTEGRATION

1. Finding areas
2. Integration: the reverse process of differentiation

Past Paper SL

Past paper/By topic

TOPIC 1 - Algebra

1.1

• Arithmetic sequences and series;
• sum of finite arithmetic series;
• geometric sequences and series;
• sum of finite and infinite geometric series.
• Sigma notation.

1.2

• Elementary treatment of exponents and logarithms.
• Laws of exponents; laws of logarithms.
• Change of base.

1.3

• The binomial theorem: expansion of (a+b)^n, n∈
• Calculation of binomial coefficients using Pascal’s triangle and (n/r)

Topic 2 - Functions and equations

2.1

• Concept of function f : x ↦ f (x) f : x ↦ f(x)
• Domain, range; image (value).
• Composite functions.
• Identity function. Inverse function f^-1

2.2

• The graph of a function; its equation :

• Function graphing skills.
• Investigation of key features of graphs, such as maximum and minimum values, intercepts, horizontal and vertical asymptotes, symmetry, and consideration of domain and range.
• Use of technology to graph a variety of functions, including ones not specifically mentioned.
• The graph of y = f^-1(x) as the reflection in the line y=x of the graph of y= f(x)

2.3

• Transformations of graphs.
• Translations: y = f(x) + b; y = f(x-1)
• Reflections (in both axes): y = - f(x); y=f(-x)
• Vertical stretch with scale factor p; y=pf(x)
• Stretch in the x-direction with scale factor 1/q; y-f(qx)
• Composite transformations.

2.4

• The quadratic function x ↦ ax^2 +bx +c; its graph, y-intercept (0,c). Axis of symmetry.
• The form x ↦ a(x-p)(x-q), x-intercepts

• The form x ↦ a(x-h)^2 +k ,vertex (h , k)

2.5

• The reciprocal function x ↦ 1x , x ≠ 0: its graph and self-inverse nature.
• The rational function x->(ax+b)/(cx+d) and its graph.

• Vertical and horizontal asymptotes.

2.6

• Exponential functions and their graphs x a^2, a>0, x ↦ e^z
• Logarithmic functions and their graphs x ↦logax, x>0, x->ln, x>0
• Relationships between these function:

2.7

• Solving equations, both graphically and analytically.
• Use of technology to solve a variety of equations, including those where there is no appropriate analytic approach.
• The quadratic formula.
• The discriminant delta=b^2-4ac and the nature of the roots, that is, two distinct real roots, two equal real roots, no real roots.
• Solving exponential equations.

2.8

• Applications of graphing skills and solving equations that relate to real-life situations.

Topic 3 - Circular functions and trigonometry

3.1

The circle: radian measure of angles; length of an arc; area of a sector.3.2

• Definition of cos theta and sin theta in terms of the unit circle.
• Definition of tan theta as sin theta /cos theta
• Exact values of trigonometric ratios of

3.3

3.4

• The circular functions sinx , cosx and tanx: their domains and ranges; amplitude, their periodic nature; and their graphs.
• Composite functions of the form f(x)=a sin (b ( x+c)) d .
• Transformations.
• Applications.

3.5

• Solving trigonometric equations in a finite interval, both graphically and analytically.
• Equations leading to quadratic equations in sin x , cos x or tan x

3.6

• Solution of triangles.
• The cosine rule.
• The sine rule, including the ambiguous case.

• Area of a triangle, 1/2 ab sin C
• Applications.

Topic 4 - Vectors

4.1

• Vectors as displacements in the plane and in three dimensions.
• Components of a vector; column representation ; v={v1 v2 v3} = v1i+v2j+v3 k
• Algebraic and geometric approaches to the sum and difference of two vectors; the zero vector, the vector -v
• Algebraic and geometric approaches to multiplication by a scalar, kv; parallel vectors.
• Algebraic and geometric approaches to magnitude of a vector, |v|
• Algebraic and geometric approaches to unit vectors; base vectors; i, j and k.
• Algebraic and geometric approaches to position vectors −−→OA=a .
• Algebraic and geometric approaches to −−→AB=−−→OB−−−→OA=b−a .

4.2

• The scalar product of two vectors.
• Perpendicular vectors; parallel vectors.
• The angle between two vectors.

4.3

4.4

Topic 5 - Statistics and probability

5.1

• Concepts of population, sample, random sample, discrete and continuous data.
• Presentation of data: frequency distributions (tables); frequency histograms with equal class intervals
• Box-and-whisker plots; outliers.
• Grouped data: use of mid-interval values for calculations; interval width; upper and lower interval boundaries; modal class.

5.2

• Statistical measures and their interpretations.
• Central tendency: mean, median, mode.
• Quartiles, percentiles.
• Dispersion: range, interquartile range, variance, standard deviation.
• Effect of constant changes to the original data.
• Applications.

5.3

• Cumulative frequency; cumulative frequency graphs; use to find median, quartiles, percentiles.

1. Representing numbers exactly and approximately
2. Angles and triangles
3. Three dimensional geometry

1. Collecting and organizing data
2. Statistical measures
3. Ways in which you can present data
4. Bivariate data

1. Coordinate geometry in 2 and 3 dimensions
2. The equation of a straight line in 2 dimensions
3. Voronoi diagrams
4. Displacement vectors
5. The scalar and vector products
6. Vector equations of lines

1. Function
2. Linear models
3. Inverse functions
4. Arithmetic sequences and series
5. Linear regression

1. Reflecting on experiences in the world of chance. First steps in the quantification of probabilities
2. Representing combined probabilities with diagrams
3. Representing combined probabilities with diagrams and formulae
4. Complete, concise and consistent representations

1. Quadratic models
2. Problems involving quadratics
3. Cubic functions and models
4. Power functions, direct and inverse variation and models

1. Geometric sequences and series
2. Financial applications of geometric sequences and series
3. Exponential functions and models
4. Laws of exponents - laws of logarithms
5. Logistic models

1. Measuring angles
2. Sinusoidal models: f(x) = a sin [b (x-c)] + d
3. Completing our number system
4. A geometrical interpretation of complex numbers
5. Using complex numbers to understand periodic models

1. Introduction to matrices and matrix operations
2. Matrix nultiplication and Properties
3. Solving systems of equations using matrices
4. Transformations of the plane
5. Representing systems
6. Representing steady state systems
7. Eigenvalues and eigenvectors

1. Limits and derivatives
2. Differentiation: further rules and techniques
3. Applications and higher derivatives

1. Finding approximate area for irregular regions
2. Indefinite integrals and techniques of integration
3. Applications of integration
4. Differential equations
5. Slope fields and differential equations

1. Vector quantities
2. Motion with variable velocity
3. Exact solutions of coupled differential equations
4. Approximate solutions to coupled linear equations

1. Modelling random behaviour
2. Modelling the number of successes in a fixed number of trials

Past Paper HL

Topic 3: Geometry & Trig

Geometry of 3D Shapes

Trigonometry

Voronoi Diagrams