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Syllabus

Section name
Prior learning topics
Number
Routine use of addition, subtraction, multiplication and division, using integers, decimals and fractions, including order of operations.
Simple positive exponents.
Simplification of expressions involving roots (surds or radicals).
Prime numbers and factors, including greatest common divisors and least common multiples.
Simple applications of ratio, percentage and proportion, linked to similarity.
Definition and elementary treatment of absolute value (modulus), \(\lvert a\rvert\)
Rounding, decimal approximations and significant figures, including appreciation of errors.
Expression of numbers in standard form (scientific notation), that is, \(a×10^k, 1 ≤ a < 10, k \in \mathbb{Z}\)
Sets and numbers
Concept and notation of sets, elements, universal (reference) set, empty (null) set, complement, subset, equality of sets, disjoint sets.
Operations on sets: union and intersection.
Commutative, associative and distributive properties.
Venn diagrams.
Number systems: natural numbers; integers,\(\mathbb{Z}\); rationals,\(\mathbb{Q}\), and irrationals; real numbers,\(\mathbb{R}\).
Intervals on the real number line using set notation and using inequalities.
Expressing the solution set of a linear inequality on the number line and in set notation.
Mappings of the elements of one set to another. Illustration by means of sets of ordered pairs, tables, diagrams and graphs.
Algebra
Manipulation of simple algebraic expressions involving factorization and expansion, including quadratic expressions.
Rearrangement, evaluation and combination of simple formulae. Examples from other subject areas, particularly the sciences, should be included.
The linear function and its graph, gradient and y-intercept.
Addition and subtraction of algebraic fractions
The properties of order relations: <,≤,>,≥.
Solution of equations and inequalities in one variable, including cases with rational coefficients.
Solution of simultaneous equations in two variables.
Trigonometry
Angle measurement in degrees. Compass directions and three figure bearings.
Right-angle trigonometry. Simple applications for solving triangles.
Pythagoras’ theorem and its converse
Geometry
Simple geometric transformations: translation, reflection, rotation, enlargement.
Congruence and similarity, including the concept of scale factor of an enlargement.
The circle, its centre and radius, area and circumference. The terms “arc”, “sector”, “chord”, “tangent” and “segment”.
Perimeter and area of plane figures. Properties of triangles and quadrilaterals,including parallelograms, rhombuses, rectangles, squares, kites and trapeziums (trapezoids); compound shapes
Volumes of prisms, pyramids, spheres, cylinders and cones
Coordinate geometry
Elementary geometry of the plane, including the concepts of dimension for point, line, plane and space.The equation of a line in the form \(y=mx+c\).
Parallel and perpendicular lines, including \(m_1=m_2\) and \(m_1m_2=−1\).
Geometry of simple plane figures.
The Cartesian plane: ordered pairs (x,y), origin, axes.
Mid-point of a line segment and distance between two points in the Cartesian plane and in three dimensions
Statistics and probability
Descriptive statistics: collection of raw data; display of data in pictorial and diagrammatic forms, including pie charts, pictograms, stem and leaf diagrams, bar graphs and line graphs.
Obtaining simple statistics from discrete and continuous data, including mean, median, mode, quartiles, range, interquartile range.
Calculating probabilities of simple events
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.
Applications.
1.2
Elementary treatment of exponents and logarithms.
Laws of exponents; laws of logarithms.
Change of base.
1.3
The binomial theorem: expansion of \({\left( {a + b} \right)^n}\), \(n \in \mathbb{N}\) .
Calculation of binomial coefficients using Pascal’s triangle and \(\left( {\begin{array}{*{20}{c}} n \\ r \end{array}} \right)\).
Topic 2 - Functions and equations
2.1
Concept of function \(f:x \mapsto f\left( x \right)\) .
Domain, range; image (value).
Composite functions.
Identity function. Inverse function \({f^{ - 1}}\) .
2.2
The graph of a function; its equation \(y = f\left( x \right)\) .
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 \(y = {f^{ - 1}}\left( x \right)\) as the reflection in the line \(y = x\) of the graph of \(y = f\left( x \right)\).
2.3
Transformations of graphs.
Translations: \(y = f\left( x \right) + b\) ; \(y = f(x - a)\) .
Reflections (in both axes): \(y = - f\left( x \right)\) ; \(y = f( - x)\) .
Vertical stretch with scale factor \(p\): \(y = pf\left( x \right)\) .
Stretch in the x-direction with scale factor \(\frac{1}{q}\): \(y = f(qx)\) .
Composite transformations.
2.4
The quadratic function \(x \mapsto a{x^2} + bx + c\) : its graph, \(y\)-intercept \((0, c)\). Axis of symmetry.
The form \(x \mapsto a\left( {x - p} \right)\left( {x - q} \right)\) , \(x\)-intercepts \((p, 0)\) and \((q,0)\) .
The form \(x \mapsto a{\left( {x - h} \right)^2} + k\) , vertex \((h, k)\) .
2.5
The reciprocal function \(x \mapsto \frac{1}{x}\) , \(x \ne 0\) : its graph and self-inverse nature.
The rational function \(x \mapsto \frac{{ax + b}}{{cx + d}}\) and its graph.
Vertical and horizontal asymptotes.
2.6
Exponential functions and their graphs: \(x \mapsto {a^x}\) , \(a > 0\) , \(x \mapsto {{\text{e}}^x}\) .
Logarithmic functions and their graphs: \(x \mapsto {\log _a}x\) , \(x > 0\) , \(x \mapsto \ln x\), \(x > 0\) .
Relationships between these functions: \({a^x} = {{\text{e}}^{x\ln a}}\) ; \({\log _a}{a^x} = x\) ; \({a^{{{\log }_a}x}} = x\) , \(x > 0\) .
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.
Solving \(a{x^2} + bx + c = 0\) , \(a \ne 0\) .
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 \(\frac{{\sin \theta }}{{\cos \theta }}\) .
Exact values of trigonometric ratios of \(0\), \(\frac{\pi }{6}\), \(\frac{\pi }{4}\), \(\frac{\pi }{3}\), \(\frac{\pi }{2}\) and their multiples.
3.3
The Pythagorean identity \({\cos ^2}\theta + {\sin ^2}\theta = 1\) .
Double angle identities for sine and cosine.
Relationship between trigonometric ratios.
3.4
The circular functions \(\sin x\) , \(\cos x\) and \(\tan x\) : 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, \(\frac{1}{2}ab\sin C\) .
Applications.
Topic 4 - Vectors
4.1
Vectors as displacements in the plane and in three dimensions.
Components of a vector; column representation; \(v = \left( {\begin{array}{*{20}{c}} {{v_1}} \\ {{v_2}} \\ {{v_3}} \end{array}} \right) = {v_1}i + {v_2}j + {v_3}k\) .
Algebraic and geometric approaches to the sum and difference of two vectors; 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, \(\left\| v \right\|\) .
Algebraic and geometric approaches to unit vectors; base vectors; \(i\), \(j\) and \(k\).
Algebraic and geometric approaches to position vectors \(\overrightarrow {OA} = a\) .
Algebraic and geometric approaches to \(\overrightarrow {AB} = \overrightarrow {OB} - \overrightarrow {OA} = b - a\) .
4.2
The scalar product of two vectors.
Perpendicular vectors; parallel vectors.
The angle between two vectors.
4.3
Vector equation of a line in two and three dimensions: \(r = a + tb\) .
The angle between two lines.
4.4
Distinguishing between coincident and parallel lines.
Finding the point of intersection of two lines.
Determining whether two lines intersect.
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.
5.4
Linear correlation of bivariate data.
Pearson’s product–moment correlation coefficient \(r\).
Scatter diagrams; lines of best fit.
Equation of the regression line of \(y\) on \(x\).
Use of the equation for prediction purposes.
Mathematical and contextual interpretation.
5.5
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 and tables of outcomes.
5.6
Combined events, \(P\left( {A \cup B} \right)\) .
Mutually exclusive events: \(P(A \cap B) = 0\) .
Conditional probability; the definition \(P\left( {\left. A \right\|B} \right) = \frac{{P\left( {A\mathop \cap \nolimits B} \right)}}{{P\left( B \right)}}\) .
Independent events; the definition \(P\left( {\left. A \right\|B} \right) = P\left( A \right) = P\left( {\left. A \right\|B'} \right)\) .
Probabilities with and without replacement.
5.7
Concept of discrete random variables and their probability distributions.
Expected value (mean), \({\text{E}}(X)\) for discrete data.
Applications.
5.8
Binomial distribution.
Mean and variance of the binomial distribution.
5.9
Normal distributions and curves.
Standardization of normal variables (\(z\)-values, \(z\)-scores).
Properties of the normal distribution.
Topic 6 - Calculus
6.1
Informal ideas of limit and convergence.
Limit notation.
Definition of derivative from first principles as \(f'\left( x \right) = {\mathop {\lim }\limits_{h \to 0 } } \left( {\frac{{f\left( {x + h} \right) - f\left( x \right)}}{h}} \right)\) .
Derivative interpreted as gradient function and as rate of change.
Tangents and normals, and their equations.
6.2
Derivative of \({x^n}\left( {n \in \mathbb{Q}} \right)\) , \(\sin x\) , \(\cos x\) , \(\tan x\) , \({{\text{e}}^x}\) and \(\ln x\) .
Differentiation of a sum and a real multiple of these functions.
The chain rule for composite functions.
The product and quotient rules.
The second derivative.
Extension to higher derivatives.
6.3
Local maximum and minimum points.
Testing for maximum or minimum.
Points of inflexion with zero and non-zero gradients.
Graphical behaviour of functions, including the relationship between the graphs of \(f\) , \({f'}\) and \({f''}\) .
Optimization.
Applications.
6.4
Indefinite integration as anti-differentiation.
Indefinite integral of \({x^n}\left( {n \in \mathbb{Q}} \right)\) , \(\sin x\) , \(\cos x\) , \(\frac{1}{x}\) and \({{\text{e}}^x}\) .
The composites of any of these with the linear function \(ax + b\) .
Integration by inspection, or substitution of the form \(\mathop \int \nolimits f\left( {g\left( x \right)} \right)g'\left( x \right){\text{d}}x\) .
6.5
Anti-differentiation with a boundary condition to determine the constant term.
Definite integrals, both analytically and using technology.
Areas under curves (between the curve and the \(x\)-axis).
Areas between curves.
Volumes of revolution about the \(x\)-axis.
6.6
Kinematic problems involving displacement \(s\), velocity \(v\) and acceleration \(a\).
Total distance travelled.

DP Mathematics SL Questionbank