Syllabus
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Unit 1: Algebra, statistics and functions
Topic 1: Arithmetic and geometric sequences and series 1
Arithmetic sequences
Unit 1: Algebra, statistics and functions > Topic 1: Arithmetic and geometric sequences and series 1 > Arithmetic sequences
- Recognise and use the recursive definition of an arithmetic sequence: \( t_{n+1} = t_n + d \)
- Use the formula \( t_n = t_1 + (n - 1)d \) for the general term of an arithmetic sequence and Recognise its linear nature
- Use arithmetic sequences in contexts involving discrete linear growth or decay, such as simple interest
- Establish and use the formula for the sum of the first \( n \) terms of an arithmetic sequence: \( S_n = \frac{n}{2} (2t_1 + (n - 1)d) = \frac{n}{2} (t_1 + t_n) \)
Topic 2: Functions and graphs
Functions
Unit 1: Algebra, statistics and functions > Topic 2: Functions and graphs > Functions
- Understand the concept of a relation as a mapping between sets, a graph and as a rule or a formula that defines one variable quantity in terms of another
- Recognise the distinction between functions and relations and use the vertical line test to determine whether a relation is a function
- Use function notation, domain and range, and independent and dependent variables
- Examine transformations of the graphs of \( f(x) \), including dilations and reflections, and the graphs of \( y = a|x| \) and \( y = f(bx) \), translations, and the graphs of \( y = f(x + c) \) and \( y = f(x) + d \); \( a, b, c, d \in \mathbb{R} \)
- Recognise and use piece-wise functions as a combination of multiple sub-functions with restricted domains
- Identify contexts suitable for modelling piece-wise functions and use them to solve practical problems (taxation, taxis, the changing velocity of a parachutist)
Review of quadratic relationships
Unit 1: Algebra, statistics and functions > Topic 2: Functions and graphs > Review of quadratic relationships
- Examine examples of quadratically related variables
- Recognise and determine features of the graphs of \( y = x^2 \), \( y = ax^2 + bx + c \), \( y = a(x - b)^2 + c \), and \( y = a(x - b)(x - c) \), including their parabolic nature, turning points, axes of symmetry and intercepts
- Solve quadratic equations algebraically using factorisation, the quadratic formula (both exact and approximate solutions), and completing the square and using technology
- Identify contexts suitable for modelling with quadratic functions and use models to solve problems with and without technology; verify and evaluate the usefulness of the model using qualitative statements and quantitative analysis
- Understand the role of the discriminant to determine the number of solutions to a quadratic equation
- Determine turning points and zeros of quadratic functions with and without technology
Inverse proportions
Unit 1: Algebra, statistics and functions > Topic 2: Functions and graphs > Inverse proportions
- Examine examples of inverse proportion
- Recognise features of the graphs of \( y = \frac{1}{x} \) and \( y = a\left(\frac{b}{x^n}\right) \), including their hyperbolic shapes, their intercepts, their asymptotes, and behaviour as \( x \to \infty \) and \( x \to -\infty \).
Powers and polynomials
Unit 1: Algebra, statistics and functions > Topic 2: Functions and graphs > Powers and polynomials
- Identify the coefficients and the degree of a polynomial
- Expand quadratic and cubic polynomials from factors
- Recognise and determine features of the graphs of \( y = x^3 \), \( y = ax^3 + bx^2 + cx + d \) and \( y = k(x - a)(x - b)(x - c) \), including shape, intercepts and behaviour as \( x \to \infty \) and \( x \to -\infty \)
- Use the factor theorem to factorise cubic polynomials in cases where a linear factor is easily obtained
- Solve cubic equations using technology, and algebraically in cases where a linear factor is easily obtained
- Recognise and determine features of the graphs \( y = ax^3 - bx^4 + c \), including shape and behaviour
- Solve equations involving combinations of the functions above, using technology where appropriate
Graphs of relations
Unit 1: Algebra, statistics and functions > Topic 2: Functions and graphs > Graphs of relations
- Recognise and determine features of the graphs of \( x^2 + y^2 = r^2 \) and \( (x - a)^2 + (y - b)^2 = r^2 \), including their circular shapes, centres and radii
- Recognise and determine features of the graph of \( y = x^2 \), including its parabolic shape and axis of symmetry
Topic 3: Counting and probability
Language of events and sets
Unit 1: Algebra, statistics and functions > Topic 3: Counting and probability > Language of events and sets
- Recall the concepts and language of outcomes, sample spaces and events as sets of outcomes
- Use set language and notation for events, including \( A \) or \( \overline{A} \) for the complement of an event, \( A \cap B \) for the intersection of events A and B, \( A \cup B \) for the union, and recognise mutually exclusive events
- Use everyday occurrences to illustrate set descriptions and representations of events, and set operations, including the use of Venn diagrams
Review of the fundamentals of probability
Unit 1: Algebra, statistics and functions > Topic 3: Counting and probability > Review of the fundamentals of probability
- Recall probability as a measure of the 'likelihood of occurrence' of an event
- Recall the probability scale: \( 0 \leq P(A) \leq 1 \) for each event A, with \( P(A) = 0 \) if A is an impossibility and \( P(A) = 1 \) if A is a certainty
- Recall the rules \( P(A) = 1 - P(\overline{A}) \) and \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \)
- Use relative frequencies obtained from data as point estimates of probabilities
Conditional probability and independence
Unit 1: Algebra, statistics and functions > Topic 3: Counting and probability > Conditional probability and independence
- Understand the notion of a conditional probability, and recognise and use language that indicates conditionality
- Use the notation \( P(A|B) \) and the formula \( P(A \cap B) = P(A|B)P(B) \) to solve problems
- Understand and use the notion of independence of an event A from an event B, as defined by \( P(A \cap B) = P(A)P(B) \)
- Establish and use the formula \( P(A \cap B) = P(A)P(B) \) for independent events A and B
- Use relative frequencies obtained from data as point estimates of conditional probabilities and as indications of possible independence of events
Binomial expansion
Unit 1: Algebra, statistics and functions > Topic 3: Counting and probability > Binomial expansion
- Understand the notion of a combination as an unordered set of \(r\) objects taken from a set of \(n\) distinct objects
- Recognise and use the link between Pascal's triangle and the notation \(\binom{n}{r}\)
- Expand \((x + y)^n\) for small positive integers \(n\)
Topic 4: Exponential functions 1
Indices and the index laws
Unit 1: Algebra, statistics and functions > Topic 4: Exponential functions 1 > Indices and the index laws
- Recall indices (including negative and fractional indices) and the index laws
- Convert radicals to and from fractional indices
- Understand and use scientific notation
Topic 5: Arithmetic and geometric sequences and series 2
Geometric sequences
Unit 1: Algebra, statistics and functions > Topic 5: Arithmetic and geometric sequences and series 2 > Geometric sequences
- Recognise and use the recursive definition of a geometric sequence: \( t_{n+1} = r t_n \)
- Use the formula \( t_n = t_1 r^{(n-1)} \) for the general term of a geometric sequence and recognise its exponential nature
- Understand the limiting behaviour as \( n \to \infty \) of the terms \( t_n \) in a geometric sequence and its dependence on the value of the common ratio \( r \)
- Establish and use the formula \( S_n = t_1 \frac{(r^n - 1)}{(r - 1)} \) for the sum of the first \( n \) terms of a geometric sequence
- Establish and use the formula \( S_\infty = \frac{t_1}{(1 - r)} \), \( |r| < 1 \) for the sum to infinity of a geometric progression
- Use geometric sequences in contexts involving geometric growth or decay, including compound interest and annuities
Unit 2: Calculus and further functions
Topic 1: Exponential functions 2
Introduction to exponential functions
Unit 2: Calculus and further functions > Topic 1: Exponential functions 2 > Introduction to exponential functions
- Recognise and determine the qualitative features of the graph of \( y = a^x \) (\( a > 0 \)), including asymptotes, and of its translations (\( y = a^x + b \) and \( y = a^{x+c} \))
- Recognise and determine the features of the graphs of ( \( y = b \cdot a^x \) ) and ( \( y = a^{kx} \) (\( k \neq 0 \)))
- Identify contexts suitable for modelling by exponential functions and use models to solve practical problems; verify and evaluate the usefulness of the model using qualitative statements and quantitative analysis
- Solve equations involving exponential functions with and without technology
Topic 2: The logarithmic function 1
Introduction to logs
Unit 2: Calculus and further functions > Topic 2: The logarithmic function 1 > Introduction to logs
- Define logarithms as indices: \( a^x = b \) is equivalent to \( x = \log_a(b) \)
- Recognise the inverse relationship between logarithms and exponentials: \( y = a^x \) is equivalent to \( x = \log_a(y) \)
- Solve equations involving indices with and without technology
Topic 3: Trigonometric functions 1
Circular measure and radian measure
Unit 2: Calculus and further functions > Topic 3: Trigonometric functions 1 > Circular measure and radian measure
- Define and use radian measure and understand its relationship with degree measure
- Calculate lengths of arcs and areas of sectors in circles
Introduction to trigonometric functions
Unit 2: Calculus and further functions > Topic 3: Trigonometric functions 1 > Introduction to trigonometric functions
- Understand the unit circle definition of \( \cos(\pi) \), \( \sin(\pi) \), and \( \tan(\pi) \) and periodicity using radians
- Recall the exact values of \( \sin(\pi) \), \( \cos(\pi) \), and \( \tan(\pi) \) at integer multiples of \( \frac{\pi}{6} \) and \( \frac{\pi}{4} \)
- Sketch the graphs of \( y = \sin(kx) \), \( y = \cos(kx) \), and \( y = \tan(kx) \) on extended domains
- Investigate the effect of the parameters A, B, C and D on the graphs of \( y = A \sin(B(x + C)) + D \), \( y = A \cos(B(x + C)) + D \) with and without technology
- Sketch the graphs of \( y = A \sin(B(x + C)) + D \), \( y = A \cos(B(x + C)) + D \) with and without technology
- Identify contexts suitable for modelling by trigonometric functions and use them to solve practical problems; verify and evaluate the usefulness of the model using qualitative statements and quantitative analysis
- Solve equations involving trigonometric functions with and without technology; including use of the Pythagorean identity \( \sin^2(\pi) + \cos^2(\pi) = 1 \)
Topic 4: Introduction to differential calculus
Rates of change and the concept of derivatives
Unit 2: Calculus and further functions > Topic 4: Introduction to differential calculus > Rates of change and the concept of derivatives
- Explore average and instantaneous rate of change in a variety of practical contexts
- Use a numerical technique to estimate a limit or an average rate of change
- Examine the behaviour of the difference quotient \(\frac{f(x+h)-f(x)}{h}\) as \(h \rightarrow 0\) as an informal introduction to the concept of a limit
- Differentiate simple power functions and polynomial functions from first principles
- Interpret the derivative as the instantaneous rate of change
- Interpret the derivative as the gradient of a tangent line of the graph of \(y = f(x)\)
Properties and computation of derivatives
Unit 2: Calculus and further functions > Topic 4: Introduction to differential calculus > Properties and computation of derivatives
- Examine examples of variable rates of change of non-linear functions
- Establish the formula \(d (x^n) = nx^{n-1}\) for positive integers
- Understand the concept of the derivative as a function
- Recognise and use properties of the derivative \(d (\frac{du}{dx} (f(x) + g(x))) = d (\frac{df}{dx} (x)) + d (\frac{dg}{dx} (x))\)
- Calculate derivatives of power and polynomial functions
Applications of derivatives
Unit 2: Calculus and further functions > Topic 4: Introduction to differential calculus > Applications of derivatives
- Determine instantaneous rates of change
- Determine the gradient of a tangent and the equation of the tangent
- Construct and interpret displacement-time graphs, with velocity as the slope of the tangent
- Sketch curves associated with power functions and polynomials up to and including degree 4; find stationary points and local and global maxima and minima with and without technology; and examine behaviour as \(x \rightarrow \infty\) and \(x \rightarrow - \infty\)
- Identify contexts suitable for modelling optimisation problems involving polynomials up to and including degree 4 and power functions on finite interval domains, and use models to solve practical problems with and without technology; verify and evaluate the usefulness of the model using qualitative statements and quantitative analysis
Topic 5: Further differentiation and applications 1
Differentiation rules
Unit 2: Calculus and further functions > Topic 5: Further differentiation and applications 1 > Differentiation rules
- Understand and apply the product rule and quotient rule for power and polynomial functions
- Understand the notion of composition of power and polynomial functions and use the chain rule for determining the derivatives of composite functions
- Select and apply the product rule, quotient rule and chain rule to differentiate power and polynomial functions; express derivative in simplest and factorised form
Topic 6: Discrete random variables 1
General discrete random variables
Unit 2: Calculus and further functions > Topic 6: Discrete random variables 1 > General discrete random variables
- Understand the concepts of a discrete random variable and its associated probability function, and its use in modelling data
- Use relative frequencies obtained from data to determine point estimates of probabilities associated with a discrete random variable
- Recognise uniform discrete random variables and use them to model random phenomena with equally likely outcomes
- Examine simple examples of non-uniform discrete random variables
- Recognise the mean or expected value of a discrete random variable as a measurement of centre, and evaluate it in simple cases
- Recognise the variance and standard deviation of a discrete random variable as a measurement of spread, and evaluate these in simple cases
- Use discrete random variables and associated probabilities to solve practical problems
Unit 3: Further calculus
view_agenda query_statsTopic 1: The logarithmic function 2
view_agenda query_statsLogarithmic laws and logarithmic functions
view_agenda query_statsUnit 3: Further calculus > Topic 1: The logarithmic function 2 > Logarithmic laws and logarithmic functions
- Establish and use logarithmic laws and definitions
- Interpret and use logarithmic scales such as decibels in acoustics, the Richter scale for earthquake magnitude, octaves in music, pH in chemistry
- Solve equations involving indices with and without technology
- Recognise the qualitative features of the graph of \( y = \log_a(x) \) (\( a > 1 \)), including asymptotes, and of its translations \( y = \log_a(x) + b \) and \( y = \log_a(x + c) \)
- Solve equations involving logarithmic functions with and without technology
- Identify contexts suitable for modelling by logarithmic functions and use them to solve practical problems; verify and evaluate the usefulness of the model using qualitative statements and quantitative analysis
Topic 2: Further differentiation and applications 2
view_agenda query_statsCalculus of exponential functions
view_agenda query_statsUnit 3: Further calculus > Topic 2: Further differentiation and applications 2 > Calculus of exponential functions
- Estimate the limit of \( \frac{a^h - 1}{h} \) as \( h \to 0 \) using technology, for various values of \( a > 0 \)
- Recognise that e is the unique number for which the above limit is 1
- Define the exponential function \( e^x \)
- Establish and use the formulas \( \frac{d}{dx} (e^x) = e^x \) and \( \frac{d}{dx} (e^{f(x)}) = f'(x)e^{f(x)} \)
- Identify contexts suitable for mathematical modelling by exponential functions and their derivatives and use the models to solve practical problems; verify and evaluate the usefulness of the model using qualitative statements and quantitative analysis
Calculus of logarithmic functions
view_agenda query_statsUnit 3: Further calculus > Topic 2: Further differentiation and applications 2 > Calculus of logarithmic functions
- Define the natural logarithm \( \ln(x) = \log_e(x) \)
- Recognise and use the inverse relationship of the functions \( y = e^x \) and \( y = \ln(x) \)
- Establish and use the formulas \( \frac{d}{dx} (\ln(x)) = \frac{1}{x} \) and \( \frac{d}{dx} (\ln(f(x))) = \frac{f'(x)}{f(x)} \)
- Use logarithmic functions and their derivatives to solve practical problems
Calculus of trigonometric functions
view_agenda query_statsUnit 3: Further calculus > Topic 2: Further differentiation and applications 2 > Calculus of trigonometric functions
- Establish the formulas \( \frac{d}{dx} (\sin(x)) = \cos(x) \), and \( \frac{d}{dx} (\cos(x)) = - \sin(x) \) by numerical estimations of the limits and informal proofs based on geometric constructions
- Identify contexts suitable for modelling by trigonometric functions and their derivatives and use the models to solve practical problems; verify and evaluate the usefulness of the model using qualitative statements and quantitative analysis
- Use trigonometric functions and their derivatives to solve practical problems; including trigonometric functions of the form \( y = \sin(f(x)) \) and \( y = \cos(f(x)) \)
Differentiation rules
view_agenda query_statsUnit 3: Further calculus > Topic 2: Further differentiation and applications 2 > Differentiation rules
- Select and apply the product rule, quotient rule and chain rule to differentiate functions; express derivatives in simplest and factorised form
Topic 3: Integrals
view_agenda query_statsAnti-differentiation
view_agenda query_statsUnit 3: Further calculus > Topic 3: Integrals > Anti-differentiation
- Recognise anti-differentiation as the reverse of differentiation
- Use the notation \(\int f(x)dx\) for anti-derivatives or indefinite integrals
- Establish and use the formula \(\int x^n dx = \frac{1}{n+1}x^{n+1} + c\) for \(n \neq -1\)
- Establish and use the formula \(\int e^x dx = e^x + c\)
- Establish and use the formulas \(\int \frac{1}{x} dx = \ln|x| + c\), for \(x > 0\) and \(\int \frac{1}{ax+b} dx = \frac{1}{a}\ln|ax+b| + c\)
- Establish and use the formulas \(\int \sin(x) dx = -\cos(x) + c\) and \(\int \cos(x)dx = \sin(x) + c\)
- Understand and use the formula for indefinite integrals of the form \(\int (f(x) + g(x))dx = \int f(x)dx + \int g(x)dx\)
- Determine indefinite integrals of the form \(\int f(ax + b)dx\)
- Determine \(f(x)\), given \(f'(x)\) and an initial condition \(f(a) = b\)
- Determine the integral of a function using information about the derivative of the given function (integration by recognition)
- Determine displacement given velocity in linear motion problems
Fundamental theorem of calculus and definite integrals
view_agenda query_statsUnit 3: Further calculus > Topic 3: Integrals > Fundamental theorem of calculus and definite integrals
- Examine the area problem, and use sums of the form \(\sum_{i} f(x_i) \Delta x_i\), to estimate the area under the curve \(y = f(x)\)
- Use the trapezoidal rule for the approximation of the value of a definite integral numerically
- Interpret the definite integral \(\int_{a}^{b} f(x)dx\) as area under the curve \(y = f(x)\) if \(f(x) > 0\)
- Recognise the definite integral \(\int_{a}^{b} f(x)dx\) as a limit of sums of the form \(\sum_{i} f(x_i) \Delta x_i\)
- Understand the formula \(\int_{a}^{b} f(x)dx = F(b) - F(a)\) and use it to calculate definite integrals
Applications of integration
view_agenda query_statsUnit 3: Further calculus > Topic 3: Integrals > Applications of integration
- Calculate the area under a curve
- Calculate total change by integrating instantaneous or marginal rate of change
- Calculate the area between curves with and without technology
- Determine displacements given acceleration and initial values of displacement and velocity
Unit 4: Further functions and statistics
view_agenda query_statsTopic 1: Further differentiation and applications 3
view_agenda query_statsThe second derivative and applications of differentiation
view_agenda query_statsUnit 4: Further functions and statistics > Topic 1: Further differentiation and applications 3 > The second derivative and applications of differentiation
- Understand the concept of the second derivative as the rate of change of the first derivative function
- Recognise acceleration as the second derivative of displacement position with respect to time
- Understand the concepts of concavity and points of inflection and their relationship with the second derivative
- Understand and use the second derivative test for finding local maxima and minima
- Sketch the graph of a function using first and second derivatives to locate stationary points and points of inflection
- Solve optimisation problems from a wide variety of fields using first and second derivatives, where the function to be optimised is both given and developed
Topic 2: Trigonometric functions 2
view_agenda query_statsCosine and sine rules
view_agenda query_statsUnit 4: Further functions and statistics > Topic 2: Trigonometric functions 2 > Cosine and sine rules
- Recall sine, cosine and tangent as ratios of side lengths in right-angled triangles
- Understand the unit circle definition of \( \cos(\pi) \), \( \sin(\pi) \), and \( \tan(\pi) \) and periodicity using degrees and radians
- Establish and use the sine (ambiguous case is required) and cosine rules and the formula \( area = \frac{1}{2}bc\sin(A) \) for the area of a triangle
- Construct mathematical models using the sine and cosine rules in two- and three-dimensional contexts (including bearings in two-dimensional context) and use the model to solve problems; verify and evaluate the usefulness of the model using qualitative statements and quantitative analysis
Topic 3: Discrete random variables 2
view_agenda query_statsBernoulli distributions
view_agenda query_statsUnit 4: Further functions and statistics > Topic 3: Discrete random variables 2 > Bernoulli distributions
- Use a Bernoulli random variable as a model for two-outcome situations
- Identify contexts suitable for modelling by Bernoulli random variables
- Recognise and determine the mean \( p \) and variance \( p(1 - p) \) of the Bernoulli distribution with parameter \( p \)
- Use Bernoulli random variables and associated probabilities to model data and solve practical problems
Binomial distributions
view_agenda query_statsUnit 4: Further functions and statistics > Topic 3: Discrete random variables 2 > Binomial distributions
- Understand the concepts of Bernoulli trials and the concept of a binomial random variable as the number of 'successes' 'n' in independent Bernoulli trials, with the same probability of success 'p' in each trial
- Identify contexts suitable for modelling by binomial random variables
- Determine and use the probabilities \( P(X = r) = \binom{n}{r} p^r (1-p)^{n-r} \) associated with the binomial distribution with parameters n and p
- Calculate the mean \( np \) and variance \( np(1 - p) \) of a binomial distribution using technology and algebraic methods
- Identify contexts suitable to model binomial distributions and associated probabilities to solve practical problems, including the language of 'at most' and 'at least'
Topic 4: Continuous random variables and the normal distribution
view_agenda query_statsGeneral continuous random variables
view_agenda query_statsUnit 4: Further functions and statistics > Topic 4: Continuous random variables and the normal distribution > General continuous random variables
- Use relative frequencies and histograms obtained from data to estimate probabilities associated with a continuous random variable
- Understand the concepts of a probability density function, cumulative distribution function, and probabilities associated with a continuous random variable given by integrals; examine simple types of continuous random variables and use them in appropriate contexts
- Calculate the expected value, variance and standard deviation of a continuous random variable in simple cases
- Understand standardised normal variables (z-values, z-scores) and use these to compare samples
Normal distributions
view_agenda query_statsUnit 4: Further functions and statistics > Topic 4: Continuous random variables and the normal distribution > Normal distributions
- Identify contexts, such as naturally occurring variations, that are suitable for modelling by normal random variables
- Recognise features of the graph of the probability density function of the normal distribution with mean \( \mu \) and standard deviation \( \sigma \) and the use of the standard normal distribution
- Calculate probabilities and quantiles associated with a given normal distribution using technology and use these to solve practical problems
Topic 5: Interval estimates for proportions
view_agenda query_statsRandom sampling
view_agenda query_statsUnit 4: Further functions and statistics > Topic 5: Interval estimates for proportions > Random sampling
- Understand the concept of a random sample
- Discuss sources of bias in samples, and procedures to ensure randomness
- Investigate the variability of random samples from various types of distributions, including uniform, normal and Bernoulli, using graphical displays of real and simulated data
Sample proportions
view_agenda query_statsUnit 4: Further functions and statistics > Topic 5: Interval estimates for proportions > Sample proportions
- Understand the concept of the sample proportion \(\hat{p}\) as a random variable whose value varies between samples, and the formulas for the mean \( p \) and standard deviation \[\sqrt{\dfrac{p(1 - p)}{n}}\] of the sample proportion \(\hat{p}\)
- Consider the approximate normality of the distribution of \(\hat{p}\) for large samples
- Simulate repeated random sampling, for a variety of values of \( p \) and a range of sample sizes, to illustrate the distribution of \(\hat{p}\) and the approximate standard normality of \( \dfrac{\hat{p} - p}{\sqrt{\dfrac{p(1 - p)}{n}}} \) where the closeness of the approximation depends on both \( n \) and \( p \)
Confidence intervals for proportions
view_agenda query_statsUnit 4: Further functions and statistics > Topic 5: Interval estimates for proportions > Confidence intervals for proportions
- Understand the concept of an interval estimate for a parameter associated with a random variable
- Use the approximate confidence interval \( \left(\hat{p} - z \sqrt{\dfrac{p(1 - p)}{n}}, \hat{p} + z \sqrt{\dfrac{p(1 - p)}{n}}\right) \), as an interval estimate for \( p \), where \( z \) is the appropriate quantile for the standard normal distribution
- Define the approximate margin of error \( E = z \sqrt{\dfrac{p(1 - p)}{n}} \) and understand the trade-off between margin of error and level of confidence
- Use simulation to illustrate variations in confidence intervals between samples and to show that most but not all confidence intervals contain \( p \)