H2 Mathematics
Lessons in h2 mathematics. This course is still being developed; independent practice and assessment coverage is incomplete.
More
Offline pages
Saved on this device only. Connect at least once every 30 days to confirm Premium. Sign-out or a change of account removes saved copies.
Lesson progress unavailable
Functions and graphs
Your saved work does not identify a current lesson. Choose where to continue.
About lesson progress
Lesson marks record what you have studied. Checks show understanding separately. Guest marks come from this device.
Find this activity in your course
Topics
H2 Mathematics
- Functions and graphsSuggested next · 5 core lessonsBuild and connect the representations and methods in functions and graphs.
- Sequences and seriesSuggested next · 2 core lessonsBuild and connect the representations and methods in sequences and series.
- Vectors in two and three dimensionsSuggested next · 3 core lessonsBuild and connect the representations and methods in vectors in two and three dimensions.
- Complex numbersSuggested next · 2 core lessonsBuild and connect the representations and methods in complex numbers.
- DifferentiationSuggested next · 6 core lessonsInterpret derivative graphs and solve problems about tangents, normals, optimisation and connected rates.
- Series approximationsSuggested next · 2 core lessonsChoose valid local expansions, derive composite and implicit coefficients, and check the accuracy of truncated approximations.
- IntegrationSuggested next · 4 core lessonsBuild and connect the representations and methods in integration.
- Differential equationsSuggested next · 1 core lessonSolve separable equations, retain initial-value domains and excluded solutions, then formulate and interpret changing-quantity models.
- ProbabilitySuggested next · 2 core lessonsCombine conditional probability with restricted, repeated and circular arrangements under an explicit sampling convention.
- Probability modelsSuggested next · 3 core lessonsBuild discrete distributions, choose binomial and normal models, and distinguish transformations from independent combinations.
- Sampling and inferenceSuggested next · 2 core lessonsConnect sampling design, mean uncertainty and unbiased variance estimates to planned population-mean tests and contextual conclusions.
- Correlation and regressionSuggested next · 2 core lessonsUse directional least-squares fits and square, reciprocal or log transformations, then assess predictions in their original setting.
No topics match your search. Try a shorter phrase, or show all topics.
Practise and check
Choose practice
- GCE A-Level H2 Mathematics: Paper 1 practice
- GCE A-Level H2 Mathematics: Paper 2 practice
- All practice for this course
Other Science combinations
Choose the syllabus edition you are studying.
H2 Mathematics 2024
Mixed practice
Practise a mix of topics from this course.
About 10 minutes
Time is up and your answers could not be submitted; check your connection and try again.
Paper instructions
This is for practice only and doesn’t count towards your course progress.
- Question count is checked before you start.
- Feedback appears after you finish and submit.
Your work
Without an account, unfinished answers stay in this tab. Signed in, they are saved on this device until the deadline shown when you start.
Review settings
Checking saved course progress…
Getting your questions…
Recent attempts
History is stored only in this browser.
No completed attempts are saved yet.
Beyond the syllabus: optional enrichment that does not count towards your progress.
GCE A-Level H2 Mathematics 2027
Mixed practice
Practise a mix of topics from this course.
About 10 minutes
Time is up and your answers could not be submitted; check your connection and try again.
Paper instructions
This is for practice only and doesn’t count towards your course progress.
- Question count is checked before you start.
- Feedback appears after you finish and submit.
Your work
Without an account, unfinished answers stay in this tab. Signed in, they are saved on this device until the deadline shown when you start.
Review settings
Checking saved course progress…
Getting your questions…
Recent attempts
History is stored only in this browser.
No completed attempts are saved yet.
Beyond the syllabus: optional enrichment that does not count towards your progress.
Course info
Course guidance
Lessons in h2 mathematics. This course is still being developed; independent practice and assessment coverage is incomplete.
Learning goals
- Determine function domains, ranges and admissible inputs
- Compose and invert functions with existence conditions
- Interpret and draw graphs of functions
- Transform graphs and relate representations
- Solve inequalities and represent their solution sets
- Formulate, use and interpret a mathematical model
- Express and use the general term of a sequence
- Find finite sums and analyse convergence of series
- Add vectors by ordered components
- Subtract vectors by ordered components
- Multiply a vector by a scalar
- Calculate the magnitude of a vector
- Produce a unit vector in a given direction
- Locate a point from a displacement parameter
- Find the directed displacement between two points
- Calculate the distance between two points
- Test all coordinates for point collinearity
- Locate an internally or externally dividing point from its ratio
- Explain vector geometry, position, displacement and ratio relationships
- Calculate a scalar product
- Calculate an ordered vector product
- Calculate the angle between directed vectors
- Find a signed scalar component along a direction
- Find the parallel vector component along a direction
- Find the perpendicular component's magnitude
- Justify the geometrical meanings and properties of vector products
- Produce a vector equation of a spatial line
- Produce independent Cartesian equalities for a spatial line
- Produce a vector equation of a plane
- Produce a Cartesian equation of a plane
- Distinguish coincident, parallel, intersecting and skew lines
- Classify a line's relationship with a plane
- Distinguish coincident, parallel and intersecting planes
- Produce the intersection line of two planes
- Find the foot of a perpendicular from a point to a line
- Find the perpendicular distance from a point to a line
- Find the foot of a perpendicular from a point to a plane
- Find the perpendicular distance from a point to a plane
- Find the unique intersection point of a line and a plane
- Find the smaller angle between spatial lines
- Find the smaller angle between planes
- Find the angle between a line and a plane
- Justify spatial vector relationships and parameter conditions
- Find the parallel component's magnitude
- Add complex numbers
- Subtract complex numbers
- Multiply complex numbers
- Divide by a nonzero complex number
- Find the conjugate of a complex number
- Calculate the modulus of a complex number
- Calculate a complex number's principal argument
- Find every distinct root of a real quadratic over the complex numbers
- Solve a unique linear complex equation, including a conjugate term
- Recover the remaining distinct roots of a real cubic or quartic from a non-real root
- Explain complex equality, the number system and real-coefficient root properties
- Represent and explain complex operations on an Argand diagram
- Relate a function to its derivative graphs
- Differentiate implicit and parametric relationships within stated scope
- Classify stationary points using justified derivative tests
- Find and interpret a tangent to a curve
- Find and interpret a normal to a curve
- Model and solve an optimisation problem
- Relate connected rates of change
- Use and derive bounded Maclaurin approximations
- Find antiderivatives within the permitted scope
- Select and apply permitted integration techniques
- Evaluate definite integrals
- Relate a definite integral to area under a curve
- Relate definite integrals to signed area and geometric area
- Use integrals to calculate volumes of revolution
- Formulate, solve and interpret separable differential equations
- Calculate event probabilities from an explicit sample space
- Distinguish independence from mutual exclusivity
- Calculate and interpret conditional probabilities
- Use counting principles, permutations and combinations
- Build a finite distribution from equally likely outcome counts
- Calculate an event probability from a finite distribution
- Calculate a finite discrete variable's expectation
- Calculate a finite discrete variable's second moment
- Calculate a finite discrete variable's variance
- Build a transformed finite distribution, combining equal values
- Calculate the expectation of a transformed finite variable
- Calculate the variance of a transformed finite variable
- Explain discrete models, probability constraints and moment interpretation
- Choose and use a binomial probability model
- Calculate an event probability in a specified binomial model
- Calculate a conditional probability in a specified binomial model
- Calculate the expectation of a specified binomial count
- Calculate the variance of a specified binomial count
- Recover the unique binomial parameters from exact moments
- Find the least trial count satisfying a binomial probability threshold
- Choose and use a normal probability model
- Calculate an event probability in a specified normal model
- Calculate a conditional probability in a specified normal model
- Recover a normal boundary from its specified tail probability
- Recover a normal mean from a probability and known variance
- Recover a normal variance from a probability and known mean
- Recover the unique normal mean and variance from two probabilities
- Calculate a linear combination's expectation from specified moments
- Calculate a linear combination's variance with justified dependence conditions
- Calculate a probability for a specified combination of normal variables
- Represent random variables, expectations and variances
- Reason about sampling and sampling distributions
- Calculate a population-mean estimate from a specified independent sample
- Calculate an unbiased population-variance estimate from a specified independent sample
- Calculate an estimated standard error from a specified independent sample
- Calculate a sample mean's expectation from specified population moments
- Calculate a sample mean's variance from specified population moments
- Calculate a sample mean's standard error from specified population moments
- Calculate a sample-mean probability under a specified exact or approximate normal model
- Calculate a minimum sample size for specified normal-model precision
- Formulate and interpret a justified hypothesis test
- Calculate a signed population-mean test statistic under a stipulated normal-test model
- Calculate a population-mean test p-value for a prespecified alternative
- Calculate all standardised critical boundaries for a planned population-mean test
- Calculate all sample-mean critical boundaries for a planned known-variance normal test
- Determine a population-mean test decision using the unrounded planned p-value and level
- Fit and interpret correlation and regression models
- Calculate product-moment correlation from specified paired data
- Calculate complete ordered intercept and gradient from specified paired data
- Calculate original-response model prediction from specified paired data
- Calculate signed residual on the fitted response scale from specified paired data
- Calculate sum of squared residuals on the fitted response scale from specified paired data
- Calculate original-input solution of a specified fitted calibration model from specified paired data
- Calculate product-moment correlation from specified paired data and transformations
- Calculate complete ordered intercept and gradient from specified paired data and transformations
- Calculate original-response model prediction from specified paired data and transformations
- Calculate signed residual on the fitted response scale from specified paired data and transformations
- Calculate sum of squared residuals on the fitted response scale from specified paired data and transformations
- Calculate original-input solution of a specified fitted calibration model from specified paired data and transformations
Syllabus statements covered
- Include: function, domain and range; inverse and composite functions; conditions for existence of inverse and composite functions; restricting domain to obtain an inverse function; relationship between a one-to-one function and its inverse graphs. Exclude (fg)⁻¹ = g⁻¹f⁻¹ and restricting domain to obtain a composite function.
- Include: graphing a given function with graphing calculator or graphing software; symmetry, intercepts, turning points and asymptotes for y² = ax, x² = by, x²/a² + y²/b² = 1, x²/a² − y²/b² = 1, y²/b² − x²/a² = 1, y = (ax + b)/(cx + d), y = (ax² + bx + c)/(dx + e); equations of asymptotes and axes of symmetry and restrictions on x and/or y; transformations af(x), f(x) + a, f(x + a), f(ax) and combinations; relating |f(x)|, f(|x|) and 1/f(x) to f(x); simple parametric equations and graphs.
- Include: formulating equations, systems of linear equations or inequalities from a problem situation; solving equations exactly or approximately and systems of linear equations using a graphing calculator or graphing software; inequalities f(x)/g(x) > 0 with linear or quadratic expressions; |x| and |x − a| < b ⇔ a − b < x < a + b, |x − a| > b ⇔ x < a − b or x > a + b; solving inequalities graphically.
- Include: finite and infinite sequences and series; sequence as y = f(n) for positive integer n; nth term u_n and sum S_n relationship; sequences defined by nth-term formula or u_(n+1) = f(u_n), including calculator or computer generation; sum and difference of two series; convergence and sum to infinity; nth term and finite sum of arithmetic and geometric series; condition for convergence and sum to infinity of an infinite geometric series.
- Include: vector addition, subtraction, scalar multiplication and geometrical interpretations; position, displacement and direction vectors; magnitude; unit vectors; distance between points; collinearity; ratio theorem in geometrical applications.
- Include: scalar and vector products and properties; angle between vectors; geometrical meanings of |a · n̂| and |a × n̂| for unit vector n̂. Exclude triple products a · (b × c) and a × (b × c).
- Include: vector and Cartesian equations of lines and planes; foot of perpendicular and distance from point to line or plane; angles between lines, line and plane, or planes; relationships between coplanar or skew lines, line and plane, and planes. Exclude shortest distance and common perpendicular between two skew lines.
- Include: extending real numbers to complex numbers; complex roots of quadratic equations; modulus, argument and conjugate; four operations; equality; conjugate roots of real-coefficient polynomials; Argand diagrams; geometrical effects of conjugation, negation, addition, subtraction and multiplication by i. Exclude polar or modulus-argument and exponential forms.
- Include: graphical interpretation of f′(x) > 0, = 0, < 0 and f″(x) > 0, < 0; relating f′ and f graphs; simple implicit and parametric differentiation; first or second derivative tests for local extrema and stationary points of inflexion in simple cases; calculator or software location of extrema and approximate derivative at a point; tangents and normals including implicit and parametric curves; local extrema and connected rates of change. Exclude non-stationary points of inflexion and second derivatives of parametric functions.
- Include: standard expansions of (1 + x)^n for rational n, e^x, sin x, cos x and ln(1 + x); first few terms by repeated differentiation (e.g. sec x), repeated implicit differentiation (e.g. y³ + y² + y = x² − 2x), or standard series (e.g. e^x cos 2x, ln((1 + x)/(1 − x))); convergence ranges; approximation of a function; small-angle approximations sin x ≈ x, cos x ≈ 1 − x²/2, tan x ≈ x. Exclude derivation of a general term.
- Include: integration of f′(x)[f(x)]^n including n = −1, f′(x)e^f(x), sin² x, cos² x, tan² x, 1/(a² + x²), 1/√(a² − x²), 1/(a² − x²), 1/(x² − a²); given substitutions; integration by parts. Exclude reduction formulae.
- Include: definite integral as limit of sum and area under a curve; evaluating definite integrals; areas bounded by curve and lines parallel to axes, curve and line, or two curves; area below x-axis; volume of revolution about x- or y-axis; approximating integrals with graphing calculator or software. Exclude area and volume of revolution where the curve is defined parametrically.
- Include: general and particular solutions of dy/dx = f(x)g(y), including reducing to this form by a given substitution; formulating a differential equation from a problem situation; interpreting an equation and its solution contextually.
- Include: addition and multiplication principles for counting; permutations nPr and combinations nCr; arrangements of objects in a line or circle, including repetition and restrictions; addition and multiplication of probabilities; mutually exclusive and independent events; tables of outcomes, Venn diagrams, tree diagrams, permutations and combinations to calculate probabilities; conditional probabilities in simple cases; P(A′) = 1 − P(A); P(A ∪ B) = P(A) + P(B) − P(A ∩ B); P(A|B) = P(A ∩ B)/P(B).
- Include: discrete random variables, probability distributions, expectations and variances; binomial distribution B(n, p) as a discrete probability model and conditions for suitability; binomial mean and variance without proof. Exclude finding a cumulative distribution function of a discrete random variable.
- Include: concept of continuous random variables; concept of normal distribution as a continuous probability model with mean and variance; use of N(μ, σ²); standard normal distribution; finding P(X < x₁) or related probabilities given x₁, μ, σ, and finding a relationship between x₁, μ, σ given a probability; symmetry and properties of the normal curve; problems using E(aX + b) and Var(aX + b), and E(aX + bY) and Var(aX + bY) for independent X and Y. Exclude normal approximation to binomial distribution. The concept of continuous random variables is for teaching and learning only.
- Include: population and simple random sample; sample mean X̄ as a random variable with E(X̄) = μ and Var(X̄) = σ²/n; distribution of sample mean from a normal population; Central Limit Theorem for sufficiently large samples (e.g. n ≥ 30); unbiased estimates of population mean and variance from a sample, including summarised data Σx and Σx², or Σ(x − a) and Σ(x − a)².
- Include: null and alternative hypotheses H₀ and H₁, test statistic, critical region and value, significance level and p-value; formulating hypotheses and testing a population mean from a normal population of known variance or a large sample from any population; one- and two-tailed tests; contextual interpretation. Exclude the term Type I error, concept of Type II error and testing the difference between two population means.
- Include: scatter diagrams and plausibility of a linear relationship; correlation coefficient as a measure of fit; interpretation of product moment correlation near −1, 0 and 1; linear regression and least squares; interpolation and extrapolation; appropriate regression line for prediction or estimation in practical situations, including how well it models the situation. Include square, reciprocal or logarithmic transformations to achieve linearity. Exclude derivation of formulae, r² = b₁b₂ for regression coefficients, hypothesis tests.
- Include: function, domain and range; inverse and composite functions; conditions for existence of inverse and composite functions; restricting domain to obtain an inverse function; relationship between a one-to-one function and its inverse graphs. Exclude (fg)⁻¹ = g⁻¹f⁻¹ and restricting domain to obtain a composite function.
- Include: graphing a given function with graphing calculator or graphing software; symmetry, intercepts, turning points and asymptotes for y² = ax, x² = by, x²/a² + y²/b² = 1, x²/a² − y²/b² = 1, y²/b² − x²/a² = 1, y = (ax + b)/(cx + d), y = (ax² + bx + c)/(dx + e); equations of asymptotes and axes of symmetry and restrictions on x and/or y; transformations af(x), f(x) + a, f(x + a), f(ax) and combinations; relating |f(x)|, f(|x|) and 1/f(x) to f(x); simple parametric equations and graphs.
- Include: formulating equations, systems of linear equations or inequalities from a problem situation; solving equations exactly or approximately and systems of linear equations using a graphing calculator or graphing software; inequalities f(x)/g(x) > 0 with linear or quadratic expressions; |x| and |x − a| < b ⇔ a − b < x < a + b, |x − a| > b ⇔ x < a − b or x > a + b; solving inequalities graphically.
- Include: finite and infinite sequences and series; sequence as y = f(n) for positive integer n; nth term u_n and sum S_n relationship; sequences defined by nth-term formula or u_(n+1) = f(u_n), including calculator or computer generation; sum and difference of two series; convergence and sum to infinity; nth term and finite sum of arithmetic and geometric series; condition for convergence and sum to infinity of an infinite geometric series.
- Include: vector addition, subtraction, scalar multiplication and geometrical interpretations; position, displacement and direction vectors; magnitude; unit vectors; distance between points; collinearity; ratio theorem in geometrical applications.
- Include: scalar and vector products and properties; angle between vectors; geometrical meanings of |a · n̂| and |a × n̂| for unit vector n̂. Exclude triple products a · (b × c) and a × (b × c).
- Include: vector and Cartesian equations of lines and planes; foot of perpendicular and distance from point to line or plane; angles between lines, line and plane, or planes; relationships between coplanar or skew lines, line and plane, and planes. Exclude shortest distance and common perpendicular between two skew lines.
- Include: extending real numbers to complex numbers; complex roots of quadratic equations; modulus, argument and conjugate; four operations; equality; conjugate roots of real-coefficient polynomials; Argand diagrams; geometrical effects of conjugation, negation, addition, subtraction and multiplication by i. Exclude polar or modulus-argument and exponential forms.
- Include: graphical interpretation of f′(x) > 0, = 0, < 0 and f″(x) > 0, < 0; relating f′ and f graphs; simple implicit and parametric differentiation; first or second derivative tests for local extrema and stationary points of inflexion in simple cases; calculator or software location of extrema and approximate derivative at a point; tangents and normals including implicit and parametric curves; local extrema and connected rates of change. Exclude non-stationary points of inflexion and second derivatives of parametric functions.
- Include: standard expansions of (1 + x)^n for rational n, e^x, sin x, cos x and ln(1 + x); first few terms by repeated differentiation (e.g. sec x), repeated implicit differentiation (e.g. y³ + y² + y = x² − 2x), or standard series (e.g. e^x cos 2x, ln((1 + x)/(1 − x))); convergence ranges; approximation of a function; small-angle approximations sin x ≈ x, cos x ≈ 1 − x²/2, tan x ≈ x. Exclude derivation of a general term.
- Include: integration of f′(x)[f(x)]^n including n = −1, f′(x)e^f(x), sin² x, cos² x, tan² x, 1/(a² + x²), 1/√(a² − x²), 1/(a² − x²), 1/(x² − a²); given substitutions; integration by parts. Exclude reduction formulae.
- Include: definite integral as limit of sum and area under a curve; evaluating definite integrals; areas bounded by curve and lines parallel to axes, curve and line, or two curves; area below x-axis; volume of revolution about x- or y-axis; approximating integrals with graphing calculator or software. Exclude area and volume of revolution where the curve is defined parametrically.
- Include: general and particular solutions of dy/dx = f(x)g(y), including reducing to this form by a given substitution; formulating a differential equation from a problem situation; interpreting an equation and its solution contextually.
- Include: addition and multiplication principles for counting; permutations nPr and combinations nCr; arrangements of objects in a line or circle, including repetition and restrictions; addition and multiplication of probabilities; mutually exclusive and independent events; tables of outcomes, Venn diagrams, tree diagrams, permutations and combinations to calculate probabilities; conditional probabilities in simple cases; P(A′) = 1 − P(A); P(A ∪ B) = P(A) + P(B) − P(A ∩ B); P(A|B) = P(A ∩ B)/P(B).
- Include: discrete random variables, probability distributions, expectations and variances; binomial distribution B(n, p) as a discrete probability model and conditions for suitability; binomial mean and variance without proof. Exclude finding a cumulative distribution function of a discrete random variable.
- Include: concept of continuous random variables; concept of normal distribution as a continuous probability model with mean and variance; use of N(μ, σ²); standard normal distribution; finding P(X < x₁) or related probabilities given x₁, μ, σ, and finding a relationship between x₁, μ, σ given a probability; symmetry and properties of the normal curve; problems using E(aX + b) and Var(aX + b), and E(aX + bY) and Var(aX + bY) for independent X and Y. Exclude normal approximation to binomial distribution. The concept of continuous random variables is for teaching and learning only.
- Include: population and simple random sample; sample mean X̄ as a random variable with E(X̄) = μ and Var(X̄) = σ²/n; distribution of sample mean from a normal population; Central Limit Theorem for sufficiently large samples (e.g. n ≥ 30); unbiased estimates of population mean and variance from a sample, including summarised data Σx and Σx², or Σ(x − a) and Σ(x − a)².
- Include: null and alternative hypotheses H₀ and H₁, test statistic, critical region and value, significance level and p-value; formulating hypotheses and testing a population mean from a normal population of known variance or a large sample from any population; one- and two-tailed tests; contextual interpretation. Exclude the term Type I error, concept of Type II error and testing the difference between two population means.
- Include: scatter diagrams and plausibility of a linear relationship; correlation coefficient as a measure of fit; interpretation of product moment correlation near −1, 0 and 1; linear regression and least squares; interpolation and extrapolation; appropriate regression line for prediction or estimation in practical situations, including how well it models the situation. Include square, reciprocal or logarithmic transformations to achieve linearity. Exclude derivation of formulae, r² = b₁b₂ for regression coefficients, hypothesis tests.