WJEC Level 3 Advanced GCE in Mathematics
Official qualification regulated by Ofqual, awarded by WJEC. Classified as a GCE A Level. Total guided learning hours: 360 hours.
Course Syllabus & Study Modules
Complete breakdown of verified curriculum topics and learning objectives.
Algebra and Functions (A2 Unit 3: Pure Mathematics B)
Decompose rational functions into partial fractions (denominators not more complicated than squared linear terms and with no more than 3 terms, numerators constant or linear).
- Decompose rational expressions with distinct linear factors into partial fractions.
- Decompose rational expressions containing repeated linear factors into partial fractions.
- Determine partial fraction constants using strategic substitution and equating coefficients.
Algebra and Functions (A2 Unit 3: Pure Mathematics B)
Simplify rational expressions, including by factorising and cancelling and by algebraic division (by linear expressions only).
- Factorise and simplify complex algebraic rational expressions.
- Divide polynomials by linear expressions using algebraic long division.
- Rewrite improper algebraic fractions as a polynomial quotient plus a proper fraction.
Algebra and Functions (A2 Unit 3: Pure Mathematics B)
Sketch curves defined by the modulus of a linear function.
- Sketch graphs of modulus functions y = |ax + b| with vertices and intercepts labelled.
- Solve equations involving modulus functions algebraically and graphically.
- Solve inequalities involving modulus functions using critical values and graph sketches.
Algebra and Functions (A2 Unit 3: Pure Mathematics B)
Understand and use composite functions; inverse functions and their graphs.
- Form and simplify composite functions fg(x) and evaluate their domain and range.
- Determine conditions under which a function has an inverse and calculate f⁻¹(x) algebraically.
- Sketch inverse functions as reflections in the line y = x and locate intersection points.
Algebra and Functions (A2 Unit 3: Pure Mathematics B)
Understand the effect of combinations of transformations on the graph of y = f (x), as represented by y = af (x), y = f (x) + a, y = f (x + a) and y = f (ax).
- Analyse and describe combinations of vertical and horizontal graph transformations.
- Determine the effect of combined transformations on coordinates of turning points and asymptotes.
- Sketch curves resulting from multiple sequential algebraic transformations.
Algebra and Functions (A2 Unit 3: Pure Mathematics B)
Use of functions in modelling, including consideration of limitations and refinements of the models.
- Select and construct appropriate mathematical functions to model physical scenarios.
- Determine practical domain and range restrictions based on physical constraints.
- Evaluate the strengths, limitations, and refinement opportunities of mathematical models.
Algebra and Functions (AS Unit 1: Pure Mathematics A)
Completing the square.
- Complete the square for monic and non-monic quadratic expressions.
- Deduce the coordinates and nature of the turning point directly from completed square form.
- Apply completed square form to solve quadratic equations and establish bounds.
Algebra and Functions (AS Unit 1: Pure Mathematics A)
Express solutions through the correct use of ‘and’ and ‘or’, or through set notation.
- Distinguish between conjunction ('and') and disjunction ('or') in algebraic solutions.
- Express continuous and disjoint inequality solutions using standard set-builder notation.
- Use set union (∪) and intersection (∩) symbols correctly to represent solution sets.
Algebra and Functions (AS Unit 1: Pure Mathematics A)
Interpret algebraic solutions of equations graphically.
- Relate the real roots of f(x) = g(x) to the geometric intersection points of y = f(x) and y = g(x).
- Use algebraic discriminants to determine whether a line is a tangent, secant, or does not intersect a curve.
- Interpret algebraic solutions geometrically in applied and pure mathematical contexts.
Algebra and Functions (AS Unit 1: Pure Mathematics A)
Manipulate polynomials algebraically, including expanding brackets and collecting like terms, factorisation and simple algebraic division; use of the Factor Theorem.
- Apply the Factor Theorem to identify linear factors of polynomials.
- Perform algebraic division to divide a polynomial by a linear expression.
- Factorise cubic and higher-order polynomials fully into linear and irreducible factors.
Algebra and Functions (AS Unit 1: Pure Mathematics A)
Represent linear and quadratic inequalities graphically.
- Graph linear and quadratic boundary curves using dashed lines for strict inequalities and solid for non-strict.
- Identify feasible planar regions using coordinates of convenient test points.
- Represent systems of simultaneous inequalities graphically and label the feasible region.
Algebra and Functions (AS Unit 1: Pure Mathematics A)
Solution of quadratic equations in a function of the unknown.
- Recognise algebraic and transcendental equations that have an underlying quadratic form.
- Apply a substitution u = g(x) to transform non-linear equations into standard quadratics.
- Filter out invalid intermediate values by checking the domain of the substitution function.
Algebra and Functions (AS Unit 1: Pure Mathematics A)
Solve linear and quadratic inequalities in a single variable and interpret such inequalities graphically, including inequalities with brackets and fractions.
- Solve linear inequalities, reversing inequality signs when multiplying or dividing by negative numbers.
- Find critical values and solve quadratic inequalities using curve sketches or sign charts.
- Handle algebraic inequalities containing brackets and fractions with variable denominators.
Algebra and Functions (AS Unit 1: Pure Mathematics A)
Solve simultaneous equations in two variables by elimination and by substitution, including one linear and one quadratic equation.
- Solve simultaneous linear and quadratic equations using the substitution method.
- Pair solutions correctly as coordinate pairs representing geometric intersection points.
- Interpret the discriminant of the resulting quadratic to determine the number of intersection points.
Algebra and Functions (AS Unit 1: Pure Mathematics A)
The discriminant of a quadratic function, including the conditions for real roots and repeated roots.
- Calculate the discriminant of a quadratic equation accurately.
- Determine the number and nature of real roots using the value of b² − 4ac.
- Form and solve equations and inequalities in an unknown parameter based on root conditions.
Algebra and Functions (AS Unit 1: Pure Mathematics A)
Understand and use graphs of functions; sketch curves defined by simple equations, including polynomials.
- Determine the global end behavior of polynomial functions from degree and leading coefficient.
- Locate all real roots and calculate the y-intercept of polynomial functions.
- Sketch polynomial graphs showing distinct crossings, tangential touches, and inflection points.
Algebra and Functions (AS Unit 1: Pure Mathematics A)
Understand and use proportional relationships and their graphs.
- Translate statements of direct and inverse variation into algebraic equations with a constant k.
- Calculate constants of proportionality from given boundary conditions or data points.
- Sketch graphs representing direct and inverse power proportionalities.
Algebra and Functions (AS Unit 1: Pure Mathematics A)
Understand and use the laws of indices for all rational exponents.
- Apply the laws of indices to simplify complex algebraic and numerical expressions.
- Convert between radical notation, reciprocal expressions, and rational exponents.
- Manipulate expressions with fractional and negative powers to solve equations.
Algebra and Functions (AS Unit 1: Pure Mathematics A)
Understand the effect of simple transformations on the graph of y = f (x) including sketching associated graphs: y = af (x), y = f (x) + a, y = f (x + a ), y = f (ax).
- Apply translations, stretches, and reflections to the graph of y = f(x).
- Describe single transformations using correct terminology, vectors, and scale factors.
- Determine the transformed coordinates of turning points, intercepts, and asymptotes.
Algebra and Functions (AS Unit 1: Pure Mathematics A)
Use and manipulate surds, including rationalising the denominator.
- Simplify surds by extracting square factors from under the radical sign.
- Multiply, divide, and expand expressions involving surds accurately.
- Rationalise monomial and binomial denominators using conjugate surd expressions.
Algebra and Functions (AS Unit 1: Pure Mathematics A)
Use intersection points of graphs of curves to solve equations.
- Rearrange complex algebraic equations into pairs of standard functions y = f(x) and y = g(x).
- Sketch pairs of curves on identical axes to locate and count intersection points.
- Deduce the number and approximate values of real solutions from graphical intersections.
Algebra and Functions (AS Unit 1: Pure Mathematics A)
Work with quadratic functions and their graphs.
- Determine roots, intercepts, and vertex coordinates of quadratic functions.
- Sketch quadratic graphs with correct orientation, symmetry, and labelled intercepts.
- Interpret the vertex of a quadratic function as a global maximum or minimum value.
Algebra and Functions (AS Unit 1: Pure Mathematics A)
y = a/x and y = a/x² , including their vertical and horizontal asymptotes.
- Sketch reciprocal curves y = a/x and y = a/x² displaying correct quadrant placement and symmetry.
- Identify vertical and horizontal asymptotes from algebraic equations of reciprocal functions.
- Apply translations and scalings to reciprocal curves and determine transformed asymptotes.
Coordinate geometry in the (x, y) plane (A2 Unit 3: Pure Mathematics B)
Understand and use the parametric equations of curves and conversion between Cartesian and parametric forms.
- Convert parametric equations into Cartesian form by algebraic substitution and trigonometric identities.
- Determine the Cartesian domain and range from given parametric intervals.
- Differentiate parametric equations using dy/dx = (dy/dt) / (dx/dt).
Coordinate geometry in the (x, y) plane (A2 Unit 3: Pure Mathematics B)
Use parametric equations in modelling in a variety of contexts.
- Formulate parametric models where the parameter represents time or angular displacement.
- Solve practical kinematic problems involving trajectories, clearances, and collision times.
- Interpret parametric derivatives as physical velocity components dx/dt and dy/dt.
Coordinate geometry in the (x, y) plane (AS Unit 1: Pure Mathematics A)
Be able to use straight line models in a variety of contexts.
- Formulate linear models y = mx + c from contextual descriptions and data pairs.
- Interpret the gradient and intercept of linear models in practical real-world terms with units.
- Evaluate the validity, domain limitations, and extrapolation risks of linear models.
Coordinate geometry in the (x, y) plane (AS Unit 1: Pure Mathematics A)
Understand and use the coordinate geometry of the circle using the equation of a circle in the form (x − a)² + (y − b)² = r²; completing the square to find the centre and radius of a circle .
- Write down the equation of a circle given its centre (a, b) and radius r.
- Complete the square on general quadratic forms to identify the centre and radius of a circle.
- Determine geometric relationships between points, lines, tangents, and circles.
Coordinate geometry in the (x, y) plane (AS Unit 1: Pure Mathematics A)
Understand and use the equation of a straight line, including the forms y = mx + c, y − y₁ = m(x − x₁) and ax + by+ c = 0; gradient conditions for two straight lines to be parallel or perpendicular.
- Calculate gradients, midpoints, and lengths of line segments joining two points in the Cartesian plane.
- Form equations of straight lines in forms y = mx + c, y − y₁ = m(x − x₁), and ax + by + c = 0.
- Apply parallel (m₁ = m₂) and perpendicular (m₁m₂ = −1) gradient conditions to solve geometric problems.
Coordinate geometry in the (x, y) plane (AS Unit 1: Pure Mathematics A)
Use of the following circle properties: (i) the angle in a semicircle is a right angle; (ii) the perpendicular from the centre to a chord bisects the chord; (iii) the radius of a circle at a given point on its circumference is perpendicular to the tangent to the circle at that point.
- Apply the angle in a semicircle theorem to verify right angles and identify diameters.
- Locate circle centres by finding the intersection of perpendicular bisectors of chords.
- Determine equations of tangents and normals to circles at specified points of contact.
Data presentation and interpretation (AS Unit 2: Applied Mathematics A)
(Calculations of coefficients of regression lines are excluded.)
- Predict values of a response variable by substituting into a given linear regression equation.
- Assess the validity and reliability of interpolated versus extrapolated regression predictions.
- Interpret the contextual meaning of regression coefficients without computing manual summation formulas.
Data presentation and interpretation (AS Unit 2: Applied Mathematics A)
Be able to calculate standard deviation, including from summary statistics.
- Compute the mean and standard deviation from summary statistics including n, Σx, and Σx².
- Calculate standard deviation from grouped frequency tables using Σf, Σfx, and Σfx².
- Maintain intermediate numerical precision to avoid rounding errors in variance calculations.
Data presentation and interpretation (AS Unit 2: Applied Mathematics A)
Be able to clean data, including dealing with missing data, errors and outliers.
- Identify data entry errors, missing records, and physically impossible values in raw datasets.
- Formulate and justify strategies for handling missing, erroneous, or extreme statistical observations.
- Analyse the impact of data cleaning procedures on calculated sample means and measures of spread.
Data presentation and interpretation (AS Unit 2: Applied Mathematics A)
Connect to probability distributions.
- Demonstrate how relative frequency histograms approximate continuous probability density functions.
- Identify candidate theoretical probability models from the shape and skewness of sample data, recognising that shape alone is not decisive.
- Compare observed sample proportions against theoretical model probabilities to assess goodness-of-fit.
Data presentation and interpretation (AS Unit 2: Applied Mathematics A)
Interpret diagrams for single-variable data, including understanding that area in a histogram represents frequency.
- Calculate frequency densities and interpret histogram bars with unequal class widths.
- Determine class frequencies and total sample size from the areas of histogram rectangles.
- Estimate medians and quartiles from grouped continuous data using linear interpolation.
Data presentation and interpretation (AS Unit 2: Applied Mathematics A)
Interpret measures of central tendency and variation, extending to standard deviation.
- Compare two datasets in context by evaluating their respective measures of central tendency and dispersion.
- Select appropriate summary metrics based on the presence of skewness and extreme outliers.
- Interpret standard deviation as a measure of variability and consistency around the sample mean.
Data presentation and interpretation (AS Unit 2: Applied Mathematics A)
Interpret scatter diagrams and regression lines for bivariate data, including recognition of scatter diagrams which include distinct sections of the population.
- Interpret the gradient and intercept of a linear regression line in practical context.
- Distinguish between explanatory and response variables when formulating regression models.
- Identify distinct sub-populations in bivariate scatter plots and assess the need for separate models.
Data presentation and interpretation (AS Unit 2: Applied Mathematics A)
Recognise and interpret possible outliers in data sets and statistical diagrams.
- Calculate lower and upper outlier boundaries using quartile and standard deviation formulas.
- Identify and plot outliers on box plots following established mathematical conventions.
- Evaluate whether extreme observations represent measurement errors or genuine rare events.
Data presentation and interpretation (AS Unit 2: Applied Mathematics A)
Select or critique data presentation techniques in the context of a statistical problem.
- Select appropriate diagrammatic techniques tailored to categorical, discrete, continuous, and bivariate data.
- Critique published statistical diagrams by spotting truncated axes, poor scaling, and distorted proportions.
- Justify the use of side-by-side box plots for comparing location, spread, and skewness across distributions.
Data presentation and interpretation (AS Unit 2: Applied Mathematics A)
Understand informal interpretation of correlation.
- Describe the direction and strength of linear correlation from visual inspection of scatter diagrams.
- Distinguish between strong linear association, weak linear association, and no linear correlation.
- Recognise when bivariate data display strong non-linear relationships that linear measures overlook.
Data presentation and interpretation (AS Unit 2: Applied Mathematics A)
Understand that correlation does not imply causation.
- Explain why observed statistical correlation does not demonstrate causal influence.
- Identify and describe plausible confounding variables that produce spurious correlations in real data.
- Critique claims of causality in scientific and media contexts based purely on observational correlation.
Differentiation (A2 Unit 3: Pure Mathematics B)
Apply differentiation to find points of inflection.
- Locate candidate points of inflection by solving f''(x) = 0.
- Test and verify inflection points by proving a change in the sign of the second derivative.
- Classify inflection points as stationary (dy/dx = 0) or non-stationary (dy/dx ≠ 0).
Differentiation (A2 Unit 3: Pure Mathematics B)
Construct simple differential equations in pure mathematics.
- Translate contextual and geometric statements of rate of change into first-order differential equations.
- Incorporate proportionality constants and appropriate signs for growth and decay processes.
- Formulate differential equations describing rates of change in pure mathematics.
Differentiation (A2 Unit 3: Pure Mathematics B)
Differentiate eᵏˣ, aᵏˣ , sinkx, coskx, tankx, and related sums, differences and constant multiples.
- Differentiate exponential functions e^(kx) and general base exponentials a^(kx).
- Differentiate sine, cosine, and tangent functions with linear composite arguments.
- Apply linearity rules to differentiate complex sums of exponential and trigonometric terms.
Differentiation (A2 Unit 3: Pure Mathematics B)
Differentiate simple functions and relations defined implicitly or parametrically, for first derivative only.
- Differentiate implicit equations term-by-term with respect to x using the chain and product rules.
- Differentiate parametric equations using dy/dx = (dy/dt) / (dx/dt).
- Determine equations of tangents and normals to implicit and parametric curves.
Differentiation (A2 Unit 3: Pure Mathematics B)
Differentiate using the product rule, the quotient rule and the chain rule, including problems involving connected rates of change and inverse functions.
- Differentiate products and quotients of algebraic, exponential and trigonometric functions.
- Apply the chain rule to composite functions and evaluate derivatives of inverse functions.
- Solve connected rates of change problems in geometric and physical contexts.
Differentiation (A2 Unit 3: Pure Mathematics B)
Differentiation from first principles for sin x and cos x.
- Prove from first principles that the derivative of sin x is cos x.
- Prove from first principles that the derivative of cos x is −sin x.
- Apply compound angle formulas and small-angle limits in formal calculus proofs.
Differentiation (A2 Unit 3: Pure Mathematics B)
Understand and use the derivative of lnx.
- Differentiate the natural logarithm function ln x.
- Apply the composite logarithmic derivative rule d/dx(ln(f(x))) = f'(x)/f(x).
- Simplify logarithmic expressions using logarithm laws prior to differentiation.
Differentiation (A2 Unit 3: Pure Mathematics B)
Understand and use the second derivative as the rate of change of gradient; connection to convex and concave sections of curves, and points of inflection.
- Identify convex and concave intervals of curves using the sign of the second derivative.
- Locate potential points of inflection by solving d²y/dx² = 0.
- Verify points of inflection by proving a change of sign in the second derivative.
Differentiation (A2 Unit 4: Applied Mathematics B)
Construct simple differential equations in context (contexts may include kinematics, population growth and modelling the relationship between price and demand).
- Translate verbal descriptions of rates of change into first-order differential equations.
- Introduce and interpret constants of proportionality with appropriate mathematical signs.
- Construct mathematical differential models for kinematics, biological growth, and economic demand.
Differentiation (AS Unit 1: Pure Mathematics A)
Apply differentiation to find gradients, tangents and normals, maxima and minima, and stationary points.
- Determine equations of tangents and normals to curves at specified points.
- Locate and classify stationary points of functions using first and second derivatives.
- Apply differentiation to solve real-world optimisation problems under constraints.
Differentiation (AS Unit 1: Pure Mathematics A)
Differentiate xⁿ for rational n, and related constant multiples, sums and differences.
- Differentiate terms of the form Axⁿ for all rational powers n.
- Prepare complex expressions for differentiation by expanding brackets and splitting algebraic fractions.
- Apply linearity rules to differentiate sums, differences, and constant multiples of power terms.
Differentiation (AS Unit 1: Pure Mathematics A)
Differentiation from first principles for small positive integer powers of x.
- State and apply the formal limit definition of the derivative.
- Differentiate small positive integer powers of x (such as x² and x³) from first principles.
- Present rigorous proofs using correct limit notation throughout.
Differentiation (AS Unit 1: Pure Mathematics A)
Identify where functions are increasing or decreasing.
- Define increasing and decreasing functions in terms of the sign of the first derivative.
- Set up and solve inequalities involving derivatives to identify monotonic intervals.
- Prove that a given function is strictly increasing or decreasing across its entire domain.
Differentiation (AS Unit 1: Pure Mathematics A)
Understand and use the derivative of f (x) as the gradient of the tangent to the graph of y = f (x) at a general point (x, y); the gradient of the tangent as a limit; interpretation as a rate of change; sketching the gradient function for a given curve; second order derivatives.
- Define the derivative geometrically as the gradient of the tangent and analytically as a limit.
- Sketch the gradient function f'(x) given the graph of a curve y = f(x).
- Interpret first and second derivatives as instantaneous rates of change.
Differentiation (AS Unit 1: Pure Mathematics A)
Understand and use the second derivative as the rate of change of gradient.
- Calculate the second derivative d²y/dx² for algebraic and polynomial functions.
- Apply the second derivative test to classify stationary points as local maxima or minima.
- Interpret the second derivative geometrically as the rate of change of gradient.
Exponentials and logarithms (AS Unit 1: Pure Mathematics A)
Consideration of limitations and refinements of exponential models.
- Identify the limitations of unconstrained exponential growth and decay models.
- Explain the physical factors that cause real-world processes to diverge from pure exponential curves.
- Propose mathematical refinements such as carrying capacities and baseline offsets.
Exponentials and logarithms (AS Unit 1: Pure Mathematics A)
Know and use lnx as the inverse function of eˣ.
- Demonstrate that y = eˣ and y = ln x are inverse functions algebraically and graphically.
- Apply natural logarithms to isolate variables in exponential equations.
- Apply exponential operations to solve logarithmic equations.
Exponentials and logarithms (AS Unit 1: Pure Mathematics A)
Know and use the definition of logₐ x as the inverse of aˣ , where a is positive and x ≥ 0.
- Define logₐ x as the inverse of the exponential function aˣ.
- Convert equations fluently between logarithmic and exponential forms.
- Sketch logarithmic curves and identify their domain, range, and asymptotes.
Exponentials and logarithms (AS Unit 1: Pure Mathematics A)
Know and use the function aˣ and its graph, where a is positive.
- Sketch exponential functions y = aˣ for a > 1 and 0 < a < 1 with key features labelled.
- Identify the domain (ℝ), range (y > 0), and horizontal asymptote (y = 0) of exponential curves.
- Apply vertical and horizontal transformations to exponential functions.
Exponentials and logarithms (AS Unit 1: Pure Mathematics A)
Know and use the function eˣ and its graph.
- Define and sketch the natural exponential function y = eˣ and identify its key properties.
- Explain the unique calculus property that the derivative of eˣ is equal to itself.
- Relate y = eˣ to its inverse function y = ln x geometrically and algebraically.
Exponentials and logarithms (AS Unit 1: Pure Mathematics A)
Know and use the function lnx and its graph.
- Sketch the graph of y = ln x, labelling its x-intercept and vertical asymptote.
- State the domain and range of the natural logarithm function and its transformations.
- Differentiate and manipulate expressions containing the natural logarithm.
Exponentials and logarithms (AS Unit 1: Pure Mathematics A)
Know that the gradient of eᵏˣ is equal to keᵏˣ and hence understand why the exponential model is suitable in many applications.
- Differentiate exponential functions of the form y = e^(kx) to obtain dy/dx = k e^(kx).
- Explain why exponential functions are natural models for systems where rate of change is proportional to size.
- Apply exponential growth and decay models to real-world contexts including populations and cooling.
Exponentials and logarithms (AS Unit 1: Pure Mathematics A)
logₐ x + logₐ y = logₐ (xy)
- Prove the product law of logarithms from the laws of indices.
- Apply the product law to combine logarithmic terms into a single logarithm.
- Solve equations using the product law and verify roots against domain restrictions.
Exponentials and logarithms (AS Unit 1: Pure Mathematics A)
logₐ x − logₐ y = logₐ (x/y) k logₐ x = logₐ (xᵏ) (including, for example k = –1, k = -½)
- Apply the quotient law to express differences of logarithms as a single logarithm of a quotient.
- Apply the power law to move coefficients inside arguments as powers, including fractional and negative exponents.
- Synthesise logarithm laws to condense multi-term expressions into a single solvable term.
Exponentials and logarithms (AS Unit 1: Pure Mathematics A)
Solve equations in the form aˣ = b.
- Solve simple exponential equations of the form aˣ = b using logarithms.
- Solve equations with differing bases by taking logarithms and rearranging linear terms.
- Express solutions to exponential equations in exact logarithmic form.
Exponentials and logarithms (AS Unit 1: Pure Mathematics A)
Understand and use exponential growth and decay; use in modelling (examples may include the use of e in continuous compound interest, radioactive decay, drug concentration decay, exponential growth as model for population growth.)
- Formulate exponential growth and decay functions of the form y = y₀ e^(kt).
- Calculate unknown parameters and rates from given contextual data points.
- Determine half-life and doubling time using natural logarithms.
Exponentials and logarithms (AS Unit 1: Pure Mathematics A)
Understand and use the laws of logarithms.
- Apply the product, quotient, and power laws to simplify logarithmic expressions.
- Combine multi-term logarithmic expressions into a single compact logarithmic term.
- Solve logarithmic equations by applying the laws of logarithms systematically.
Exponentials and logarithms (AS Unit 1: Pure Mathematics A)
Use logarithmic graphs to estimate parameters in relationships of the form y = axⁿ and y = kbˣ , given data for x and y.
- Transform power laws y = axⁿ into linear relationships ln y = n ln x + ln a.
- Transform exponential relationships y = kbˣ into linear relationships ln y = x ln b + ln k.
- Estimate constants and powers from the gradient and vertical intercept of logarithmic graphs.
Forces and Newton’s laws (A2 Unit 4: Applied Mathematics B)
Extend Newton’s second law to situations where forces need to be resolved (restricted to two dimensions).
- Select and align optimal orthogonal reference axes for two-dimensional accelerated motion.
- Resolve all applied forces along chosen axes to formulate parallel and perpendicular equations.
- Solve dynamic equations to determine normal reactions, friction, and accelerations on slopes.
Forces and Newton’s laws (A2 Unit 4: Applied Mathematics B)
Limiting friction and statics.
- Identify limiting equilibrium states where friction reaches its maximum static value F = μR.
- Formulate simultaneous equilibrium equations combining force balance and limiting friction.
- Determine critical forces, tilt angles, and coefficients of friction at the threshold of slipping.
Forces and Newton’s laws (A2 Unit 4: Applied Mathematics B)
Resolve forces in two dimensions.
- Decompose coplanar forces into perpendicular components using sine and cosine functions.
- Formulate Cartesian vector representations from force magnitudes and inclination angles.
- Sum resolved components across multiple forces to determine total resultant force vectors.
Forces and Newton’s laws (A2 Unit 4: Applied Mathematics B)
The coefficient of friction.
- Define the coefficient of friction μ as a dimensionless measure of contact surface roughness.
- Calculate μ from limiting static equilibrium measurements on horizontal and inclined planes.
- Explain why the coefficient of friction is independent of contact surface area.
Forces and Newton’s laws (A2 Unit 4: Applied Mathematics B)
The motion of a body on a rough surface.
- Formulate equations of motion for particles sliding across rough horizontal and inclined surfaces.
- Calculate normal reaction and kinetic friction forces opposing particle motion.
- Determine acceleration, velocity, and stopping distances for bodies decelerating on rough surfaces.
Forces and Newton’s laws (A2 Unit 4: Applied Mathematics B)
Understand and use addition of forces; resultant forces; dynamics for motion in a plane.
- Calculate the resultant of multiple coplanar forces using Cartesian component addition.
- Determine the magnitude and direction of resultant force vectors using trigonometry and Pythagoras.
- Apply Newton's second law in two dimensions (R = ma) to evaluate particle acceleration.
Forces and Newton’s laws (A2 Unit 4: Applied Mathematics B)
Understand and use the equilibrium of a particle under coplanar forces.
- Formulate and solve orthogonal equilibrium equations (ΣF_x = 0, ΣF_y = 0) for coplanar forces.
- Construct and solve closed vector triangles for three coplanar forces in equilibrium.
- Determine unknown force magnitudes, tensions, and angles in static mechanical systems.
Forces and Newton’s laws (A2 Unit 4: Applied Mathematics B)
Understand and use the F ≤ μR model for friction.
- Apply the Coulomb friction inequality F ≤ μR to model static and dynamic surface resistance.
- Distinguish between static non-limiting friction (F < μR) and limiting or kinetic friction (F = μR).
- Evaluate normal reaction forces by resolving perpendicular to rough horizontal and inclined surfaces.
Forces and Newton’s laws (AS Unit 2: Applied Mathematics A)
(The inverse square law for gravitation is not required and g may be assumed to be constant, but learners should be aware that g is not a universal constant but depends on location.)
- Justify the modelling assumption that gravitational acceleration g is constant near Earth's surface.
- Explain why g varies with geographic altitude, latitude, and planetary body.
- Assess the limitations of assuming constant gravitational acceleration over extreme vertical distances.
Forces and Newton’s laws (AS Unit 2: Applied Mathematics A)
Applications to problems involving smooth pulleys and connected particles.
- Formulate separate equations of motion for connected particles linked over smooth pulleys.
- Solve simultaneous dynamic equations to determine common acceleration and string tension.
- Explain the mathematical consequences of modelling assumptions: light, inextensible, and smooth.
Forces and Newton’s laws (AS Unit 2: Applied Mathematics A)
Equilibrium of forces on a particle and motion in a straight line (restricted to forces in two perpendicular directions or simple cases of forces given as 2-D vectors)
- Resolve coplanar forces into two perpendicular axes to establish static equilibrium equations.
- Formulate and solve combined equilibrium and dynamic equations (ΣF_perp = 0, ΣF_parallel = ma).
- Solve for unknown vector forces and accelerations using two-dimensional Cartesian components.
Forces and Newton’s laws (AS Unit 2: Applied Mathematics A)
Understand and use Newton’s first law.
- State Newton's first law and explain the physical meaning of translational equilibrium.
- Formulate and solve force balance equations (ΣF = 0) for particles moving at constant velocity.
- Recognise that zero acceleration implies balanced forces across all orthogonal directions.
Forces and Newton’s laws (AS Unit 2: Applied Mathematics A)
Understand and use Newton’s second law for motion in a straight line (restricted to forces in two perpendicular directions or simple cases of forces given as 2-D vectors).
- Formulate and solve the equation of motion ΣF = ma for particles in rectilinear motion.
- Balance forces in the direction perpendicular to motion to evaluate normal contact forces.
- Apply Newton's second law in vector component form to solve two-dimensional particle dynamics.
Forces and Newton’s laws (AS Unit 2: Applied Mathematics A)
Understand and use Newton’s third law.
- State Newton's third law and identify the four criteria defining action-reaction force pairs.
- Distinguish between balanced forces acting on a single body and third-law pairs acting on distinct bodies.
- Apply Newton's third law to analyse internal contact forces in interacting mechanical systems.
Forces and Newton’s laws (AS Unit 2: Applied Mathematics A)
Understand and use weight and motion in a straight line under gravity; gravitational acceleration, g, and its value in S.I. units to varying degrees of accuracy.
- Calculate the weight of a particle from its mass using gravitational acceleration g = 9.8 m s⁻².
- Model vertical motion under gravity using consistent sign conventions and suvat formulas.
- Determine key vertical flight characteristics including time of flight and maximum height reached.
Forces and Newton’s laws (AS Unit 2: Applied Mathematics A)
Understand the concept of a force.
- Identify contact and non-contact forces acting on a particle in mechanical equilibrium or motion.
- Construct clear free-body force diagrams displaying all applied forces with correct directions.
- Calculate the resultant force acting on a particle by summing collinear and orthogonal force vectors.
Integration (A2 Unit 3: Pure Mathematics B)
(Separation of variables may require factorisation involving a common factor.)
- Factorise algebraic expressions to express differential equations in separable product form.
- Separate variables and integrate the resulting expressions term by term.
- Solve initial value problems where differential equations require initial factorisation.
Integration (A2 Unit 3: Pure Mathematics B)
Carry out simple cases of integration by substitution and integration by parts.
- Perform integration by substitution, transforming integrands, differentials, and limits.
- Perform integration by parts on products of algebraic, exponential, and trigonometric functions.
- Select the optimal integration method for a given integrand structure.
Integration (A2 Unit 3: Pure Mathematics B)
Evaluate the analytical solution of simple first order differential equations with separable variables, including finding particular solutions.
- Separate variables in first-order differential equations to set up solvable integrals.
- Find general solutions containing arbitrary constants of integration.
- Determine unique particular solutions by substituting specified initial conditions.
Integration (A2 Unit 3: Pure Mathematics B)
Integrate eᵏˣ , 1/x , sinkx, coskx and related sums, differences and constant multiples.
- Integrate exponential, reciprocal, and trigonometric functions with linear arguments.
- Divide by the coefficient k when reversing the chain rule for linear composites.
- Include absolute value signs in logarithmic integrals ln|x| and manage integration constants.
Integration (A2 Unit 3: Pure Mathematics B)
Integrate using partial fractions that are linear in the denominator.
- Decompose rational expressions and integrate them term-by-term.
- Integrate linear fractional terms to obtain logarithmic functions scaled by leading coefficients.
- Combine logarithmic limits into exact single logarithms using the laws of logarithms.
Integration (A2 Unit 3: Pure Mathematics B)
Integration by parts includes more than one application of the method but excludes reduction formulae.
- Apply integration by parts more than once to integrate expressions like x² e^(kx) and x² sin kx.
- Manage negative signs and algebraic constants across nested applications of the parts formula.
- Evaluate definite integrals requiring repeated integration by parts.
Integration (A2 Unit 3: Pure Mathematics B)
Integration by substitution includes finding a suitable substitution and is limited to cases where one substitution will lead to a function which can be integrated.
- Identify and select effective substitutions to evaluate algebraic and trigonometric integrals.
- Express all occurrences of the original variable and differential in terms of the new variable u.
- Integrate the transformed expression using standard polynomial or power rules.
Integration (A2 Unit 3: Pure Mathematics B)
Understand and use integration as the limit of a sum.
- Explain the definite integral as the limit of a Riemann sum as strip width δx approaches zero.
- Convert limits of summation expressions into equivalent definite integrals.
- Apply the limit of a sum concept to model physical and geometric quantities.
Integration (A2 Unit 3: Pure Mathematics B)
Understand these methods as the reverse processes of the chain rule and the product rule respectively.
- Recognise and evaluate integrals of the form ∫ f'(x)/f(x) dx and ∫ f'(x)[f(x)]ⁿ dx by inspection.
- Explain the theoretical connection between integration by substitution and the chain rule.
- Explain the theoretical derivation of integration by parts from the product rule.
Integration (A2 Unit 3: Pure Mathematics B)
Use a definite integral to find the area between two curves.
- Calculate intersection points of pairs of curves to determine limits of integration.
- Set up and evaluate definite integrals of the form ∫ [f(x) − g(x)] dx to find enclosed areas.
- Partition integration intervals when curves cross multiple times within a domain.
Integration (A2 Unit 4: Applied Mathematics B)
Evaluate the analytical solution of simple first order differential equations with separable variables, including finding particular solutions.
- Separate variables to solve first-order differential equations analytically.
- Evaluate the arbitrary constant of integration using given initial boundary conditions.
- Express particular solutions explicitly in terms of the independent variable.
Integration (A2 Unit 4: Applied Mathematics B)
Interpret the solution of a differential equation in the context of solving a problem, including identifying limitations of the solution; includes links to kinematics.
- Interpret analytical differential equation solutions within kinematics and population contexts.
- Determine asymptotic limiting behaviours and terminal velocities as time approaches infinity.
- Identify physical constraints and practical limitations of differential equation models.
Integration (AS Unit 1: Pure Mathematics A)
Evaluate definite integrals.
- Evaluate definite integrals using the Fundamental Theorem of Calculus.
- Handle negative limits and algebraic signs accurately during limit substitution.
- Apply integral properties including limits reversal and interval splitting.
Integration (AS Unit 1: Pure Mathematics A)
Integrate xⁿ (excluding n = − 1) and related sums, differences and constant multiples.
- Integrate powers of x for any rational index n ≠ −1 using the power rule.
- Simplify products and algebraic fractions into separate power terms prior to integration.
- Include and manage the arbitrary constant of integration + c correctly.
Integration (AS Unit 1: Pure Mathematics A)
Know and use the Fundamental Theorem of Calculus.
- Explain the Fundamental Theorem of Calculus linking differentiation and integration as inverse processes.
- Evaluate indefinite integrals including the arbitrary constant of integration + c.
- Determine specific constants of integration from boundary and initial conditions.
Integration (AS Unit 1: Pure Mathematics A)
Use a definite integral to find the area under a curve.
- Calculate the geometric area between a curve and the x-axis using definite integration.
- Locate x-intercepts and split integrals into separate intervals when curves cross the axis.
- Combine areas of regions above and below the x-axis correctly to determine total enclosed area.
Kinematics (A2 Unit 4: Applied Mathematics B)
Extend the use of calculus in kinematics for motion in a straight line to 2 dimensions using vectors.
- Differentiate two-dimensional position and velocity vectors with respect to time.
- Integrate vector acceleration and velocity functions, determining vector constants of integration.
- Determine times and conditions under which a particle moves parallel to coordinate axes.
Kinematics (A2 Unit 4: Applied Mathematics B)
Extend, use and derive the formulae for constant acceleration for motion in a straight line to 2 dimensions using vectors.
- Derive two-dimensional vector equations of motion under constant acceleration by integration.
- Apply vector suvat equations to compute position and velocity vectors at specified times.
- Determine speed and direction of travel from two-dimensional velocity vector components.
Kinematics (A2 Unit 4: Applied Mathematics B)
Model motion under gravity in a vertical plane using vectors; projectiles.
- Model projectile trajectories by decoupling horizontal constant-speed and vertical gravitational motions.
- Calculate projectile characteristics including time of flight, maximum altitude, and horizontal range.
- Derive and apply the Cartesian trajectory equation linking horizontal and vertical coordinates.
Kinematics (AS Unit 2: Applied Mathematics A)
Understand and use the language of kinematics: position, displacement, distance travelled, velocity, speed, acceleration.
- Differentiate between vector quantities (displacement, velocity, acceleration) and scalar measures (distance, speed).
- Calculate total distance travelled by identifying turning points where velocity equals zero.
- Interpret the physical meaning of positive and negative signs for displacement, velocity, and acceleration.
Kinematics (AS Unit 2: Applied Mathematics A)
Understand, use and derive the formulae for constant acceleration for motion in a straight line.
- Derive the five constant acceleration formulae using velocity-time graphs and algebraic elimination.
- Select and apply the appropriate suvat equation to solve one-dimensional motion problems.
- Enforce rigorous directional sign conventions when modelling vertical motion under gravity.
Kinematics (AS Unit 2: Applied Mathematics A)
Understand, use and interpret graphs in kinematics for motion in a straight line: displacement against time and interpretation of the gradient; velocity against time and interpretation of the gradient and the area under the graph
- Determine velocity and acceleration from the gradients of displacement-time and velocity-time graphs.
- Calculate displacement and total distance travelled from enclosed areas on velocity-time graphs.
- Sketch accurate kinematic graphs to represent multi-stage rectilinear particle journeys.
Kinematics (AS Unit 2: Applied Mathematics A)
Use calculus in kinematics for motion in a straight line.
- Determine velocity and acceleration functions by differentiating displacement with respect to time.
- Calculate velocity and displacement functions by integrating variable acceleration expressions.
- Evaluate constants of integration using initial spatial and temporal boundary conditions.
Moments (A2 Unit 4: Applied Mathematics B)
Understand and use moments in simple static contexts.
- Apply the Principle of Moments (ΣM_clockwise = ΣM_anticlockwise) to rigid bodies in static equilibrium.
- Combine moment equations with vertical force equilibrium to determine multiple support reactions.
- Solve tilting problems by identifying support lift-off conditions where reaction forces become zero.
Numerical Methods (A2 Unit 3: Pure Mathematics B)
Locate roots of f (x) = 0 by considering changes in sign of f (x) in an interval of x in which f (x) is sufficiently well-behaved.
- Apply change of sign tests to locate roots of continuous functions in given intervals.
- State formal conclusions citing both sign change and continuity of the function.
- Verify that an approximation to a root is correct to a specified degree of accuracy using boundary tests.
Numerical Methods (A2 Unit 3: Pure Mathematics B)
Solve equations approximately using simple iterative methods; be able to draw associated cobweb and staircase diagrams.
- Rearrange equations into fixed-point form x = g(x) and generate iterative sequences.
- Construct staircase and cobweb diagrams to illustrate convergence or divergence of iterations.
- Relate the convergence of iterative methods to the gradient of g(x) near the root.
Numerical Methods (A2 Unit 3: Pure Mathematics B)
Solve equations using the Newton-Raphson method and other recurrence relations of the form xₙ₊₁ = g (xₙ).
- Derive the Newton-Raphson formula geometrically from the equation of the tangent line.
- Apply the Newton-Raphson method to calculate successive approximations to roots.
- Solve equations using recurrence relations of the form xₙ₊₁ = g(xₙ) and evaluate iterates until consecutive values agree to a specified precision.
Numerical Methods (A2 Unit 3: Pure Mathematics B)
Understand and use numerical integration of functions, including the use of the trapezium rule and estimating the approximate area under a curve and limits that it must lie between.
- Apply the trapezium rule to approximate definite integrals from algebraic functions or data tables.
- Determine whether the trapezium rule produces an overestimate or underestimate from the curvature, and hence state a bound for the integral.
- Explain how increasing the number of strips improves the accuracy of the numerical approximation.
Numerical Methods (A2 Unit 3: Pure Mathematics B)
Understand how change of sign methods can fail.
- Explain how discontinuities and vertical asymptotes cause sign-change root-finding methods to fail.
- Explain why intervals containing an even number of roots or repeated roots may show no sign change.
- Use graphical sketches to identify scenarios where sign-change tests fail.
Numerical Methods (A2 Unit 3: Pure Mathematics B)
Understand how such methods can fail.
- Explain why the Newton-Raphson method fails when f'(xₙ) = 0 or f'(xₙ) is near zero.
- Identify geometric conditions that lead to divergence, cycling, or convergence to the wrong root.
- Illustrate failure modes of numerical root-finding methods using graphical sketches.
Numerical Methods (A2 Unit 3: Pure Mathematics B)
Use numerical methods to solve problems in context.
- Select and apply appropriate numerical methods to solve real-world contextual problems.
- Execute numerical algorithms to attain specified precision tolerances.
- Interpret numerical solutions in terms of practical operational limits and units.
Overarching themes - Mathematical argument, language and proof
Apply to solutions of inequalities and probability
- Formulate solutions to quadratic and linear inequalities using correct set union and intersection notation.
- Distinguish between conjunction ("and") and disjunction ("or") in algebraic solution sets.
- Apply set theory operations to calculate probabilities of compound events including unions and intersections.
Overarching themes - Mathematical argument, language and proof
Comprehend and critique mathematical arguments, proofs and justifications of methods and formulae, including those relating to applications of mathematics
- Analyse step-by-step mathematical deductions to identify invalid operations such as division by zero.
- Evaluate justifications of formulae and theorems for circular reasoning and hidden domain assumptions.
- Critique applied mathematical models by assessing the physical validity of their underlying assumptions.
Overarching themes - Mathematical argument, language and proof
Construct and present mathematical arguments through appropriate use of diagrams; sketching graphs; logical deduction; precise statements involving correct use of symbols and connecting language, including: constant, coefficient, expression, equation, function, identity, index, term, variable
- Distinguish clearly between expressions, equations, functions, and identities using precise mathematical terminology.
- Construct coherent multi-step deductions supported by labelled sketches and accurate logical connectives.
- Evaluate algebraic arguments to identify invalid reasoning, false equivalences, or omitted conditions.
Overarching themes - Mathematical argument, language and proof
Understand and use language and symbols associated with set theory, as set out in the content.
- Use standard set symbols including ∈, ∉, ∪, ∩, ⊆, ⊂, ξ or S, and A' or Aᶜ correctly in pure mathematics.
- Represent solution sets of equations and inequalities using rigorous set-builder and interval notation.
- Identify standard sets of numbers including natural numbers ℕ, integers ℤ, rationals ℚ, and reals ℝ.
Overarching themes - Mathematical argument, language and proof
Understand and use mathematical language and syntax as set out in the content
- Apply mathematical syntax and symbols correctly to express deductive relationships without ambiguity.
- Differentiate between implication (⇒), converse implication (⇐), and logical equivalence (⇔).
- Structure step-by-step mathematical reasoning cleanly using standard conventions and notation.
Overarching themes - Mathematical argument, language and proof
Understand and use the definition of a function; domain and range of functions
- Define a function formally as a single-valued mapping from a specified domain to a codomain.
- Determine the maximal valid domain and corresponding range for algebraic, exponential, and trigonometric functions.
- Identify conditions required for a function to be one-to-one and thus have a well-defined inverse.
Overarching themes - Mathematical modelling
Interpret the outputs of a mathematical model in the context of the original situation (for a given model or a model constructed or selected by the learner)
- Translate abstract mathematical outputs into meaningful conclusions within the original problem context.
- Interpret the physical significance of model features including stationary points, asymptotes, and rates of change.
- Communicate model conclusions with correct contextual units and appropriate degrees of precision.
Overarching themes - Mathematical modelling
Translate a situation in context into a mathematical model, making simplifying assumptions
- Translate real-world contexts into symbolic mathematical models using algebra and calculus.
- Identify and articulate simplifying assumptions required to make real-world problems mathematically tractable.
- Formulate governing relationships such as proportionalities and differential equations from contextual descriptions.
Overarching themes - Mathematical modelling
Understand and use modelling assumptions
- Identify and state standard modelling assumptions used in pure and applied mathematics.
- Explain the specific mathematical consequences of modelling assumptions on equations of motion and equilibrium.
- Evaluate how relaxing assumptions impacts the complexity and accuracy of the resulting mathematical model.
Overarching themes - Mathematical modelling
Understand that a mathematical model can be refined by considering its outputs and simplifying assumptions; evaluate whether the model is appropriate
- Evaluate the appropriateness of mathematical models by comparing theoretical predictions with physical reality.
- Identify specific failure points in models where simplifying assumptions cause divergence from real observations.
- Propose constructive mathematical refinements, including non-linear terms or constraints, to improve model accuracy.
Overarching themes - Mathematical modelling
Use a mathematical model with suitable inputs to engage with and explore situations (for a given model or a model constructed or selected by the learner)
- Substitute empirical or theoretical inputs into mathematical models to calculate intermediate and final quantities.
- Determine unknown model parameters by applying initial conditions and boundary constraints.
- Explore model sensitivity and limiting behavior as independent variables approach extreme values.
Overarching themes - Mathematical problem solving
Construct extended arguments to solve problems presented in an unstructured form, including problems in context
- Deconstruct complex unstructured problems into manageable intermediate mathematical sub-goals.
- Synthesise techniques from algebra, coordinate geometry, and calculus to form extended logical arguments.
- Present complete, coherent mathematical solutions with fully justified transitions between stages.
Overarching themes - Mathematical problem solving
Evaluate, including by making reasoned estimates, the accuracy or limitations of solutions, including those obtained using numerical methods
- Assess the accuracy of numerical solutions by analysing algorithm convergence and error bounds.
- Determine whether numerical integration produces an overestimate or underestimate based on curve convexity.
- Identify conditions under which numerical root-finding algorithms fail, cycle, or diverge.
Overarching themes - Mathematical problem solving
Interpret and communicate solutions in the context of the original problem
- Evaluate abstract mathematical outputs against physical and practical real-world constraints.
- Select and reject algebraic roots based on contextual domain limitations such as non-negative lengths.
- Communicate final conclusions clearly using appropriate contextual language, units, and precision.
Overarching themes - Mathematical problem solving
Recognise the underlying mathematical structure in a situation and simplify and abstract appropriately to enable problems to be solved
- Translate non-routine contextual scenarios into abstract mathematical equations or geometric models.
- Select appropriate mathematical structures (such as quadratics, calculus, or vectors) to represent physical relationships.
- Apply simplifying assumptions systematically to render complex contextual problems analytically solvable.
Overarching themes - Mathematical problem solving
Understand that many mathematical problems cannot be solved analytically, but numerical methods permit solution to a required level of accuracy
- Differentiate between exact analytical solutions and approximate numerical solutions to mathematical problems.
- Identify scenarios where transcendental equations or non-elementary integrals necessitate numerical methods.
- Apply numerical approximation techniques systematically to determine solutions to a required tolerance.
Overarching themes - Mathematical problem solving
Understand the concept of a mathematical problem solving cycle, including specifying the problem, collecting information, processing and representing information and interpreting results, which may identify the need to repeat the cycle
- Describe the four distinct phases of the mathematical problem-solving cycle.
- Apply algebraic and graphical techniques to process information and deduce initial solutions.
- Evaluate calculated outcomes critically to identify the need for model recalibration and cycle repetition.
Overarching themes - Mathematical problem solving
Understand, interpret and extract information from diagrams and construct mathematical diagrams to solve problems, including in mechanics
- Construct clear, labelled mathematical diagrams and sketches to represent geometric and physical scenarios.
- Extract geometric constraints, coordinates, and force components accurately from visual representations.
- Translate visual information from diagrams into formal algebraic and trigonometric equations.
Probability (A2 Unit 4: Applied Mathematics B)
Modelling with probability, including critiquing assumptions made and the likely effect of more realistic assumptions.
- Identify underlying mathematical assumptions in real-world probabilistic models.
- Critique the validity of independence and constant probability assumptions in practical contexts.
- Predict the qualitative and quantitative effects of adopting more realistic modelling assumptions.
Probability (A2 Unit 4: Applied Mathematics B)
Understand and use conditional probability, including the use of tree diagrams, Venn diagrams and two-way tables.
- Calculate conditional probabilities from two-way tables by restricting to row or column totals.
- Construct tree diagrams and calculate compound probabilities along sequential branches.
- Evaluate conditional probabilities using Venn diagrams by comparing intersection to subset area.
Probability (A2 Unit 4: Applied Mathematics B)
Understand and use the conditional probability formula: P(A ∩ B) = P(A) P(B|A) = P(B) P(A|B).
- Apply the conditional probability formula P(A|B) = P(A ∩ B) / P(B) to evaluate probabilities.
- Combine the multiplication rule and addition rule to solve complex multi-stage probability problems.
- Use conditional probability formulas to test whether two events are statistically independent.
Probability (AS Unit 2: Applied Mathematics A)
Link to discrete and continuous distributions.
- Apply the total probability condition Σ P(X = x) = 1 to solve for unknown parameters in discrete distributions.
- Contrast the evaluation of interval probabilities in discrete distributions versus continuous models.
- Explain why individual point probabilities equal zero for all continuous random variables.
Probability (AS Unit 2: Applied Mathematics A)
Understand and use mutually exclusive and independent events when calculating probabilities.
- Calculate probabilities for mutually exclusive events using the addition rule P(A ∪ B) = P(A) + P(B).
- Verify whether two events are independent by testing whether P(A ∩ B) = P(A) × P(B).
- Distinguish clearly between the concepts of mutual exclusivity and statistical independence.
Probability (AS Unit 2: Applied Mathematics A)
Use Venn diagrams to calculate probabilities.
- Construct and populate two-event and three-event Venn diagrams from probability data.
- Calculate union, intersection, and complement probabilities using partitioned Venn diagram regions.
- Evaluate conditional probabilities by restricting the sample space to a specified subset in a Venn diagram.
Proof (A2 Unit 3: Pure Mathematics B)
Proof by contradiction (including proof of the irrationality of √2 and the infinity of primes, and application to unfamiliar proofs).
- Structure and write rigorous proofs by contradiction for unfamiliar mathematical statements.
- Prove the irrationality of √2 using divisibility and parity arguments.
- Prove the infinity of primes using Euclid's construction and proof by contradiction.
Proof (AS Unit 1: Pure Mathematics A)
Understand and use the structure of mathematical proof, proceeding from given assumptions through a series of logical steps to a conclusion; use methods of proof, including (a) proof by deduction, (b) proof by exhaustion, (c) disproof by counter example.
- Construct algebraic proofs using deduction, exhaustion, and counterexample.
- Formulate general algebraic expressions for even, odd, and consecutive integers.
- Identify and state clear counterexamples to disprove false universal conjectures.
Quantities and units in mechanics (A2 Unit 4: Applied Mathematics B)
Understand and use derived quantities and units for moments.
- Define the moment of a force as the product of force and perpendicular distance to the pivot.
- Apply the derived SI unit of newton metres (N m) and assign clockwise or anticlockwise sense.
- Determine perpendicular distances from pivot points to inclined force lines of action.
Quantities and units in mechanics (AS Unit 2: Applied Mathematics A)
Understand and use derived quantities and units: velocity, acceleration, force, weight.
- Define derived mechanical units: m s⁻¹ for velocity, m s⁻² for acceleration, and N for force.
- Calculate weight in newtons from mass in kilograms using gravitational acceleration g.
- Distinguish between scalar and vector quantities across derived mechanical variables.
Quantities and units in mechanics (AS Unit 2: Applied Mathematics A)
Understand and use fundamental quantities and units in the S.I. system; length, time and mass.
- Identify and use SI base units for length (m), time (s), and mass (kg) in mechanical models.
- Convert measurements with metric prefixes and temporal units into standard SI base values.
- Verify dimensional consistency across algebraic and numerical equations in mechanics.
Sequences and Series - The Binomial Theorem (AS Unit 1: Pure Mathematics A)
Link to binomial probabilities.
- Connect the terms of the algebraic binomial expansion (q + p)ⁿ to binomial probability distributions.
- Calculate individual and cumulative binomial probabilities using expansion terms.
- Explain the combinatorial role of (n choose r) in determining the number of paths to r successes.
Sequences and Series - The Binomial Theorem (AS Unit 1: Pure Mathematics A)
The notations n!, (n choose r) and nCr.
- Evaluate factorials and simplify algebraic expressions containing factorial terms.
- Calculate binomial coefficients using the formula n! / (r!(n − r)!) and calculator functions.
- Apply combinations and Pascal's triangle properties to determine binomial expansion coefficients.
Sequences and Series - The Binomial Theorem (AS Unit 1: Pure Mathematics A)
Understand and use the binomial expansion of (a + bx)ⁿ for positive integer n .
- Expand (a + bx)ⁿ for positive integers n up to a specified power of x.
- Calculate individual coefficients in a binomial expansion without expanding fully.
- Use binomial expansions to approximate numerical powers by substituting small values of x.
Sequences and Series (A2 Unit 3: Pure Mathematics B)
Be aware that the expansion is valid for |bx/a| < 1 (proof not required).
- State the convergence condition |X| < 1 for infinite binomial series.
- Determine the range of validity for expansions of the form (a + bx)ⁿ.
- Explain why substituting values outside the range of validity produces invalid results.
Sequences and Series (A2 Unit 3: Pure Mathematics B)
Increasing sequences, decreasing sequences, periodic sequences.
- Classify sequences as increasing, decreasing, periodic, or oscillating.
- Prove monotonicity of sequences by analysing differences or ratios of consecutive terms.
- Determine the order of periodic sequences and evaluate large-index terms.
Sequences and Series (A2 Unit 3: Pure Mathematics B)
The sum to infinity of a convergent geometric series, including the use of | r | < 1; modulus notation.
- State and apply the convergence condition |r| < 1 for infinite geometric series.
- Calculate the sum to infinity using S_∞ = a / (1 − r).
- Convert recurring decimals to exact rational fractions using infinite geometric series.
Sequences and Series (A2 Unit 3: Pure Mathematics B)
Understand and use sigma notation for sums of series.
- Interpret and expand series expressed in sigma notation.
- Apply linearity and splitting properties to evaluate sums with non-unit lower limits.
- Identify arithmetic and geometric series in sigma notation and calculate their sums.
Sequences and Series (A2 Unit 3: Pure Mathematics B)
Understand and use the binomial expansion of (a + bx)n , for any rational n, including its use for approximation.
- Expand (1 + x)ⁿ for negative and fractional rational powers up to a specified power of x.
- Transform (a + bx)ⁿ into aⁿ(1 + (b/a)x)ⁿ prior to series expansion.
- State the validity condition |(b/a)x| < 1 for the infinite binomial expansion.
- Use binomial series expansions to compute decimal approximations to roots and fractions.
Sequences and Series (A2 Unit 3: Pure Mathematics B)
Understand and work with arithmetic sequences and series, including the formulae for the nth term and the sum to n terms.
- Calculate the nth term and sum of n terms of an arithmetic progression.
- Prove the formula for the sum of an arithmetic series using Gaussian pairing.
- Solve simultaneous linear and quadratic equations to determine AP parameters.
Sequences and Series (A2 Unit 3: Pure Mathematics B)
Understand and work with geometric sequences and series, including the formulae for the nth term and the sum of a finite geometric series.
- Calculate terms and finite sums of geometric sequences and series.
- Prove the formula for the sum of a finite geometric series algebraically.
- Solve inequalities for the number of terms n in a geometric series using logarithms.
Sequences and Series (A2 Unit 3: Pure Mathematics B)
Use sequences and series in modelling.
- Translate contextual problems into arithmetic or geometric sequences and series.
- Distinguish between individual term values and cumulative series totals in applications.
- Solve practical financial, biological, and physical problems using progression formulas.
Sequences and Series (A2 Unit 3: Pure Mathematics B)
Work with sequences, including those given by a formula for the nth term and those generated by a simple relation of the form xₙ₊₁ = f (xₙ).
- Generate sequence terms from explicit nth term formulas and inductive recurrence relations.
- Identify patterns and formulate closed-form expressions for sequence terms.
- Calculate the limiting value of convergent recurrence relations using fixed-point equations.
Statistical distributions (A2 Unit 4: Applied Mathematics B)
Find probabilities using the Normal distribution.
- Calculate probabilities for Normal distributions using statistical calculator functions and standardisation.
- Determine values corresponding to given percentiles using inverse Normal calculator operations.
- Formulate and solve simultaneous equations to find unknown mean μ and standard deviation σ.
Statistical distributions (A2 Unit 4: Applied Mathematics B)
Link to histograms, mean, standard deviation, points of inflection and the binomial distribution.
- Locate points of inflection on a Normal probability curve at one standard deviation from the mean (μ ± σ).
- Apply the percentage properties of Normal distributions (68%, 95%, 99.7%) to estimate data spreads.
- Explain how large sample sizes allow discrete binomial distributions to approximate continuous Normal curves.
Statistical distributions (A2 Unit 4: Applied Mathematics B)
Select an appropriate probability distribution for a context, with appropriate reasoning, including recognising when the continuous uniform or Normal model may not be appropriate.
- Select and justify continuous uniform or Normal probability distributions for specific physical phenomena.
- Identify violations of normality such as skewness, multi-modality, and impossible negative predictions.
- Critique continuous model suitability and recommend appropriate refinements.
Statistical distributions (A2 Unit 4: Applied Mathematics B)
Understand and use the continuous uniform distribution and Normal distributions as models.
- Define and sketch the probability density function for a continuous uniform distribution X ~ U(a, b).
- Calculate interval probabilities for uniform distributions using geometric rectangular areas.
- Describe the fundamental characteristics of the Normal distribution: bell shape, symmetry, and parameters μ and σ².
Statistical distributions (AS Unit 2: Applied Mathematics A)
Calculate probabilities using • the binomial distribution. • the Poisson distribution. • the discrete uniform distribution.
- Compute exact and cumulative probabilities for binomial distributions using formulae and calculator functions.
- Calculate Poisson probabilities, adjusting the rate parameter λ proportionally for different intervals.
- Convert discrete inequality ranges into cumulative expressions suitable for calculator inputs.
- Recognise and use the discrete uniform distribution with probability 1/n for each of n outcomes.
Statistical distributions (AS Unit 2: Applied Mathematics A)
Select an appropriate probability distribution for a context, with appropriate reasoning, including recognising when the binomial, Poisson or discrete uniform model may not be appropriate.
- Select the appropriate probability distribution model for a given scenario with stated parameters.
- Justify distribution choice by matching physical context conditions to mathematical axioms.
- Critique and reject inappropriate probability models by identifying violated theoretical assumptions.
Statistical distributions (AS Unit 2: Applied Mathematics A)
Understand and use simple, discrete probability distributions.
- Represent and interpret discrete probability distributions using tables and formulas.
- Determine unknown parameters in discrete distributions using the condition that total probability equals 1.
- Calculate probabilities for compound events by summing individual probability mass terms.
Statistical distributions (AS Unit 2: Applied Mathematics A)
Understand and use, • the binomial distribution, as a model • the Poisson distribution, as a model • the discrete uniform distribution, as a model (Calculation of mean and variance of discrete random variables is excluded.)
- State and verify the four assumptions required to model a scenario with a binomial distribution.
- Explain the conditions necessary for a Poisson process: constant average rate, independence, and events occurring singly.
- Identify scenarios where equally likely discrete outcomes justify a discrete uniform distribution.
Statistical hypothesis testing (A2 Unit 4: Applied Mathematics B)
(The calculation of correlation coefficients is excluded.)
- Interpret provided PMCC values in terms of the direction and strength of linear association.
- Execute hypothesis tests on correlation using given r values and statistical tables.
- Explain why correlation coefficients close to zero do not preclude strong non-linear relationships.
Statistical hypothesis testing (A2 Unit 4: Applied Mathematics B)
Conduct a statistical hypothesis test for the mean of a Normal distribution with known, given or assumed variance, and interpret the results in context.
- Formulate hypotheses for population mean tests using correct parameter notation μ.
- Calculate test statistics using the sampling distribution of the mean X̄ ~ N(μ, σ²/n).
- Execute one-tailed and two-tailed Normal mean tests and interpret findings in context.
Statistical hypothesis testing (A2 Unit 4: Applied Mathematics B)
Understand and apply statistical hypothesis testing to correlation coefficients as measures of how close data points lie to a straight line and be able to interpret a given correlation coefficient using a given p-value or critical value.
- Formulate null and alternative hypotheses for correlation tests using population parameter ρ.
- Evaluate sample correlation coefficients against critical values or p-values at given significance levels.
- Interpret test outcomes in practical context, avoiding claims of causal determination.
Statistical hypothesis testing (AS Unit 2: Applied Mathematics A)
Conduct a statistical hypothesis test for the proportion in the binomial distribution and interpret the results in context.
- Conduct a complete hypothesis test for a binomial proportion using p-values or critical regions.
- Determine critical values for one-tailed and two-tailed binomial tests at specified significance levels.
- Formulate nuanced, non-dogmatic contextual conclusions based on statistical test outcomes.
Statistical hypothesis testing (AS Unit 2: Applied Mathematics A)
Interpret and calculate Type I and Type II errors, and know their practical meaning.
- Define and distinguish between Type I (false positive) and Type II (false negative) statistical errors.
- Calculate the numerical probabilities of Type I and Type II errors for a specified binomial test.
- Assess the practical and financial consequences of Type I and Type II errors in real-world scenarios.
Statistical hypothesis testing (AS Unit 2: Applied Mathematics A)
Understand and apply the language of statistical hypothesis testing, developed through a binomial model: null hypothesis, alternative hypothesis, significance level, test statistic, 1-tail test, 2-tail test, critical value, critical region, acceptance region, p-value.
- Formulate null and alternative hypotheses using correct mathematical parameter notation.
- Distinguish between critical regions, acceptance regions, critical values, and significance levels.
- Explain how p-values and critical regions provide equivalent criteria for evaluating hypotheses.
Statistical hypothesis testing (AS Unit 2: Applied Mathematics A)
Understand that a sample is being used to make an inference about the population and appreciate that the significance level is the probability of incorrectly rejecting the null hypothesis.
- Explain the significance level as the probability of committing a Type I error.
- Calculate the actual significance level of a discrete test from its critical region.
- Evaluate the balance between Type I and Type II error probabilities when selecting a significance threshold.
Statistical Sampling (AS Unit 2: Applied Mathematics A)
Select or critique sampling techniques in the context of solving a statistical problem, including understanding that different samples can lead to different conclusions about the population.
- Critique proposed sampling designs by identifying coverage errors and selection bias.
- Explain why natural sampling variation leads different random samples to produce different inferences.
- Select and justify appropriate sampling techniques for complex real-world statistical problems.
Statistical Sampling (AS Unit 2: Applied Mathematics A)
Understand and use sampling techniques, including simple random sampling, systematic sampling and opportunity sampling.
- Implement simple random sampling using a numbered sampling frame and random number generation.
- Calculate the sampling interval and random starting index for systematic sampling procedures.
- Compare the operational trade-offs between probability sampling and opportunity sampling.
Statistical Sampling (AS Unit 2: Applied Mathematics A)
Understand and use the terms 'population' and 'sample'.
- Distinguish between an entire population and a representative sample in statistical investigations.
- Evaluate the practical advantages and limitations of a census compared to taking a sample.
- Identify real-world scenarios where destructive testing makes a full census impossible.
Statistical Sampling (AS Unit 2: Applied Mathematics A)
Use samples to make informal inferences about the population.
- Estimate unknown population parameters using sample proportions and sample means.
- Explain how sample size influences the variability and reliability of informal inferences.
- Formulate cautious, evidence-based statements about a population from sample data.
Trigonometry (A2 Unit 3: Pure Mathematics B)
Construct proofs involving trigonometric functions and identities.
- Construct rigorous deductive proofs of trigonometric identities starting from one side.
- Select and deploy compound angle, double angle, and Pythagorean identities strategically.
- Simplify complex trigonometric fractions using common denominators and factorisation.
Trigonometry (A2 Unit 3: Pure Mathematics B)
Know and use exact values of sin and cos for 0, π/6, π/4, π/3, π/2, π and multiples thereof, and exact values of tan for 0, π/6, π/4, π/3, π and multiples thereof.
- State exact values of sine, cosine, and tangent for special angles 0, π/6, π/4, π/3, π/2, and π.
- Evaluate exact trigonometric ratios for any integer multiple of special angles across all quadrants.
- Derive exact values using equilateral and isosceles right-angled reference triangles.
Trigonometry (A2 Unit 3: Pure Mathematics B)
sin θ ≈ θ , cos θ ≈ 1 − θ²/2 and tan θ ≈ θ , where θ is in radians.
- Apply the explicit formulas sin θ ≈ θ, cos θ ≈ 1 − θ²/2, and tan θ ≈ θ.
- Evaluate limits of rational trigonometric expressions as θ approaches zero.
- Solve approximate algebraic equations derived from small-angle substitutions.
Trigonometry (A2 Unit 3: Pure Mathematics B)
Understand and use double angle formulae.
- State and apply double angle formulas for sine, cosine, and tangent.
- Use power-reduction forms of cos(2A) to integrate sin²x and cos²x.
- Solve trigonometric equations involving mixed single and double angle arguments.
Trigonometry (A2 Unit 3: Pure Mathematics B)
Understand and use expressions for acosθ + bsinθ in the equivalent forms of r cos(θ ± α) or r sin(θ ± α).
- Express a cos θ + b sin θ in the equivalent forms R cos(θ ± α) and R sin(θ ± α).
- Determine maximum and minimum values of harmonic expressions and the angles where they occur.
- Solve trigonometric equations of the form a cos θ + b sin θ = c.
Trigonometry (A2 Unit 3: Pure Mathematics B)
Understand and use sec²θ ≡ 1 + tan²θ and cosec²θ ≡ 1 + cot²θ.
- Derive and apply the identities sec²θ ≡ 1 + tan²θ and cosec²θ ≡ 1 + cot²θ.
- Transform equations involving secant, cosecant, and cotangent into single-variable quadratics.
- Solve quadratic trigonometric equations involving reciprocal ratios in degrees and radians.
Trigonometry (A2 Unit 3: Pure Mathematics B)
Understand and use the definitions of sec, cosec, cot, sin⁻¹, cos⁻¹ and tan⁻¹. Understand the relationships of all of these to sin, cos and tan and understand their graphs, ranges and domains.
- Define secant, cosecant, and cotangent and sketch their graphs with asymptotes.
- Define arcsin, arccos, and arctan and state their domains and principal value ranges.
- Solve equations involving reciprocal and inverse trigonometric functions.
Trigonometry (A2 Unit 3: Pure Mathematics B)
Understand and use the standard small angle approximations of sine, cosine and tangent.
- State the small-angle approximations for sine, cosine, and tangent in radians.
- Apply small-angle approximations to composite arguments kθ.
- Simplify trigonometric fractions and evaluate limits using small-angle formulas.
Trigonometry (A2 Unit 3: Pure Mathematics B)
Understand geometric proofs of these formulae.
- Construct geometric diagrams demonstrating the derivation of compound angle formulas.
- Derive sin(A + B) and cos(A + B) using right-angled triangle dissections.
- Explain the geometric origin of the sign difference between sine and cosine addition formulas.
Trigonometry (A2 Unit 3: Pure Mathematics B)
Use of formulae for sin(A ± B), cos(A ± B) and tan(A ± B).
- Apply compound angle formulas for sine, cosine, and tangent to expand expressions.
- Evaluate exact trigonometric ratios for non-standard angles using sums and differences.
- Prove trigonometric identities and solve equations using compound angle expansions.
Trigonometry (A2 Unit 3: Pure Mathematics B)
Work with radian measure, including use for arc length, area of sector and area of segment.
- Convert angles fluently between degrees and radians.
- Calculate arc lengths and sector areas using s = rθ and Area = ½r²θ.
- Calculate areas of circular segments using Area = ½r²(θ − sin θ).
Trigonometry (A2 Unit 4: Applied Mathematics B)
Use trigonometric functions to solve problems in context, including problems involving vectors, kinematics and forces.
- Resolve force and velocity vectors into orthogonal components using trigonometric functions.
- Analyse forces on inclined planes using parallel (mg sin α) and perpendicular (mg cos α) components.
- Apply trigonometric identities to eliminate time and angle parameters in kinematic models.
Trigonometry (AS Unit 1: Pure Mathematics A)
Solve simple trigonometric equations in a given interval, including quadratic equations in sin, cos and tan, and equations involving multiples of the unknown angle.
- Find principal and secondary solutions to trigonometric equations in degrees and radians.
- Solve equations involving multiples of angles by transforming the interval first.
- Solve quadratic trigonometric equations and identify all valid roots in a given interval.
Trigonometry (AS Unit 1: Pure Mathematics A)
Understand and use cos²θ + sin²θ = 1.
- Apply the identity cos²θ + sin²θ = 1 to simplify trigonometric expressions.
- Transform mixed trigonometric equations into single-variable quadratic equations.
- Solve trigonometric quadratics over specified intervals, rejecting invalid roots.
Trigonometry (AS Unit 1: Pure Mathematics A)
Understand and use tan θ = sin θ / cos θ .
- Apply the identity tan θ = sin θ / cos θ to simplify trigonometric expressions.
- Transform equations involving linear combinations of sine and cosine into tangent equations.
- Solve trigonometric equations accurately while preserving roots where cos θ = 0.
Trigonometry (AS Unit 1: Pure Mathematics A)
Understand and use the definitions of sine, cosine and tangent for all arguments.
- Define sine, cosine, and tangent for any real angle using the unit circle.
- Determine the sign and magnitude of trigonometric ratios in all four quadrants.
- Calculate exact trigonometric values for negative angles and angles exceeding 360°.
Trigonometry (AS Unit 1: Pure Mathematics A)
Understand and use the sine and cosine rules, and the area of a triangle in the form ½ absinC.
- Apply the sine rule to find unknown lengths and angles in non-right-angled triangles.
- Apply the cosine rule to calculate side lengths and interior angles.
- Calculate the area of general triangles using Area = ½ab sin C.
Trigonometry (AS Unit 1: Pure Mathematics A)
Understand and use the sine, cosine and tangent functions.
- State the domain and range of the sine, cosine, and tangent functions.
- Identify the locations and equations of vertical asymptotes for the tangent function.
- Apply symmetry and parity properties of trigonometric functions to evaluate expressions.
Trigonometry (AS Unit 1: Pure Mathematics A)
Understand and use their graphs, symmetries and periodicity.
- Sketch the graphs of sine, cosine, and tangent over specified angular intervals.
- Identify lines of symmetry, rotational symmetry, amplitude, and periodicity of trigonometric curves.
- Use graphical symmetries to deduce all solutions to trigonometric equations in an interval.
Use of data in statistics
Learners should be able to demonstrate the ability to explore large data sets, and associated contexts, during their course of study to enable them to perform tasks, and understand ways in which technology can help explore the data. Learners should be able to demonstrate the ability to analyse a subset or features of the data using a calculator with standard statistical functions.
- Calculate sample means and standard deviations from a data subset using calculator statistical functions.
- Assess whether an extracted subset reflects the characteristic patterns of the wider data set.
- Communicate analytical methods clearly when using calculator technology to evaluate sample features.
Use of data in statistics
This specification requires learners, during the course of their study, to: • develop skills relevant to exploring and analysing large data sets (these data must be real and sufficiently rich to enable the concepts and skills of data presentation and interpretation in the specification to be explored); • use technology such as spreadsheets or specialist statistical packages to explore data sets; • interpret real data presented in summary or graphical form; • use data to investigate questions arising in real contexts.
- Explore multivariate real data using spreadsheet filters, pivot tables, and statistical tools.
- Interpret summary statistics and graphical displays produced from large real-world data sets.
- Investigate contextual questions by selecting and analysing relevant variables from large data sets.
Vectors (A2 Unit 4: Applied Mathematics B)
Understand and use vectors in three dimensions.
- Represent 3D vectors using unit vectors i, j, k and column vector notation.
- Calculate magnitudes and distances between points in three-dimensional Cartesian space.
- Construct unit vectors parallel to given three-dimensional vectors.
- Use the scalar product to find the angle between two 3D vectors and to test perpendicularity.
Vectors (A2 Unit 4: Applied Mathematics B)
Use vectors to solve problems in context, including forces and kinematics.
- Solve 3D static equilibrium and dynamic problems by applying ΣF = 0 and ΣF = ma.
- Apply three-dimensional vector kinematics and calculus to model spatial particle trajectories.
- Compute scalar speeds, travel times, and spatial separations from 3D velocity and position vectors.
Vectors (AS Unit 1: Pure Mathematics A)
Add vectors diagrammatically and perform the algebraic operations of vector addition and multiplication by scalars, and understand their geometrical interpretations.
- Perform vector addition, subtraction, and scalar multiplication algebraically and diagrammatically.
- Apply the triangle and parallelogram laws to determine resultant vectors.
- Use scalar multiples to test and prove that two vectors are parallel.
Vectors (AS Unit 1: Pure Mathematics A)
Understand and use position vectors; calculate the distance between points represented by position vectors.
- Distinguish between position vectors relative to an origin and displacement vectors between points.
- Calculate displacement vectors using the relation AB = OB − OA.
- Calculate distances between points by finding the magnitude of their displacement vector.
Vectors (AS Unit 1: Pure Mathematics A)
Use vectors in two dimensions.
- Represent two-dimensional vectors using column notation and unit base vectors i and j.
- Add, subtract and multiply vectors by scalars, and use position vectors.
- Calculate the magnitude of a 2D vector and determine its direction angle and unit vector.
Vectors (AS Unit 1: Pure Mathematics A)
Use vectors to solve problems in pure mathematics.
- Express geometric paths through shapes as linear combinations of base vectors.
- Prove that three points are collinear using vector parallelism and common points.
- Solve geometric intersection and ratio problems by equating coefficients of independent vectors.
Vectors (AS Unit 2: Applied Mathematics A)
Calculate the magnitude and direction of a vector and convert between component form and magnitude/direction form.
- Compute the magnitude of two-dimensional vectors using the Pythagorean formula |v| = √(x² + y²).
- Determine vector direction relative to coordinate axes or as a three-figure compass bearing.
- Convert fluently between Cartesian component form (xi + yj) and magnitude-direction form.
Vectors (AS Unit 2: Applied Mathematics A)
Use vectors to solve problems in context, including forces.
- Calculate resultant force vectors by summing individual coplanar force vectors.
- Apply Newton's second law in vector form (ΣF = ma) to determine particle accelerations.
- Solve contextual navigation and projectile problems using vector kinematics formulas.
Regulatory & Verification Details
- Qualification Number (QN)
- 603/1977/X
- Regulated Status
- Available to learners
- Awarding Body
- WJEC
- Qualification Type
- GCE A Level
- Operational Start
- Sept 2017