Cambridge OCR Level 3 Advanced GCE in Mathematics B (MEI)

Comprehend and critique mathematical arguments, proofs and justifications of methods and formulae, including those relating to applications of mathematics

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

Understand and use language and symbols associated with set theory, as set out in the content. Apply to solutions of inequalities and probability

Understand and use mathematical language and syntax as set out in the content

Understand and use the definition of a function; domain and range of functions

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 student)

Translate a situation in context into a mathematical model, making simplifying assumptions

Understand and use modelling assumptions

Understand that a mathematical model can be refined by considering its outputs and simplifying assumptions; evaluate whether the model is appropriate

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 student)

Construct extended arguments to solve problems presented in an unstructured form, including problems in context

Evaluate, including by making reasoned estimates, the accuracy or limitations of solutions, including those obtained using numerical methods

Interpret and communicate solutions in the context of the original problem

Recognise the underlying mathematical structure in a situation and simplify and abstract appropriately to enable problems to be solved

Understand that many mathematical problems cannot be solved analytically, but numerical methods permit solution to a required level of accuracy

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

Understand, interpret and extract information from diagrams and construct mathematical diagrams to solve problems, including in mechanics

Be able to find the resultant of several concurrent forces when the forces are parallel or in two perpendicular directions or in simple cases of forces given as 2-D vectors in component form.

Be able to identify the forces acting on a system and represent them in a force diagram. Understand the difference between external and internal forces and be able to identify the forces acting on part of the system.

Know that the acceleration due to gravity is not a universal constant but depends on location in the universe. Know that on earth, the acceleration due to gravity is often modelled to be a constant, g m s⁻².

Understand the concept of equilibrium and know that a particle is in equilibrium if and only if the vector sum of the forces acting on it is zero in the cases where the forces are parallel or in two perpendicular directions or in simple cases of forces given as 2-D vectors in component form.

Understand the language relating to forces.

Be able to apply Newton’s Laws to problems involving friction.

Be able to formulate and solve equations for a particle in equilibrium: by resolving forces in suitable directions; by drawing and using a polygon of forces.

Be able to resolve a force into components and be able to select suitable directions for resolution. Be able to find the resultant of several concurrent forces by resolving and adding components.

Know that a particle is in equilibrium if and only if the resultant of the forces acting on it is zero. Know that a body is in equilibrium under a set of concurrent forces if and only if their resultant is zero.

Know that vectors representing a set of forces in equilibrium sum to zero. Know that a closed figure may be drawn to represent the addition of the forces on an object in equilibrium.

Understand that the frictional force may be modelled by F ≤ μR and that friction acts in the direction to oppose sliding. Model friction using F = μR when sliding occurs.

Understand that the overall contact force between surfaces may be expressed in terms of a frictional force and a normal contact force and be able to draw an appropriate force diagram. Understand that the normal contact force cannot be negative.

Be able to differentiate position and velocity with respect to time and know what measures result.

Be able to draw and interpret kinematics graphs for motion in a straight line, knowing the significance (where appropriate) of their gradients and the areas underneath them.

Be able to integrate acceleration and velocity with respect to time and know what measures result.

Be able to recognise when the use of constant acceleration formulae is appropriate.

Be able to solve kinematics problems using constant acceleration formulae and calculus for motion in a straight line.

Know the difference between position, displacement, distance and distance travelled.

Know the difference between velocity and speed, and between acceleration and magnitude of acceleration.

Understand and use the language of kinematics.

Be able to extend the scope of techniques from motion in 1 dimension to that in 2 dimensions by using vectors.

Be able to find the cartesian equation of the path of a particle when the components of its position vector are given in terms of time.

Be able to use vectors to solve problems in kinematics.

Understand the language of kinematics appropriate to motion in 2 dimensions. Know the difference between, displacement, distance from and distance travelled; velocity and speed, and between acceleration and magnitude of acceleration.

Know the language used to describe simplifying assumptions in mechanics.

Understand and use derived quantities and units: velocity, acceleration, force, weight.

Understand and use fundamental quantities and units in the S.I. system: length, time, mass.

Understand and use the particle model.

Understand and use derived quantities and units: moment.

Be able to formulate the equation of motion for a particle moving in a straight line when the forces acting are parallel or in two perpendicular directions or in simple cases of forces given as 2-D vectors in component form.

Be able to formulate the equations of motion for the individual particles within the system.

Be able to model a system as a set of connected particles.

Know and understand the meaning of Newton’s three laws.

Know that a system in which none of its components have any relative motion may be modelled as a single particle with the mass of the system.

Understand the term equation of motion.

Be able to formulate the equation of motion for a particle moving in a straight line or in a plane.

Be able to eliminate time from the component equations that give the horizontal and vertical displacement in terms of time to obtain the equation of the trajectory.

Be able to find the initial velocity of a projectile given sufficient information.

Be able to model motion under gravity in a vertical plane using vectors. Be able to formulate the equations of motion of a projectile using vectors.

Be able to solve simple problems involving projectiles.

Know how to find the position and velocity at any time of a projectile and find range and maximum height.

Be able to calculate the moment of a force about a point or axis.

Know that, for the purpose of calculating its moment, the weight of a body can be taken as acting through a point.

Understand that a rigid body is in equilibrium when the resultant force is zero and the sum of the moments about any one point is zero.

Understand that a system of forces can have a turning effect on a rigid body.

Be able to change the subject of a formula.

Be able to express solutions of inequalities through correct use of ‘and’ and ‘or’, or by using set notation.

Be able to find the discriminant of a quadratic function and understand its significance.

Be able to rationalise the denominator of a surd.

Be able to solve linear equations in one unknown.

Be able to solve linear inequalities in one variable. Be able to represent and interpret linear inequalities graphically e.g. y > x + 1.

Be able to solve linear simultaneous equations in two unknowns.

Be able to solve quadratic equations.

Be able to solve quadratic inequalities in one variable. Be able to represent and interpret quadratic inequalities graphically e.g. y > ax² + bx + c.

Be able to solve simultaneous equations in two unknowns with one equation linear and one quadratic.

Be able to use and manipulate surds.

Know and be able to use vocabulary and notation appropriate to the subject at this level.

Know the significance of points of intersection of two graphs with relation to the solution of equations.

Understand and be able to use negative, fractional and zero indices.

Understand and be able to use the laws of indices for all rational exponents.

Understand and use proportional relationships and their graphs.

Be able to express algebraic fractions as partial fractions.

Be able to simplify rational expressions.

Be able to differentiate y = kxⁿ where k is a constant and n is rational, including related sums and differences.

Be able to find a constant of integration given relevant information.

Be able to find the equation of the tangent and normal at a point on a curve.

Be able to integrate functions of the form kxⁿ where k is a constant and n ≠ −1.

Be able to sketch the gradient function for a given curve.

Be able to use differentiation to find stationary points on a curve: maxima and minima.

Be able to use integration to find the area between a graph and the x-axis.

Know and use that the gradient of a curve at a point is given by the gradient of the tangent at the point.

Know and use that the gradient of the tangent at a point A on a curve is given by the limit of the gradient of chord AP as P approaches A along the curve.

Know that integration is the reverse of differentiation.

Know what is meant by indefinite and definite integrals. Be able to evaluate definite integrals.

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). Know that the gradient function dy/dx gives the gradient of the curve and measures the rate of change of y with respect to x.

Understand and use the second derivative as the rate of change of gradient.

Understand the terms increasing function and decreasing function and be able to find where the function is increasing or decreasing.

Be able to differentiate a function or relation defined implicitly.

Be able to differentiate composite functions using the chain rule.

Be able to differentiate eᵏˣ, aᵏˣ and ln x.

Be able to differentiate the product of two functions.

Be able to differentiate the quotient of two functions.

Be able to differentiate the trigonometrical functions: sin kx; cos kx; tan kx for x in radians.

Be able to find general or particular solutions of first order differential equations analytically by separating variables.

Be able to find rates of change using the chain rule, including connected rates of change and differentiation of inverse functions.

Be able to formulate first order differential equations using information about rates of change.

Be able to integrate eᵏˣ, 1/x, sin kx, cos kx and related sums, differences and constant multiples.

Be able to integrate using partial fractions that are linear in the denominator.

Be able to interpret the solution of a differential equation in the context of solving a problem, including identifying limitations of the solution.

Be able to use integration by substitution in cases where the process is the reverse of the chain rule (including finding a suitable substitution).

Be able to use integration by substitution in other cases.

Be able to use integration to find the area between two curves.

Be able to use the method of integration by parts in simple cases.

Understand integration as the limit of a sum.

Understand that a point of inflection on a curve is where the curve changes from concave upwards to concave downwards (or vice versa) and hence that the second derivative at a point of inflection is zero. Be able to use differentiation to find stationary and non-stationary points of inflection.

Understand that a section of curve which has increasing gradient (and so positive second derivative) is concave upwards. Understand that a section of curve which has decreasing gradient (and so negative second derivative) is concave downwards.

Be able to calculate the distance between two points.

Be able to draw a line given its equation.

Be able to find the coordinates of the midpoint of a line segment joining two points.

Be able to find the point of intersection of two lines.

Be able to find the point(s) of intersection of a line and a circle.

Be able to find the point(s) of intersection of a line and a curve or of two curves.

Be able to form the equation of a straight line.

Be able to use straight line models.

Know and be able to use the following properties: the angle in a semicircle is a right angle; the perpendicular from the centre of a circle to a chord bisects the chord; the radius of a circle at a given point on its circumference is perpendicular to the tangent to the circle at that point.

Know and be able to use the relationship between the gradients of parallel lines and perpendicular lines.

Understand and use the equation of a circle in the form (x − a)² + (y − b)² = r².

Understand and use the equation y = mx + c.

Be able to convert between cartesian and parametric forms of equations.

Be able to find the gradient of a curve defined in terms of a parameter by differentiation.

Be able to use parametric equations in modelling.

Understand and use the equation of a circle written in parametric form.

Understand the meaning of the terms parameter and parametric equations.

Be able to convert from an index to a logarithmic form and vice versa.

Be able to solve an equation of the form aˣ = b.

Be able to solve problems involving exponential growth and decay; be able to consider limitations and refinements of exponential growth and decay models.

Know and be able to use the function y = eˣ and its graph.

Know and be able to use the function y = ln x and its graph. Know the relationship between ln x and eˣ.

Know and use the function y = aˣ and its graph.

Know and use the values of logₐ a and logₐ 1.

Know how to reduce the equations y = axⁿ and y = abˣ to linear form and, using experimental data, to use a graph to estimate values of the parameters.

Know that the gradient of eᵏˣ is keᵏˣ and hence understand why the exponential model is suitable in many applications.

Understand a logarithm as the inverse of the appropriate exponential function and be able to sketch the graphs of exponential and logarithmic functions.

Understand the laws of logarithms and be able to apply them, including to taking logarithms of both sides of an equation.

Be able to add, subtract, multiply and divide polynomials.

Understand the factor theorem and be able to use it to factorise a polynomial or to determine its zeros.

Be able to solve simple inequalities containing a modulus sign.

Be able to use functions in modelling.

Understand and be able to use inverse functions and their graphs. Know the conditions necessary for the inverse of a function to exist and how to find it.

Understand and be able to use the modulus function.

Understand and use composite functions.

Understand the definition of a function, and be able to use the associated language.

Be able to sketch and interpret the graphs of simple functions including polynomials.

Be able to sketch and interpret the graphs of y = a/x and y = a/x².

Be able to sketch curves of the forms y = af(x), y = f(x) + a, y = f(x + a) and y = f(ax), given the curve of y = f(x) and describe the associated transformations. Be able to form the equation of a graph following a single transformation.

Be able to use stationary points when curve sketching.

Understand and be able to use the method of completing the square to find the line of symmetry and turning point of the graph of a quadratic function and to sketch a quadratic curve (parabola).

Understand and use graphs of functions.

Understand how to find intersection points of a curve with coordinate axes.

Be able to use stationary points of inflection when curve sketching.

Understand the effect of combined transformations on a graph and be able to form the equation of the new graph and to sketch it. Be able to recognise the transformations that have been applied to a graph from the graph or its equation.

Be able to carry out a fixed point iteration after rearranging an equation into the form x = g(x) and be able to draw associated staircase and cobweb diagrams.

Be able to find an approximate value of a definite integral using the trapezium rule, and decide whether it is an over- or an under-estimate.

Be able to locate the roots of f(x) = 0 by considering changes of sign of f(x) in an interval of x in which f(x) is sufficiently well-behaved.

Be able to use the Newton-Raphson method to find a root of an equation and represent the process on a graph.

Be aware of circumstances under which change of sign methods may fail.

Understand that not all iterations converge to a particular root of an equation.

Use numerical methods to solve problems in context.

Use the sum of a series of rectangles to find an upper and/or lower bound on the area under a curve.

Be able to disprove a conjecture by the use of a counter example.

Understand and be able to use the structure of mathematical proof. Use methods of proof, including proof by deduction and proof by exhaustion.

Understand and be able to use proof by contradiction.

Know the notations n! and ₙCᵣ and that ₙCᵣ is the number of ways of selecting r distinct objects from n.

Understand and use the binomial expansion of (a + bx)ⁿ where n is a positive integer.

Be able to generate a sequence using a formula for the kth term, or a recurrence relation of the form aₖ₊₁ = f(aₖ).

Be able to recognise increasing, decreasing and periodic sequences.

Be able to use binomial expansions with n rational to find polynomials which approximate (a + bx)ⁿ.

Be able to use sequences and series in modelling.

Be able to use the standard formulae associated with arithmetic sequences and series.

Be able to use the standard formulae associated with geometric sequences and series.

Be able to write (a + bx)ⁿ in the form aⁿ(1 + bx/a)ⁿ and hence expand (a + bx)ⁿ.

Know that a series is the sum of consecutive terms of a sequence.

Know the condition for a geometric series to be convergent and be able to find its sum to infinity.

Know the difference between convergent and divergent sequences.

Know what a sequence of numbers is and the meaning of finite and infinite with reference to sequences.

Understand and use arithmetic sequences and series.

Understand and use geometric sequences and series.

Understand and use sigma notation.

Use the binomial expansion of (1 + x)ⁿ where n is any rational number.

Be able to solve simple trigonometric equations in given intervals and know the principal values from the inverse trigonometric functions.

Be able to use the definitions of sin θ, cos θ and tan θ for any angle.

Know and be able to use the exact values of sin θ and cos θ for θ = 0°, 30°, 45°, 60° and 90° and the exact values of tan θ for θ = 0°, 30°, 45° and 60°.

Know and be able to use the fact that the area of a triangle is given by ½ab sin C.

Know and be able to use the sine and cosine rules.

Know and use the graphs of sin θ, cos θ and tan θ for all values of θ, their symmetries and periodicities.

Know how to solve right-angled triangles using trigonometry.

Understand and be able to use tan θ = sin θ/cos θ.

Understand and be able to use the identity sin²θ + cos²θ = 1.

Construct proofs involving trigonometric functions and identities.

Know and be able to find the arc length and area of a sector of a circle, when the angle is given in radians.

Know and be able to use exact values of sin θ, cos θ, tan θ for θ = 0, π/6, π/4, π/3, π and multiples thereof and sin θ, cos θ for θ = π/2 and multiples thereof.

Know and use identities for sin 2θ, cos 2θ, tan 2θ.

Understand and use expressions for a cos θ ± b sin θ in the equivalent forms R sin(θ ± α) and R cos(θ ± α).

Understand and use the definition of a radian and be able to convert between radians and degrees.

Understand and use the definitions of the functions arcsin, arccos and arctan, their relationship to sin, cos and tan, their graphs and their ranges and domains.

Understand and use the definitions of the sec, cosec and cot functions.

Understand and use the identities for sin(θ ± φ), cos(θ ± φ), tan(θ ± φ).

Understand and use the relationships tan²θ + 1 = sec²θ and cot²θ + 1 = cosec²θ.

Understand and use the standard small angle approximations of sine, cosine and tangent.

Understand relationships between the graphs of the sin, cos, tan, cosec, sec and cot functions.

Use trigonometric functions to solve problems in context, including problems involving vectors, kinematics and forces.

Use trigonometric identities, relationships and definitions in solving equations.

Be able to add and subtract vectors using a diagram or algebraically, multiply a vector by a scalar, and express a vector as a combination of others.

Be able to calculate the distance between two points represented by position vectors.

Be able to calculate the magnitude and direction of a vector and convert between component form and magnitude-direction form.

Be able to use vectors to solve problems in pure mathematics and in context, including problems involving forces.

Understand and use position vectors.

Understand the language of vectors in two dimensions.

Understand the language of vectors in three dimensions.

Be able to clean data including dealing with missing data, errors and outliers.

Be able to describe frequency distributions.

Be able to interpret a cumulative frequency diagram.

Be able to interpret a scatter diagram for bivariate data, interpret a regression line or other best fit model, including interpolation and extrapolation, understanding that extrapolation might not be justified.

Be able to recognise and describe correlation in a scatter diagram and understand that correlation does not imply causation.

Be able to recognise and work with categorical, discrete, continuous and ranked data. Be able to interpret standard diagrams for grouped and ungrouped single-variable data.

Be able to recognise when a scatter diagram appears to show distinct sections in the population. Be able to recognise and comment on outliers in a scatter diagram.

Be able to select or critique data presentation techniques in the context of a statistical problem.

Know how to calculate and interpret variance and standard deviation for raw data, frequency distributions, grouped frequency distributions. Be able to use the statistical functions of a calculator to find mean and standard deviation.

Know simple measures of spread and be able to use and interpret them appropriately.

Know the standard measures of central tendency and be able to calculate and interpret them and to decide when it is most appropriate to use one of them.

Understand that diagrams representing unbiased samples become more representative of theoretical probability distributions with increasing sample size.

Understand that the area of each bar in a histogram is proportional to frequency. Be able to calculate proportions from a histogram and understand them in terms of estimated probabilities.

Understand the term outlier and be able to identify outliers. Know that the term outlier can be applied to an item of data which is: at least 2 standard deviations from the mean; OR at least 1.5 × IQR beyond the nearer quartile.

Be able to calculate the expected frequency of an event given its probability.

Be able to calculate the probability of an event.

Be able to use appropriate diagrams to assist in the calculation of probabilities.

Know to add probabilities for mutually exclusive events.

Know to multiply probabilities for independent events.

Understand and use mutually exclusive events and independent events.

Understand the concept of a complementary event and know that the probability of an event may be found by means of finding that of its complementary event.

Be able to calculate conditional probabilities by formula, from tree diagrams, two-way tables, Venn diagrams or sample space diagrams.

Be able to use Venn diagrams to assist in the calculations of probabilities. Know how to calculate probabilities for two events which are not mutually exclusive.

Know that P(B | A) = P(B) ⇔ B and A are independent.

Understand and use mutually exclusive events and independent events and associated notation and definitions.

Be able to calculate expected frequencies associated with the binomial distribution.

Be able to calculate probabilities using the binomial distribution.

Be able to calculate the numerical probabilities for a simple distribution. Understand the term discrete uniform distribution.

Be able to identify the probability of success, p, for the binomial distribution.

Be able to use probability functions, given algebraically or in tables. Know the term discrete random variable.

Recognise situations which give rise to a binomial distribution.

Understand and use mean = np.

Be able to calculate and use probabilities from a Normal distribution.

Be able to model with probability and probability distributions, including recognising when the binomial or Normal model may not be appropriate.

Be able to use the Normal distribution as a model.

Know that linear transformation of a Normal variable gives another Normal variable and know how the mean and standard deviation are affected. Be able to standardise a Normal variable.

Know that the line of symmetry of the Normal curve is located at the mean and the points of inflection are located one standard deviation away from the mean.

Know the shape of the Normal curve and understand that histograms from increasingly large samples from a Normal distribution tend to the Normal curve.

Be able to select or evaluate sampling techniques in the context of solving a statistical problem.

Be able to use samples to make informal inferences about a population, recognising that different samples might lead to different conclusions.

Understand and be able to use a variety of sampling techniques.

Understand and be able to use the concept of random sampling.

Understand and use the terms population and sample.

Be able to conduct a hypothesis test at a given level of significance. Be able to draw a correct conclusion from the results of a hypothesis test based on a binomial probability model and interpret the results in context.

Be able to identify null and alternative hypotheses (H₀ and H₁) when setting up a hypothesis test based on a binomial probability model.

Be able to identify the critical and acceptance regions.

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.

Understand the process of hypothesis testing and the associated language.

Understand when to apply 1-tail and 2-tail tests.

Be able to carry out a hypothesis test for a single mean using the Normal distribution and be able to interpret the results in context.

Be able to identify the critical and acceptance regions.

Be able to use a given correlation coefficient for a sample to make an inference about correlation or association in the population for given p-value or critical value.

Know that random samples of size n from X ~ N(μ, σ²) have the sample mean Normally distributed with mean μ and variance σ²/n.

Understand correlation as a measure of how close data points lie to a straight line. Understand that a rank correlation coefficient measures the correlation between the data ranks rather than actual data values.