Upper secondaryCode 9709
Mathematics
38 topics and 153 objectives, each one something you should be able to do.
Written by Kindred in its own plain words. Your syllabus is still the official list, so check it for anything exact.
What this subject asks you to be able to do
1.1Quadratics5 objectives
- carry out completing the square on the quadratic polynomial ax² + bx + c, then employ the resulting completed square form
- work out the discriminant of the quadratic ax² + bx + c and apply what it tells you
- solve quadratic equations as well as quadratic inequalities, each involving a single unknown
- solve two simultaneous equations by substitution when one is linear and the other is quadratic
- recognise and solve those equations in x that take the form of a quadratic in some function of x
1.2Functions5 objectives
- know what is meant by a function, its domain and its range, a one-one function, an inverse function and the composition of functions
- identify, for simple examples, the range of any function you are given, and form the composition of two functions given to you
- determine whether the function you are given is one-one or not, and, in simple cases, obtain the inverse of a one-one function
- show graphically how a one-one function relates to its inverse
- understand and apply the transformations that map the graph of y = f(x) to y = f(x) + a, y = f(x + a), y = af(x) and y = f(ax), along with simple combinations of them
1.3Coordinate geometry5 objectives
- work out a straight line's equation when given enough information about it
- interpret the forms y = mx + c, y – y1 = m(x – x1) and ax + by + c = 0, and solve problems with any one of them
- appreciate that (x – a)² + (y – b)² = r² is the equation of the circle centred at (a, b) with radius r
- apply algebraic reasoning to resolve problems concerning lines and circles
- appreciate how a graph is linked to its algebraic equation, and apply the connection between the points where graphs intersect and the solutions of equations
1.4Circular measure2 objectives
- understand how a radian is defined, and apply the relationship linking radians to degrees
- apply the formulae s = rθ and A = ½ r² θ to problems involving both the length of an arc and the area of a sector of a circle
1.5Trigonometry5 objectives
- sketch the sine, cosine and tangent graphs, whatever the size of the angle and whether it is measured in degrees or radians, and make use of them
- apply exact values for the sine, cosine and tangent of 30°, 45°, 60°, and angles related to them
- use sin⁻¹x, cos⁻¹x, tan⁻¹x as notation for the principal values of each inverse trigonometric relation
- apply the identities sin²θ + cos²θ = 1 and cosθ/sinθ = tanθ
- find every solution of a simple trigonometric equation that lies within a given interval
1.6Series4 objectives
- apply the result for expanding (a + b)^n when n is any positive integer
- recognise when a sequence is an arithmetic progression and when it is a geometric progression
- apply the formula for the nth term of an arithmetic or geometric progression, and the one for the sum of its first n terms, when solving problems
- apply the condition for a geometric progression to converge, and, when it converges, use the formula for its sum to infinity
1.7Differentiation4 objectives
- understand the gradient at a point on a curve as the limit approached by chord gradients over a suitable sequence of chords, and write first derivatives as f ′(x) or dy/dx and second derivatives as f ″(x) or d²y/dx²
- apply the derivative of x^n, where n is any rational number, along with constant multiples, sums and differences of functions, and use the chain rule for composite functions
- use differentiation in working with gradients, tangents and normals, functions that are increasing or decreasing, and rates of change
- find the stationary points of a curve, decide what type each one is, and use what you know about them when sketching graphs
1.8Integration4 objectives
- understand that integration reverses differentiation, and integrate (ax + b)^n for every rational n except –1, along with constant multiples, sums and differences
- solve problems in which a constant of integration has to be found
- evaluate definite integrals
- apply definite integration to work out the area of a region enclosed by a curve together with lines parallel to the axes, by a curve and a line, or by two curves, and to work out a volume of revolution about either axis
2.1Algebra3 objectives
- understand what |x| means, sketch y = |ax + b|, and apply relations such as |a| = |b| ⇔ a² = b² and |x – a| < b ⇔ a – b < x < a + b to solve equations and inequalities
- divide a polynomial whose degree is not above 4 by a linear or a quadratic polynomial, then state the quotient and the remainder (the remainder can be zero)
- apply the factor theorem together with the remainder theorem
2.2Logarithmic and exponential functions4 objectives
- understand how logarithms and indices relate to each other, and apply the laws of logarithms (leaving out change of base)
- understand how e^x and lnx are defined and what properties they possess, including the link between them as inverse functions, together with their graphs
- apply logarithms when solving equations and inequalities where the unknown is in indices
- apply logarithms to put a given relationship into linear form, and from its gradient and/or intercept work out the unknown constants
2.3Trigonometry2 objectives
- understand how secant, cosecant and cotangent relate to cosine, sine and tangent, and work with the graphs and properties of the six trigonometric functions for an angle of any size
- use trigonometrical identities to simplify expressions, to evaluate them exactly and to solve equations, choosing in each case the identity or identities that suit the context, and be especially fluent with sect²θ + 1 = tan²θ and cosec²θ + 1 = cot²θ; the compound-angle expansions of sin(A ± B), cos(A ± B) and tan(A ± B); writing a sinθ + b cosθ both as R sin(θ + α) and as R cos(θ + α); and the double-angle formulae giving sin 2A, cos 2A and tan 2A
2.4Differentiation3 objectives
- apply knowledge of the derivatives of e^x, lnx, sin x, cos x and tan x, along with constant multiples, sums, differences and composite functions
- differentiate products and quotients
- find and apply the first derivative of a function given in parametric or implicit form
2.5Integration3 objectives
- extend the idea of 'reverse differentiation' so that it also covers integrating e^(ax + b), 1 / (ax + b), sin(ax + b), cos(ax + b) and sec²(ax + b)
- use relationships between trigonometric functions to help with integration
- understand and apply the trapezium rule to obtain an approximate figure for a definite integral
2.6Numerical solution of equations3 objectives
- pin down an approximate position for a root of an equation, drawing on graphs and/or hunting for a change in sign
- understand what is meant by a sequence of approximations that converges to a root of the equation being solved, and use the notation for such a sequence
- understand how a supplied straightforward iterative rule of the form xn + 1 = F(xn) ties in with the equation under study, and apply either a supplied iteration or one built from a supplied rearrangement of an equation, in order to find a root correct to a specified level of precision
3.1Algebra5 objectives
- understand what |x| represents, draw a sketch of the graph of y = |ax + b|, and apply relations such as |a| = |b| ⇔ a2 = b2 and |x – a| < b ⇔ a – b < x < a + b when working through equations and inequalities
- carry out a division of a polynomial having a degree that does not exceed 4 by a linear or quadratic polynomial, and state the quotient together with the remainder (the remainder may equal zero)
- apply the factor theorem together with the remainder theorem
- recall the right form of partial fractions for a rational function and decompose the function accordingly, for denominators no more complex than (ax + b)(cx + d)(ex + f), (ax + b)(cx + d)2, (ax + b)(cx2 + d)
- apply the expansion of (1 + x)n for any rational value of n, provided x 1 1
3.2Logarithmic and exponential functions4 objectives
- know how logarithms and indices are linked, and apply the laws of logarithms, leaving aside change of base
- understand what ex and lnx are defined to be and how they behave, including how they relate as inverse functions and what their graphs look like
- apply logarithms when solving equations and inequalities whose unknown sits in the indices
- apply logarithms to rewrite a given relationship in linear form, and so find unknown constants by looking at the gradient and/or the intercept
3.3Trigonometry2 objectives
- understand how secant, cosecant and cotangent relate to cosine, sine and tangent, and work with the graphs and properties of the six trigonometric functions for an angle of any size
- use trigonometric identities to simplify expressions, to find their exact values and when solving equations, choosing whichever identity or identities suit the context; in particular, be at ease with sect2 ii + 1 and cosecc2 ii + 1, with expanding sin(A ± B), cos(A ± B) and tan(A ± B), with the formulae giving sin 2A, cos2A and tan 2A, and with writing ab sinc ii + os as Rsin^h i ! a and as Rcos^h i ! a
3.4Differentiation3 objectives
- apply the derivatives of ex, lnx, sin x, cos x, tanx and tan–1x, even when scaled by a constant, added, subtracted or composed into composite functions
- differentiate products and quotients
- work out the first derivative when a function is given parametrically or implicitly, and use it
3.5Integration6 objectives
- broaden the notion of ‘reverse differentiation’ so that it covers integrating eax + b, ax b 1 + , sin(ax + b), cos(ax + b), sec2(ax + b) and xa 1 22 +
- apply trigonometrical relationships while performing integration
- integrate a rational function by first splitting it into partial fractions
- recognise an integrand having the form xk x f f l ^ ^ h h, and integrate functions of this kind
- recognise when it helps to treat an integrand as a product, and then integrate by parts
- apply a supplied substitution to streamline and evaluate integrals, be they definite or indefinite
3.6Numerical solution of equations3 objectives
- pin down an approximate position for a root of an equation, drawing on graphs and/or hunting for a change in sign
- understand what is meant by a sequence of approximations that converges to a root of the equation being solved, and use the notation for such a sequence
- understand how a supplied straightforward iterative rule of the form xn + 1 = F(xn) ties in with the equation under study, and apply either a supplied iteration or one built from a supplied rearrangement of an equation, in order to find a root correct to a specified level of precision
3.7Vectors6 objectives
- apply the standard notations associated with vectors
- carry out vector addition and subtraction, multiply a vector by a scalar, and give each operation a geometrical interpretation
- find a vector's magnitude, and work with unit, displacement and position vectors
- understand what every symbol means when a straight line's equation is written as r = a + tb, and determine the equation of a line when enough information is supplied
- determine if a pair of lines is parallel, intersecting or skew, and find where they intersect when they do
- apply formulae to work out the scalar product of a pair of vectors, and employ scalar products when solving problems that concern lines and points
3.8Differential equations4 objectives
- write a differential equation that expresses a simple statement about the rate at which a quantity changes
- work out, by integrating, a general form of solution to a first order differential equation whose variables can be separated
- apply a given initial condition to find the particular solution that fits it
- interpret what the solution of a differential equation tells us about the problem that the equation models
3.9Complex numbers8 objectives
- understand what a complex number is, recall what the terms real part, imaginary part, modulus, argument and conjugate mean, and use the fact that a pair of complex numbers is equal if and only if their real parts are equal and their imaginary parts are equal
- carry out each of the four operations, adding, subtracting, multiplying and dividing, on a pair of complex numbers written in the Cartesian form x + iy
- apply the fact that non-real roots of a polynomial equation whose coefficients are real come in conjugate pairs
- represent complex numbers in geometric form on an Argand diagram
- carry out both multiplication and division on a pair of complex numbers in polar form, written r(cosθ + i sinθ) or re^{iθ}
- work out the two square roots that a complex number has
- understand, in plain language, what happens geometrically when a complex number is conjugated, and what happens when two complex numbers are added, subtracted, multiplied or divided
- use loci on an Argand diagram to show simple equations and inequalities that involve complex numbers
4.1Forces and equilibrium7 objectives
- identify every force at work in a stated situation
- appreciate that force is a vector quantity, and work out and apply components and resultants
- apply the principle that a particle in equilibrium has forces acting on it whose vector sum is zero, or equivalently that, in any direction, the components of those forces add up to zero
- understand that the contact force where two surfaces touch can be treated as two components, one normal and one frictional
- apply the ‘smooth’ contact model, and understand its limitations
- appreciate what is meant by limiting friction and limiting equilibrium, state how the coefficient of friction is defined, and apply the relationship FR = n or FR G n as the situation requires
- use Newton’s third law
4.2Kinematics of motion in a straight line4 objectives
- understand what distance and speed mean as scalar quantities, together with displacement, velocity and acceleration, which are vector quantities
- sketch and interpret displacement–time and velocity–time graphs, knowing in particular that a displacement–time graph's gradient gives velocity, a velocity–time graph's gradient gives acceleration, and the area beneath a velocity–time graph gives displacement
- apply calculus to simple problems about displacement, velocity and acceleration, differentiating and integrating with respect to time as needed
- apply suitable formulae for straight-line motion with constant acceleration
4.3Momentum2 objectives
- apply the definition of linear momentum, showing that you understand it is a vector quantity
- apply the principle that linear momentum is conserved to tackle questions that can be modelled as a direct impact between two bodies.
4.4Newton’s laws of motion4 objectives
- apply Newton’s laws of motion to a particle in linear motion whose mass is constant and which is acted on by forces that are constant; these can include friction, the tension of an inextensible string and the thrust from a connecting rod
- use W = mg to link weight and mass
- solve simple problems in which a particle with constant acceleration, moving either vertically or along an inclined plane, is a suitable model
- solve simple problems in which a system of connected particles in motion is a suitable model
4.5Energy, work and power5 objectives
- understand what work done by a force means, and calculate how much work a constant force does when its point of application is displaced in a direction not necessarily parallel to the force
- know what kinetic energy and gravitational potential energy are, and apply the right formula for each
- understand and apply the link between the change in a system's energy and the work done on it by external forces, and, where appropriate, use the principle that energy is conserved
- apply the definition of power as a force's rate of doing work, and, for a force acting along the direction of motion, use the relationship linking power, force and velocity
- solve problems involving, for instance, a car's instantaneous acceleration as it moves up or down a hill against a resistance
5.1Representation of data5 objectives
- select an appropriate method for displaying raw statistical data, and discuss the strengths and/or weaknesses of particular representations
- draw each of the following and interpret it: a stem-and-leaf diagram, a box-and-whisker plot, a histogram, a cumulative frequency graph
- know the mean, median and mode as measures of central tendency, and the range, interquartile range and standard deviation as measures of variation, and use each of them
- apply a cumulative frequency graph to estimate, e.g., medians, quartiles and percentiles, or what proportion of a distribution is greater (or less) than a given value, or lies between two values
- work out and make use of the mean and standard deviation of a data set (including grouped data), either straight from the data or from the given totals /x and x / 2, or from coded totals /^h xa − and xa 2 /^h − , then apply those totals to solve problems involving one or two data sets
5.2Permutations and combinations2 objectives
- know what the terms permutation and combination mean, and work through straightforward problems that involve making selections
- solve problems on arranging objects in a line, including ones with repetition (e.g. how many ways the letters of the word ‘NEEDLESS’ can be arranged) and ones with restriction (e.g. how many ways several people can line up when two given people must, or must not, be side by side)
5.3Probability4 objectives
- evaluate probabilities, in simple cases, by listing the equiprobable elementary events or by working them out with permutations or combinations
- apply the addition and multiplication of probabilities, where suitable, in simple situations
- understand what exclusive and independent events mean, including testing if A and B are independent by setting P(A ∪ B) beside P(A)P(B) and comparing the two
- work out and apply conditional probabilities in simple situations
5.4Discrete random variables3 objectives
- draw up a table showing the probability distribution of a discrete random variable X in a given situation, and find E(X) and Var(X)
- apply the probability formulae for binomial and geometric distributions, and recognise real situations for which these distributions make suitable models
- apply the formulae that give the expectation and variance of a binomial distribution, along with the expectation of a geometric distribution
5.5The normal distribution3 objectives
- understand how a normal distribution can serve as a model for a continuous random variable, and make use of normal distribution tables
- solve problems about a variable X, where X ~ N(μ, σ²), including working out P(X > x₁) or a similar probability from given values of x₁, μ and σ, and deducing a link between x₁, μ and σ from a given probability for P(X > x₁) or a similar probability
- state the conditions under which a normal distribution may serve as an approximation to a binomial distribution, and apply that approximation with a continuity correction to solve problems
6.1The Poisson distribution5 objectives
- apply formulae to find the probabilities of outcomes under Po(λ)
- apply the result that when X ~ Po(λ), both the mean and the variance of X equal λ
- appreciate how random events are linked to the Poisson distribution, and employ that distribution as a model
- use a Poisson distribution as an approximation for a binomial distribution in cases where this is appropriate
- apply a normal distribution, together with a continuity correction, as an approximation for a Poisson distribution in suitable cases
6.2Linear combinations of random variables1 objective
- apply, when tackling questions, the facts that E(aX + b) = aE(X) + b and Var(aX + b) = a²Var(X), that E(aX + bY) = aE(X) + bE(Y), and that Var(aX + bY) = a²Var(X) + b²Var(Y) holds for independent X and Y, that a normal X gives a normal aX + b, that independent normal X and Y give a normal aX + bY, and that independent Poisson X and Y give a Poisson X + Y.
6.3Continuous random variables2 objectives
- know what a continuous random variable means, and be able to recall and use the properties that a probability density function has
- apply a probability density function to tackle questions involving probabilities, and to work out, for a distribution, its mean and variance
6.4Sampling and estimation8 objectives
- understand how a sample differs from a population, and appreciate why samples have to be picked at random
- account for, in plain language, why a particular sampling method might not be suitable
- recognise that the mean of a sample can be thought of as a random variable in its own right, and apply the results E_X i = n and Var X n 2 = v _ i
- apply the result that _X i follows a normal distribution whenever X is normally distributed
- apply the Central Limit Theorem in situations where it is suitable to do so
- work out unbiased estimates of a population's mean and variance from a sample, whether the data are raw or summarised
- work out a confidence interval for the mean of a population, and say what it tells you, in two cases: the population is normal with known variance, or the sample is large
- work out an approximate confidence interval for the proportion in a population, based on a large sample
6.5Hypothesis tests5 objectives
- understand what a hypothesis test is and how one-tailed tests differ from two-tailed ones, and know what is meant by the null and alternative hypotheses, significance level, rejection (or critical) region, acceptance region and test statistic
- set up hypotheses and carry out a hypothesis test based on one observation from a population with a binomial or Poisson distribution, working out probabilities directly or, where appropriate, using a normal approximation for the binomial or Poisson distribution
- set up hypotheses and perform a test on the population mean for situations in which the population follows a normal distribution with known variance, or in which a large sample is taken
- understand what is meant by Type I error and Type II error when hypothesis tests are considered
- work out the probability of a Type I or a Type II error in given situations, for tests that use a normal distribution or that evaluate binomial or Poisson probabilities directly
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