Upper secondaryCode 9231
Further Mathematics
24 topics and 85 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
1.1Roots of polynomial equations2 objectives
- recall how the roots of a polynomial equation are related to its coefficients, and apply these relations
- use a substitution to turn a given equation into a new one whose roots are simply linked to the roots of the starting equation
1.2Rational functions and graphs2 objectives
- sketch the graph of a simple rational function whose numerator and denominator each have degree 2 or less, including working out any oblique asymptotes
- understand how the graphs of y = f(x), y² = f(x), y = 1/f(x), y = f(hx) and y = f(x + h) are connected to one another, and make use of these links
1.3Summation of series3 objectives
- use the standard formulae for ∑r, ∑r² and ∑r³ to work out other sums that can be built from them
- apply the method of differences to find the total of a series that has a finite number of terms
- recognise, by looking directly at the partial sum of n terms, whether a series converges, and work out the sum to infinity where it does
1.4Matrices5 objectives
- carry out the operations of adding, subtracting and multiplying matrices, and identify the zero matrix and the identity (or unit) matrix
- state what is meant by 'singular' and 'non-singular' when these describe square matrices, and, for 2×2 and 3×3 matrices, evaluate determinants and obtain the inverses of non-singular matrices
- understand the rule (AB)⁻¹ = B⁻¹A⁻¹ for matrices that are non-singular, and use it
- understand how 2×2 matrices can stand for some geometric transformations of the x-y plane, and in particular: understand how the transformations given by A and A⁻¹ are related; recognise that the product AB stands for doing the transformation given by B first and then the one given by A; recall the link between a transformation's area scale factor and the determinant of its matrix; find the matrix for a given transformation or for a sequence of transformations
- know what 'invariant' signifies for points and lines when transformations are described by matrices, and solve simple questions on invariant points and invariant lines
1.5Polar coordinates3 objectives
- understand how Cartesian and polar coordinates are linked, and change a curve's equation from Cartesian into polar form or from polar into Cartesian form
- sketch simple polar curves over the interval 0 ≤ θ < 2π, or −π ≤ θ ≤ π, or over any part of one of these ranges
- state the formula ½ ∫ r² dθ giving the area of a sector, and apply it in straightforward cases
1.6Vectors1 objective
- use a plane's equation in any one of the forms r = a + λb + μc, r.n = p or ax + by + cz = d, and switch between these forms whenever solving a problem calls for it
1.7Proof by induction2 objectives
- use proof by mathematical induction to show that a given result is true
- recognise when it helps to make a conjecture from a few trial cases and then confirm it by inductive proof, and do this in simple cases
2.1Hyperbolic functions4 objectives
- understand how the exponential function is used to define each hyperbolic function: sinh x, cosh x, tanhx, sech x, cosech x and cothx
- draw sketches of the graphs of the hyperbolic functions
- show and apply identities that involve hyperbolic functions
- understand how the inverse hyperbolic functions are defined and work with those definitions, and derive their logarithmic forms and use them
2.2Matrices6 objectives
- translate a situation needing 3 linear simultaneous equations in 3 unknowns to be solved into one needing a matrix equation to be solved, or the reverse
- understand the possible situations regarding whether 3 linear simultaneous equations are consistent or inconsistent, link these to whether the associated matrix is singular or not, solve the consistent systems, and describe each situation geometrically using lines and planes
- understand what is meant by the words ‘characteristic equation’, ‘eigenvalue’ and ‘eigenvector’ when they relate to square matrices
- find the eigenvalues and the eigenvectors of any 2 × 2 or 3 × 3 matrix
- express a square matrix as QDQ–1, where D is diagonal and contains the eigenvalues while the columns of Q are eigenvectors, then apply that expression
- apply the result that every square matrix obeys its own characteristic equation
2.3Differentiation3 objectives
- differentiate sin–1x, cos–1x, sinh–1x, cosh–1x and tanh–1x, along with the hyperbolic functions in general
- obtain a formula for d²y/dx² when y and x are connected implicitly or parametrically
- derive a Maclaurin’s series for a function up to its first few terms, and make use of the result
2.4Integration5 objectives
- integrate hyperbolic functions, identify integrals of the forms 1/(√(a² - x²)), 1/(√(a² + x²)) and 1/(√(x² - a²)), and perform integration of associated functions by making trigonometric or hyperbolic substitutions where suitable
- derive and apply reduction formulae to work out definite integrals
- understand the manner in which rectangles can give an approximate value for the area beneath a curve, and apply rectangles to estimate, or to establish bounds on, the area beneath a curve, or to obtain inequalities or limits involving sums
- apply integration to obtain the lengths of arcs for curves given by equations in Cartesian coordinates, including where a parameter is used, or in polar coordinates
- apply integration to find the surface area of revolution when a curve with a Cartesian equation, including one given in terms of a parameter, is rotated about one of the axes
2.5Complex numbers6 objectives
- understand de Moivre’s theorem for an integer exponent, positive or negative, through the geometric meaning of multiplying and dividing complex numbers
- prove de Moivre’s theorem when the exponent is a positive integer
- use de Moivre’s theorem, with a rational exponent that is positive or negative, to write multiple-angle trigonometrical ratios as powers of the trigonometrical ratios at the fundamental angle
- apply de Moivre’s theorem, with a rational exponent that is positive or negative, to write sin i and cos i raised to a power as expressions involving multiple angles
- apply de Moivre’s theorem, with a rational exponent that is either positive or negative, when summing series
- apply de Moivre’s theorem, with a positive or negative rational exponent, to locate and make use of the nth roots of unity
2.6Differential equations6 objectives
- work out an integrating factor for a linear differential equation of first order, then apply that factor to obtain the general solution
- state what is meant by the terms ‘complementary function’ and ‘particular integral’ as they apply to linear differential equations, and state that the general solution equals the complementary function plus a particular integral
- work out the complementary function of a linear differential equation of first or second order whose coefficients are constant
- recall the form a particular integral takes, and find one, for a linear differential equation of first or second order when the right form to try is acos px + b sin px, a polynomial, or a e^bx; and, for other simple cases in which a suitable form for the particular integral is given, find the coefficient(s) that are needed
- apply a substitution provided to transform a differential equation into either a linear equation of first or second order with constant coefficients, or a first order equation whose variables are separable
- use initial conditions to pin down a particular solution of a differential equation, and say what a solution means in the setting of the problem that the equation models
3.1Motion of a projectile3 objectives
- model how a projectile moves by treating it as a particle under constant acceleration, and understand what restrictions the model has
- apply the equations of motion in the horizontal and vertical directions to projectile problems, including finding the velocity's magnitude and direction at a stated time or position, the projectile's range on horizontal ground and the maximum height it reaches
- derive the Cartesian equation for a projectile's trajectory and use it, including in problems where the initial speed and/or the angle of projection is unknown
3.2Equilibrium of a rigid body6 objectives
- work out what the moment of a force is when taken about a point
- apply the fact that gravity's influence on a rigid body amounts to one force acting at its centre of mass, and identify where the centre of mass sits in a uniform body by reasoning from symmetry
- apply supplied data on where the centre of mass lies for a triangular lamina and for other basic shapes
- find where a composite body's centre of mass lies by treating the body as a set of particles that is equivalent to it
- use the principle that a rigid body acted on by coplanar forces is in equilibrium if and only if the forces have a zero vector sum and their moments about any point total zero
- solve questions that concern a single rigid body kept in equilibrium by coplanar forces, including those where toppling or sliding occurs
3.3Circular motion4 objectives
- understand angular speed as it applies to a particle going round a circle, and work with the relation v = rω
- understand that a particle travelling round a circle at constant speed has an acceleration pointing to the circle's centre, and apply the formulae a = rω² and a = v²/r
- solve problems in which the model is a particle moving at constant speed round a horizontal circle
- solve problems in which the model is a particle travelling round a vertical circle without any energy being lost
3.4Hooke's law3 objectives
- apply Hooke’s law as a model connecting the force generated by an elastic string or spring with how far it is stretched or compressed, and know what the term modulus of elasticity means
- apply the formula that gives a string's or a spring's elastic potential energy
- solve problems about the forces that elastic strings or springs exert, including ones where work and energy have to be taken into account
3.5Linear motion under a variable force1 objective
- solve problems that can be represented as the linear motion of a particle acted on by a force that varies, by forming and then solving a suitable differential equation
3.6Momentum1 objective
- state Newton’s experimental law, how the coefficient of restitution is defined, the property 01 GGe, and what is meant by the terms ‘perfectly elastic’ (e = 1) and ‘inelastic’ (e = 0)
4.1Continuous random variables4 objectives
- apply a probability density function that may be given piecewise
- use the result E_g^Xhi = yfg ^^ xx hhdx, true in general, where X is a continuous random variable with probability density function f(x) and g(X) is any function of X
- understand, and make use of, how the probability density function (PDF) is related to the cumulative distribution function (CDF), and work out probabilities or percentiles from either one
- apply cumulative distribution functions (CDFs) to variables that are related to each other, in simple cases
4.2Inference using normal and t-distributions5 objectives
- set up the hypotheses and carry out a hypothesis test on the population mean by means of a t-test, working from a small sample taken from a normal population whose variance is not known
- work out the pooled estimate for the variance of a population, drawing on two samples
- set up hypotheses about the difference between two population means, then carry out whichever test suits the case: a paired-sample t-test, or a 2-sample t-test, or a test that uses a normal distribution
- find, by means of a t-distribution, a confidence interval for the mean of a normal population whose variance is unknown, working from a small sample
- find a confidence interval for the difference between two population means, choosing a normal distribution or a t-distribution to suit the situation
4.3| -tests3 objectives
- fit to the supplied data a theoretical distribution dictated by a given hypothesis
- use a χ²-test to assess the goodness of fit of the fitted distribution to the data, taking the correct number of degrees of freedom
- apply a χ²-test to check whether the variables in a contingency table are independent, employing the suitable degrees of freedom value
4.4Non-parametric tests4 objectives
- understand what is meant by a non-parametric test, and appreciate the circumstances in which such a test may prove helpful
- understand the ideas behind the sign test and behind the Wilcoxon signed-rank and rank-sum tests
- apply both a sign test and a Wilcoxon signed-rank test to data from a single sample when testing a hypothesis about a population's median
- apply, as the case requires, a sign test on paired samples, a Wilcoxon signed-rank test for matched pairs or a Wilcoxon rank-sum test, to test whether populations are identical
4.5Probability generating functions3 objectives
- understand the idea behind a probability generating function (PGF), and form the PGF of a given distribution and work with it
- apply the expressions for a discrete random variable's mean and variance written using its PGF, and use those expressions to work out the mean and the variance for a stated probability distribution
- use the result that, for independent random variables, the PGF of their sum is found by multiplying their separate PGFs together
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