11-12.ACMSM035 When two chords of a circle intersect, the product of the lengths of the intervals on one chord equals the product of the lengths of the intervals on the other chord
11-12.ACMSM036 When a secant (meeting the circle at A and B) and a tangent (meeting the circle at T) are drawn to a circle from an external point M, the square of the length of the tangent equals the product of the lengths to the circle on the secant. (AM × BM = TM2)
11-12.ACMSM037 Suitable converses of some of the above results
11-12.ACMSM055 define and use basic linear transformations: dilations of the form (x,y)→(λ 1x,λ 2y), rotations about the origin and reflection in a line which passes through the origin, and the representations of these transformations by 2 × 2 matrices
11-12.ACMSM110 examine the three cases for solutions of systems of equations – a unique solution, no solution, and infinitely many solutions – and the geometric interpretation of a solution of a system of equations with three variables.
11-12.ACMSM120 find and use the derivative of the inverse trigonometric functions: arcsine, arccosine and arctangent
11-12.ACMSM121 integrate expressions of the form ±1/√a² –x² and a/(a² + x²)
11-12.ACMSM122 use partial fractions where necessary for integration in simple cases
11-12.ACMSM123 integrate by parts.
11-12. Applications of integral calculus:
11-12.ACMSM124 calculate areas between curves determined by functions
11-12.ACMSM125 determine volumes of solids of revolution about either axis
11-12.ACMSM126 use numerical integration using technology
11-12.ACMSM127 use and apply the probability density function, f(t) = λe to the –λt power for t ≥ 0, of the exponential random variable with parameter λ > 0, and use the exponential random variables and associated probabilities and quantiles to model data and solve practical problems.
11-12.4.2 Rates of change and differential equations
11-12. Rates of change and differential equations
11-12.ACMSM128 use implicit differentiation to determine the gradient of curves whose equations are given in implicit form
11-12.ACMSM129 Related rates as instances of the chain rule: dy/dx = dy/du × du/dx
11-12.ACMSM130 solve simple first-order differential equations of the form dy/dx = f(x), differential equations of the form dy/dx = g(y) and, in general, differential equations of the form dy/dx = f(x)g(y) using separation of variables
11-12.ACMSM131 examine slope (direction or gradient) fields of a first order differential equation
11-12.ACMSM132 formulate differential equations including the logistic equation that will arise in, for example, chemistry, biology and economics, in situations where rates are involved
11-12. Modelling motion:
11-12.ACMSM133 examine momentum, force, resultant force, action and reaction
11-12.ACMSM134 consider constant and non-constant force
11-12.ACMSM135 understand motion of a body under concurrent forces
11-12.ACMSM136 consider and solve problems involving motion in a straight line with both constant and nonconstant acceleration, including simple harmonic motion and the use of expressions dv/dt,v(dv/dx) and (d(½v²)/dx for acceleration.
11-12.4.3 Statistical inference
11-12. Sample means:
11-12.ACMSM137 examine the concept of the sample mean X as a random variable whose value varies between samples where X is a random variable with mean μ and the standard deviation σ
11-12.ACMSM138 simulate repeated random sampling, from a variety of distributions and a range of sample sizes, to illustrate properties of the distribution of X across samples of a fixed size n, including its mean μ, its standard deviation σ/√n(where μ and σ are the mean and standard deviation of X), and its approximate normality if n is large
11-12.ACMSM139 simulate repeated random sampling, from a variety of distributions and a range of sample sizes, to illustrate the approximate standard normality of X–μ/s √n for large samples (n ≥ 30), where s is the sample standard deviation.
11-12. Confidence intervals for means:
11-12.ACMSM140 understand the concept of an interval estimate for a parameter associated with a random variable
11-12.ACMSM141 examine the approximate confidence interval (X – zs/√n, X + zs/√n), as an interval estimate for μ, the population mean, where z is the appropriate quantile for the standard normal distribution
11-12.ACMSM142 use simulation to illustrate variations in confidence intervals between samples and to show that most but not all confidence intervals contain μ
11-12.ACMSM143 use x and s to estimate μ and σ, to obtain approximate intervals covering desired proportions of values of a normal random variable and compare with an approximate confidence interval for μ
11-12.ACMSM144 collect data and construct an approximate confidence interval to estimate a mean and to report on survey procedures and data quality.