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ACMSM065

prove results for sums, such as \(1+4+9\dots+n^2=\frac{n(n+1)(2n+1)}6\) for any positive integer n

ACMEM102

recognise relations between volume and capacity, recognising that \(1\mathrm c\mathrm m^3=1\mathrm m\mathrm L\) and \(1\mathrm m^3=1\mathrm k\mathrm L\)

ACMMM116

establish and use the formula \(\int x^ndx=\frac1{n+1}x^{n+1}+c\) for \(n\neq-1\)

ACMGM028

classify a numerical variable as discrete, such as the number of rooms in a house, or continuous, such as the temperature in degrees Celsius

Green chemistry principles include the design of chemical synthesis processes that use renewable raw materials, limit the use of potentially harmful solvents and minimise the amount of unwanted products

ACMMM099

recognise that \(e\) is the unique number \(a\) for which the above limit is 1

ACMSM067

define the imaginary number i as a root of the equation \(x^2=-1\)

ACHAH356

The reign of Nero and the role of key events, including Rome’s relationship with Parthia, the Great Fire, the Pisonian Conspiracy, the rebellion of Vindex and Galba, Nero’s Golden House, and the role of influential individuals, for example Agrippina the …

ACMMM068

recognise and use the recursive definition of an arithmetic sequence: \(t_{n+1}=t_n+d\)

ACMMM072

recognise and use the recursive definition of a geometric sequence:\(t_{n+1}=rt_n\)

ACMSM101

review the concepts of vectors from Unit 1 and extend to three dimensions including introducing the unit vectors i, j and k.

ACSCH028

Nanomaterials are substances that contain particles in the size range 1–100 nm and have specific properties relating to the size of these particles

ACMMM178

use the approximate confidence interval \(\left(\widehat p-z\sqrt[{}]{(\widehat p(1-\widehat p)/n},\;\;\widehat p+z\sqrt[{}]{(\widehat p(1-\widehat p)/n}\right),\) as an interval estimate for \(p\), where \(z\) is the appropriate quantile for the standard …

ACMMM053

review the probability scale: \(0\leq P(A)\leq1\) for each event \(A,\) with \(P\left(A\right)=0\) if \(A\) is an impossibility and \(P\left(A\right)=1\) if \(A\) is a certaint

ACMSM116

integrate using the trigonometric identities \(\mathrm s\mathrm i\mathrm n^2x=\frac12(1-\mathrm c\mathrm o\mathrm s\;2x)\), \(\mathrm c\mathrm o\mathrm s^2x=\frac12(1+\mathrm c\mathrm o\mathrm s\;2x)\) and \(1+\;\mathrm t\mathrm a\mathrm n^2x=\mathrm …

ACMMM014

recognise features of the graphs of \(y=x^n\) for \(n\in\boldsymbol N,\) \(n=-1\) and \(n=½\), including shape, and behaviour as \(x\rightarrow\infty\) and \(x\rightarrow-\infty\)

ACMMM156

recognise the qualitative features of the graph of \(y=\log_ax\) \((a>1)\) including asymptotes, and of its translations \(y=\log_ax+b\) and \(y=\log_a{(x+c)}\)

ACMMM179

define the approximate margin of error \(E=z\sqrt[{}]{(\widehat p(1-\widehat p)/n}\) and understand the trade-off between margin of error and level of confidence

ACMMM176

simulate repeated random sampling, for a variety of values of \(p\) and a range of sample sizes, to illustrate the distribution of \(\widehat p\) and the approximate standard normality of \(\frac{\widehat p\;-p}{\sqrt[{}]{(\widehat p(1-\widehat p)/n}}\) …

ACMSM055

define and use basic linear transformations: dilations of the form \((\mathrm x,\mathrm y)\longrightarrow({\mathrm\lambda}_1\mathrm x,{\mathrm\lambda}_2\mathrm y)\) , rotations about the origin and reflection in a line which passes through the origin, …

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