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ACMSM067

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

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

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, …

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 \(n\neq-1\)

ACMMM020

recognise features of the graphs of \(x^2+y^2=r^2\) and \(\left(x-a\right)^2+\left(y-b\right)^2=r^2\), including their circular shapes, their centres and their radii

ACMSM121

integrate expressions of the form \(\frac{\pm1}{\sqrt[{}]{a^2-x^2}}\) and \(\frac a{a^2+x^2}\)

ACMMM099

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

ACMSM063

prove irrationality by contradiction for numbers such as \(\sqrt[{}]2\) and \(\log_25\)

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

ACMMM011

recognise features of the graph of the general quadratic \(y=ax^2+bx+c\)

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 …

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