11–12Mathematics Extension 2 11–12 Syllabus (2024)

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Implementation from 2026

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Year 12

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The nature of proof

MAO-WM-01
develops understanding and fluency in mathematics through exploring and connecting mathematical concepts, choosing and applying mathematical techniques to solve problems, and communicating their thinking and reasoning coherently and clearly
ME2-12-01
selects and applies the language, notation and methods of proof to prove results

The nature of proof

The language and notation of proof

Use the formal language of Loading , including the terms ‘Loading ’, ‘Loading ’, ‘Loading ’, ‘Loading ’, ‘Loading ’, ‘contradiction’, ‘Loading ’, ‘equivalence’ and ‘Loading ’
Define a statement or proposition as a sentence that is either true or false, but not both
Use the notation $P \land Q$ to represent the statement ‘ $P and Q$ ’ and the notation $P \lor Q$ to represent the statement ‘ $P or Q$ ’
Define and use the negation of $P$ as ‘not $P$ ’, denoted $\neg P$ or $\sim P$
Define an implication as an ‘if–then’ statement, where ‘if $P$ then $Q$ ’ is denoted $P \Rightarrow Q$ or $P \to Q$ , read as ‘ $P$ implies $Q$ ’
Use the quantifiers ‘for all’ $(\forall)$ , and ‘there exists’ $(\exists)$ in formulating statements
Negate statements including the negation of a negation $\neg (\neg P) = P$ , the negation of an implication $\neg (P \Rightarrow Q) = (P and ¬ Q) = (P \land ¬ Q)$ , the negation $\neg (P and Q) = (\neg P or \neg Q)$ , that is $\neg (P \land Q) = (\neg P \lor \neg Q)$ , and the negation $\neg (P or Q) = (\neg P and \neg Q)$ , that is $\neg (P \lor Q) = (\neg P \land \neg Q)$ , noting that $P \Rightarrow Q = \neg (P and ¬ Q) = (\neg P or Q)$ , that is $P \Rightarrow Q = \neg (P \land ¬ Q) = (\neg P \lor Q)$
Define and use the converse of ‘if $P$ then $Q$ ’ as ‘if $Q$ then $P$ ’, denoted $Q \Rightarrow P$
Recognise that the converse of a true implication may or may not be true
Define equivalence of $P$ and $Q$ , as both $P \Rightarrow Q$ and $Q \Rightarrow P$ , denoted $P ⟺ Q$ or $P \leftrightarrow Q$ , read as ‘ $P$ if and only if $Q$ ’, commonly abbreviated ‘ $P$ iff $Q$ ’
Define the contrapositive of ‘if $P$ then $Q$ ’ as ‘if not $Q$ then not $P$ ’, denoted $\neg Q \Rightarrow \neg P$
Recognise that an implication is equivalent to its contrapositive, that is $(P \Rightarrow Q) ⟺ (\neg Q \Rightarrow \neg P)$ , and use this to prove results

Illustrations of proofs

Use Loading to prove the truth of mathematical statements
Use examples and counterexamples to test the truth of mathematical statements
Prove results involving Loading

Proof of inequalities

Prove results involving inequalities using the definition of $a > b$ for real $a$ and $b$ , that is $a > b$ if and only if $a – b > 0$
Prove results involving inequalities using the property that squares of real numbers are non-negative, in particular ${(a \pm b)}^{2} \geq 0$
Prove and use results for numbers: if $a > b$ then $a \pm c > b \pm c$ ; if $a > b > 0$ then $\frac{1}{b} > \frac{1}{a} > 0$ and vice versa; if $(a| > (b|$ then $a^{2} > b^{2}$ and vice versa; if $a > b$ and $b > c$ then $a > c$ ; if $a > b$ and $c > d$ then $a + c > b + d$ ; if $a > b$ and $c > 0$ then $ac > bc$ ; if $a > b$ and $c < 0$ then $ac < bc$
Prove and use the triangle inequality $(a + b| \leq (a| + (b|$ and interpret the inequality geometrically
Establish and use the relationship between the arithmetic mean and geometric mean for two non-negative numbers, that is $\forall a, b \geq 0, \frac{a + b}{2} \geq \sqrt{ab}$
Prove results involving inequalities using previously obtained or known inequalities
Prove inequalities involving geometry
Prove results using the squeeze theorem: if $f (x) \leq g (x) \leq h (x)$ for all $x$ that are near $k$ , but not necessarily at $k$ , and $\lim_{x \to k} f (x) = \lim_{x \to k} h (x) = L$ , then $\lim_{x \to k} g (x) = L$
Prove inequalities using graphical or calculus techniques or a combination of both

Further proof by mathematical induction

Prove results involving trigonometric, logarithmic, exponential, Loading or other identities, including the Loading , using Loading
Prove Loading results using mathematical induction
Prove results in calculus using mathematical induction
Explain that a Loading , or Loading , is a formula that defines each term of a Loading using a preceding term
Prove results involving first-order recursive formulas using mathematical induction
Prove geometric results using mathematical induction

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11–12Mathematics Extension 2 11–12 Syllabus (2024)

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