Year 12 · HSC / Mathematics Extension 2
Flashcards and recall.
Make a short deck from material you need to recall without notes.
Try this with Mathematics Extension 2
- Write one prompt per card. Use definitions, explanations or examples from your current topic.
- Answer aloud before turning the card. Split a card if it asks for several unrelated facts.
- Set aside missed cards and revisit them in a later session.
Recall topics
- State the difference between an implication (=>) and an equivalence (<=>).
- Define the converse, negation and contrapositive of the statement 'P ⇒ Q'.
- What do the quantifiers ∀ and ∃ mean, and how is 'iff' used?
- Outline the structure of a proof by contradiction.
- Sketch the proof that √2 is irrational.
- Sketch the proof that log<sub>2</sub> 5 is irrational.
- State the definition of a > b for real numbers a, b used in inequality proofs.
- State the triangle inequality for real numbers and its geometric meaning.
- State the AM–GM inequality for two non-negative numbers and prove it.
- Quick check: prove a<sup>2</sup> + b<sup>2</sup> ≥ 2ab for all real a, b.
- In a proof by mathematical induction, what are the two essential steps?
- How is induction adapted when the initial value exceeds 1 or n does not increase by 1?
- What does sigma notation Σ<sub>r=1</sub><sup>n</sup> a_r represent, and why is it useful in induction?
- Outline proving the inequality 2<sup>n</sup> > n<sup>2</sup> for integers n > 4 by induction.
- Outline proving 3^(2n+4) − 2^(2n) is divisible by 5 for all positive integers n.
- How is induction used to prove d/dx(x<sup>n</sup>) = n·x^(n−1) for positive integers n?
- State the binomial theorem that can be proven by induction.
- State the geometric result about exterior angles provable by induction.
- State the standard unit vectors in three dimensions and the component form of a vector.
- State the magnitude of a 3D vector and how to form a unit vector.
- State the scalar (dot) product of two 3D vectors in component and geometric form.
- State the equation of a sphere centred at (a, b, c) with radius r.
- State the vector equation of the line through point A (position vector a) with direction b.
- Define i (the imaginary unit) and how it is used to solve x<sup>2</sup> = −1.
- Define the Cartesian form of a complex number and the condition for two to be equal.
- Define the complex conjugate z̄ and state z·z̄.
- State the polar (modulus–argument) form, with modulus and principal argument.
- State the key modulus and argument identities for products and powers.
- State Euler's formula and the exponential form of a complex number.
- Geometrically, what does multiplying a complex number by z = r·e^(iθ) do?
- State De Moivre's theorem.
- How is De Moivre's theorem used to derive multiple-angle trig identities?
- For a polynomial with real coefficients, what is true about complex roots?
- State the solutions of a real quadratic ax<sup>2</sup> + bx + c = 0 with negative discriminant.
- How do you solve a quadratic az<sup>2</sup> + bz + c = 0 where a, b, c are complex?
- Geometrically, how are z<sub>1</sub> + z<sub>2</sub> and z<sub>1</sub> − z<sub>2</sub> represented as vectors?
- Describe the points z̄, cz (c real) and iz relative to z on the Argand plane.
- State the nth roots of unity and where they lie in the complex plane.
- How do you find the nth roots of a complex number w?
- Describe the region |z − 3i| ≤ 4 in the complex plane.
Resource inventory inspected 4 November 2026. The main ATARMAXXING study platform is separate from tutoring. Topic labels show what was found, not a guarantee of complete curriculum coverage or tutor availability.
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