Free VCE Specialist Mathematics Flashcard Review

A conjecture is a statement proposed as true but not yet proved; a proof establishes it is true in all stated cases; a single counter-example disproves it. One counter-example refutes a universal claim, but no number of supporting examples proves one.

P ⇒ Q ('if P then Q') means whenever P is true, Q is true; P is sufficient for Q and Q is necessary for P. Its converse is Q ⇒ P, which is a logically different statement and need not be true.

P ⇔ Q means both P ⇒ Q and Q ⇒ P hold, so P and Q are logically equivalent. P is then both necessary and sufficient for Q; proving an iff requires proving both implications.

Split the situation into a finite set of exhaustive, mutually relevant cases and prove the statement separately in each; if it holds in every case, it holds in general. Example: proving a property for all integers by treating even and odd separately.

Assume the statement is false (assume its negation), then derive a logical contradiction. Since the assumption leads to a contradiction, the original statement must be true. Classic example: proving √2 is irrational.

The contrapositive is ¬Q ⇒ ¬P ('if not Q then not P'). A statement and its contrapositive are logically equivalent, so proving the contrapositive proves the original implication.

∀ means 'for all' (the claim holds for every element); ∃ means 'there exists' (at least one element satisfies it). A '∀' statement is disproved by one counter-example; a '∃' statement is disproved by showing no element works.

Base step: prove the statement P(n) is true for the first value (often n=1). Inductive step: assume P(k) is true (the inductive hypothesis) and use it to prove P(k+1). Together these show P(n) holds for all integers n ≥ the base value.

The inductive hypothesis is the assumption that P(k) is true for an arbitrary k. It is used as a given within the inductive step to derive P(k+1); the proof must explicitly invoke it, otherwise the induction is invalid.

Show the base case holds (n=1 gives 0, divisible by 6). Assume k³ − k = 6m, then expand (k+1)³ − (k+1) and rewrite it as (k³ − k) plus a multiple of 6, so it is divisible by 6, completing the induction.

It expresses a rational function P(x)/Q(x) (with deg P < deg Q) as a sum of simpler fractions whose denominators are the linear and irreducible quadratic factors of Q(x). If deg P ≥ deg Q, first divide to get a polynomial plus a proper rational part.

A vertical asymptote occurs at x = a where the denominator is zero but the numerator is non-zero (after cancelling common factors). Near it the function magnitude → ∞; the graph approaches the line x = a without crossing it.

If deg(numerator) < deg(denominator): horizontal asymptote y = 0. If degrees are equal: horizontal asymptote y = ratio of leading coefficients. If numerator degree is exactly one more: an oblique (linear) asymptote found by polynomial division.

At a stationary point (f'(x)=0): if f''(x) > 0 it is a local minimum (curve concave up); if f''(x) < 0 it is a local maximum (concave down). If f''(x)=0 the test is inconclusive and other methods (e.g. sign of f') are needed.

A point of inflection is where the concavity of the curve changes (concave up to down or vice versa). It requires f''(x) = 0 (or undefined) at that point AND a sign change of f'' across it; f''=0 alone is not sufficient.

A complex number is z = x + yi with x = Re(z), y = Im(z). Its modulus is |z| = √(x² + y²), the distance from the origin to the point (x, y) in the Argand plane.

z = r cis θ = r(cos θ + i sin θ), where r = |z| ≥ 0 and θ = arg(z) is the angle from the positive real axis. The principal argument is usually taken in (−π, π].

For z₁z₂: moduli multiply and arguments add, i.e. |z₁z₂| = |z₁||z₂| and arg(z₁z₂) = arg(z₁) + arg(z₂) (mod 2π). For division, moduli divide and arguments subtract.

For z = r cis θ and integer n, zⁿ = rⁿ cis(nθ) = rⁿ(cos nθ + i sin nθ). It is used to compute powers and roots of complex numbers in polar form.

The nth roots of unity are z = cis(2πk/n) for k = 0, 1, ..., n−1. They lie on the unit circle, equally spaced 2π/n apart, forming the vertices of a regular n-gon with one vertex at 1.