QCAA Specialist Mathematics flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Contrapositive proof

To prove P implies Q, prove the equivalent statement not Q implies not P. This is often easier when the hypothesis is a negative or inequality condition that is hard to use directly.

Mathematical induction steps

Establish the base case, assume the result for an arbitrary n = k, then use that assumption to derive the result for n = k + 1. Conclude the statement holds for all integers from the base onward.

Counterexample disproof

A universal statement claiming every element has a property is disproved by one valid case where the hypothesis holds but the conclusion fails. Verify the case satisfies all stated conditions.

Necessary and sufficient condition

P is necessary for Q when Q implies P. P is sufficient for Q when P implies Q. A condition that is both is called necessary and sufficient, written P if and only if Q.

Modulus and argument

For z = a + bi, the modulus is the square root of a squared plus b squared, and the argument is the angle from the positive real axis, found using arctan of b over a with quadrant adjustment.

De Moivre's theorem

For z = r cis theta and integer n, z to the power n equals r to the power n cis(n theta). This converts powers and roots of complex numbers into simple modulus and argument operations.

Conjugate product identity

The product of a complex number and its conjugate equals the square of its modulus: (a + bi)(a - bi) = a squared + b squared. This is used to rationalise complex denominators.

Complex nth roots

The n distinct nth roots of r cis theta have modulus r to the power 1/n and arguments (theta + 2k pi)/n for k from zero to n minus one, spaced equally around a circle.

Complex locus circle

The equation modulus of z minus z0 equals r describes all points at distance r from the fixed point z0, which is a circle centred at z0 with radius r.

Dot product formula

The dot product of vectors a and b equals the product of their magnitudes times the cosine of the angle between them, also computed as the sum of componentwise products.

Cross product area

The magnitude of the cross product of two vectors equals the area of the parallelogram they span. Half this value gives the area of the triangle formed by the two vectors.

Scalar triple product

The scalar triple product a dot (b cross c) gives the signed volume of the parallelepiped defined by the three vectors. It equals zero exactly when the vectors are coplanar.

Matrix inverse condition

A square matrix has an inverse if and only if its determinant is nonzero. A zero determinant means the matrix is singular and the associated linear system lacks a unique solution.

Eigenvalue characteristic equation

Eigenvalues of A are the values of lambda satisfying the determinant of A minus lambda I equals zero. Each eigenvalue then yields eigenvectors by solving (A minus lambda I)v = 0.

Chain rule

When y depends on u and u depends on x, dy/dx equals dy/du multiplied by du/dx. This composes derivatives through nested functions and underlies implicit and parametric differentiation.

Integration by parts

The integral of u dv equals uv minus the integral of v du. Choose u as the function that simplifies when differentiated, following the standard priority order of algebraic, trigonometric, exponential, logarithmic.

Separable differential equation

An equation of the form dy/dx equals f(x)g(y) is solved by separating variables, integrating both sides, and applying the initial condition to determine the constant of integration.

Integrating factor method

For dy/dx plus P(x)y equals Q(x), multiply through by the integrating factor e to the integral of P(x) dx. The left side then becomes the derivative of a product.

Volume of revolution

Rotating y = f(x) about the x-axis from a to b gives volume equal to pi times the integral of y squared dx. Each thin slice contributes a disc of area pi y squared.

Binomial distribution formulas

For n independent trials with success probability p, the probability of exactly k successes is C(n,k) p to the k (1 minus p) to the n minus k, with mean np and variance np(1 minus p).

Standard normal transformation

Any normal variable X with mean mu and standard deviation sigma converts to a standard normal score by z equals (x minus mu) divided by sigma, allowing use of tables for the standard normal.

Combinations versus permutations

Permutations count ordered arrangements: n! divided by (n minus r)!. Combinations count unordered selections: n! divided by r!(n minus r)!. Use combinations when order does not matter.

Pigeonhole principle

If more than n objects are placed into n containers, at least one container holds two or more objects. This proves existence results without constructing the specific arrangement.

Euler's planar graph formula

For a connected planar graph, the number of vertices minus the number of edges plus the number of faces equals two. This constrains how many edges such a graph can have.

Pythagorean trigonometric identity

Sine squared x plus cosine squared x equals one for all x. Dividing by cosine squared gives the tangent form, and dividing by sine squared gives the cotangent form.

Compound angle formulas

Sine of A plus B equals sin A cos B plus cos A sin B. Cosine of A plus B equals cos A cos B minus sin A sin B. These generate double-angle and sum-to-product identities.

Double angle identities

Sine of 2A equals 2 sin A cos A. Cosine of 2A equals cos squared A minus sin squared A, equivalently 1 minus 2 sin squared A or 2 cos squared A minus 1.

Auxiliary angle form

The expression a sin x plus b cos x rewrites as R sin(x plus alpha), where R equals the square root of a squared plus b squared and tan alpha equals b over a.

General solution for sine

The equation sin x equals k has solutions x equals arcsin k plus 2n pi and x equals pi minus arcsin k plus 2n pi for integer n, covering all angles with that sine value.

Exponential growth model

The model y equals A e to the kt grows when k is positive and decays when k is negative. Doubling time equals ln 2 divided by k; half-life equals ln 2 divided by the magnitude of k.

Logistic growth model

The differential equation dy/dt equals ky(1 minus y/L) describes constrained growth. The population approaches carrying capacity L and follows an S-shaped curve whose steepest growth occurs at half capacity.

Optimisation procedure

Find stationary points by setting the derivative to zero, classify each using the second derivative or a sign change, then compare values at endpoints of the stated domain.