QCAA Specialist Mathematics cheat sheet

Direct proof: assume the hypothesis, apply definitions and known results in sequence, and arrive at the conclusion.

Contrapositive: P implies Q is logically equivalent to (not Q) implies (not P); prove the latter when the direct route is awkward.

Proof by contradiction: assume the negation of the statement and derive a contradiction with a known truth or an earlier line.

Mathematical induction: prove the base case, assume the statement for n = k, then show it holds for n = k + 1; conclude for all integers n at or above the base.

Counterexample: a single case where the hypothesis holds but the conclusion fails disproves a universal statement.

Disproof of existence claims requires showing the negation holds for every case, not merely one failed attempt.

Necessary versus sufficient: 'P only if Q' means P implies Q; 'P if Q' means Q implies P; 'P iff Q' requires both directions.

For a rational function p(x)/q(x), vertical asymptotes occur where q(x) = 0 and p(x) is nonzero; horizontal behaviour follows the degree comparison.

Cartesian form z = a + bi with i squared equal to minus one; modulus |z| = sqrt(a squared + b squared).

Polar form z = r(cos theta + i sin theta) = r cis theta, where r is the modulus and theta the argument measured from the positive real axis.

Euler form z = r e to the i theta; multiplication multiplies moduli and adds arguments, division divides moduli and subtracts arguments.

De Moivre: (r cis theta) to the power n = r to the n cis(n theta), valid for integer n and extended to rational powers with care about multiple roots.

Conjugate z-bar = a - bi; z times z-bar = |z| squared, and z + z-bar = 2a, z - z-bar = 2bi.

nth roots of a complex number are equally spaced on a circle of radius r to the power 1/n, separated by 2 pi/n radians.

Argument is multivalued modulo 2 pi; the principal argument lies in the interval from minus pi to pi.

Loci: |z - z0| = r is a circle, |z - z0| = |z - z1| is the perpendicular bisector, and arg(z - z0) = alpha is a ray.

Dot product a dot b = |a||b| cos theta = a1b1 + a2b2 + a3b3; perpendicular vectors have zero dot product.

Cross product magnitude |a cross b| = |a||b| sin theta equals the area of the parallelogram spanned; direction follows the right-hand rule.

Scalar triple product a dot (b cross c) gives the signed volume of the parallelepiped; zero value means the vectors are coplanar.

Line in vector form r = a + t d; plane in vector form r = a + s u + t v, or Cartesian n dot (r - a) = 0.

Matrix multiplication is associative but not commutative; the identity I satisfies AI = IA = A.

Inverse exists iff det(A) is nonzero; for a 2x2 matrix, the inverse is (1/det) times the matrix with a and d swapped and b, c negated.

A system Ax = b has a unique solution when det(A) is nonzero; otherwise check consistency by row reduction.

Eigenvalues satisfy det(A - lambda I) = 0; eigenvectors solve (A - lambda I)v = 0 for each eigenvalue.

Product rule: (uv)' = u'v + uv'; quotient rule: (u/v)' = (u'v - uv')/v squared.

Chain rule: dy/dx = dy/du times du/dx; for parametric curves dy/dx = (dy/dt)/(dx/dt).

Integration by parts: integral of u dv = uv - integral of v du; choose u by the LIATE priority order.

Fundamental theorem: the integral from a to b of f(x) dx = F(b) - F(a) where F' = f.

Separable differential equations: separate variables, integrate both sides, then apply the initial condition to fix the constant.

First-order linear equations dy/dx + P(x)y = Q(x) are solved with integrating factor e to the integral of P(x) dx.

Second-order homogeneous linear equations with constant coefficients: solve the characteristic equation; distinct, repeated, or complex roots give different solution forms.

Volume of revolution about the x-axis: V = pi times the integral of y squared dx; about the y-axis use shells or dy slices.

Binomial probability: P(X = k) = C(n,k) p to the k (1 - p) to the n - k, with mean np and variance np(1 - p).

Normal approximation to the binomial is reasonable when np and n(1 - p) both exceed about 5 to 10; apply a continuity correction.

Standardising: z = (x - mu)/sigma; the standard normal has mean 0 and variance 1.

Counting: permutations nPr = n!/(n - r)!; combinations nCr = n!/(r!(n - r)!).

Pigeonhole principle: placing more than n objects into n boxes forces at least one box to hold two or more.

Recurrence relations define sequences by earlier terms; solve linear recurrences by finding a closed form or iterating.

Graph theory: the sum of all vertex degrees equals twice the number of edges; a tree on n vertices has exactly n - 1 edges.

Euler's formula for connected planar graphs: V - E + F = 2.

Pythagorean identity: sin squared x + cos squared x = 1; divide by cos squared or sin squared for the tangent and cotangent forms.

Compound angles: sin(A + B) = sin A cos B + cos A sin B; cos(A + B) = cos A cos B - sin A sin B.

Double angle: sin 2A = 2 sin A cos A; cos 2A = cos squared A - sin squared A = 1 - 2 sin squared A = 2 cos squared A - 1.

General solutions: sin x = k gives x = arcsin k + 2n pi or x = pi - arcsin k + 2n pi; cos x = k gives x = plus or minus arccos k + 2n pi.

Auxiliary angle: a sin x + b cos x = R sin(x + alpha) with R = sqrt(a squared + b squared) and tan alpha = b/a.

Derivatives: d/dx sin x = cos x, d/dx cos x = -sin x, d/dx tan x = sec squared x.

Small-angle approximations: sin x is approximately x and cos x is approximately 1 - x squared/2 for x in radians near zero.

Reciprocal identities: sec x = 1/cos x, cosec x = 1/sin x, cot x = 1/tan x; watch for undefined points where denominators vanish.

Modelling cycle: formulate assumptions, build the model, solve it, interpret results against the real situation, then refine.

Check units and dimensions at each step; a dimensional mismatch signals an algebraic error.

For optimisation, set the derivative to zero, classify with the second derivative or a sign table, and check endpoints of the domain.

Related rates: differentiate the linking equation with respect to time and substitute known rates and values.

Exponential models y = A e to the kt describe growth when k is positive and decay when k is negative; doubling time is ln 2 divided by k.

Logistic growth dy/dt = ky(1 - y/L) has carrying capacity L and an S-shaped solution curve.

Sensitivity analysis: vary a parameter and observe how the output changes to judge how robust the model's predictions are.

When reporting a model, state its assumptions and limitations explicitly and compare predictions with observed data.

Direct proof: assume the hypothesis, apply definitions and known results in sequence, and arrive at the conclusion.

Contrapositive: P implies Q is logically equivalent to (not Q) implies (not P); prove the latter when the direct route is awkward.

Proof by contradiction: assume the negation of the statement and derive a contradiction with a known truth or an earlier line.

Mathematical induction: prove the base case, assume the statement for n = k, then show it holds for n = k + 1; conclude for all integers n at or above the base.

Counterexample: a single case where the hypothesis holds but the conclusion fails disproves a universal statement.

Disproof of existence claims requires showing the negation holds for every case, not merely one failed attempt.

Necessary versus sufficient: 'P only if Q' means P implies Q; 'P if Q' means Q implies P; 'P iff Q' requires both directions.

For a rational function p(x)/q(x), vertical asymptotes occur where q(x) = 0 and p(x) is nonzero; horizontal behaviour follows the degree comparison.

Cartesian form z = a + bi with i squared equal to minus one; modulus |z| = sqrt(a squared + b squared).

Polar form z = r(cos theta + i sin theta) = r cis theta, where r is the modulus and theta the argument measured from the positive real axis.

Euler form z = r e to the i theta; multiplication multiplies moduli and adds arguments, division divides moduli and subtracts arguments.

De Moivre: (r cis theta) to the power n = r to the n cis(n theta), valid for integer n and extended to rational powers with care about multiple roots.

Conjugate z-bar = a - bi; z times z-bar = |z| squared, and z + z-bar = 2a, z - z-bar = 2bi.

nth roots of a complex number are equally spaced on a circle of radius r to the power 1/n, separated by 2 pi/n radians.

Argument is multivalued modulo 2 pi; the principal argument lies in the interval from minus pi to pi.

Loci: |z - z0| = r is a circle, |z - z0| = |z - z1| is the perpendicular bisector, and arg(z - z0) = alpha is a ray.

Dot product a dot b = |a||b| cos theta = a1b1 + a2b2 + a3b3; perpendicular vectors have zero dot product.

Cross product magnitude |a cross b| = |a||b| sin theta equals the area of the parallelogram spanned; direction follows the right-hand rule.

Scalar triple product a dot (b cross c) gives the signed volume of the parallelepiped; zero value means the vectors are coplanar.

Line in vector form r = a + t d; plane in vector form r = a + s u + t v, or Cartesian n dot (r - a) = 0.

Matrix multiplication is associative but not commutative; the identity I satisfies AI = IA = A.

Inverse exists iff det(A) is nonzero; for a 2x2 matrix, the inverse is (1/det) times the matrix with a and d swapped and b, c negated.

A system Ax = b has a unique solution when det(A) is nonzero; otherwise check consistency by row reduction.

Eigenvalues satisfy det(A - lambda I) = 0; eigenvectors solve (A - lambda I)v = 0 for each eigenvalue.

Product rule: (uv)' = u'v + uv'; quotient rule: (u/v)' = (u'v - uv')/v squared.

Chain rule: dy/dx = dy/du times du/dx; for parametric curves dy/dx = (dy/dt)/(dx/dt).

Integration by parts: integral of u dv = uv - integral of v du; choose u by the LIATE priority order.

Fundamental theorem: the integral from a to b of f(x) dx = F(b) - F(a) where F' = f.

Separable differential equations: separate variables, integrate both sides, then apply the initial condition to fix the constant.

First-order linear equations dy/dx + P(x)y = Q(x) are solved with integrating factor e to the integral of P(x) dx.

Second-order homogeneous linear equations with constant coefficients: solve the characteristic equation; distinct, repeated, or complex roots give different solution forms.

Volume of revolution about the x-axis: V = pi times the integral of y squared dx; about the y-axis use shells or dy slices.

Binomial probability: P(X = k) = C(n,k) p to the k (1 - p) to the n - k, with mean np and variance np(1 - p).

Normal approximation to the binomial is reasonable when np and n(1 - p) both exceed about 5 to 10; apply a continuity correction.

Standardising: z = (x - mu)/sigma; the standard normal has mean 0 and variance 1.

Counting: permutations nPr = n!/(n - r)!; combinations nCr = n!/(r!(n - r)!).

Pigeonhole principle: placing more than n objects into n boxes forces at least one box to hold two or more.

Recurrence relations define sequences by earlier terms; solve linear recurrences by finding a closed form or iterating.

Graph theory: the sum of all vertex degrees equals twice the number of edges; a tree on n vertices has exactly n - 1 edges.

Euler's formula for connected planar graphs: V - E + F = 2.

Pythagorean identity: sin squared x + cos squared x = 1; divide by cos squared or sin squared for the tangent and cotangent forms.

Compound angles: sin(A + B) = sin A cos B + cos A sin B; cos(A + B) = cos A cos B - sin A sin B.

Double angle: sin 2A = 2 sin A cos A; cos 2A = cos squared A - sin squared A = 1 - 2 sin squared A = 2 cos squared A - 1.

General solutions: sin x = k gives x = arcsin k + 2n pi or x = pi - arcsin k + 2n pi; cos x = k gives x = plus or minus arccos k + 2n pi.

Auxiliary angle: a sin x + b cos x = R sin(x + alpha) with R = sqrt(a squared + b squared) and tan alpha = b/a.

Derivatives: d/dx sin x = cos x, d/dx cos x = -sin x, d/dx tan x = sec squared x.

Small-angle approximations: sin x is approximately x and cos x is approximately 1 - x squared/2 for x in radians near zero.

Reciprocal identities: sec x = 1/cos x, cosec x = 1/sin x, cot x = 1/tan x; watch for undefined points where denominators vanish.

Modelling cycle: formulate assumptions, build the model, solve it, interpret results against the real situation, then refine.

Check units and dimensions at each step; a dimensional mismatch signals an algebraic error.

For optimisation, set the derivative to zero, classify with the second derivative or a sign table, and check endpoints of the domain.

Related rates: differentiate the linking equation with respect to time and substitute known rates and values.

Exponential models y = A e to the kt describe growth when k is positive and decay when k is negative; doubling time is ln 2 divided by k.

Logistic growth dy/dt = ky(1 - y/L) has carrying capacity L and an S-shaped solution curve.

Sensitivity analysis: vary a parameter and observe how the output changes to judge how robust the model's predictions are.

When reporting a model, state its assumptions and limitations explicitly and compare predictions with observed data.