HSC Maths Ext 1 cheat sheet

A function is one-to-one on its domain if f(a)=f(b) implies a=b; only one-to-one functions possess an inverse function.

Inverse relation: swap x and y, so the graph of y=f⁻¹(x) is the reflection of y=f(x) in the line y=x.

Domain of f⁻¹ equals range of f; range of f⁻¹ equals domain of f.

Odd function: f(−x)=−f(x) for all x in the domain; even function: f(−x)=f(x); test by substitution.

Absolute value: |x| equals x for x≥0 and −x for x<0; |a−b| is the distance between a and b on the number line.

Composition (f∘g)(x)=f(g(x)) is defined only where x is in the domain of g and g(x) is in the domain of f.

Vertical asymptote of a rational function occurs where the denominator is zero and the numerator is nonzero.

Horizontal asymptote of a rational function is found by comparing degrees of numerator and denominator polynomials.

Reciprocal identities: sec θ=1/cos θ, cosec θ=1/sin θ, cot θ=1/tan θ, valid wherever the denominator is nonzero.

Pythagorean identities: sin²θ+cos²θ=1, tan²θ+1=sec²θ, 1+cot²θ=cosec²θ.

Auxiliary angle: a sin x+b cos x=R sin(x+α) with R=√(a²+b²) and tan α=b/a.

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=cos²A−sin²A=1−2sin²A=2cos²A−1.

General solution: sin θ=k gives θ=nπ+(−1)ⁿ arcsin k; cos θ=k gives θ=2nπ±arccos k, n an integer.

t-formula: with t=tan(θ/2), sin θ=2t/(1+t²) and cos θ=(1−t²)/(1+t²).

For y=a sin(bx+c), amplitude is |a| and period is 2π/|b|; for y=a tan(bx), period is π/|b|.

Product rule: d/dx(uv)=u′v+uv′; quotient rule: d/dx(u/v)=(u′v−uv′)/v².

Chain rule: d/dx f(g(x))=f′(g(x))·g′(x); differentiate the outer function then multiply by the inner derivative.

Standard derivatives: d/dx sin x=cos x, d/dx cos x=−sin x, d/dx tan x=sec²x, d/dx eˣ=eˣ, d/dx ln x=1/x.

Integration by substitution reverses the chain rule; change the limits when the integral is definite.

Definite integral gives signed area; total area between curve and x-axis requires splitting where the curve crosses the axis.

Fundamental theorem: ∫ₐᵇ f(x)dx=F(b)−F(a) where F′=f.

Stationary points satisfy f′(x)=0; classify with the sign change of f′ or with f″(x).

Volume of solid of revolution about the x-axis: V=π∫ₐᵇ y² dx.

Multiplication principle: if a task has successive stages with m and n choices, total outcomes equal m×n.

Permutations: number of ordered arrangements of r objects from n distinct objects is ⁿPᵣ=n!/(n−r)!.

Combinations: number of unordered selections of r from n is ⁿCᵣ=n!/(r!(n−r)!); order does not matter.

ⁿCᵣ=ⁿCₙ₋ᵣ and ⁿCᵣ+ⁿCᵣ₊₁=ⁿ⁺¹Cᵣ₊₁ (Pascal's identity).

Arrangements of n objects with repeated types: n!/(n₁!n₂!…nₖ!).

Circular arrangements of n distinct objects: (n−1)!; if reflections are identical, divide by 2.

Binomial theorem: (a+b)ⁿ=Σ ⁿCᵣ aⁿ⁻ʳ bʳ; general term is ⁿCᵣ aⁿ⁻ʳ bʳ.

Stars and bars: number of non-negative integer solutions of x₁+…+xₖ=n is ⁿ⁺ᵏ⁻¹Cₖ₋₁.

Direct proof: assume the hypothesis and derive the conclusion by valid logical steps.

Proof by contradiction: assume the negation of the statement and derive a contradiction, then conclude the original statement.

Proof by contrapositive: proving ¬Q⇒¬P establishes P⇒Q; the two are logically equivalent.

Mathematical induction: prove the base case, assume true for n=k, then prove true for n=k+1.

Counterexample disproves a universal statement; one example suffices to show a claim is false.

Parity arguments: an integer is even if it equals 2k and odd if it equals 2k+1 for some integer k.

Irrationality proof: assume √p=a/b in lowest terms, square, and derive a common factor contradiction.

A statement and its contrapositive are equivalent, but the converse and inverse are not automatically true.

Magnitude of a=(x,y) is |a|=√(x²+y²); for (x,y,z) use √(x²+y²+z²).

Dot product: a·b=|a||b|cos θ=a₁b₁+a₂b₂; zero dot product means perpendicular vectors.

Projection of a onto b has scalar component (a·b)/|b| and vector component ((a·b)/|b|²)b.

Parallel vectors satisfy a=λb for some scalar λ; check by comparing component ratios.

Unit vector in direction of a is a/|a|; it has magnitude 1.

In two dimensions, a perpendicular vector to (x,y) is (−y,x) or (y,−x).

Vector equation of a line: r=a+td, where a is a point on the line and d is a direction vector.

cos θ=(a·b)/(|a||b|) gives the angle between two vectors; θ lies between 0 and π.

Mean of a discrete random variable: μ=E(X)=Σ x·P(X=x).

Variance: Var(X)=E(X²)−[E(X)]²; standard deviation is its positive square root.

For independent X and Y: E(X+Y)=E(X)+E(Y) and Var(X+Y)=Var(X)+Var(Y).

Binomial distribution: P(X=k)=ⁿCₖ pᵏ(1−p)ⁿ⁻ᵏ with mean np and variance np(1−p).

Standard normal: z=(x−μ)/σ converts a normal variable to Z with mean 0 and standard deviation 1.

Sample mean of n independent observations has standard deviation σ/√n, called the standard error.

Correlation coefficient r lies between −1 and 1; r near ±1 indicates strong linear association, not causation.

For a transformation Y=aX+b: E(Y)=aE(X)+b and Var(Y)=a²Var(X).