Mathematics Specialised Level 4 cheat sheet

Factor theorem: (x − a) is a factor of P(x) iff P(a) = 0; remainder theorem: P(x) divided by (x − a) leaves remainder P(a).

Quadratic discriminant Δ = b² − 4ac: Δ > 0 two distinct real roots, Δ = 0 one repeated real root, Δ < 0 no real roots (two complex conjugates).

Rational function asymptotes: vertical where denominator = 0 (after cancellation); horizontal y = ratio of leading coefficients when degrees equal, y = 0 when denominator degree is larger, none when numerator degree is larger.

Completing the square: ax² + bx + c = a(x + b/(2a))² + (c − b²/(4a)); vertex at (−b/(2a), c − b²/(4a)).

Partial fractions: for distinct linear factors use A/(x−a) + B/(x−b); for a repeated factor (x−a)² include A/(x−a) + B/(x−a)²; for an irreducible quadratic include (Cx+D)/(x²+px+q).

Absolute value inequalities: |f(x)| < k ⇔ −k < f(x) < k; |f(x)| > k ⇔ f(x) < −k or f(x) > k (k > 0).

Domain of a function: exclude values making denominators zero, radicands negative (for even roots), or log arguments non-positive; range follows from the transformed parent function.

Exponential and logarithm laws: aˣ aʸ = aˣ⁺ʸ, (aˣ)ʸ = aˣʸ, logₐ(xy) = logₐx + logₐy, logₐ(xⁿ) = n logₐx, logₐx = ln x / ln a.

Graph transformations: y = af(b(x − h)) + k scales vertically by a, horizontally by 1/b, shifts right h and up k; order of transformations matters for non-commuting steps.

Distance between (x₁,y₁) and (x₂,y₂): √((x₂−x₁)² + (y₂−y₁)²); midpoint ((x₁+x₂)/2, (y₁+y₂)/2).

Gradient m = (y₂−y₁)/(x₂−x₁); parallel lines have equal gradients; perpendicular lines satisfy m₁m₂ = −1.

Line forms: point-gradient y − y₁ = m(x − x₁); general ax + by + c = 0 has gradient −a/b.

Circle centre (h,k) radius r: (x−h)² + (y−k)² = r²; expanded form x² + y² + 2gx + 2fy + c = 0 has centre (−g,−f) and radius √(g² + f² − c).

Perpendicular distance from (x₀,y₀) to ax + by + c = 0 is |ax₀ + by₀ + c| / √(a² + b²).

Tangent to a circle at point P is perpendicular to the radius at P; the normal at P passes through the centre.

Parametric parabola y² = 4ax: point (at², 2at); tangent ty = x + at²; chord joining t₁,t₂ has equation (t₁+t₂)y = 2x + 2at₁t₂.

Ellipse x²/a² + y²/b² = 1 has eccentricity e = √(1 − b²/a²) when a > b; foci at (±ae, 0).

Area of triangle with vertices (x₁,y₁),(x₂,y₂),(x₃,y₃) = ½|x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|.

Arithmetic sequence: nth term aₙ = a + (n−1)d; sum Sₙ = n/2 [2a + (n−1)d] = n/2 (a + l).

Geometric sequence: nth term aₙ = arⁿ⁻¹; sum Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1; infinite sum converges to a/(1 − r) iff |r| < 1.

Binomial theorem: (a + b)ⁿ = Σ_{k=0}^{n} C(n,k) aⁿ⁻ᵏ bᵏ; general term C(n,k) aⁿ⁻ᵏ bᵏ.

C(n,k) = n!/(k!(n−k)!); identities C(n,k) = C(n,n−k) and C(n,k) + C(n,k+1) = C(n+1,k+1).

Method of differences: if uₙ = f(n) − f(n+1), then Σ_{n=1}^{N} uₙ = f(1) − f(N+1).

Sigma notation: Σ_{k=1}^{n} k = n(n+1)/2; Σ k² = n(n+1)(2n+1)/6; Σ k³ = [n(n+1)/2]².

Maclaurin series: f(x) = f(0) + f′(0)x + f″(0)x²/2! + …; standard expansions eˣ = Σ xⁿ/n!, sin x = Σ (−1)ⁿ x²ⁿ⁺¹/(2n+1)!, cos x = Σ (−1)ⁿ x²ⁿ/(2n)!.

Convergence tests: ratio test compares |aₙ₊₁/aₙ| to 1; alternating series converges if terms decrease in magnitude to zero.

Recurrence relations: solve linear first-order uₙ₊₁ = auₙ + b by finding fixed point L = b/(1−a) and writing uₙ − L = aⁿ(u₀ − L).

Power rule: d/dx (xⁿ) = nxⁿ⁻¹; d/dx (eˣ) = eˣ; d/dx (ln x) = 1/x; d/dx (sin x) = cos x; d/dx (cos x) = −sin x; d/dx (tan x) = sec²x.

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

Chain rule: dy/dx = dy/du · du/dx; for y = f(g(x)), y′ = f′(g(x))·g′(x).

Implicit differentiation: differentiate both sides with respect to x, applying the chain rule to y terms, then solve for dy/dx.

Parametric curves: dy/dx = (dy/dt)/(dx/dt); second derivative d²y/dx² = [d/dt(dy/dx)]/(dx/dt).

Inverse trigonometric derivatives: d/dx(arcsin x) = 1/√(1−x²); d/dx(arctan x) = 1/(1+x²); d/dx(arccos x) = −1/√(1−x²).

Stationary points occur where dy/dx = 0; classify by sign change of dy/dx or by second derivative (positive = minimum, negative = maximum, zero = inconclusive).

L'Hôpital's rule: if lim f/g is 0/0 or ∞/∞, then lim f/g = lim f′/g′ provided the latter limit exists.

Newton–Raphson iteration: xₙ₊₁ = xₙ − f(xₙ)/f′(xₙ) for solving f(x) = 0.

Standard integrals: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1); ∫1/x dx = ln|x| + C; ∫eˣ dx = eˣ + C; ∫sin x dx = −cos x + C; ∫cos x dx = sin x + C; ∫sec²x dx = tan x + C.

Substitution: ∫f(g(x))g′(x) dx = ∫f(u) du with u = g(x); change limits when evaluating a definite integral.

Integration by parts: ∫u dv = uv − ∫v du; choose u by LIATE (logarithmic, inverse trig, algebraic, trigonometric, exponential) where possible.

Partial fractions enable integration of rational functions; split into linear and irreducible quadratic factors before integrating.

Definite integral properties: ∫ₐᵇ f = −∫ᵦᵃ f; ∫ₐᵇ f + ∫ᵦᶜ f = ∫ₐᶜ f; area between curves = ∫ₐᵇ |f(x) − g(x)| dx.

Volume of revolution about x-axis: V = π∫ₐᵇ y² dx; about y-axis: V = π∫ x² dy (with appropriate limits).

Trapezium rule: ∫ₐᵇ f dx ≈ h/2 [f₀ + 2(f₁ + … + fₙ₋₁) + fₙ] with h = (b−a)/n.

Improper integrals: ∫ₐ^∞ f dx = lim_{t→∞} ∫ₐᵗ f dx; converges if the limit is finite.

Reduction formulae relate Iₙ to Iₙ₋₁ via integration by parts; e.g., Iₙ = ∫xⁿ eˣ dx gives Iₙ = xⁿ eˣ − nIₙ₋₁.

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

Compound angles: sin(A±B) = sinA cosB ± cosA sinB; cos(A±B) = cosA cosB ∓ sinA sinB; tan(A±B) = (tanA ± tanB)/(1 ∓ tanA tanB).

Double angle: sin2A = 2 sinA cosA; cos2A = cos²A − sin²A = 1 − 2sin²A = 2cos²A − 1; tan2A = 2tanA/(1 − tan²A).

R-form: a sinθ + b cosθ = R sin(θ + α) where R = √(a² + b²) and tanα = b/a; max value R, min −R.

General solutions: sinθ = k gives θ = (−1)ⁿ arcsin k + nπ; cosθ = k gives θ = ±arccos k + 2nπ; tanθ = k gives θ = arctan k + nπ.

Sine rule: a/sinA = b/sinB = c/sinC = 2R; cosine rule: a² = b² + c² − 2bc cosA.

Area of triangle = ½ab sinC; also = √(s(s−a)(s−b)(s−c)) with semiperimeter s.

Inverse trig principal ranges: arcsin returns [−π/2, π/2]; arccos returns [0, π]; arctan returns (−π/2, π/2).

Small-angle approximations: sinθ ≈ θ, tanθ ≈ θ, cosθ ≈ 1 − θ²/2 for θ in radians and |θ| small.

Dot product: a·b = |a||b|cosθ = a₁b₁ + a₂b₂ + a₃b₃; perpendicular vectors have a·b = 0.

Cross product magnitude |a×b| = |a||b|sinθ gives area of parallelogram; direction is perpendicular to both a and b by right-hand rule.

Line equation: r = a + λd; plane equation: r·n = d or ax + by + cz = d with normal n = (a,b,c).

Angle between line and plane: sinθ = |d·n|/(|d||n|); angle between two planes equals angle between their normals.

Shortest distance from point P to plane r·n = d is |(p − a)·n|/|n| where a is any point on the plane.

Matrix multiplication is associative but not commutative; (AB)ᵀ = BᵀAᵀ; (AB)⁻¹ = B⁻¹A⁻¹ when inverses exist.

2×2 inverse: for M = [[a,b],[c,d]], M⁻¹ = (1/(ad−bc))[[d,−b],[−c,a]] provided det M = ad − bc ≠ 0.

Determinant properties: det(AB) = detA·detB; det(kA) = kⁿ detA for n×n; det(Aᵀ) = detA.

Eigenvalues solve det(A − λI) = 0; eigenvectors satisfy (A − λI)v = 0 for each eigenvalue λ.

Cartesian form z = a + bi with i² = −1; modulus |z| = √(a² + b²); argument arg z = atan2(b, a).

Complex conjugate z̄ = a − bi; z z̄ = |z|²; division: (a+bi)/(c+di) = (a+bi)(c−di)/(c²+d²).

Polar form: z = r(cosθ + i sinθ) = re^{iθ}; multiplication multiplies moduli and adds arguments; division divides moduli and subtracts arguments.

De Moivre's theorem: [r(cosθ + i sinθ)]ⁿ = rⁿ(cos nθ + i sin nθ); valid for integer n.

Roots of zⁿ = w: if w = ρe^{iφ}, roots are ρ^{1/n} e^{i(φ+2kπ)/n} for k = 0, 1, …, n−1; they lie on a circle of radius ρ^{1/n} equally spaced by 2π/n.

Euler's formula: e^{iθ} = cosθ + i sinθ; hence cosθ = (e^{iθ} + e^{−iθ})/2 and sinθ = (e^{iθ} − e^{−iθ})/(2i).

Loci: |z − a| = r is a circle centre a radius r; |z − a| = |z − b| is the perpendicular bisector of segment ab; arg(z − a) = θ is a half-line from a at angle θ.

Triangle inequality: |z₁ + z₂| ≤ |z₁| + |z₂|; equality holds when the arguments differ by a multiple of 2π.

Fundamental theorem of algebra: every non-constant polynomial of degree n has exactly n complex roots counted with multiplicity; real polynomials have complex roots in conjugate pairs.

Conditional probability: P(A|B) = P(A ∩ B)/P(B); multiplication rule P(A ∩ B) = P(A|B)P(B); independence means P(A ∩ B) = P(A)P(B).

Bayes' theorem: P(A|B) = P(B|A)P(A)/P(B); law of total probability P(B) = Σ P(B|Aᵢ)P(Aᵢ) over a partition.

Binomial distribution: P(X = k) = C(n,k) pᵏ(1−p)ⁿ⁻ᵏ; mean np; variance np(1−p).

Poisson distribution: P(X = k) = e^{−λ} λᵏ/k!; mean and variance both equal λ; suitable for counts of rare events in a fixed interval.

Normal distribution: standardise Z = (X − μ)/σ; use tables for Φ(z); sums of independent normals are normal with means and variances added.

Discrete random variable: E(X) = Σ x P(X = x); Var(X) = E(X²) − [E(X)]²; E(aX + b) = aE(X) + b; Var(aX + b) = a² Var(X).

Continuous random variable: ∫f(x) dx = 1; P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx; E(X) = ∫x f(x) dx.

Hypothesis testing: test statistic compared with critical value at significance level α; p-value is the probability of a result at least as extreme as observed under H₀.

Correlation coefficient r lies in [−1, 1]; |r| near 1 indicates strong linear association; r² is the proportion of variance explained by linear regression.

Direct proof: assume hypotheses and derive conclusion via definitions and known theorems; e.g., sum of two odd integers is even because (2m+1)+(2n+1) = 2(m+n+1).

Proof by contradiction: assume the negation of the statement and derive a contradiction; classic example is the irrationality of √2.

Proof by contrapositive: prove ¬Q ⇒ ¬P to establish P ⇒ Q; logically equivalent statements.

Mathematical induction: prove base case P(1); assume P(k) (inductive hypothesis); prove P(k+1); conclude P(n) for all n ≥ 1.

Strong induction: assume P(j) for all j ≤ k and prove P(k+1); useful when P(k+1) depends on multiple earlier cases.

Proof by exhaustion: check all possible cases, e.g., prove a property for each residue class modulo n.

Disproof by counterexample: a single example satisfying the hypothesis but not the conclusion disproves a universal statement.

Necessary and sufficient: P ⇒ Q means P is sufficient for Q; Q ⇒ P means P is necessary for Q; P ⇔ Q means both.

Logical equivalences: ¬(P ∧ Q) ≡ ¬P ∨ ¬Q; ¬(P ∨ Q) ≡ ¬P ∧ ¬Q; P ⇒ Q ≡ ¬P ∨ Q ≡ ¬Q ⇒ ¬P.