All🔢 Arithmetic & Percentages✳️ Algebra & Identities📈 Quadratics & Sequences📐 Geometry🧊 Mensuration🗺️ Coordinate Geometry📊 Trigonometry🎲 Statistics & Probability∫ Calculus➡️ Vectors & Matrices🧩 Sets & Logic🧿 Number Theory🌀 Complex Numbers🛰️ Conics & Polar➰ Series & Expansions🗻 Multivariable Calculus🌊 Vector Calculus🧮 Linear Algebra🎲 Probability Distributions📈 Inference & Regression📉 Differential Equations🎚️ Laplace & Fourier💻 Numerical Methods🎯 Optimisation & OR🕸️ Discrete Maths & Graphs⚙️ Engineering Mathematics

🔢ARITHMETIC & PERCENTAGES

Percentages, profit, interest and averages

Percentage% = (part / whole) × 100Middle

Percentage changechange % = (new − old) / old × 100Middle

Profit and lossprofit % = (SP − CP) / CP × 100Middle

Simple interestSI = P R T / 100Middle

Compound interestA = P (1 + R/100)ᵀHigh

Average (mean)mean = sum of values / number of valuesMiddle

Ratio sharingshare = (part of ratio / total ratio) × amountMiddle

Speed, distance, timedistance = speed × timeMiddle

✳️ALGEBRA & IDENTITIES

Expanding, factorising and the standard identities

Square of a sum(a + b)² = a² + 2ab + b²Middle

Square of a difference(a − b)² = a² − 2ab + b²Middle

Difference of two squaresa² − b² = (a + b)(a − b)Middle

Cubesa³ ± b³ = (a ± b)(a² ∓ ab + b²)High

Laws of indicesaᵐ·aⁿ = aᵐ⁺ⁿ ; (aᵐ)ⁿ = aᵐⁿ ; a⁻ⁿ = 1/aⁿMiddle

Laws of logarithmslog(xy) = log x + log y ; log(xⁿ) = n log xHigh

Change of baselog_b x = log x / log bAdvanced

📈QUADRATICS & SEQUENCES

Roots, discriminant, AP and GP

Quadratic formulax = [−b ± √(b² − 4ac)] / 2aHigh

DiscriminantD = b² − 4acHigh

Sum and product of rootsα + β = −b/a ; αβ = c/aHigh

AP: nth termaₙ = a + (n − 1)dHigh

AP: sum of n termsSₙ = n/2 [2a + (n − 1)d]High

GP: nth term and sumaₙ = a rⁿ⁻¹ ; Sₙ = a(rⁿ − 1)/(r − 1)High

Infinite GP sumS∞ = a / (1 − r), |r| < 1Advanced

Binomial theorem(a + b)ⁿ = Σ ⁿCᵣ aⁿ⁻ʳ bʳAdvanced

📐GEOMETRY

Angles, triangles and circles

Angle sumstriangle 180° · quadrilateral 360° · polygon (n − 2)180°Middle

Pythagoras' theorema² + b² = c²Middle

Exterior angleexterior = 360° / nMiddle

Circle theoremsangle at centre = 2 × angle at circumferenceHigh

Similar trianglescorresponding sides are in the same ratioHigh

Midpoint theoremthe line joining two midpoints = ½ the third sideHigh

🧊MENSURATION

Area, perimeter, surface area and volume

Rectangle and squareA = l × b ; P = 2(l + b)Middle

Area of a triangleA = ½ b h ; Heron: √[s(s−a)(s−b)(s−c)]High

CircleA = πr² ; C = 2πrMiddle

Sector and arcarc = (θ/360) 2πr ; area = (θ/360) πr²High

Cuboid and cubeV = l b h ; SA = 2(lb + bh + hl)Middle

CylinderV = πr²h ; SA = 2πr(r + h)High

ConeV = ⅓πr²h ; CSA = πrlHigh

SphereV = 4/3 πr³ ; SA = 4πr²High

🗺️COORDINATE GEOMETRY

Lines, distance and circles on the grid

Distance between two pointsd = √[(x₂ − x₁)² + (y₂ − y₁)²]High

MidpointM = ((x₁+x₂)/2 , (y₁+y₂)/2)High

Gradient of a linem = (y₂ − y₁) / (x₂ − x₁)High

Equation of a liney = mx + c ; y − y₁ = m(x − x₁)High

Parallel and perpendicularparallel: m₁ = m₂ ; perpendicular: m₁m₂ = −1High

Equation of a circle(x − h)² + (y − k)² = r²Advanced

Area from coordinatesA = ½ |x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|Advanced

📊TRIGONOMETRY

Ratios, identities and the sine & cosine rules

Basic ratiossin θ = opp/hyp ; cos θ = adj/hyp ; tan θ = opp/adjHigh

Pythagorean identitysin²θ + cos²θ = 1High

Sine rulea/sin A = b/sin B = c/sin CHigh

Cosine rulea² = b² + c² − 2bc cos AHigh

Area with sineA = ½ ab sin CHigh

Compound anglessin(A ± B) = sinA cosB ± cosA sinBAdvanced

Double anglesin2A = 2 sinA cosA ; cos2A = 1 − 2sin²AAdvanced

Radiansπ radians = 180° ; arc s = rθHigh

🎲STATISTICS & PROBABILITY

Averages, spread, counting and chance

Mean, median, modemean = Σx/n ; median = middle value ; mode = most commonMiddle

Standard deviationσ = √[Σ(x − x̄)² / n]High

ProbabilityP(A) = favourable outcomes / total outcomesMiddle

Addition and multiplication rulesP(A∪B) = P(A) + P(B) − P(A∩B) ; P(A∩B) = P(A)P(B) if independentHigh

Conditional probabilityP(A|B) = P(A∩B) / P(B)Advanced

Permutations and combinationsⁿPᵣ = n!/(n−r)! ; ⁿCᵣ = n!/[r!(n−r)!]High

Normal distributionz = (x − μ) / σAdvanced

∫CALCULUS

Limits, differentiation and integration

Definition of the derivativef′(x) = lim(h→0) [f(x+h) − f(x)] / hAdvanced

Power ruled/dx (xⁿ) = n xⁿ⁻¹Advanced

Product and quotient rules(uv)′ = u′v + uv′ ; (u/v)′ = (u′v − uv′)/v²Advanced

Chain ruledy/dx = dy/du × du/dxAdvanced

Standard derivativessin→cos, cos→−sin, eˣ→eˣ, ln x→1/xAdvanced

Integration of a power∫xⁿ dx = xⁿ⁺¹/(n+1) + C , n ≠ −1Advanced

Definite integral∫ₐᵇ f(x) dx = F(b) − F(a)Advanced

Integration by parts∫u dv = uv − ∫v duAdvanced

➡️VECTORS & MATRICES

Vectors in 2D/3D, determinants and complex numbers

Magnitude of a vector|v| = √(x² + y² + z²)Advanced

Dot producta · b = |a||b| cos θAdvanced

Cross product|a × b| = |a||b| sin θAdvanced

2×2 determinant and inversedet = ad − bc ; A⁻¹ = (1/det)[d −b; −c a]Advanced

Matrix multiplication(AB)ᵢⱼ = Σ Aᵢₖ BₖⱼAdvanced

Complex numbersz = a + bi ; |z| = √(a² + b²)Advanced

Euler's formulae^(iθ) = cos θ + i sin θAdvanced

🧩SETS & LOGIC

Sets, counting and number theory basics

Union of setsn(A∪B) = n(A) + n(B) − n(A∩B)High

Number of subsetssubsets = 2ⁿHigh

HCF and LCMHCF × LCM = a × bMiddle

Divisibility quick testsby 3: digit sum ÷ 3 · by 9: digit sum ÷ 9 · by 11: alternating sumMiddle

Standard series sumsΣn = n(n+1)/2 ; Σn² = n(n+1)(2n+1)/6High

🧿NUMBER THEORY

Divisibility, modular arithmetic, primes

Division algorithma = bq + r, 0 ≤ r < bHigh

Bézout identitygcd(a, b) = ax + byAdvanced

Euler totientφ(n) = n ∏(1 − 1/p) over primes p | nAdvanced

Fermat's little theorema^(p−1) ≡ 1 (mod p), p prime, p ∤ aAdvanced

Chinese remainder theoremx ≡ aᵢ (mod mᵢ) has one solution mod ∏mᵢAdvanced

Prime counting estimateπ(x) ≈ x / ln xAdvanced

🌀COMPLEX NUMBERS

Polar form, De Moivre, roots of unity

Modulus and argumentz = x + iy = r(cos θ + i sin θ), r = √(x² + y²)High

Euler formulae^{iθ} = cos θ + i sin θAdvanced

De Moivre theorem(cos θ + i sin θ)ⁿ = cos nθ + i sin nθAdvanced

n-th roots of unityz_k = e^{2πik/n}, k = 0 … n−1Advanced

Cauchy–Riemann equationsu_x = v_y, u_y = −v_xEngineering

Residue theorem∮_C f dz = 2πi Σ Res(f, a_k)Engineering

🛰️CONICS & POLAR

Parabola, ellipse, hyperbola, polar curves

Parabolay² = 4ax · focus (a, 0) · directrix x = −aHigh

Ellipsex²/a² + y²/b² = 1, e = √(1 − b²/a²)High

Hyperbolax²/a² − y²/b² = 1, asymptotes y = ±(b/a)xAdvanced

Polar equation of a conicr = l / (1 + e cos θ)Advanced

Area in polar coordinatesA = ½ ∫ r² dθAdvanced

Polar arc lengths = ∫ √(r² + (dr/dθ)²) dθAdvanced

➰SERIES & EXPANSIONS

Taylor, Maclaurin and convergence tests

Taylor seriesf(x) = Σ f⁽ⁿ⁾(a)(x − a)ⁿ / n!Advanced

Standard expansionssin x = x − x³/3! + … · cos x = 1 − x²/2! + … · ln(1+x) = x − x²/2 + …Advanced

Binomial series(1 + x)ⁿ = 1 + nx + n(n−1)x²/2! + … , |x| < 1Advanced

Ratio testL = lim |a_{n+1}/a_n| : L < 1 converges, L > 1 divergesAdvanced

p-seriesΣ 1/n^p converges ⟺ p > 1High

Infinite geometric sumS = a / (1 − r), |r| < 1High

🗻MULTIVARIABLE CALCULUS

Partial derivatives, gradients, multiple integrals

Chain rule (several variables)dz/dt = (∂z/∂x)(dx/dt) + (∂z/∂y)(dy/dt)Advanced

Gradient and directional derivative∇f = (f_x, f_y, f_z) ; D_u f = ∇f · ûAdvanced

Second derivative testD = f_xx f_yy − f_xy² ; D > 0, f_xx > 0 → minimumAdvanced

Lagrange multipliers∇f = λ∇g subject to g = 0Advanced

Change of variables∬ f dx dy = ∬ f |J| du dvAdvanced

Spherical volume elementdV = ρ² sin φ dρ dφ dθAdvanced

🌊VECTOR CALCULUS

Divergence, curl and the integral theorems

Divergence∇·F = ∂F₁/∂x + ∂F₂/∂y + ∂F₃/∂zAdvanced

Curl∇×F = (F₃_y − F₂_z, F₁_z − F₃_x, F₂_x − F₁_y)Advanced

Green's theorem∮_C (P dx + Q dy) = ∬_R (Q_x − P_y) dAAdvanced

Stokes' theorem∮_C F·dr = ∬_S (∇×F)·n dSEngineering

Divergence theorem∯_S F·n dS = ∭_V (∇·F) dVEngineering

Laplacian∇²f = f_xx + f_yy + f_zzAdvanced

🧮LINEAR ALGEBRA

Eigenvalues, rank and decompositions

Eigenvalues and eigenvectorsAv = λv, det(A − λI) = 0Advanced

Trace and determinanttr A = Σλᵢ , det A = ∏λᵢAdvanced

Rank–nullityrank(A) + nullity(A) = nAdvanced

Cayley–HamiltonA satisfies its own characteristic equationAdvanced

Gram–Schmidtu_k = v_k − Σ (⟨v_k, u_j⟩/⟨u_j, u_j⟩) u_jEngineering

Singular value decompositionA = UΣVᵀEngineering

🎲PROBABILITY DISTRIBUTIONS

Binomial, Poisson, normal and friends

Binomial distributionP(X = k) = C(n,k) p^k (1−p)^{n−k}, μ = np, σ² = np(1−p)High

Poisson distributionP(X = k) = e^{−λ} λ^k / k!, μ = σ² = λAdvanced

Normal distributionf(x) = e^{−(x−μ)²/2σ²} / (σ√(2π))Advanced

Exponential distributionf(x) = λe^{−λx}, μ = 1/λAdvanced

Central limit theoremx̄ ≈ N(μ, σ²/n) for large nAdvanced

Bayes theoremP(A|B) = P(B|A)P(A) / P(B)High

📈INFERENCE & REGRESSION

Confidence intervals, tests and least squares

Confidence interval for a meanx̄ ± z σ/√n (or t s/√n)Advanced

Test statisticz = (x̄ − μ₀) / (σ/√n)Advanced

Chi-square goodness of fitχ² = Σ (O − E)² / EAdvanced

Least squares lineb = Σ(x−x̄)(y−ȳ) / Σ(x−x̄)² , a = ȳ − b x̄Advanced

Correlation coefficientr = Σ(x−x̄)(y−ȳ) / √(Σ(x−x̄)² Σ(y−ȳ)²)High

Coefficient of determinationR² = 1 − SS_res/SS_totAdvanced

📉DIFFERENTIAL EQUATIONS

First and second order, plus systems

Separable equation∫ dy/g(y) = ∫ f(x) dxAdvanced

Linear first ordery' + P y = Q , μ = e^{∫P dx} , (μy)' = μQAdvanced

Constant-coefficient second orderay'' + by' + cy = 0 , am² + bm + c = 0Advanced

Method of undetermined coefficientsy = y_complementary + y_particularAdvanced

WronskianW = y₁y₂' − y₂y₁' ≠ 0 → independentAdvanced

Linear systemx' = Ax → x(t) = e^{At} x(0)Engineering

🎚️LAPLACE & FOURIER

Transforms for signals and circuits

Laplace transformF(s) = ∫₀^∞ f(t) e^{−st} dtEngineering

Transform of a derivativeL{f'} = sF(s) − f(0)Engineering

Shifting theoremsL{e^{at}f(t)} = F(s−a) ; L{f(t−a)u(t−a)} = e^{−as}F(s)Engineering

Convolution theoremL{f * g} = F(s)G(s)Engineering

Fourier seriesf(x) = a₀/2 + Σ (aₙ cos nx + bₙ sin nx)Engineering

Fourier transformF(ω) = ∫ f(t) e^{−iωt} dtEngineering

💻NUMERICAL METHODS

Roots, interpolation, quadrature and step methods

Newton–Raphsonx_{n+1} = x_n − f(x_n)/f'(x_n)Advanced

Bisectionerror after n steps ≤ (b − a)/2ⁿHigh

Trapezoidal rule∫ ≈ (h/2)[y₀ + 2(y₁ + … + y_{n−1}) + yₙ]Advanced

Simpson's rule∫ ≈ (h/3)[y₀ + 4(odd) + 2(even) + yₙ]Advanced

Lagrange interpolationP(x) = Σ yᵢ ∏_{j≠i} (x − xⱼ)/(xᵢ − xⱼ)Advanced

Runge–Kutta 4y_{n+1} = y_n + (h/6)(k₁ + 2k₂ + 2k₃ + k₄)Engineering

🎯OPTIMISATION & OR

Linear programming, gradients and queues

Linear programmemaximise cᵀx subject to Ax ≤ b, x ≥ 0Advanced

Weak dualitycᵀx ≤ bᵀy for feasible primal x and dual yEngineering

Gradient descentx_{k+1} = x_k − α ∇f(x_k)Engineering

KKT conditions∇f + Σλᵢ∇gᵢ = 0, λᵢgᵢ = 0, λᵢ ≥ 0Engineering

Economic order quantityQ* = √(2DS/H)Engineering

M/M/1 queueL = ρ/(1 − ρ), ρ = λ/μEngineering

🕸️DISCRETE MATHS & GRAPHS

Graphs, recurrences and counting arguments

Handshaking lemmaΣ deg(v) = 2EHigh

Euler's formula for planar graphsV − E + F = 2High

Linear recurrenceaₙ = c₁a_{n−1} + c₂a_{n−2} → characteristic r² = c₁r + c₂Advanced

Inclusion–exclusion|A ∪ B ∪ C| = ΣA − Σ pairs + |A∩B∩C|High

Pigeonhole principlen items in k boxes → some box holds ≥ ⌈n/k⌉Middle

Master theoremT(n) = aT(n/b) + f(n) → compare f(n) with n^{log_b a}Engineering

⚙️ENGINEERING MATHEMATICS

Control, signals, tensors and reliability

Transfer functionG(s) = Y(s)/U(s)Engineering

Second-order responseG = ωₙ² / (s² + 2ζωₙs + ωₙ²)Engineering

PID controlleru = K_p e + K_i ∫e dt + K_d de/dtEngineering

Sampling theoremf_s > 2BEngineering

Z transformX(z) = Σ x[n] z^{−n}Engineering

Euler–Lagrange equation∂F/∂y − d/dx(∂F/∂y') = 0Engineering

Reliability and MTBFR(t) = e^{−λt}, MTBF = 1/λEngineering

Shannon entropyH = −Σ pᵢ log₂ pᵢEngineering

Finite element stiffnessK u = FEngineering