Adv. Algebra & Quadratics (1-25)Comp Math Part 1
- Quadratic Formulax = [-b ± √(b² – 4ac)] / 2a
- Sum of Roots (α + β)-b / a
- Product of Roots (αβ)c / a
- Difference of Roots (|α – β|)√(D) / |a|
- Condition for Common Root(c₁a₂ – c₂a₁) / (a₁b₂ – a₂b₁) = (b₁c₂ – b₂c₁)
- Max/Min Value of Quadratic-D / (4a)
- AM ≥ GM Inequality(a + b)/2 ≥ √(ab)
- GM ≥ HM Inequality√(ab) ≥ 2ab/(a+b)
- Cauchy-Schwarz Inequality(a₁b₁ + a₂b₂)² ≤ (a₁²+a₂²)(b₁²+b₂²)
- Logarithm Base Changelog_b(a) = log_c(a) / log_c(b)
- Log Property (Power)log_a(xⁿ) = n · log_a(x)
- Exponential-Log Identitya^(log_a(x)) = x
- Sum of n terms APSₙ = n/2 [2a + (n-1)d]
- Sum of n terms GPa(rⁿ – 1)/(r – 1)
- Sum of Infinite GPS = a / (1 – r)
- Sum of first n cubes[n(n+1)/2]²
- Sum of first n squaresn(n+1)(2n+1)/6
- (a + b + c)²a² + b² + c² + 2ab + 2bc + 2ca
- (a + b)³a³ + b³ + 3ab(a + b)
- (a – b)³a³ – b³ – 3ab(a – b)
- a³ + b³ + c³ – 3abc(a+b+c)(a²+b²+c²-ab-bc-ca)
- Condition for Real Rootsb² – 4ac ≥ 0
- Remainder TheoremP(a) is remainder when P(x) is div by (x-a)
- Factor TheoremIf P(a)=0, (x-a) is a factor
- Descartes’ Rule of SignsBounds on positive/negative roots
Complex Numbers (26-50)Comp Math Part 2
- Complex Number Formz = x + iy
- Modulus of Complex No|z| = √(x² + y²)
- Argument of zarg(z) = tan⁻¹(y / x)
- Polar Formz = r(cosθ + isinθ) = re^(iθ)
- Conjugate of zz̄ = x – iy
- Modulus Property|z₁z₂| = |z₁||z₂|
- Triangle Inequality|z₁ + z₂| ≤ |z₁| + |z₂|
- De Moivre’s Theorem(cosθ + isinθ)ⁿ = cos(nθ) + isin(nθ)
- Cube roots of unity1, ω, ω² where 1+ω+ω²=0
- Properties of ωω³ = 1
- Distance between two complex nos|z₁ – z₂|
- Equation of Circle in Complex|z – z₀| = r
- Rotation Theorem(z₃ – z₁)/(z₂ – z₁) = |.|e^(iθ)
- Arg(z₁/z₂)arg(z₁) – arg(z₂)
- z · z̄|z|²
- Multiplicative Inversez⁻¹ = z̄ / |z|²
- Real part of z(z + z̄) / 2
- Imaginary part of z(z – z̄) / 2i
- Square root of a + ib±[√((|z|+x)/2) + i sgn(y)√((|z|-x)/2)]
- Arg of Negative Realπ
- Arg of Pure Imaginary (+i)π / 2
- Euler’s Formulae^(ix) = cos x + i sin x
- Modulus of z₁/z₂|z₁| / |z₂|
- Conjugate of Sumz̄₁ + z̄₂
- Conjugate of Productz̄₁ · z̄₂
Matrices & Determinants (51-75)Comp Math Part 3
- Determinant of 2×2 Matrixad – bc
- Inverse of Matrix AA⁻¹ = adj(A) / |A|
- Transpose Property(AB)ᵀ = BᵀAᵀ
- Inverse Property(AB)⁻¹ = B⁻¹A⁻¹
- Determinant of Product|AB| = |A||B|
- Determinant of Adjoint|adj(A)| = |A|ⁿ⁻¹
- Adjoint of Adjoint|A|ⁿ⁻² · A
- Symmetric MatrixA = Aᵀ
- Skew-Symmetric MatrixA = -Aᵀ
- Orthogonal MatrixA · Aᵀ = I
- Idempotent MatrixA² = A
- Involutory MatrixA² = I
- Nilpotent MatrixAᵏ = 0
- Trace of MatrixSum of principal diagonal elements
- Cramer’s Rule for 2 varsx = Dₓ/D, y = Dᵧ/D
- Consistent System ConditionRank(A) = Rank(A|B)
- Trivial Solution Condition|A| ≠ 0
- Non-trivial Solution Condition|A| = 0
- Scalar Multiple Determinant|kA| = kⁿ|A|
- Inverse of Diagonal MatrixReciprocal of diagonal elements
- Sum of Symmetric & SkewA = ½(A+Aᵀ) + ½(A-Aᵀ)
- Determinant of Inverse1 / |A|
- Skew-Symmetric Odd Order detAlways 0
- Elementary Row OperationsDoes not change determinant value
- Identity Matrix PropertyAI = IA = A
Permutations & Binomial (76-100)Comp Math Part 4
- Permutation (nPr)n! / (n – r)!
- Combination (nCr)n! / [r!(n – r)!]
- nCr Symmetry RuleⁿCᵣ = ⁿCₙ₋ᵣ
- Pascal’s IdentityⁿCᵣ + ⁿCᵣ₋₁ = ⁿ⁺¹Cᵣ
- Circular Permutation(n – 1)!
- Necklace Permutation(n – 1)! / 2
- Selection of r items from nⁿCᵣ
- Total combinations (at least 1)2ⁿ – 1
- Binomial Expansion general(x+a)ⁿ = Σ ⁿCᵣ xⁿ⁻ʳ aʳ
- Binomial General Term (Tᵣ₊₁)ⁿCᵣ xⁿ⁻ʳ aʳ
- Middle term (n even)(n/2 + 1)th term
- Sum of Binomial CoefficientsⁿC₀ + ⁿC₁ + … + ⁿCₙ = 2ⁿ
- Sum of Odd Binomial Coeffs2ⁿ⁻¹
- Sum of Even Binomial Coeffs2ⁿ⁻¹
- Number of divisors formula(p+1)(q+1)… for N = pᵃ qᵇ…
- Sum of divisors[(pᵃ⁺¹-1)/(p-1)] · [(qᵇ⁺¹-1)/(q-1)]
- Derangements formulan! [1 – 1/1! + 1/2! – … + (-1)ⁿ/n!]
- Distributing identical objectsⁿ⁺ʳ⁻¹Cᵣ
- Multinomial TheoremCoefficient of xᵃyᵇzᶜ is n!/(a!b!c!)
- Greatest Binomial CoefficientⁿCₙ/₂ (n even) or ⁿC₍ₙ₋₁₎/₂ (n odd)
- Divisibility problems shortcut(1+x)ⁿ expansion use
- Exponent of prime p in n![n/p] + [n/p²] + [n/p³] + …
- Geometric CombinationsPoints forming triangles = ⁿC₃
- Diagonals in n-sided polygonⁿC₂ – n
- Rectangles in grid m x nᵐC₂ · ⁿC₂
Adv. Trigonometry & ITF (101-130)Comp Math Part 5
- Sin(A+B)Sin(A-B)sin²A – sin²B = cos²B – cos²A
- Cos(A+B)Cos(A-B)cos²A – sin²B = cos²B – sin²A
- SinC + SinD2sin[(C+D)/2]cos[(C-D)/2]
- CosC + CosD2cos[(C+D)/2]cos[(C-D)/2]
- CosC – CosD-2sin[(C+D)/2]sin[(C-D)/2]
- Maximum value of Asinθ + Bcosθ√(A² + B²)
- Minimum value of Asinθ + Bcosθ-√(A² + B²)
- Sin 3θ3sinθ – 4sin³θ
- Cos 3θ4cos³θ – 3cosθ
- Tan 3θ(3tanθ – tan³θ) / (1 – 3tan²θ)
- Sin⁻¹(-x)-sin⁻¹x
- Cos⁻¹(-x)π – cos⁻¹x
- Tan⁻¹(-x)-tan⁻¹x
- Sin⁻¹x + Cos⁻¹xπ / 2
- Tan⁻¹x + Cot⁻¹xπ / 2
- Sec⁻¹x + Csc⁻¹xπ / 2
- Tan⁻¹x + Tan⁻¹ytan⁻¹[(x+y)/(1-xy)]
- 2Tan⁻¹x (in Sin)sin⁻¹[2x/(1+x²)]
- 2Tan⁻¹x (in Cos)cos⁻¹[(1-x²)/(1+x²)]
- Sine Rule in Trianglea/sinA = b/sinB = c/sinC = 2R
- Cosine RulecosA = (b²+c²-a²)/(2bc)
- Projection Rulea = b cosC + c cosB
- Half-angle formula (sin A/2)√[(s-b)(s-c)/bc]
- Area of Triangle (Heron/Trig)½ bc sinA = √(s(s-a)(s-b)(s-c))
- Inradius (r) of TriangleΔ / s
Coordinate Geometry & Conics (131-160)Comp Math Part 6
- Angle between two linestanθ = |(m₁ – m₂)/(1 + m₁m₂)|
- Pair of Straight Linesax² + 2hxy + by² + 2gx + 2fy + c = 0
- Angle between pair of linestanθ = 2√(h² – ab) / (a + b)
- Condition for parallel linesh² = ab
- Length of intercept by circle2√(r² – d²)
- Equation of Tangent to circlexx₁ + yy₁ = r²
- Condition of tangency for linec² = r²(1 + m²)
- Length of Tangent from point√(x₁² + y₁² + 2gx₁ + 2fy₁ + c)
- Equation of Polar of circlexx₁ + yy₁ + g(x+x₁) + f(y+y₁) + c = 0
- Parabola Tangent parametricyt = x + at²
- Parabola Normal parametricy + tx = 2at + at³
- Ellipse Latus Rectum2b² / a
- Ellipse Directrix equationx = ±a / e
- Auxiliary Circle of Ellipsex² + y² = a²
- Hyperbola Asymptotesx²/a² – y²/b² = 0
- Rectangular Hyperbolaxy = c²
- Eccentricity of Rectangular Hyperbola√2
- Director Circle of Circlex² + y² = 2r²
- Director Circle of Parabolax = -a (Directrix)
- Director Circle of Ellipsex² + y² = a² + b²
Adv. Calculus & Diff. Equations (161-200)Comp Math Part 7
- L’Hopital’s Rulelim f(x)/g(x) = lim f'(x)/g'(x)
- Rolle’s Theoremf'(c) = 0 for some c in (a,b) if f(a)=f(b)
- Lagrange’s Mean Value Theoremf'(c) = [f(b) – f(a)] / (b – a)
- Derivative of Inverse Function(f⁻¹)'(x) = 1 / [f'(f⁻¹(x))]
- Leibniz Differentiation Ruled/dx[∫_{u(x)}^{v(x)} f(t)dt]
- Radius of Curvatureρ = [1 + (dy/dx)²]^(3/2) / |d²y/dx²|
- Subtangent Length|y / (dy/dx)|
- Subnormal Length|y · (dy/dx)|
- Angle between curvestanθ = |(m₁ – m₂)/(1 + m₁m₂)|
- Standard Definite Integral 0 to π/2∫ sinⁿx dx = [(n-1)/n] · [n-2/…]*π/2
- Newton-Leibniz Formula∫ₐᵇ f(x)dx = F(b) – F(a)
- King Property of Definite Integrals∫ₐᵇ f(x)dx = ∫ₐᵇ f(a+b-x)dx
- Even/Odd Function Integral2∫₀ᵃ f(x)dx (even) or 0 (odd)
- Integration of eᵃˣcos(bx)[eᵃˣ/(a²+b²)] [a cosbx + b sinbx]
- Integration of √(a² – x²)½[x√(a²-x²) + a²sin⁻¹(x/a)]
- Order of Differential EquationHighest derivative present
- Degree of Differential EquationPower of highest derivative
- Variable Separable Methodf(x)dx = g(y)dy
- Homogeneous Diff Equation substitutey = vx
- Linear Diff Equation IFe^(∫ P dx)
- Bernoulli Differential Equationdy/dx + Py = Qyⁿ
- Orthogonal TrajectoriesReplace dy/dx with -dx/dy
- Area bounded by two curves∫ₐᵇ [f(x) – g(x)] dx
- RMS value of function√[ (1/(b-a)) ∫ₐᵇ f²(x)dx ]
- Mean value of function(1/(b-a)) ∫ₐᵇ f(x)dx
General Maths CoreMaths
- Quadratic Formulax = [-b ± √(b² – 4ac)] / 2a
- (a + b)²a² + b² + 2ab
- Pythagorean Identitysin²θ + cos²θ = 1
- Derivative of xⁿn · xⁿ⁻¹
- Integration of xⁿxⁿ⁺¹ / (n + 1)
Physics CorePhysics
- Equation of Motionv = u + at
- Newton’s Second LawF = ma
- Ideal Gas EquationPV = nRT
- Ohm’s LawV = IR
- Lens Formula1/f = 1/v – 1/u
Chemistry CoreChemistry
- Mole Conceptn = Mass / Molar Mass
- MolarityMoles / Volume (L)
- pH Scale-log[H⁺]
- First Order Half-lifet₁/₂ = 0.693 / k
- Nernst EquationE = E° – (0.0591/n)log(Q)