Mathematics Formula Sheet
77 formulas across 13 chapters — with variables explained and exam tips where needed.
Ch 1Relations and Functions(3 formulas)
Reflexive relation
(a, a) ∈ R for all a ∈ A
Symmetric relation
(a,b) ∈ R ⟹ (b,a) ∈ R
Transitive relation
(a,b) ∈ R and (b,c) ∈ R ⟹ (a,c) ∈ R
Ch 2Inverse Trigonometric Functions(9 formulas)
Domain and Range of sin⁻¹
Domain: [−1,1] | Range: [−π/2, π/2]
Domain and Range of cos⁻¹
Domain: [−1,1] | Range: [0, π]
Domain and Range of tan⁻¹
Domain: ℝ | Range: (−π/2, π/2)
sin⁻¹(−x)
= −sin⁻¹x
cos⁻¹(−x)
= π − cos⁻¹x
tan⁻¹x + cot⁻¹x
= π/2
sin⁻¹x + cos⁻¹x
= π/2
tan⁻¹x + tan⁻¹y
= tan⁻¹[(x+y)/(1−xy)] if xy < 1
2 tan⁻¹x
= sin⁻¹(2x/(1+x²)) = cos⁻¹((1−x²)/(1+x²)) = tan⁻¹(2x/(1−x²))
Ch 3Matrices(4 formulas)
Order of product AB
If A is m×n and B is n×p, then AB is m×p
Transpose properties
(AB)ᵀ = BᵀAᵀ | (Aᵀ)ᵀ = A
Symmetric matrix
Aᵀ = A
Skew-symmetric matrix
Aᵀ = −A
Ch 4Determinants(4 formulas)
2×2 Determinant
|A| = ad − bc for A = [[a,b],[c,d]]
Area of triangle (determinant)
Area = ½|x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|
Cramer's Rule (2 variables)
x = D_x/D, y = D_y/D
Adjoint and Inverse
A⁻¹ = adj(A)/|A|
|A| ≠ 0 for inverse to exist
Ch 5Continuity and Differentiability(14 formulas)
Chain Rule
dy/dx = (dy/du) × (du/dx)
Product Rule
d/dx(uv) = u(dv/dx) + v(du/dx)
Quotient Rule
d/dx(u/v) = [v(du/dx) − u(dv/dx)] / v²
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
d/dx(xⁿ)
nxⁿ⁻¹
d/dx(sin⁻¹x)
1/√(1−x²)
d/dx(cos⁻¹x)
−1/√(1−x²)
d/dx(tan⁻¹x)
1/(1+x²)
Rolle's Theorem condition
f(a) = f(b), f continuous on [a,b], differentiable on (a,b) ⟹ ∃c such that f'(c) = 0
Mean Value Theorem
f'(c) = [f(b)−f(a)]/(b−a)
Ch 6Application of Derivatives(5 formulas)
Increasing function condition
f'(x) > 0 on interval
Decreasing function condition
f'(x) < 0 on interval
Equation of tangent at (x₁,y₁)
y − y₁ = m(x − x₁) where m = dy/dx at (x₁,y₁)
Equation of normal at (x₁,y₁)
y − y₁ = −(1/m)(x − x₁)
Maxima/Minima — Second derivative test
f''(c) < 0 ⟹ local max | f''(c) > 0 ⟹ local min
Ch 7Integrals(12 formulas)
∫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
∫1/√(1−x²) dx
sin⁻¹x + C
∫1/(1+x²) dx
tan⁻¹x + C
∫1/√(x²+a²) dx
ln|x + √(x²+a²)| + C
Integration by parts
∫u dv = uv − ∫v du (ILATE rule)
ILATE: Inverse trig, Logarithm, Algebraic, Trig, Exponential
Definite integral property
∫ₐᵇ f(x) dx = ∫ₐᵇ f(a+b−x) dx
King's property
∫₀ᵃ f(x) dx = ∫₀ᵃ f(a−x) dx
Ch 8Application of Integrals(2 formulas)
Area between curve and x-axis
A = ∫ₐᵇ |f(x)| dx
Area between two curves
A = ∫ₐᵇ |f(x) − g(x)| dx
Ch 9Differential Equations(3 formulas)
Order of DE
Highest derivative present
Variable separable method
f(y) dy = g(x) dx ⟹ integrate both sides
Linear DE (first order)
dy/dx + Py = Q ⟹ IF = e^(∫P dx)
Solution: y × IF = ∫(Q × IF) dx + C
Ch 10Vector Algebra(7 formulas)
Magnitude of vector
|a⃗| = √(a₁² + a₂² + a₃²)
Dot product
a⃗ · b⃗ = |a||b| cosθ = a₁b₁ + a₂b₂ + a₃b₃
Cross product magnitude
|a⃗ × b⃗| = |a||b| sinθ
Unit vector
â = a⃗/|a⃗|
Projection of a⃗ on b⃗
(a⃗ · b⃗)/|b⃗|
Area of parallelogram
|a⃗ × b⃗|
Area of triangle
½|a⃗ × b⃗|
Ch 11Three Dimensional Geometry(6 formulas)
Direction cosines
l² + m² + n² = 1 where l = cosα, m = cosβ, n = cosγ
Distance between two points
d = √[(x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²]
Equation of line through (x₁,y₁,z₁) with direction (a,b,c)
(x−x₁)/a = (y−y₁)/b = (z−z₁)/c
Angle between two lines
cosθ = |l₁l₂ + m₁m₂ + n₁n₂|
Distance from point to plane ax+by+cz+d=0
d = |ax₁+by₁+cz₁+d| / √(a²+b²+c²)
Angle between line and plane
sinθ = |al+bm+cn| / √(a²+b²+c²)·√(l²+m²+n²)
Ch 12Linear Programming(2 formulas)
Feasible region
Set of all points satisfying all constraints (including non-negativity)
Corner point theorem
Optimal value of objective function occurs at a corner (vertex) of feasible region
Always evaluate Z at all corner points
Ch 13Probability(6 formulas)
Conditional probability
P(A|B) = P(A∩B) / P(B)
Multiplication rule
P(A∩B) = P(A) × P(B|A)
Bayes' Theorem
P(Aᵢ|B) = P(Aᵢ)·P(B|Aᵢ) / Σ P(Aⱼ)·P(B|Aⱼ)
Binomial distribution mean
μ = np
Binomial distribution variance
σ² = npq where q = 1−p
P(X=r) in binomial
P(X=r) = ⁿCᵣ × pʳ × qⁿ⁻ʳ