Composite
(f ∘ g)(x) = f(g(x))
One sheet · NCERT · Board revision
Class 12 Mathematics Formula Sheet
66 formulas from Mathematics and Science, chapter by chapter. Search, filter, copy, or print. This is a revision sheet — not a replacement for the textbook derivation.
Practice questions aligned to NCERT topics used in CBSE classrooms. These are original exam-oriented questions for revision — not official CBSE board papers and not a guarantee of any exam question. Always match the textbook issued by your school and the latest PDFs on ncert.nic.in and cbseacademic.nic.in.
➗ Class 12 · Mathematics
Relations and Functions
Inverse
f(f⁻¹(x)) = x = f⁻¹(f(x))
Number of relations
2^(m n) from a set of m to a set of n elements
➗ Class 12 · Mathematics
Inverse Trigonometric Functions
Principal range sin⁻¹
[−π/2, π/2]
Principal range cos⁻¹
[0, π]
Principal range tan⁻¹
(−π/2, π/2)
sin⁻¹x + cos⁻¹x
π/2, x ∈ [−1, 1]
tan⁻¹x + cot⁻¹x
π/2
tan⁻¹x + tan⁻¹y
tan⁻¹[(x+y)/(1−xy)] (xy < 1)
➗ Class 12 · Mathematics
Matrices
Order
m × n has m rows and n columns
Multiplication
(AB)ᵢⱼ = Σₐ Aᵢₐ Bₐⱼ
Inverse uniqueness
(AB)⁻¹ = B⁻¹ A⁻¹
Transpose
(AB)ᵀ = Bᵀ Aᵀ
Symmetric
Aᵀ = A
Skew-symmetric
Aᵀ = −A
➗ Class 12 · Mathematics
Determinants
2 × 2
|A| = ad − bc
Inverse via adjoint
A⁻¹ = (1/|A|) adj A, |A| ≠ 0
A (adj A)
A (adj A) = |A| I
Area of triangle
Δ = (1/2) |det of coordinates|
Cramer’s rule
x = Δx/Δ, y = Δy/Δ, z = Δz/Δ
➗ Class 12 · Mathematics
Continuity and Differentiability
Derivative definition
f′(x) = limₕ→₀ [f(x+h) − f(x)] / h
Product rule
(uv)′ = u′v + uv′
Quotient rule
(u/v)′ = (u′v − uv′) / v²
Chain rule
dy/dx = (dy/du)(du/dx)
d/dx (xⁿ)
n xⁿ⁻¹
d/dx (sin x)
cos x
d/dx (cos x)
−sin x
d/dx (eˣ)
eˣ
d/dx (ln x)
1/x
d/dx (tan x)
sec² x
➗ Class 12 · Mathematics
Application of Derivatives
Increasing
f′(x) ≥ 0 on the interval
Tangent slope
m = f′(x₀)
Tangent line
y − y₀ = f′(x₀)(x − x₀)
Normal slope
−1 / f′(x₀) (if f′ ≠ 0)
First derivative test
f′ changes + to − ⇒ local max
➗ Class 12 · Mathematics
Integrals
Power
∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, n ≠ −1
Log
∫ dx/x = ln|x| + C
Exponential
∫ eˣ dx = eˣ + C
sin / cos
∫ sin x dx = −cos x + C, ∫ cos x dx = sin x + C
sec²
∫ sec² x dx = tan x + C
By parts
∫ u dv = uv − ∫ v du
Definite (FTC)
∫ₐᵇ f(x) dx = F(b) − F(a)
Property
∫ₐᵇ f(x) dx = ∫ₐᵇ f(a+b−x) dx
From chapter list
∫x^n dx = x^{n+1}/(n+1)+C
➗ Class 12 · Mathematics
Application of Integrals
Area under y = f(x)
A = ∫ₐᵇ y dx
Area between curves
A = ∫ₐᵇ |f(x) − g(x)| dx
➗ Class 12 · Mathematics
Differential Equations
Order / degree
Order = highest derivative; degree = power of that derivative (poly form)
Variable separable
dy/dx = f(x) g(y) ⇒ ∫ dy/g(y) = ∫ f(x) dx
Linear
dy/dx + P y = Q, IF = e^{∫P dx}
➗ Class 12 · Mathematics
Vector Algebra
Unit vector
â = a / |a|
Dot product
a · b = |a| |b| cos θ
Cross product magnitude
|a × b| = |a| |b| sin θ
Scalar triple
[a b c] = a · (b × c)
i, j, k
i · i = 1, i · j = 0, i × j = k
➗ Class 12 · Mathematics
Three Dimensional Geometry
Direction cosines
cos²α + cos²β + cos²γ = 1
Distance (3D)
d = √[(x₂−x₁)²+(y₂−y₁)²+(z₂−z₁)²]
Angle between lines
cos θ = |d₁ · d₂| / (|d₁| |d₂|)
Plane
ax + by + cz + d = 0
Distance point to plane
|ax₁+by₁+cz₁+d| / √(a²+b²+c²)
➗ Class 12 · Mathematics
Linear Programming
Corner-point method
Optimum of a linear objective on a closed polygon is at a vertex
➗ Class 12 · Mathematics
Probability
Conditional
P(A|B) = P(A ∩ B) / P(B)
Bayes
P(Eᵢ|A) = P(A|Eᵢ) P(Eᵢ) / Σ P(A|Eₖ) P(Eₖ)
Independent
P(A ∩ B) = P(A) P(B)
Binomial P(X=r)
ⁿCᵣ pʳ (1−p)ⁿ⁻ʳ
Binomial mean / variance
Mean = np, Variance = npq
From chapter list
P(A|B)=P(B|A)P(A)/P(B)
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FAQs
Are these all official board formulas?
They are the standard NCERT relations used in CBSE Maths and Science papers. Always match the edition issued by your school.
Why are some Biology items short?
Biology is mostly definitions. Only equations and standard ratios (photosynthesis, Hardy–Weinberg, growth models) sit on this sheet.
Will a formula come as-is in the exam?
Often the paper asks you to use a formula, not recite it. Practise a numerical or a derivation after you memorise the line.