Composite
(f ∘ g)(x) = f(g(x))
One sheet · NCERT · Board revision
Class 12 Formula Sheet
187 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)
⚛️ Class 12 · Physics
Electric Charges and Fields
Coulomb
F = (1/(4πε₀)) (q₁ q₂ / r²)
1/(4πε₀)
9 × 10⁹ N m² C⁻²
Electric field
E = F / q₀ = (1/(4πε₀)) (q / r²)
Dipole field (axial)
E = (1/(4πε₀)) (2p / r³)
Dipole field (equatorial)
E = (1/(4πε₀)) (p / r³)
Torque on dipole
τ = p E sin θ
Gauss’s law
∮ E · dA = Q_enclosed / ε₀
Infinite sheet
E = σ / (2ε₀)
From chapter list
F=q1q2/(4πε0 r²)
⚛️ Class 12 · Physics
Electrostatic Potential and Capacitance
Potential
V = W / q₀ = (1/(4πε₀)) (q / r)
Relation
E = −dV/dr (radial)
Capacitance
C = Q / V
Parallel-plate
C = ε₀ A / d
Series
1/C = 1/C₁ + 1/C₂
Parallel
C = C₁ + C₂
Energy
U = ½ C V² = Q² / (2C)
From chapter list
U=½CV²
⚛️ Class 12 · Physics
Current Electricity
Current
I = q / t = n e v_d A
Ohm (microscopic)
J = σ E
Resistance
R = ρ l / A
EMF and internal r
ε = I (R + r), V = ε − I r
Kirchhoff junction
Σ I = 0
Kirchhoff loop
Σ ε = Σ I R
Wheatstone
P/Q = R/S (balanced)
From chapter list
ε=I(R+r)
⚛️ Class 12 · Physics
Moving Charges and Magnetism
Lorentz force
F = q (v × B) + q E
Force on wire
F = I l B sin θ
Biot–Savart (wire idea)
dB = (μ₀ / 4π) (I dl sinθ / r²)
Long straight wire
B = μ₀ I / (2π r)
Centre of loop
B = μ₀ I / (2R)
Solenoid
B = μ₀ n I
Cyclotron frequency
f = qB / (2π m)
⚛️ Class 12 · Physics
Magnetism and Matter
Bar magnet dipole
B_axial = (μ₀ / 4π) (2M / r³)
Torque
τ = m B sin θ
⚛️ Class 12 · Physics
Electromagnetic Induction
Faraday–Lenz
ε = − dφ_B / dt
Motional emf
ε = B l v (rod on rails)
Self inductance
ε = −L dI/dt, φ = L I
Energy in inductor
U = ½ L I²
From chapter list
ε=−dφB/dt
⚛️ Class 12 · Physics
Alternating Current
RMS
I_rms = I₀ / √2, V_rms = V₀ / √2
Inductive reactance
X_L = ω L = 2π f L
Capacitive reactance
X_C = 1 / (ω C)
Series LCR impedance
Z = √[R² + (X_L − X_C)²]
Resonance
ω₀ = 1 / √(LC)
Power
P = V_rms I_rms cos φ
Transformer
Vₛ / Vₚ = Nₛ / Nₚ = Iₚ / Iₛ (ideal)
From chapter list
XL=ωL; XC=1/ωC
⚛️ Class 12 · Physics
Electromagnetic Waves
Speed
c = 1 / √(μ₀ ε₀) = 3 × 10⁸ m/s
c and fields
c = E₀ / B₀
⚛️ Class 12 · Physics
Ray Optics and Optical Instruments
Mirror / lens formula
1/v − 1/u = 1/f (lens sign convention)
Refraction at spherical surface
n₂/v − n₁/u = (n₂ − n₁)/R
Lens maker
1/f = (n−1)(1/R₁ − 1/R₂)
Prism
μ = sin[(A+δ_m)/2] / sin(A/2)
TIR
i > i_c, sin i_c = 1/μ
Simple microscope
m = 1 + D/f
Telescope (normal)
m = f_o / f_e
From chapter list
1/v−1/u=1/f
⚛️ Class 12 · Physics
Wave Optics
YDSE fringe
β = λ D / d
Path difference (bright)
d sinθ = n λ
Brewster
μ = tan i_p
⚛️ Class 12 · Physics
Dual Nature of Radiation and Matter
Photon energy
E = h ν = hc / λ
Photoelectric
K_max = hν − φ
de Broglie
λ = h / p = h / (mv)
From chapter list
Kmax=hν−φ
⚛️ Class 12 · Physics
Atoms
Bohr radius
rₙ = n² a₀, a₀ ≈ 0.53 Å
Energy levels
Eₙ = −13.6 / n² eV (H-atom)
Wave number
1/λ = R (1/n₁² − 1/n₂²)
From chapter list
En=−13.6/n² eV
⚛️ Class 12 · Physics
Nuclei
Decay law
N = N₀ e^{−λ t}
Half-life
T½ = 0.693 / λ
Mean life
τ = 1 / λ
Mass–energy
E = Δm c²
From chapter list
N=N0 e^{−λt}
⚛️ Class 12 · Physics
Semiconductor Electronics: Materials, Devices and Simple Circuits
NAND / NOR
Universal gates
AND, OR, NOT
Basic Boolean operations as in NCERT truth tables
From chapter list
NAND is universal
🧪 Class 12 · Chemistry
Solutions
Mole fraction
xᵢ = nᵢ / Σ n
Molarity
M = moles of solute / litres of solution
Molality
m = moles of solute / kg of solvent
Raoult (volatile)
p₁ = x₁ p₁°
Relative lowering
(p° − p) / p° = x₂
Elevation of boiling point
ΔT_b = K_b m
Depression of freezing point
ΔT_f = K_f m
Osmotic pressure
π = i C R T
van’t Hoff
i = observed / calculated colligative
🧪 Class 12 · Chemistry
Electrochemistry
Nernst
E = E° − (RT/nF) ln Q ≈ E° − (0.059/n) log Q (25°C)
ΔG°
ΔG° = −n F E°_cell
Conductivity
κ, Λ_m = κ / C (C in mol/cm³ or as in book units)
Kohlrausch
Λ_m° = ν₊ λ₊° + ν₋ λ₋°
Faraday (1st)
w = Z I t, Z = M / (n F)
From chapter list
E=E°−(RT/nF)lnQ
🧪 Class 12 · Chemistry
Chemical Kinetics
Rate law
rate = k [A]ᵐ [B]ⁿ
First order
k = (2.303/t) log ([A]₀ / [A])
Half-life (1st order)
t½ = 0.693 / k
Arrhenius
k = A e^{−Ea/RT}
Two-temperature
log (k₂/k₁) = (Ea/2.303 R) (1/T₁ − 1/T₂)
🧪 Class 12 · Chemistry
The d- and f-Block Elements
Lanthanoid contraction
Steady decrease in size from La to Lu
🧪 Class 12 · Chemistry
Coordination Compounds
Coordination number
Number of ligand donor atoms bound to the metal
Werner primary / secondary
Oxidation number vs coordination number
🧪 Class 12 · Chemistry
Haloalkanes and Haloarenes
SN2
One-step, inversion, preferred by 1°
SN1
Two-step via carbocation, racemisation, preferred by 3°
🧪 Class 12 · Chemistry
Alcohols, Phenols and Ethers
Acidity order (idea)
phenol > water > 1° alcohol (as taught)
Williamson
R−ONa + R′−X → R−O−R′
🧪 Class 12 · Chemistry
Aldehydes, Ketones and Carboxylic Acids
Nucleophilic addition
C=O is the site; HCN, Grignard, etc.
Tollens / Fehling
Aldehydes (not ketones, generally) reduce these reagents
🧪 Class 12 · Chemistry
Amines
Basicity (aqueous idea)
Compare 2°, 1°, 3° as in NCERT discussion
🧪 Class 12 · Chemistry
Biomolecules
Peptide bond
−CO−NH− between amino acids
🧬 Class 12 · Biology
Sexual Reproduction in Flowering Plants
Double fertilisation
Syngamy + triple fusion → zygote + endosperm (3n)
🧬 Class 12 · Biology
Human Reproduction
Menstrual cycle
≈ 28 days; ovulation around day 14
🧬 Class 12 · Biology
Principles of Inheritance and Variation
Monohybrid
F2 3 : 1 phenotype
Dihybrid
F2 9 : 3 : 3 : 1
Hardy–Weinberg (also Evolution)
p² + 2pq + q² = 1, p + q = 1
🧬 Class 12 · Biology
Molecular Basis of Inheritance
Chargaff
A = T, G ≡ C (DNA)
Base pairing
A–T two H-bonds; G–C three H-bonds
From chapter list
A=T, G≡C pairing
🧬 Class 12 · Biology
Evolution
Hardy–Weinberg
p² + 2pq + q² = 1
🧬 Class 12 · Biology
Organisms and Populations
Exponential growth
dN/dt = r N
Logistic growth
dN/dt = r N (K − N)/K
🧬 Class 12 · Biology
Ecosystem
NPP
NPP = GPP − R
10% law
≈ 10% energy to next trophic level
🧬 Class 12 · Biology
Biotechnology: Principles and Processes
PCR idea
Denaturation → annealing → extension (repeat)
Also open
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.