formula sheet ✦

All your CBSE Class 12 formulas in one place — tap any card to drill it with Arya

Electric Charges and Fields

Electrostatic Potential and Capacitance

Current Electricity

Moving Charges and Magnetism

Magnetism and Matter

Electromagnetic Induction

Alternating Current

Electromagnetic Waves

Ray Optics and Optical Instruments

Wave Optics

Dual Nature of Radiation and Matter

Atoms

Nuclei

Semiconductor Electronics

Solutions

Electrochemistry

Chemical Kinetics

d and f Block Elements

Coordination Compounds

Haloalkanes and Haloarenes

Alcohols, Phenols and Ethers

Aldehydes, Ketones and Carboxylic Acids

Amines

Biomolecules

Relations and Functions

Inverse Trigonometric Functions

Matrices

Determinants

Continuity and Differentiability

Application of Derivatives

Integrals

Application of Integrals

Differential Equations

Vector Algebra

Three Dimensional Geometry

Linear Programming

Probability

Physics153 formulas

Ch 1 · Electric Charges and Fields

Coulomb's law★★★

F = \frac{k q _{1} q _{2}}{r ^{2}}

(k = 9×10⁹ N m² C⁻²)

board favourite

Electric field due to point charge★★★

E = \frac{k q}{r ^{2}}

E = \frac{q}{4 π ε _{0} r ^{2}}

board favourite

Electric field from force

E = \frac{F}{q _{0}}

Electric dipole moment★★

p = q \times 2 a

(direction: −q to +q)

frequently asked

E on axial point of dipole★★

E_{a x} = \frac{2 k p}{r ^{3}}

(r >> a)

frequently asked

E on equatorial point of dipole★★

E_{e q} = \frac{k p}{r ^{3}}

(r >> a)

frequently asked

Torque on dipole in uniform field★★

τ = pE sin θ

τ = p \times E

frequently asked

Gauss's law★★★

Φ_{E} = \oint E \cdot d A = \frac{q _{e n c}}{ε _{0}}

board favourite

E due to infinite line charge (λ)★★

E = \frac{λ}{2 π ε _{0} r}

frequently asked

E due to infinite plane sheet (σ)★★

E = \frac{σ}{2 ε _{0}}

frequently asked

E outside uniformly charged sphere

E = \frac{q}{4 π ε _{0} r ^{2}}

Electric flux★★

Φ = E \cdot A = E A cos θ

frequently asked

📐 See NCERT diagram

Diagram: Field lines (point charges, dipole, planes)

Ch 2 · Electrostatic Potential and Capacitance

Electric potential due to point charge★★

V = \frac{k q}{r}

frequently asked

Potential difference

V_{A} - V_{B} = \frac{W _{A B}}{q}

E and V relation★★

E = - \frac{d V}{d r}

frequently asked

Potential due to dipole (axial)

V = \frac{k p c o s θ}{r ^{2}}

(axial: θ=0 →

V = k p / r^{2}

)

Potential due to dipole (equatorial)

V = 0

(θ = 90°)

PE of system of two charges★★

U = \frac{k q _{1} q _{2}}{r}

frequently asked

PE of dipole in external field★★

U = - pE cos θ = - p \cdot E

frequently asked

Capacitance definition★★★

C = \frac{Q}{V}

board favourite

Parallel plate capacitor★★★

C = \frac{ε _{0} A}{d}

board favourite

With dielectric slab (thickness t)★★

C = \frac{ε _{0} A}{d - t + t / K}

frequently asked

Capacitors in series★★★

\frac{1}{C} = \frac{1}{C _{1}} + \frac{1}{C _{2}} + \frac{1}{C _{3}}

board favourite

Capacitors in parallel★★★

C = C_{1} + C_{2} + C_{3}

board favourite

Energy stored in capacitor★★★

U = \frac{1}{2} C V^{2} = \frac{Q ^{2}}{2 C} = \frac{Q V}{2}

board favourite

Energy density in electric field★★

u = \frac{1}{2} ε_{0} E^{2}

frequently asked

📐 See NCERT diagram

Diagram: Equipotential surfaces, parallel plate capacitor

Ch 3 · Current Electricity

Electric current★★

I = \frac{Q}{t} = n A e v_{d}

frequently asked

Ohm's law★★★

V = I R

board favourite

Resistance from geometry★★★

R = \frac{ρ L}{A}

board favourite

Temperature dependence of resistance★★

R_{t} = R_{0} (1 + α Δ T)

frequently asked

Drift velocity★★

v_{d} = \frac{e E τ}{m} = \frac{I}{n A e}

frequently asked

Mobility

μ = \frac{v _{d}}{E} = \frac{e τ}{m}

Resistors in series★★★

R_{e q} = R_{1} + R_{2} + R_{3}

board favourite

Resistors in parallel★★★

\frac{1}{R _{e q}} = \frac{1}{R _{1}} + \frac{1}{R _{2}} + \frac{1}{R _{3}}

board favourite

EMF and terminal voltage★★★

V = ε - I r

(during discharge)

board favourite

Cells in series★★

ε_{e q} = ε_{1} + ε_{2}

;

r_{e q} = r_{1} + r_{2}

frequently asked

Cells in parallel (identical)★★

ε_{e q} = ε

;

r_{e q} = r / n

frequently asked

Kirchhoff's Current Law (KCL)★★★

Σ I_{in} = Σ I_{o u t}

at junction

board favourite

Kirchhoff's Voltage Law (KVL)★★★

Σ V = 0

around closed loop

board favourite

Wheatstone bridge balance condition★★★

\frac{P}{Q} = \frac{R}{S}

board favourite

Electric power★★★

P = V I = I^{2} R = \frac{V ^{2}}{R}

board favourite

Electrical energy★★

W = V I t = I^{2} R t

frequently asked

📐 See NCERT diagram

Diagram: Metre bridge, Potentiometer

Ch 4 · Moving Charges and Magnetism

Biot-Savart law★★★

d B = \frac{μ _{0}}{4 π} \cdot \frac{I d l s i n θ}{r ^{2}}

board favourite

B at centre of circular loop★★★

B = \frac{μ _{0} I}{2 R}

board favourite

B on axis of circular loop★★

B = \frac{μ _{0} I R ^{2}}{2 ( R ^{2} + x ^{2} ) ^{3/2}}

frequently asked

Ampere's circuital law★★★

\oint B \cdot d l = μ_{0} I_{e n c}

board favourite

B inside solenoid★★★

B = μ_{0} n I

(n = N/L turns/metre)

board favourite

B inside toroid★★

B = \frac{μ _{0} N I}{2 π r}

frequently asked

B due to long straight wire★★★

B = \frac{μ _{0} I}{2 π r}

board favourite

Lorentz force on moving charge★★★

F = q (v \times B)

;

F = q v B sin θ

board favourite

Force on current-carrying conductor★★★

F = B I L sin θ

;

F = I L \times B

board favourite

Force per unit length between two wires★★

\frac{F}{L} = \frac{μ _{0} I _{1} I _{2}}{2 π d}

frequently asked

Torque on current loop in B field★★

τ = N I A B sin θ

;

τ = M \times B

frequently asked

Magnetic moment of current loop★★

M = N I A

frequently asked

Radius of circular motion in B field★★

r = \frac{m v}{q B}

frequently asked

Time period in cyclotron★★

T = \frac{2 π m}{q B}

;

ν = \frac{q B}{2 π m}

frequently asked

Galvanometer → ammeter (shunt S)★★

S = \frac{I _{g} G}{I - I _{g}}

frequently asked

Galvanometer → voltmeter (series R)★★

R = \frac{V}{I _{g}} - G

frequently asked

Ch 5 · Magnetism and Matter

Bar magnet: B on axial point★★

B = \frac{μ _{0}}{4 π} \cdot \frac{2 M}{r ^{3}}

frequently asked

Bar magnet: B on equatorial point★★

B = \frac{μ _{0}}{4 π} \cdot \frac{M}{r ^{3}}

frequently asked

Torque on bar magnet in field★★

τ = M B sin θ = M \times B

frequently asked

PE of bar magnet in external field

U = - M B cos θ

Magnetic susceptibility

χ = \frac{M}{H}

Relative permeability

μ_{r} = 1 + χ

B in magnetic material

B = μ_{0} (H + M) = μ_{0} μ_{r} H

Curie's law (paramagnetism)★★

M = \frac{C B}{T}

(C = Curie constant)

frequently asked

Earth's magnetic field components★★

B_{v} = B sin δ

;

B_{H} = B cos δ

frequently asked

📐 See NCERT diagram

Diagram: B-H hysteresis loop

Ch 6 · Electromagnetic Induction

Faraday's law★★★

ε = - \frac{d Φ}{d t}

;

ε = - N \frac{d Φ}{d t}

board favourite

Magnetic flux★★★

Φ = B \cdot A = B A cos θ

board favourite

Lenz's law★★★

Induced current opposes change causing it

board favourite

Motional EMF★★★

ε = B v l

board favourite

Self-inductance (flux linkage)★★

N Φ = L I

→

ε = - L \frac{d I}{d t}

frequently asked

Self-inductance of solenoid★★

L = μ_{0} n^{2} A l = \frac{μ _{0} N ^{2} A}{l}

frequently asked

Mutual inductance★★

ε_{2} = - M \frac{d I _{1}}{d t}

frequently asked

Energy stored in inductor★★★

U = \frac{1}{2} L I^{2}

board favourite

Peak EMF in AC generator★★

ε = N B A ω sin ω t = ε_{0} sin ω t

frequently asked

Ch 7 · Alternating Current

Instantaneous values & RMS★★★

e = E_{0} sin ω t

;

E_{r m s} = \frac{E _{0}}{2}

;

I_{r m s} = \frac{I _{0}}{2}

board favourite

Inductive reactance★★★

X_{L} = ω L = 2 π f L

(unit: Ω)

board favourite

Capacitive reactance★★★

X_{C} = \frac{1}{ω C} = \frac{1}{2 π f C}

(unit: Ω)

board favourite

Impedance of series LCR★★★

Z = R^{2} + (X_{L} - X_{C})^{2}

board favourite

Phase angle φ★★★

tan ϕ = \frac{X _{L} - X _{C}}{R}

board favourite

Resonance condition★★★

X_{L} = X_{C}

→

ω_{0} = \frac{1}{L C}

board favourite

Resonant frequency★★★

f_{0} = \frac{1}{2 π L C}

board favourite

Average power in AC circuit★★★

P = E_{r m s} \cdot I_{r m s} \cdot cos ϕ = I_{r m s}^{2} \cdot R

board favourite

Power factor★★

cos ϕ = \frac{R}{Z}

frequently asked

Quality factor (Q-factor)★★

Q = \frac{ω _{0} L}{R} = \frac{1}{ω _{0} C R} = \frac{1}{R} \frac{L}{C}

frequently asked

Transformer turns ratio★★★

\frac{V _{2}}{V _{1}} = \frac{N _{2}}{N _{1}} = \frac{I _{1}}{I _{2}}

board favourite

Transformer efficiency★★

η = \frac{P _{o u t}}{P _{in}} = \frac{V _{2} I _{2}}{V _{1} I _{1}}

frequently asked

📐 See NCERT diagram

Diagram: Phasor diagrams, resonance curve

Ch 8 · Electromagnetic Waves

Speed of EM waves in vacuum★★

c = \frac{1}{μ _{0} ε _{0}} = 3 \times 1 0^{8}

m/s

frequently asked

Speed in a medium

v = \frac{1}{μ ε} = \frac{c}{n}

E and B amplitudes★★

\frac{E _{0}}{B _{0}} = c

frequently asked

Displacement current★★

I_{d} = ε_{0} \frac{d Φ _{E}}{d t}

frequently asked

Energy density of EM wave

u = \frac{1}{2} ε_{0} E^{2} + \frac{B ^{2}}{2 μ _{0}}

Generalised Ampere's law★★

\oint B \cdot d l = μ_{0} (I_{c} + I_{d})

frequently asked

📐 See NCERT diagram

Diagram: E⃗ ⊥ B⃗ ⊥ v⃗, EM spectrum

Ch 9 · Ray Optics and Optical Instruments

Mirror formula★★★

\frac{1}{v} + \frac{1}{u} = \frac{1}{f}

;

f = \frac{R}{2}

board favourite

Magnification (mirror)★★★

m = - \frac{v}{u} = \frac{h ^{'}}{h}

board favourite

Snell's law of refraction★★★

n_{1} sin θ_{1} = n_{2} sin θ_{2}

board favourite

Critical angle for TIR★★★

sin C = \frac{n _{2}}{n _{1}}

(n₁ > n₂)

board favourite

Lens formula★★★

\frac{1}{v} - \frac{1}{u} = \frac{1}{f}

board favourite

Magnification (lens)★★

m = \frac{v}{u} = \frac{h ^{'}}{h}

frequently asked

Lens maker's formula★★★

\frac{1}{f} = (n - 1) [\frac{1}{R _{1}} - \frac{1}{R _{2}}]

board favourite

Power of a lens★★

P = \frac{1}{f ( m )}

[unit: dioptre, D]

frequently asked

Lenses in contact (equivalent)★★

\frac{1}{f} = \frac{1}{f _{1}} + \frac{1}{f _{2}}

;

P = P_{1} + P_{2}

frequently asked

Refraction at spherical surface★★

\frac{n _{2}}{v} - \frac{n _{1}}{u} = \frac{n _{2} - n _{1}}{R}

frequently asked

Prism: angle of deviation★★

δ = (i + e) - A

frequently asked

Prism: refractive index (min deviation)★★★

n = \frac{s i n ( \frac{A + D}{2} )}{s i n ( \frac{A}{2} )}

board favourite

Simple microscope magnification★★

m = 1 + \frac{D}{f}

(D = 25 cm)

frequently asked

Compound microscope magnification★★

M = - \frac{L}{f _{0}} \times (1 + \frac{D}{f _{e}})

frequently asked

Astronomical telescope magnification★★

M = - \frac{f _{0}}{f _{e}}

frequently asked

📐 See NCERT diagram

Diagram: Mirror, lens, prism, TIR, telescope ray diagrams

Ch 10 · Wave Optics

Constructive interference (bright fringe)★★★

Δ x = nλ

(n = 0, 1, 2, …)

board favourite

Destructive interference (dark fringe)★★★

Δ x = \frac{( 2 n - 1 ) λ}{2}

board favourite

YDSE: fringe width β★★★

β = \frac{λ D}{d}

board favourite

nth bright fringe position★★

y_{n} = \frac{nλ D}{d}

frequently asked

nth dark fringe position★★

y_{n} = \frac{( 2 n - 1 ) λ D}{2 d}

frequently asked

Single slit: first minima★★

a sin θ = λ

→

y = \frac{λ D}{a}

frequently asked

Angular width of central maximum★★

2 θ = \frac{2 λ}{a}

frequently asked

Malus's law (polarisation)★★★

I = I_{0} cos^{2} θ

board favourite

Brewster's angle★★

tan i_{B} = n

frequently asked

📐 See NCERT diagram

Diagram: YDSE setup, single slit diffraction pattern

Ch 11 · Dual Nature of Radiation and Matter

Photon energy★★★

E = h f = \frac{h c}{λ}

board favourite

Photon momentum★★

p = \frac{h}{λ} = \frac{E}{c}

frequently asked

Einstein's photoelectric equation★★★

K E_{ma x} = h f - ϕ = h (f - f_{0})

board favourite

Work function★★

ϕ = h f_{0}

(f₀ = threshold frequency)

frequently asked

Stopping potential★★

e V_{0} = K E_{ma x}

frequently asked

de Broglie wavelength★★★

λ = \frac{h}{m v} = \frac{h}{p}

board favourite

de Broglie λ for accelerated electron★★★

λ = \frac{h}{2 m e V}

board favourite

de Broglie λ in terms of KE★★

λ = \frac{h}{2 m \cdot K E}

frequently asked

📐 See NCERT diagram

Diagram: Photoelectric effect, KE vs f graph

Ch 12 · Atoms

Bohr's radius of nth orbit★★★

r_{n} = \frac{n ^{2} a _{0}}{Z}

(a₀ = 0.529 Å)

board favourite

Velocity in nth orbit★★

v_{n} = \frac{2.18 \times 1 0 ^{6} Z}{n}

m/s

frequently asked

Energy of electron in nth orbit★★★

E_{n} = - \frac{13.6 Z ^{2}}{n ^{2}}

board favourite

Frequency of emitted photon★★

h ν = E_{2} - E_{1}

(E₂ > E₁)

frequently asked

Rydberg formula★★★

\frac{1}{λ} = R Z^{2} (\frac{1}{n _{1}^{2}} - \frac{1}{n _{2}^{2}})

[R = 1.097×10⁷ m⁻¹]

board favourite

Bohr's angular momentum condition★★

m v r = \frac{nh}{2 π} = n ℏ

frequently asked

📐 See NCERT diagram

Diagram: Rutherford/Bohr model, hydrogen spectrum series

Ch 13 · Nuclei

Nuclear radius★★

R = R_{0} A^{1/3}

(R₀ = 1.2×10⁻¹⁵ m = 1.2 fm)

frequently asked

Mass defect★★★

Δ m = [Z m_{p} + (A - Z) m_{n}] - M_{n u c l e u s}

board favourite

Binding energy★★★

B E = Δ m \times c^{2} = Δ m \times 931.5

MeV/u

board favourite

Binding energy per nucleon★★

B E / A

frequently asked

Radioactive decay law★★★

N = N_{0} e^{- λ t}

board favourite

Half-life★★★

T_{1/2} = \frac{0.693}{λ}

board favourite

Activity★★★

A = λ N = A_{0} e^{- λ t}

[unit: Becquerel]

board favourite

Mean life (average life)★★

τ = \frac{1}{λ} = \frac{T _{1/2}}{0.693}

frequently asked

Mass-energy equivalence★★★

E = m c^{2}

board favourite

Q-value of nuclear reaction★★

Q = (m_{r e a c t an t s} - m_{p r o d u c t s}) c^{2}

frequently asked

📐 See NCERT diagram

Diagram: BE/A vs A curve, N vs t decay graph

Ch 14 · Semiconductor Electronics

Intrinsic carrier concentration★★

n_{i}^{2} = n_{e} \times n_{h}

frequently asked

Conductivity of semiconductor

σ = n_{i} e (μ_{e} + μ_{h})

Diode current equation

I = I_{0} (e^{e V / k_{B} T} - 1)

Transistor current relation★★★

I_{E} = I_{B} + I_{C}

board favourite

Current gain α (CB configuration)★★★

α = \frac{I _{C}}{I _{E}}

(< 1)

board favourite

Current gain β (CE configuration)★★★

β = \frac{I _{C}}{I _{B}}

;

β = \frac{α}{1 - α}

board favourite

Relation between α and β★★★

α = \frac{β}{1 + β}

;

β = \frac{α}{1 - α}

board favourite

Voltage gain of CE amplifier★★

A_{v} = - β \times \frac{R _{C}}{R _{B}}

frequently asked

De Morgan's theorems★★★

(A \cdot B)^{'} = A^{'} + B^{'}

;

(A + B)^{'} = A^{'} \cdot B^{'}

board favourite

NAND gate output★★

Y = (A \cdot B)^{'}

frequently asked

NOR gate output★★

Y = (A + B)^{'}

frequently asked

📐 See NCERT diagram

Diagram: Energy bands, p-n junction, transistor circuits

Chemistry90 formulas

Ch 1 · Solutions

Raoult's law (ideal solution)★★★

p_{A} = x_{A} \cdot p_{A}^{\circ}

(Raoult's law)

board favourite

Total pressure of ideal binary solution★★★

P_{t o t a l} = x_{A} p_{A}^{\circ} + x_{B} p_{B}^{\circ}

board favourite

Henry's law (gas in liquid)★★

p = K_{H} \cdot x

(K_H = Henry's constant)

frequently asked

Elevation in boiling point★★★

Δ T_{b} = K_{b} \cdot m

(Kb = molal elevation constant)

board favourite

Depression in freezing point★★★

Δ T_{f} = K_{f} \cdot m

(Kf = molal depression constant)

board favourite

Osmotic pressure★★★

π = C R T = \frac{n}{V} R T

board favourite

van't Hoff factor★★

i

= observed colligative property / calculated value (no assoc./dissoc.)

frequently asked

Molality★★

m = \frac{moles of solute}{kg of solvent}

frequently asked

Mole fraction★★

x_{A} = \frac{n _{A}}{n _{A} + n _{B}}

frequently asked

📐 See NCERT diagram

Diagram: P-x graph (ideal, +ve, −ve deviation), osmosis setup

Ch 2 · Electrochemistry

Standard cell potential★★★

E_{ce l l}^{\circ} = E_{c a t h o d e}^{\circ} - E_{an o d e}^{\circ}

board favourite

Nernst equation★★★

E_{ce l l} = E_{ce l l}^{\circ} - \frac{0.0592}{n} lo g Q

(at 25°C)

board favourite

Gibbs energy and cell potential★★★

Δ G^{\circ} = - n F E_{ce l l}^{\circ}

board favourite

Equilibrium constant from E°★★★

lo g K = \frac{n E _{ce l l}^{\circ}}{0.0592}

board favourite

Kohlrausch's law (strong electrolyte)★★

Λ_{m} = Λ_{m}^{\circ} - b c

frequently asked

Molar conductivity at infinite dilution★★

Λ_{m}^{\circ} = ν_{+} λ_{+}^{\circ} + ν_{-} λ_{-}^{\circ}

frequently asked

Degree of dissociation from conductivity★★

α = \frac{Λ _{m}}{Λ _{m}^{\circ}}

frequently asked

Faraday's first law of electrolysis★★★

m = Z I t = \frac{M}{n F} \cdot I t

board favourite

Faraday's second law★★

\frac{m _{1}}{m _{2}} = \frac{E _{1}}{E _{2}}

(E = equivalent weight)

frequently asked

Electrochemical equivalent Z

Z = \frac{M}{n F}

📐 See NCERT diagram

Diagram: Daniell cell, electrolytic cell, lead storage battery

Ch 3 · Chemical Kinetics

Rate of reaction★★

Rate = - \frac{1}{a} \frac{d [ A ]}{d t} = + \frac{1}{b} \frac{d [ B ]}{d t}

frequently asked

Rate law★★★

Rate = k [A]^{m} [B]^{n}

(m, n = order w.r.t. A, B)

board favourite

Zero order: integrated rate law★★

[A] = [A]_{0} - k t

;

t_{1/2} = \frac{[ A ] _{0}}{2 k}

frequently asked

First order: integrated rate law★★★

ln [A] = ln [A]_{0} - k t

;

t_{1/2} = \frac{0.693}{k}

board favourite

First order k from concentration★★★

k = \frac{2.303}{t} lo g \frac{[ A ] _{0}}{[ A ]}

board favourite

Second order: integrated rate law★★

\frac{1}{[ A ]} = \frac{1}{[ A ] _{0}} + k t

;

t_{1/2} = \frac{1}{k [ A ] _{0}}

frequently asked

Arrhenius equation★★★

k = A e^{- E_{a} / R T}

ln k = ln A - \frac{E _{a}}{R T}

board favourite

Arrhenius: k from two temperatures★★★

lo g \frac{k _{2}}{k _{1}} = \frac{E _{a}}{2.303 R} (\frac{1}{T _{1}} - \frac{1}{T _{2}})

board favourite

📐 See NCERT diagram

Diagram: PE vs reaction coordinate (Ea), concentration vs time plots

Ch 4 · d and f Block Elements

Electronic config of transition metals (general)★★

(n - 1) d^{1 - 10} n s^{1 - 2}

frequently asked

Oxidation states: variable due to d-electrons★★

e.g. Mn: +2,+3,+4,+6,+7

frequently asked

Ionisation enthalpy trend across 3d series

Generally increases (irregularities at Cr, Cu)

Atomic radius trend across 3d series

Decreases slightly then levels off

Crystal field splitting: octahedral

Δ_{o}

= energy gap between t₂g and eᵍ sets

Crystal field splitting: tetrahedral

Δ_{t} = \frac{4}{9} Δ_{o}

Magnetic moment formula★★★

μ = n (n + 2)

BM (n = unpaired electrons)

board favourite

KMnO₄: structure and oxidising action★★

MnO₄⁻ → Mn²⁺ (acidic); MnO₄⁻ → MnO₂ (neutral/basic)

frequently asked

K₂Cr₂O₇: dichromate structure and reactions★★

Cr₂O₇²⁻ + 14H⁺ + 6e⁻ → 2Cr³⁺ + 7H₂O

frequently asked

Ch 5 · Coordination Compounds

Werner's theory: primary & secondary valency★★

Primary = oxidation state; Secondary = coordination number

frequently asked

IUPAC naming rule order

[cation name][anion name]; ligands alphabetical

VBT: inner orbital complexes (low spin)★★

Use inner (n−1)d orbitals; diamagnetic/low spin

frequently asked

VBT: outer orbital complexes (high spin)★★

Use outer nd orbitals; paramagnetic/high spin

frequently asked

Octahedral geometry — hybridisation★★★

s p^{3} d^{2}

d^{2} s p^{3}

board favourite

Tetrahedral geometry — hybridisation★★★

s p^{3}

board favourite

Square planar geometry — hybridisation★★★

d s p^{2}

board favourite

Geometrical isomerism: cis and trans★★

cis-[Pt(NH₃)₂Cl₂]: both Cl same side

frequently asked

Optical isomerism: d and l enantiomers★★

Non-superimposable mirror images; rotate plane-polarised light

frequently asked

Linkage isomerism

e.g. −NO₂ (nitro) vs −ONO (nitrito)

📐 See NCERT diagram

Diagram: Octahedral, tetrahedral, square planar geometries

Ch 6 · Haloalkanes and Haloarenes

SN1 mechanism (unimolecular)★★★

Step 1: ionisation → carbocation; Step 2: Nu attack; Rate = k[RX]

board favourite

SN2 mechanism (bimolecular)★★★

One step; backside attack; Walden inversion; Rate = k[RX][Nu]

board favourite

Optical isomers: R and S configuration (CIP)★★

Priorities by atomic number; R = clockwise; S = anticlockwise

frequently asked

Resonance of haloarenes (−I but +R effect)★★

Lone pair on X delocalises into ring → less reactive to Nu

frequently asked

Finkelstein reaction★★

R−Cl + NaI → R−I + NaCl (in dry acetone)

frequently asked

Swarts reaction★★

R−Cl + AgF → R−F + AgCl

frequently asked

Grignard reagent formation★★★

R−X + Mg → R−Mg−X (in dry ether)

board favourite

Wurtz reaction★★

2R−X + 2Na → R−R + 2NaX

frequently asked

📐 See NCERT diagram

Diagram: SN1 energy profile, SN2 transition state, Walden inversion

Ch 7 · Alcohols, Phenols and Ethers

H-bonding in alcohols (high bp vs ethers)★★

O−H···O bonding; intermolecular; raises bp

frequently asked

Resonance of phenol★★

−OH lone pair delocalises into ring; ring activated; acidic

frequently asked

Lucas test (1°, 2°, 3° alcohols)★★★

3° → immediate turbidity; 2° → turbid in 5 min; 1° → no reaction (cold)

board favourite

Acid catalysed dehydration order★★

3° > 2° > 1° (stability of carbocation)

frequently asked

Victor Meyer test★★

1° → red; 2° → blue; 3° → colourless

frequently asked

Kolbe's reaction (phenol + CO₂)★★

Phenol + NaOH → sodium phenoxide + CO₂ → salicylate

frequently asked

Riemer-Tiemann reaction★★

Phenol + CHCl₃ + NaOH → o-hydroxybenzaldehyde

frequently asked

Williamson synthesis (preparation of ethers)★★

R−ONa + R'−X → R−O−R' + NaX

frequently asked

Ch 8 · Aldehydes, Ketones and Carboxylic Acids

Nucleophilic addition to C=O★★

Nu⁻ attacks C of C=O → tetrahedral intermediate → product

frequently asked

Aldol condensation★★★

2CH₃CHO → (NaOH) CH₃CH(OH)CH₂CHO → (−H₂O) CH₃CH=CHCHO

board favourite

Cannizzaro reaction (non-enolisable)★★

2HCHO → (NaOH) CH₃OH + HCOONa

frequently asked

Clemmensen reduction★★

C=O → CH₂ (Zn-Hg/HCl)

frequently asked

Wolff-Kishner reduction★★

C=O → CH₂ (NH₂NH₂/KOH/Δ)

frequently asked

Tollens' test (silver mirror)★★★

RCHO + 2[Ag(NH₃)₂]⁺ → RCOO⁻ + 2Ag↓

board favourite

Fehling's test★★★

RCHO + 2Cu²⁺ → RCOO⁻ + Cu₂O↓ (red ppt)

board favourite

Iodoform test (methyl ketones)★★★

RCOCH₃ + I₂/NaOH → CHI₃↓ (yellow) + RCOONa

board favourite

Hell-Volhard-Zelinsky (HVZ) reaction★★

RCOOH + Br₂/(P) → RCH(Br)COOH

frequently asked

Resonance of carboxylate anion (RCOO⁻)

C−O bonds equivalent (1.5 bond order); charge delocalised

📐 See NCERT diagram

Diagram: Nucleophilic addition mechanism, aldol mechanism

Ch 9 · Amines

Structure of amines: sp³ N with lone pair★★

Pyramidal shape; lone pair responsible for basic character

frequently asked

Resonance of aniline (reduced basicity)★★

Lone pair delocalised into ring; less available for protonation

frequently asked

Basicity order★★★

Aliphatic: 2° > 1° > 3° > NH₃; Aromatic: NH₃ > aniline

board favourite

Gabriel phthalimide synthesis (primary amine)★★

Phthalimide + KOH → K-salt + RX → N-alkyl phthalimide → (H₂NNH₂) 1° amine

frequently asked

Hoffmann bromamide degradation★★

RCONH₂ + Br₂ + 4NaOH → RNH₂ + Na₂CO₃ + 2NaBr + 2H₂O

frequently asked

Carbylamine reaction (ID test for 1° amine)★★★

RNH₂ + CHCl₃ + 3KOH → RNC (isocyanide, foul smell)

board favourite

Diazonium salt formation (Sandmeyer)★★★

ArNH₂ + NaNO₂ + HCl → ArN₂⁺Cl⁻ → ArCl (with CuCl)

board favourite

Azo dye coupling reaction★★

ArN₂⁺ + ArOH (alkaline) → Ar−N=N−Ar (coloured azo dye)

frequently asked

📐 See NCERT diagram

Diagram: Pyramidal amine, aniline resonance, diazonium coupling

Ch 10 · Biomolecules

Glucose open chain structure (aldohexose)★★

CHO−(CHOH)₄−CH₂OH (D-glucose)

frequently asked

Haworth (ring) structure of glucose★★

α-D-glucopyranose: −OH on C1 below ring; β: above

frequently asked

Fructose: ketohexose ring structure

Furanose ring (5-membered); α and β anomers

Disaccharide linkage★★★

Sucrose: α-1,β-2 glycosidic bond (non-reducing); Maltose: α-1,4 (reducing)

board favourite

Peptide bond formation★★★

−COOH + H₂N− → −CO−NH− + H₂O (condensation)

board favourite

Amino acid general structure★★

H₂N−CHR−COOH (R = side chain); zwitterion at isoelectric pH

frequently asked

Secondary protein structure: α-helix★★

Intramolecular H-bonds (C=O···H−N) every 4th residue

frequently asked

Secondary protein structure: β-pleated sheet

Intermolecular H-bonds between adjacent polypeptide chains

DNA double helix base pairing★★★

A−T (2 H-bonds); G−C (3 H-bonds); antiparallel strands

board favourite

Nucleotide = base + sugar + phosphate★★

Purine bases: Adenine, Guanine; Pyrimidine: Cytosine, Thymine, Uracil

frequently asked

📐 See NCERT diagram

Diagram: Haworth projections, α-helix, DNA double helix

Mathematics148 formulas

Ch 1 · Relations and Functions

Reflexive relation★★

a R a \forall a \in A

frequently asked

Symmetric relation★★

a R b \Rightarrow b R a \forall a, b \in A

frequently asked

Transitive relation★★

a R b & b R c \Rightarrow a R c

frequently asked

Equivalence relation★★★

Reflexive + Symmetric + Transitive

board favourite

One-one (injective)★★

f (x_{1}) = f (x_{2}) \Rightarrow x_{1} = x_{2}

frequently asked

Onto (surjective)★★

Range of f = Codomain

frequently asked

Number of relations from A to B

2^{n (A) \times n (B)}

Number of functions from A to B

[n (B)]^{n (A)}

Ch 2 · Inverse Trigonometric Functions

sin⁻¹(−x) = −sin⁻¹x★★★

sin^{- 1} (- x) = - sin^{- 1} x

board favourite

cos⁻¹(−x) = π − cos⁻¹x★★★

cos^{- 1} (- x) = π - cos^{- 1} x

board favourite

tan⁻¹(−x) = −tan⁻¹x★★

tan^{- 1} (- x) = - tan^{- 1} x

frequently asked

sin⁻¹x + cos⁻¹x = π/2★★★

sin^{- 1} x + cos^{- 1} x = \frac{π}{2}, x \in [- 1, 1]

board favourite

tan⁻¹x + cot⁻¹x = π/2★★★

tan^{- 1} x + cot^{- 1} x = \frac{π}{2}

board favourite

sec⁻¹x + cosec⁻¹x = π/2★★

sec^{- 1} x + csc^{- 1} x = \frac{π}{2}

frequently asked

tan⁻¹x + tan⁻¹y (|xy| < 1)★★★

tan^{- 1} \frac{x + y}{1 - x y}

board favourite

tan⁻¹x − tan⁻¹y★★

tan^{- 1} \frac{x - y}{1 + x y}

frequently asked

2tan⁻¹x (as sin⁻¹ and cos⁻¹)★★

sin^{- 1} \frac{2 x}{1 + x ^{2}} = cos^{- 1} \frac{1 - x ^{2}}{1 + x ^{2}} = 2 tan^{- 1} x

frequently asked

sin⁻¹x = cosec⁻¹(1/x)

sin^{- 1} x = csc^{- 1} \frac{1}{x}

cos⁻¹x = sec⁻¹(1/x)

cos^{- 1} x = sec^{- 1} \frac{1}{x}

tan⁻¹x = cot⁻¹(1/x) for x > 0

tan^{- 1} x = cot^{- 1} \frac{1}{x}, x > 0

Ch 3 · Matrices

Order of matrix: m rows × n columns

A = [a_{ij}]_{m \times n}

(Aᵀ)ᵀ = A★★

(A^{T})^{T} = A

frequently asked

(A+B)ᵀ = Aᵀ + Bᵀ★★

(A + B)^{T} = A^{T} + B^{T}

frequently asked

(AB)ᵀ = BᵀAᵀ★★★

(A B)^{T} = B^{T} A^{T}

board favourite

Symmetric matrix★★

A^{T} = A

frequently asked

Skew-symmetric matrix★★

A^{T} = - A

(diagonal elements = 0)

frequently asked

Matrix multiplication★★

(A B)_{ij} = \sum_{k} a_{ik} \cdot b_{k j}

frequently asked

Inverse of A★★★

A^{- 1} = \frac{adj ( A )}{∣ A ∣}

(|A| ≠ 0)

board favourite

(AB)⁻¹ = B⁻¹A⁻¹★★

(A B)^{- 1} = B^{- 1} A^{- 1}

frequently asked

AA⁻¹ = A⁻¹A = I★★

A A^{- 1} = A^{- 1} A = I

frequently asked

Ch 4 · Determinants

2×2 determinant★★★

a c b d = a d - b c

board favourite

Area of triangle (coordinate geometry)★★★

\frac{1}{2} ∣ x_{1} (y_{2} - y_{3}) + x_{2} (y_{3} - y_{1}) + x_{3} (y_{1} - y_{2}) ∣

board favourite

Adjoint of A★★

adj (A) = [C_{j i}]

(transpose of cofactor matrix)

frequently asked

Inverse of matrix★★★

A^{- 1} = \frac{adj ( A )}{∣ A ∣}, ∣ A ∣ \neq = 0

board favourite

A·adj(A) = |A|·I★★

A \cdot adj (A) = adj (A) \cdot A = ∣ A ∣ \cdot I

frequently asked

AX = B has unique solution if |A| ≠ 0★★

X = A^{- 1} B

frequently asked

|AB| = |A|·|B|★★

∣ A B ∣ = ∣ A ∣ \cdot ∣ B ∣

frequently asked

|kA| = kⁿ|A| for n×n matrix★★

∣ k A ∣ = k^{n} ∣ A ∣

frequently asked

|Aᵀ| = |A|

∣ A^{T} ∣ = ∣ A ∣

Ch 5 · Continuity and Differentiability

Continuity at x = a★★

lim_{x \to a^{-}} f (x) = lim_{x \to a^{+}} f (x) = f (a)

frequently asked

Chain rule★★★

\frac{d}{d x} [f (g (x))] = f^{'} (g (x)) \cdot g^{'} (x)

board favourite

d/dx(sin⁻¹x)★★★

\frac{1}{1 - x ^{2}}

board favourite

d/dx(cos⁻¹x)★★★

- \frac{1}{1 - x ^{2}}

board favourite

d/dx(tan⁻¹x)★★★

\frac{1}{1 + x ^{2}}

board favourite

d/dx(cot⁻¹x)★★

- \frac{1}{1 + x ^{2}}

frequently asked

d/dx(sec⁻¹x)★★

\frac{1}{x x ^{2} - 1}

frequently asked

d/dx(cosec⁻¹x)★★

- \frac{1}{x x ^{2} - 1}

frequently asked

d/dx(eˣ)★★

e^{x}

frequently asked

d/dx(aˣ)★★

a^{x} ln a

frequently asked

d/dx(ln x)★★

\frac{1}{x}

frequently asked

Parametric differentiation★★

\frac{d y}{d x} = \frac{d y / d t}{d x / d t}

frequently asked

Logarithmic differentiation★★

y = f (x)^{g (x)} \Rightarrow ln y = g (x) \cdot ln f (x)

, differentiate both sides

frequently asked

Rolle's theorem★★★

f (a) = f (b)

, f cont., f' exists

\Rightarrow \exists c \in (a, b) : f^{'} (c) = 0

board favourite

Mean Value Theorem (Lagrange)★★★

f^{'} (c) = \frac{f ( b ) - f ( a )}{b - a}

board favourite

Ch 6 · Application of Derivatives

Rate of change★★

\frac{d y}{d t} = \frac{d y}{d x} \cdot \frac{d x}{d t}

frequently asked

Increasing on (a,b)★★★

f^{'} (x) > 0 \forall x \in (a, b)

board favourite

Decreasing on (a,b)★★★

f^{'} (x) < 0 \forall x \in (a, b)

board favourite

First derivative test (maxima/minima)★★★

f^{'} (c) = 0

; +→− gives max; −→+ gives min

board favourite

Second derivative test★★★

f^{'} (c) = 0

f^{''} (c) < 0

→ max;

f^{''} (c) > 0

→ min

board favourite

Equation of tangent at (x₁,y₁)★★★

y - y_{1} = f^{'} (x_{1}) (x - x_{1})

board favourite

Equation of normal at (x₁,y₁)★★

y - y_{1} = - \frac{1}{f ^{'} ( x _{1} )} (x - x_{1})

frequently asked

Small approximations★★

Δ y \approx d y = f^{'} (x) \cdot Δ x

frequently asked

Ch 7 · Integrals

∫xⁿ dx★★★

\frac{x ^{n + 1}}{n + 1} + C

(n ≠ −1)

board favourite

∫eˣ dx★★★

e^{x} + C

board favourite

∫aˣ dx★★

\frac{a ^{x}}{l n a} + C

frequently asked

∫(1/x) dx★★★

ln ∣ x ∣ + C

board favourite

∫sin x dx★★★

- cos x + C

board favourite

∫cos x dx★★★

sin x + C

board favourite

∫sec²x dx★★★

tan x + C

board favourite

∫cosec²x dx★★★

- cot x + C

board favourite

∫sec x tan x dx★★

sec x + C

frequently asked

∫cosec x cot x dx★★

- csc x + C

frequently asked

∫tan x dx★★

ln ∣ sec x ∣ + C

frequently asked

∫cot x dx★★

ln ∣ sin x ∣ + C

frequently asked

∫sec x dx★★

ln ∣ sec x + tan x ∣ + C

frequently asked

∫cosec x dx★★

ln ∣ csc x - cot x ∣ + C

frequently asked

∫dx/(x²+a²)★★★

\frac{1}{a} tan^{- 1} \frac{x}{a} + C

board favourite

∫dx/√(a²−x²)★★★

sin^{- 1} \frac{x}{a} + C

board favourite

∫dx/√(x²+a²)★★

ln x + x^{2} + a^{2} + C

frequently asked

∫dx/√(x²−a²)★★

ln x + x^{2} - a^{2} + C

frequently asked

∫dx/(x²−a²)★★

\frac{1}{2 a} ln \frac{x - a}{x + a} + C

frequently asked

∫dx/(a²−x²)★★

\frac{1}{2 a} ln \frac{a + x}{a - x} + C

frequently asked

∫√(a²−x²) dx★★

\frac{x}{2} a^{2} - x^{2} + \frac{a ^{2}}{2} sin^{- 1} \frac{x}{a} + C

frequently asked

∫√(x²+a²) dx

\frac{x}{2} x^{2} + a^{2} + \frac{a ^{2}}{2} ln x + x^{2} + a^{2} + C

∫√(x²−a²) dx

\frac{x}{2} x^{2} - a^{2} - \frac{a ^{2}}{2} ln x + x^{2} - a^{2} + C

Integration by parts (ILATE)★★★

\int u \cdot v d x = u \cdot \int v d x - \int (u^{'} \cdot \int v d x) d x

board favourite

∫eˣ[f(x)+f'(x)] dx = eˣf(x) + C★★★

\int e^{x} [f (x) + f^{'} (x)] d x = e^{x} f (x) + C

board favourite

Fundamental Theorem of Calculus★★★

\int_{a}^{b} f (x) d x = F (b) - F (a)

board favourite

∫ₐᵇ f(x)dx = −∫ᵦₐ f(x)dx★★

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x

frequently asked

∫ₐᵇ = ∫ₐᶜ + ∫ᶜᵦ★★

\int_{a}^{b} f (x) d x = \int_{a}^{c} f (x) d x + \int_{c}^{b} f (x) d x

frequently asked

∫₀ᵃ f(x)dx = ∫₀ᵃ f(a−x)dx★★★

\int_{0}^{a} f (x) d x = \int_{0}^{a} f (a - x) d x

board favourite

Even/Odd function integral property★★★

\int_{- a}^{a} f (x) d x = 2 \int_{0}^{a} f (x) d x

(even); = 0 (odd)

board favourite

Ch 8 · Application of Integrals

Area under curve (above x-axis)★★★

A = \int_{a}^{b} f (x) d x

[f(x) ≥ 0]

board favourite

Area between two curves★★★

A = \int_{a}^{b} [f (x) - g (x)] d x

[f(x) ≥ g(x)]

board favourite

Area w.r.t. y-axis★★

A = \int_{c}^{d} g (y) d y

frequently asked

Area of full circle x²+y²=a²★★

A = π a^{2}

frequently asked

Area of ellipse x²/a²+y²/b²=1★★

A = π ab

frequently asked

Ch 9 · Differential Equations

Order: highest derivative present

Order = n of highest

\frac{d ^{n} y}{d x ^{n}}

Degree: power of highest derivative

Degree = exponent of highest order derivative (polynomial in derivatives)

Variable separable method★★★

f (y) d y = g (x) d x \Rightarrow \int f (y) d y = \int g (x) d x + C

board favourite

Homogeneous DE: substitute y = vx★★★

\frac{d y}{d x} = F (\frac{y}{x})

→ y = vx → separate variables

board favourite

Linear DE: dy/dx + Py = Q★★★

IF = e^{\int P d x}

; solution:

y \cdot IF = \int Q \cdot IF d x + C

board favourite

Linear DE in x: dx/dy + Px = Q★★

IF = e^{\int P d y}

; solution:

x \cdot IF = \int Q \cdot IF d y + C

frequently asked

Ch 10 · Vector Algebra

Magnitude of vector★★

∣ a ∣ = a_{1}^{2} + a_{2}^{2} + a_{3}^{2}

frequently asked

Unit vector★★

\overset{a}{^} = \frac{a}{∣ a ∣}

frequently asked

Section formula (internal division m:n)★★

r = \frac{m b + n a}{m + n}

frequently asked

Dot (scalar) product★★★

a \cdot b = ∣ a ∣∣ b ∣ cos θ = a_{1} b_{1} + a_{2} b_{2} + a_{3} b_{3}

board favourite

Angle between two vectors★★★

cos θ = \frac{a \cdot b}{∣ a ∣∣ b ∣}

board favourite

Perpendicular vectors★★★

a ⊥ b \Leftrightarrow a \cdot b = 0

board favourite

Cross (vector) product magnitude★★★

∣ a \times b ∣ = ∣ a ∣∣ b ∣ sin θ

board favourite

Cross product (determinant form)★★★

a \times b = \hat{i} a_{1} b_{1} \hat{j} a_{2} b_{2} \hat{k} a_{3} b_{3}

board favourite

Parallel vectors★★

a ∥ b \Leftrightarrow a \times b = 0

frequently asked

Area of parallelogram★★

∣ a \times b ∣

frequently asked

Area of triangle★★

\frac{1}{2} ∣ a \times b ∣

frequently asked

Scalar triple product★★

[a b c] = a \cdot (b \times c)

frequently asked

Coplanar vectors★★

[a b c] = 0

frequently asked

Projection of a⃗ on b⃗★★

\frac{a \cdot b}{∣ b ∣}

frequently asked

Ch 11 · Three Dimensional Geometry

Direction cosines identity★★

l^{2} + m^{2} + n^{2} = 1

(l=cosα, m=cosβ, n=cosγ)

frequently asked

DC from direction ratios a,b,c★★

l = \frac{a}{a ^{2} + b ^{2} + c ^{2}}, m = \frac{b}{a ^{2} + b ^{2} + c ^{2}}, n = \frac{c}{a ^{2} + b ^{2} + c ^{2}}

frequently asked

Vector equation of line★★★

r = a + λ b

board favourite

Cartesian equation of line★★★

\frac{x - x _{1}}{a} = \frac{y - y _{1}}{b} = \frac{z - z _{1}}{c}

board favourite

Angle between two lines★★★

cos θ = \frac{∣ a _{1} a _{2} + b _{1} b _{2} + c _{1} c _{2} ∣}{∣ b _{1} ∣∣ b _{2} ∣}

board favourite

Lines perpendicular★★★

a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2} = 0

board favourite

Lines parallel★★

\frac{a _{1}}{a _{2}} = \frac{b _{1}}{b _{2}} = \frac{c _{1}}{c _{2}}

frequently asked

Shortest distance (skew lines)★★★

d = \frac{∣ ( a _{2} - a _{1} ) \cdot ( b _{1} \times b _{2} ) ∣}{∣ b _{1} \times b _{2} ∣}

board favourite

Distance between parallel lines★★

d = \frac{∣ b \times ( a _{2} - a _{1} ) ∣}{∣ b ∣}

frequently asked

Vector equation of plane★★★

r \cdot \overset{n}{^} = d

board favourite

Cartesian equation of plane★★★

a x + b y + cz = d

board favourite

Plane through point with normal (a,b,c)★★★

a (x - x_{1}) + b (y - y_{1}) + c (z - z_{1}) = 0

board favourite

Distance from point to plane★★★

dist = \frac{∣ a x _{1} + b y _{1} + c z _{1} - d ∣}{a ^{2} + b ^{2} + c ^{2}}

board favourite

Angle between plane and line★★

sin θ = \frac{∣ b \cdot n ∣}{∣ b ∣∣ n ∣}

frequently asked

Intercept form of plane★★

\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1

frequently asked

Angle between two planes★★★

cos θ = \frac{∣ n _{1} \cdot n _{2} ∣}{∣ n _{1} ∣∣ n _{2} ∣}

board favourite

Ch 12 · Linear Programming

Objective function★★

Maximise/Minimise

Z = a x + b y

frequently asked

Feasible region satisfies all constraints★★

Feasible region = intersection of all constraint half-planes

frequently asked

Corner point theorem★★★

Optimal Z always occurs at a corner (vertex) of feasible region

board favourite

Unbounded region: check if optimal exists★★

Draw half-plane

a x + b y > M

(or < m); if no common point, optimal exists

frequently asked

Ch 13 · Probability

Conditional probability★★★

P (A ∣ B) = \frac{P ( A \cap B )}{P ( B )}, P (B) \neq = 0

board favourite

Multiplication theorem★★★

P (A \cap B) = P (A) \cdot P (B ∣ A) = P (B) \cdot P (A ∣ B)

board favourite

Independent events★★★

P (A \cap B) = P (A) \cdot P (B)

board favourite

Total Probability theorem★★★

P (A) = \sum_{i} P (E_{i}) \cdot P (A ∣ E_{i})

board favourite

Bayes' theorem★★★

P (E_{i} ∣ A) = \frac{P ( E _{i} ) P ( A ∣ E _{i} )}{\sum _{j} P ( E _{j} ) P ( A ∣ E _{j} )}

board favourite

Binomial distribution P(X = r)★★★

(r n) p^{r} q^{n - r}

(q = 1−p)

board favourite

Mean of Binomial distribution★★★

μ = E (X) = n p

board favourite

Variance of Binomial distribution★★★

σ^{2} = n pq

board favourite

Standard deviation★★

σ = n pq

frequently asked

Expected value E(X)★★★

E (X) = \sum x_{i} \cdot P (x_{i})

board favourite

Variance Var(X)★★★

Var (X) = E (X^{2}) - [E (X)]^{2} = \sum x_{i}^{2} P (x_{i}) - [E (X)]^{2}

board favourite