CBSE Math and Science Formula Sheet

Rational exponent

a^{p/q} = (q√a)^p

Laws of exponents

a^m · a^n = a^{m+n}, (a^m)^n = a^{mn}

From chapter list

a^(p/q) = (q√a)^p

Remainder theorem

p(x) divided by (x − a) leaves remainder p(a)

Factor theorem

x − a is a factor iff p(a) = 0

From chapter list

p(x) = (x−a)q(x) + p(a)

Standard form

ax + by + c = 0

Linear pair

Adjacent angles on a straight line sum to 180°

Vertically opposite

Vertically opposite angles are equal

Angle sum

A + B + C = 180°

Exterior angle

Exterior = sum of two remote interiors

Angle sum

Sum of interior angles = 360°

Mid-point theorem

Segment joining mid-points of two sides ∥ third side and half of it

Angle in a semicircle

90°

Cyclic quadrilateral

Opposite angles sum to 180°

Semi-perimeter

s = (a + b + c) / 2

Heron

Δ = √[s(s−a)(s−b)(s−c)]

Cuboid

TSA = 2(lb+bh+hl), V = lbh

Cube

TSA = 6a², V = a³

Cylinder

CSA = 2πrh, TSA = 2πr(h+r), V = πr²h

From chapter list

V_cuboid = l×b×h

Ungrouped mean

x̄ = Σx / n

From chapter list

mean = Σx/n for ungrouped

Kelvin

K = °C + 273

Mass percent

mass % = (mass of solute / mass of solution) × 100

Mole

1 mol = 6.022 × 10²³ particles

Molar mass idea

Mass of 1 mole in grams = formula mass in u

Mass number

A = Z + N

Atomic number

Z = number of protons

Speed

speed = distance / time

Velocity

v = displacement / time

Acceleration

a = (v − u) / t

Equation 1

v = u + at

Equation 2

s = ut + ½ at²

Equation 3

v² = u² + 2as

From chapter list

v = u + at; s = ut + ½at²; v² = u²+2as

Newton II

F = m a

Momentum

p = m v

Impulse

F Δt = Δp

From chapter list

F = ma; p = mv

Universal law

F = G m₁ m₂ / r²

Weight

W = m g

Pressure in liquid

P = h ρ g

Density

ρ = m / V

From chapter list

W = mg; P = hρg

Work

W = F s cos θ

Kinetic energy

KE = ½ m v²

Potential energy

PE = m g h

Power

P = W / t

From chapter list

KE = ½mv²; PE = mgh

Wave relation

v = f λ

HCF–LCM product

HCF(a, b) × LCM(a, b) = a × b

Positive integers

Euclid’s division

a = bq + r, 0 ≤ r < b

Terminating decimal

Denominator of simplest form is 2^m × 5^n

Non-terminating repeating

Prime factors other than 2 and 5 in the denominator

Quadratic sum of zeros

α + β = −b / a

Quadratic product of zeros

αβ = c / a

Cubic sum of zeros

α + β + γ = −b / a

Cubic sum of products two at a time

αβ + βγ + γα = c / a

Cubic product of zeros

αβγ = −d / a

Quadratic polynomial from zeros

k[x² − (α+β)x + αβ]

From chapter list

α+β = −b/a; αβ = c/a

Unique solution

a1/a2 ≠ b1/b2

Intersecting lines, consistent

No solution

a1/a2 = b1/b2 ≠ c1/c2

Parallel, inconsistent

Infinitely many

a1/a2 = b1/b2 = c1/c2

Coincident

Cross-multiplication (x)

x = (b1c2 − b2c1) / (a1b2 − a2b1)

Cross-multiplication (y)

y = (c1a2 − c2a1) / (a1b2 − a2b1)

From chapter list

a1/a2, b1/b2, c1/c2 comparison

Standard form

ax² + bx + c = 0, a ≠ 0

Discriminant

D = b² − 4ac

Quadratic formula

x = [−b ± √(b² − 4ac)] / (2a)

Two distinct real roots

D > 0

Two equal real roots

D = 0, x = −b / (2a)

No real roots

D < 0

nth term

aₙ = a + (n − 1)d

Sum of n terms

Sₙ = n/2 [2a + (n − 1)d]

Sum using last term

Sₙ = n/2 (a + l)

Common difference

d = aₙ − aₙ₋₁

Number of terms

n = [(l − a) / d] + 1

From chapter list

an=a+(n−1)d; Sn=n/2[2a+(n−1)d]

Basic Proportionality (Thales)

If DE ∥ BC in ΔABC, then AD/DB = AE/EC

AA / SSS / SAS similarity

Corresponding angles equal ⇒ sides proportional

Areas of similar triangles

ar(Δ1) / ar(Δ2) = (side1 / side2)²

Pythagoras

In a right triangle, (hyp)² = (base)² + (perp)²

Converse of Pythagoras

If a² + b² = c², the angle opposite c is 90°

From chapter list

ΔABC ~ ΔDEF ⇒ ar(ABC)/ar(DEF)=(AB/DE)²

Distance

d = √[(x₂ − x₁)² + (y₂ − y₁)²]

Section formula (internal)

P = ( (m x₂ + n x₁)/(m+n) , (m y₂ + n y₁)/(m+n) )

Mid-point

M = ( (x₁+x₂)/2 , (y₁+y₂)/2 )

Area of triangle

Δ = (1/2) | x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂) |

Collinear points

Area of triangle = 0

From chapter list

d=√[(x2−x1)²+(y2−y1)²]

Sine

sin θ = opposite / hypotenuse

Cosine

cos θ = adjacent / hypotenuse

Tangent

tan θ = opposite / adjacent = sin θ / cos θ

Reciprocals

csc θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ

Identity 1

sin²θ + cos²θ = 1

Identity 2

1 + tan²θ = sec²θ

Identity 3

1 + cot²θ = csc²θ

Standard angles

sin 0°=0, 30°=1/2, 45°=1/√2, 60°=√3/2, 90°=1

cos standard

cos 0°=1, 30°=√3/2, 45°=1/√2, 60°=1/2, 90°=0

tan standard

tan 0°=0, 30°=1/√3, 45°=1, 60°=√3, 90° undefined

From chapter list

sin²A+cos²A=1

Height from elevation

height = base × tan θ

θ = angle of elevation

Distance from depression

Use tan of angle of depression in the right triangle

Two elevations from same point

Draw two right triangles; subtract or add heights

Tangent ⊥ radius

Radius to the point of contact is perpendicular to the tangent

Two tangents from an external point

They are equal in length

Number of tangents

From a point: 0 (inside), 1 (on circle), 2 (outside)

Circumference

C = 2πr = πd

Area of circle

A = πr²

Length of arc

l = (θ/360°) × 2πr

Area of sector

A_sector = (θ/360°) × πr²

Area of segment

A_segment = area of sector − area of triangle

From chapter list

A=πr²; arc=(θ/360)×2πr

Cuboid volume

V = l × b × h

Cube

TSA = 6a², V = a³

Cylinder CSA

CSA = 2πrh

Cylinder TSA

TSA = 2πr(h + r)

Cylinder volume

V = πr²h

Cone CSA

CSA = πrl, l = √(r² + h²)

Cone TSA

TSA = πr(l + r)

Cone volume

V = (1/3) πr²h

Sphere CSA / TSA

4πr²

Sphere volume

V = (4/3) πr³

Hemisphere CSA (curved)

2πr²

Hemisphere TSA

3πr²

Hemisphere volume

V = (2/3) πr³

Recast solids

Volume remains constant when a solid is melted and recast

From chapter list

V_sphere=4/3πr³; V_cone=⅓πr²h

Mean (direct)

x̄ = Σ fᵢ xᵢ / Σ fᵢ

Mean (assumed mean)

x̄ = a + (Σ fᵢ dᵢ / Σ fᵢ), dᵢ = xᵢ − a

Mean (step-deviation)

x̄ = a + h (Σ fᵢ uᵢ / Σ fᵢ), uᵢ = (xᵢ − a)/h

Median (grouped)

Median = l + [(n/2 − cf) / f] × h

Mode (grouped)

Mode = l + [(f₁ − f₀) / (2f₁ − f₀ − f₂)] × h

Empirical relation

3 Median = Mode + 2 Mean

Approximate, for moderately skewed data

Classical probability

P(E) = n(E) / n(S)

Equally likely outcomes

Range

0 ≤ P(E) ≤ 1

Complement

P(not E) = 1 − P(E)

Sure event

P(S) = 1

Impossible event

P(∅) = 0

Conservation of mass

A balanced equation has equal atoms of each element on both sides

Water formation

2H₂ + O₂ → 2H₂O

Combination

A + B → AB

Decomposition

AB → A + B

Displacement

A + BC → AC + B (A more reactive than B)

Double displacement

AB + CD → AD + CB

Neutralisation

Acid + Base → Salt + Water

Oxidation (oxygen)

Gain of oxygen or loss of hydrogen / electrons

From chapter list

e.g. 2H2 + O2 → 2H2O

pH

pH = −log₁₀ [H⁺] (idea as taught)

pH scale

pH < 7 acidic; pH = 7 neutral; pH > 7 basic

Water ion product (idea)

H₂O ⇌ H⁺ + OH⁻

Washing soda

Na₂CO₃ · 10H₂O

Baking soda

NaHCO₃

Bleaching powder

CaOCl₂

Plaster of Paris

CaSO₄ · ½H₂O

Gypsum

CaSO₄ · 2H₂O

From chapter list

pH = −log10[H+] (idea)

Ionic compound (idea)

Metal atom loses e⁻; non-metal gains e⁻

Rusting

4Fe + 3O₂ + x H₂O → 2Fe₂O₃ · x H₂O

Thermite (example)

Fe₂O₃ + 2Al → 2Fe + Al₂O₃ + heat

Aqua regia

conc. HCl + conc. HNO₃ (3 : 1 by volume)

Tetravalency

Carbon forms 4 covalent bonds

Homologous series

Successive members differ by −CH₂−

General alkane

CₙH₂ₙ₊₂

General alkene

CₙH₂ₙ

General alkyne

CₙH₂ₙ₋₂

Alcohol general

CₙH₂ₙ₊₁OH

Combustion (alkane idea)

CₓHᵧ + O₂ → CO₂ + H₂O + heat

Photosynthesis

6CO₂ + 6H₂O → C₆H₁₂O₆ + 6O₂ (light, chlorophyll)

Aerobic respiration

C₆H₁₂O₆ + 6O₂ → 6CO₂ + 6H₂O + energy

Anaerobic (muscle)

Glucose → lactic acid + energy

Anaerobic (yeast)

Glucose → ethanol + CO₂ + energy

Reflex path

Receptor → sensory neuron → spinal cord → motor neuron → effector

Sexual outline

Gametes fuse → zygote → embryo

Monohybrid F2

Phenotypic ratio 3 : 1

Dihybrid F2 (if asked)

Phenotypic ratio 9 : 3 : 3 : 1

Human sex

XX female, XY male

From chapter list

F2 monohybrid 3:1

Laws of reflection

∠i = ∠r; incident ray, reflected ray and normal are coplanar

Mirror formula

1/v + 1/u = 1/f

Magnification (mirror)

m = h′/h = −v/u

Focal length (spherical mirror)

f = R/2

Snell’s law

n = sin i / sin r (= n₂/n₁)

Absolute refractive index

n = c / v

Lens formula

1/v − 1/u = 1/f

Magnification (lens)

m = h′/h = v/u

Power of a lens

P = 1/f (f in metres, P in dioptre)

Combination of thin lenses

P = P₁ + P₂ + …

From chapter list

1/v + 1/u = 1/f (mirror)

Power of accommodation

Ability of the eye lens to change focal length

Myopia correction

Concave (diverging) lens

Hypermetropia correction

Convex (converging) lens

Prism deviation (idea)

White light splits into VIBGYOR

Current

I = Q / t

Ohm’s law

V = I R (temperature constant)

Resistance (material)

R = ρ l / A

Series

Rₛ = R₁ + R₂ + R₃ + …

Parallel (two)

1/Rₚ = 1/R₁ + 1/R₂

Parallel (two shortcut)

Rₚ = (R₁ R₂) / (R₁ + R₂)

Electric power

P = V I = I² R = V² / R

Electrical energy

E = P t = V I t

Heat (Joule)

H = I² R t

Commercial unit

1 kWh = 3.6 × 10⁶ J

From chapter list

V=IR; P=VI

Right-hand thumb rule

Thumb → current; curled fingers → field

Fleming’s left-hand rule

Force on a current in a magnetic field (motor)

Fleming’s right-hand rule

Induced current (generator)

Domestic fuse

Fuse is in the live wire, in series

10% law

About 10% of energy passes to the next trophic level

n(A ∪ B)

n(A) + n(B) − n(A ∩ B)

De Morgan

(A ∪ B)′ = A′ ∩ B′, (A ∩ B)′ = A′ ∪ B′

Radian

π rad = 180°

Identity

sin²x + cos²x = 1

sin(A±B)

sin A cos B ± cos A sin B

cos(A±B)

cos A cos B ∓ sin A sin B

Modulus

|z| = √(a² + b²) for z = a + ib

i²

i² = −1

Product of moduli

|z₁ z₂| = |z₁| |z₂|

From chapter list

|z1 z2|=|z1||z2|

nPr

ⁿPᵣ = n! / (n − r)!

nCr

ⁿCᵣ = n! / [r! (n − r)!]

Symmetry

ⁿCᵣ = ⁿCₙ₋ᵣ

From chapter list

nCr = n! / r!(n−r)!

General term

Tᵣ₊₁ = ⁿCᵣ aⁿ⁻ʳ bʳ

AP nth / sum

aₙ = a+(n−1)d, Sₙ = n/2 [2a+(n−1)d]

GP nth

aₙ = a rⁿ⁻¹

GP sum

Sₙ = a(rⁿ − 1)/(r − 1), r ≠ 1

Infinite GP

S∞ = a / (1 − r), |r| < 1

Sum of first n naturals

Σn = n(n+1)/2

Sum of squares

Σn² = n(n+1)(2n+1)/6

Sum of cubes

Σn³ = [n(n+1)/2]²

From chapter list

Sn=a(r^n−1)/(r−1)

Slope

m = tan θ = (y₂−y₁)/(x₂−x₁)

Point–slope

y − y₁ = m(x − x₁)

Distance point to line

d = |ax₁ + by₁ + c| / √(a² + b²)

From chapter list

d=|ax1+by1+c|/√(a²+b²)

Circle

(x − h)² + (y − k)² = r²

Parabola

y² = 4ax (standard)

Ellipse

x²/a² + y²/b² = 1

Hyperbola

x²/a² − y²/b² = 1

Distance 3D

d = √[(x₂−x₁)²+(y₂−y₁)²+(z₂−z₁)²]

d/dx (xⁿ)

n xⁿ⁻¹

d/dx (sin x)

cos x

d/dx (cos x)

−sin x

From chapter list

d/dx (sin x)=cos x

Variance

σ² = Σ(x − x̄)² / n

Standard deviation

σ = √variance

From chapter list

σ=√(Σ(x−x̄)²/n)

Addition

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

Conditional

P(A|B) = P(A ∩ B) / P(B)

Force dimension

[F] = M L T⁻²

Relative error

Δa / a

Constant a

v = u + at, s = ut + ½at², v² = u² + 2as

From chapter list

v=u+at

Projectile time of flight

T = 2 u sinθ / g

Maximum height

H = u² sin²θ / (2g)

Range

R = u² sin 2θ / g

Centripetal acceleration

a = v² / r = ω² r

Newton II

F = dp/dt = m a

Friction

f ≤ μ N, f_k = μ_k N

From chapter list

f ≤ μN

Work

W = F · s

KE

½ m v²

Power

P = F · v

From chapter list

P=F v; KE=½mv²

Torque

τ = r × F = I α

Angular momentum

L = I ω

From chapter list

τ=Iα

Escape speed

v_esc = √(2GM / R)

Orbital speed

v_o = √(GM / r)

g at height

g_h = g (R / (R+h))²

Young’s modulus

Y = (F/A) / (ΔL/L)

Pressure

P = h ρ g

Bernoulli

P + ρ g h + ½ ρ v² = constant

Surface tension

T = F / l

Linear expansion

ΔL = α L ΔT

Heat

Q = m c ΔT, Q = m L

First law (NCERT)

ΔU = Q − W

Ideal gas

Cₚ − Cᵥ = R

rms speed

v_rms = √(3RT / M)

SHM period (spring)

T = 2π √(m / k)

Simple pendulum

T = 2π √(l / g)

Wave speed

v = f λ

Molarity

M = n / V_L

Mole

n = given mass / molar mass

Photon

E = h ν = hc / λ

From chapter list

E=hν

First law (chemistry)

ΔU = q + w, w = −PΔV

Gibbs

ΔG = ΔH − TΔS

Kc

K_c = [C]^c [D]^d / [A]^a [B]^b

Water

K_w = 10⁻¹⁴ at 25°C

pH

pH = −log₁₀ [H⁺]

From chapter list

Kc = [C]^c[D]^d/[A]^a[B]^b

Overall

6CO₂ + 12H₂O → C₆H₁₂O₆ + 6O₂ + 6H₂O (as in NCERT)

RQ

RQ = CO₂ evolved / O₂ consumed

Vital capacity

VC = TV + IRV + ERV

Enzyme (idea)

V depends on [S]; K_m is [S] at ½ V_max

Composite

(f ∘ g)(x) = f(g(x))

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

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)

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

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

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

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

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

Area under y = f(x)

A = ∫ₐᵇ y dx

Area between curves

A = ∫ₐᵇ |f(x) − g(x)| dx

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}

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

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)

Corner-point method

Optimum of a linear objective on a closed polygon is at a vertex

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)

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)

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

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)

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)

Bar magnet dipole

B_axial = (μ₀ / 4π) (2M / r³)

Torque

τ = m B sin θ

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

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

Speed

c = 1 / √(μ₀ ε₀) = 3 × 10⁸ m/s

c and fields

c = E₀ / B₀

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

YDSE fringe

β = λ D / d

Path difference (bright)

d sinθ = n λ

Brewster

μ = tan i_p

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

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

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}

NAND / NOR

Universal gates

AND, OR, NOT

Basic Boolean operations as in NCERT truth tables

From chapter list

NAND is universal

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

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

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)

Lanthanoid contraction

Steady decrease in size from La to Lu

Coordination number

Number of ligand donor atoms bound to the metal

Werner primary / secondary

Oxidation number vs coordination number

SN2

One-step, inversion, preferred by 1°

SN1

Two-step via carbocation, racemisation, preferred by 3°

Acidity order (idea)

phenol > water > 1° alcohol (as taught)

Williamson

R−ONa + R′−X → R−O−R′

Nucleophilic addition

C=O is the site; HCN, Grignard, etc.

Tollens / Fehling

Aldehydes (not ketones, generally) reduce these reagents

Basicity (aqueous idea)

Compare 2°, 1°, 3° as in NCERT discussion

Peptide bond

−CO−NH− between amino acids

Double fertilisation

Syngamy + triple fusion → zygote + endosperm (3n)

Menstrual cycle

≈ 28 days; ovulation around day 14

Monohybrid

F2 3 : 1 phenotype

Dihybrid

F2 9 : 3 : 3 : 1

Hardy–Weinberg (also Evolution)

p² + 2pq + q² = 1, p + q = 1

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

Hardy–Weinberg

p² + 2pq + q² = 1

Exponential growth

dN/dt = r N

Logistic growth

dN/dt = r N (K − N)/K

NPP

NPP = GPP − R

10% law

≈ 10% energy to next trophic level

PCR idea

Denaturation → annealing → extension (repeat)