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ISC · Class 12

ISC Class 12 Formula Sheets

Chapter-wise key formulas for ISC Class 12 Physics, Chemistry, Maths, and Biology. Compact reference for revision — organised by chapter with variable notation.

Physics

Electrostatics

  • F = kq₁q₂/r² = q₁q₂/4πε₀r²
  • E = F/q = kq/r²
  • V = W/q = kq/r
  • C = Q/V ; C_parallel = ε₀A/d
  • C_series: 1/C = 1/C₁+1/C₂ ; C_parallel: C = C₁+C₂
  • U = ½CV² = Q²/2C = QV/2
  • With dielectric: C = Kε₀A/d

Current Electricity

  • V = IR ; R = ρl/A
  • P = VI = I²R = V²/R
  • R_series = R₁+R₂ ; 1/R_parallel = 1/R₁+1/R₂
  • EMF: ε = V + Ir = I(R+r)
  • Kirchhoff: ΣI = 0 (junction) ; ΣV = 0 (loop)
  • Wheatstone bridge: P/Q = R/S (balanced)
  • Meter bridge: R/S = l/(100−l)
  • Drift velocity: I = nAevd

Magnetic Effects of Current and Magnetism

  • Biot-Savart: dB = μ₀Idl sinθ / 4πr²
  • Straight wire: B = μ₀I/2πr
  • Centre of circular loop: B = μ₀I/2R
  • Solenoid: B = μ₀nI
  • Force on conductor: F = BIl sinθ
  • Force on charge: F = qvB sinθ
  • Torque on coil: τ = nBIA sinθ
  • Galvanometer → Ammeter: S = Ig×G/(I−Ig)
  • Galvanometer → Voltmeter: R = V/Ig − G

Electromagnetic Induction and Alternating Currents

  • Flux: Φ = BAcosθ
  • Faraday: e = −dΦ/dt = −N dΦ/dt
  • Motional EMF: e = Blv
  • Self-inductance: e = −L dI/dt
  • Solenoid: L = μ₀n²Al
  • Mutual inductance: e₂ = −M dI₁/dt
  • Energy in inductor: U = ½LI²
  • Transformer: Vs/Vp = Ns/Np = Ip/Is

Alternating Currents

  • Vrms = V₀/√2 ; Irms = I₀/√2
  • XL = ωL = 2πfL
  • XC = 1/ωC = 1/2πfC
  • Impedance: Z = √[R² + (XL−XC)²]
  • Phase angle: tanφ = (XL−XC)/R
  • Resonance: f₀ = 1/2π√(LC) ; XL = XC
  • Power: P = VrmsIrmscosφ = I²rmsR
  • Power factor: cosφ = R/Z

Optics — Ray Optics

  • Mirror: 1/f = 1/v + 1/u ; f = R/2
  • Mirror magnification: m = −v/u = −f/(f−u)
  • Snell's law: n₁sinθ₁ = n₂sinθ₂
  • Critical angle: sinC = n₂/n₁ = 1/n (n₂=1)
  • Lens: 1/f = 1/v − 1/u
  • Lens maker's: 1/f = (n−1)[1/R₁ − 1/R₂]
  • Lens magnification: m = v/u
  • Power: P = 1/f(m) ; P_combined = P₁+P₂
  • Microscope: m = L/f₀ × D/fe
  • Telescope: m = f₀/fe

Optics — Wave Optics

  • Fringe width: β = λD/d
  • Bright fringe: d sinθ = nλ
  • Dark fringe: d sinθ = (2n−1)λ/2
  • Single slit dark: a sinθ = nλ
  • Refractive index: n = c/v = λ_air/λ_medium
  • Brewster's law: tanθp = n₂/n₁

Dual Nature of Radiation and Matter

  • Photon energy: E = hν = hc/λ
  • Photoelectric: KEmax = hν − φ = h(ν−ν₀)
  • Stopping potential: eV₀ = hν − φ
  • de Broglie: λ = h/p = h/mv
  • λ for electron: λ = h/√(2meV)
  • Einstein's mass-energy: E = mc²

Atoms & Nuclei

  • Bohr radius: rn = n²a₀ ; a₀ = 0.529 Å
  • Energy levels: En = −13.6/n² eV
  • Rydberg: 1/λ = R[1/n₁² − 1/n₂²]
  • Radioactive decay: N = N₀e^(−λt)
  • Half-life: T½ = 0.693/λ ; N = N₀(½)^(t/T½)
  • Activity: A = λN = A₀e^(−λt)
  • Mass defect: Δm = [Zmp + (A−Z)mn − M]
  • Binding energy: BE = Δm × 931.5 MeV/u

Electronic Devices

  • Transistor: IC = β×IB ; IE = IC + IB
  • Voltage gain: Av = −β×RC/RB
  • Zener regulation: VZ = constant (in breakdown)
  • Logic: AND (Y=A·B) ; OR (Y=A+B) ; NOT (Y=Ā)
  • NAND: Y = A̅·̅B̅ ; NOR: Y = A̅+̅B̅

Chemistry

Solutions

  • Raoult's Law: p = x₂P₁° (dilute) ; p₁ = x₁P₁°
  • Relative lowering VP: (P°−p)/P° = x_solute
  • Boiling point elevation: ΔTb = Kb × m
  • Freezing point depression: ΔTf = Kf × m
  • Osmotic pressure: π = iMRT
  • Molality: m = (w₂×1000) / (M₂×w₁)
  • van't Hoff factor: i = observed/expected colligative property
  • Degree of dissociation: α = (i−1)/(n−1)

Electrochemistry

  • E°cell = E°cathode − E°anode
  • Nernst (25°C): E = E° − (0.0592/n)log Q
  • ΔG° = −nFE° = −RT ln K
  • Faraday's 1st law: w = ZIt = (M/nF)×Q
  • Specific conductance: κ = 1/ρ
  • Molar conductance: Λm = κ×1000/M
  • Kohlrausch: Λ°m = Σλ°_ions
  • Kohlrausch law: Λm = Λ°m − K√C

Chemical Kinetics

  • Rate: r = −d[A]/dt = k[A]ᵐ[B]ⁿ
  • Zero order: [A]t = [A]₀ − kt ; t½ = [A]₀/2k
  • First order: k = (2.303/t)log([A]₀/[A]t)
  • First order t½ = 0.693/k
  • Arrhenius: k = Ae^(−Ea/RT)
  • log(k₂/k₁) = Ea/2.303R × (T₂−T₁)/T₁T₂
  • Activation energy from graph: slope = −Ea/2.303R

d- and f-Block Elements

  • d-block: 3d, 4d, 5d series ; f-block: 4f (lanthanides), 5f (actinides)
  • Magnetic moment: μ = √[n(n+2)] BM (n = unpaired electrons)
  • KMnO₄: purple; K₂Cr₂O₇: orange; colour due to d-d transitions
  • E°(Mn³⁺/Mn²⁺) = +1.51V (high, Mn²⁺ very stable)
  • Cr₂O₇²⁻ + 14H⁺ + 6e⁻ → 2Cr³⁺ + 7H₂O (E° = +1.33V)
  • MnO₄⁻ + 8H⁺ + 5e⁻ → Mn²⁺ + 4H₂O (acidic; E° = +1.51V)

Coordination Chemistry

  • EAN = (Z − electrons donated by metal) + 2×coordination number
  • Oxidation state: charge = sum of ligand charges + metal OS
  • d-orbital splitting: Δ₀ for octahedral ; Δt for tetrahedral
  • CFSE for strong field: pairing energy vs Δ
  • Effective atomic number rule for stability

Mathematics

Matrices & Determinants

  • det(AB) = det A × det B
  • det(Aᵀ) = det A ; det(kA) = kⁿ det A
  • A·(adj A) = det(A)·I
  • A⁻¹ = (adj A)/det A ; det A ≠ 0
  • Cramer's rule: x = D₁/D, y = D₂/D, z = D₃/D
  • 3×3 det expansion along any row/column

Differentiation

  • d/dx(xⁿ) = nxⁿ⁻¹ ; d/dx(eˣ) = eˣ ; d/dx(aˣ) = aˣ ln a
  • d/dx(ln x) = 1/x ; d/dx(log_a x) = 1/(x ln a)
  • d/dx(sin x) = cos x ; d/dx(cos x) = −sin x
  • d/dx(sin⁻¹x) = 1/√(1−x²) ; d/dx(tan⁻¹x) = 1/(1+x²)
  • Product: (uv)' = u'v + uv'
  • Quotient: (u/v)' = (u'v − uv')/v²
  • Chain rule: dy/dx = (dy/du)·(du/dx)
  • Parametric: dy/dx = (dy/dt)/(dx/dt)

Applications of Derivatives

  • Increasing: f'(x) > 0 ; Decreasing: f'(x) < 0
  • Local max/min: f'(x) = 0 (critical point)
  • 2nd derivative test: f''(c)<0 → max ; f''(c)>0 → min
  • Rate of change: dy/dt = (dy/dx)·(dx/dt)
  • Approximation: Δy ≈ f'(x)·Δx = dy
  • Rolle's theorem: f(a)=f(b) → ∃c: f'(c)=0
  • LMVT: f'(c) = [f(b)−f(a)]/(b−a)
  • Tangent slope = f'(x₁) ; Normal slope = −1/f'(x₁)

Integration

  • ∫xⁿdx = xⁿ⁺¹/(n+1)+C (n≠−1) ; ∫(1/x)dx = ln|x|+C
  • ∫eˣdx = eˣ+C ; ∫aˣdx = aˣ/ln a + C
  • ∫sin x dx = −cos x+C ; ∫cos x dx = sin x+C
  • ∫sec²x dx = tan x+C ; ∫cosec²x dx = −cot x+C
  • ∫1/√(a²−x²)dx = sin⁻¹(x/a)+C
  • ∫1/(a²+x²)dx = (1/a)tan⁻¹(x/a)+C
  • By parts: ∫uv dx = u∫v dx − ∫[u'∫v dx]dx (ILATE)
  • ∫f(ax+b)dx = F(ax+b)/a + C

Definite Integrals & Area

  • ∫[a→b] f(x)dx = F(b) − F(a) (Newton-Leibniz)
  • ∫[a→b] f(x)dx = −∫[b→a] f(x)dx
  • ∫[0→a] f(x)dx = ∫[0→a] f(a−x)dx
  • ∫[0→2a] f(x)dx = 2∫[0→a] f(x)dx if f(2a−x)=f(x)
  • Area = ∫[a→b] |f(x)| dx
  • Area between curves = ∫[a→b] [f(x)−g(x)]dx
  • Area using y-axis = ∫[c→d] x dy = ∫[c→d] g(y) dy

Differential Equations

  • Order: highest derivative ; Degree: power of highest derivative
  • Variable separable: f(y)dy = g(x)dx → integrate both sides
  • Homogeneous: put y = vx → dv separable
  • Linear 1st order: dy/dx + P(x)y = Q(x)
  • Integrating factor: IF = e^∫P dx
  • Solution: y·IF = ∫Q·IF dx + C
  • Particular solution: use initial conditions to find C

Vectors

  • |a| = √(a₁²+a₂²+a₃²) ; â = a/|a|
  • a·b = |a||b|cosθ = a₁b₁+a₂b₂+a₃b₃
  • |a×b| = |a||b|sinθ ; a×b = |i j k / a₁ a₂ a₃ / b₁ b₂ b₃|
  • Scalar triple product: [a b c] = a·(b×c)
  • Projection of a on b: a·b̂ = a·b/|b|
  • Section formula: r = (mb+na)/(m+n)
  • a and b parallel: a×b = 0 ; perpendicular: a·b = 0

3D Geometry

  • Distance: √[(x₂−x₁)²+(y₂−y₁)²+(z₂−z₁)²]
  • DCs: l²+m²+n²=1 ; l=cosα, m=cosβ, n=cosγ
  • DRs (a,b,c): l=a/√(a²+b²+c²) etc.
  • Line through (x₁,y₁,z₁) with DRs a,b,c: (x−x₁)/a=(y−y₁)/b=(z−z₁)/c
  • Angle between lines: cosθ = |l₁l₂+m₁m₂+n₁n₂|
  • Plane: ax+by+cz+d=0 ; normal = (a,b,c)
  • Distance from point to plane: |ax₁+by₁+cz₁+d|/√(a²+b²+c²)
  • Angle between line and plane: sinθ = |al+bm+cn|/√(a²+b²+c²)

Probability

  • P(A∪B) = P(A)+P(B)−P(A∩B)
  • P(A|B) = P(A∩B)/P(B)
  • Multiplication: P(A∩B) = P(A)·P(B|A)
  • Independence: P(A∩B) = P(A)·P(B)
  • Bayes: P(Aᵢ|B) = P(B|Aᵢ)P(Aᵢ) / ΣP(B|Aⱼ)P(Aⱼ)
  • Binomial: P(X=r) = ⁿCr·pʳ·qⁿ⁻ʳ
  • Mean of B(n,p) = np ; Variance = npq

Linear Regression (Section C)

  • Regression line of y on x: y − ȳ = byx(x − x̄)
  • byx = r × (σy/σx) = Σ(xi−x̄)(yi−ȳ) / Σ(xi−x̄)²
  • Regression line of x on y: x − x̄ = bxy(y − ȳ)
  • bxy = r × (σx/σy)
  • r² = byx × bxy (r = correlation coefficient)
  • If r = 0: no linear correlation ; r = ±1: perfect linear
  • Lines of regression intersect at (x̄, ȳ)

Linear Programming (Section C)

  • Objective function: Z = ax + by (maximise or minimise)
  • Constraints: linear inequalities in x and y
  • Feasible region: common region satisfying all constraints
  • Optimal solution occurs at a corner (vertex) of feasible region
  • Corner point method: evaluate Z at each vertex
  • No feasible region → no solution ; Unbounded → check if optimal exists

Biology

Reproduction in Plants

  • Double fertilization: 1 sperm + egg → zygote (2n) ; 1 sperm + 2 polar nuclei → endosperm (3n)
  • Embryo sac = megagametophyte (7 cells, 8 nuclei)
  • Microsporogenesis: 2 meiotic divisions → 4 microspores
  • Megasporogenesis: 1 functional megaspore (chalazal end)
  • Germination: water absorption (imbibition) → enzyme activation

Reproduction in Animals

  • Spermatogenesis: 1 primary spermatocyte → 4 spermatids → 4 sperm
  • Oogenesis: 1 primary oocyte → 1 ovum + 3 polar bodies
  • Menstrual cycle: 28 days ; ovulation ≈ day 14
  • Gestation period (human): ~280 days / 40 weeks
  • Cleavage: rapid mitosis without growth; cell volume constant

Genetics — Mendelian

  • Monohybrid ratio: F₂ = 3:1 (phenotype) ; 1:2:1 (genotype)
  • Dihybrid ratio: F₂ = 9:3:3:1
  • Test cross: Aa × aa → 1:1 ; AaBb × aabb → 1:1:1:1
  • χ² = Σ[(O−E)²/E] ; p<0.05 → significant deviation
  • Incomplete dominance F₂: 1:2:1 phenotype
  • Codominance: both alleles expressed (e.g., ABO blood groups)

Molecular Basis of Inheritance

  • Chargaff's rules: A=T ; G=C ; (A+G)=(T+C) i.e., purines=pyrimidines
  • Nucleotide = phosphate + pentose sugar + nitrogenous base
  • DNA replication: semi-conservative (Meselson-Stahl experiment)
  • Translation: 3 bases (codon) → 1 amino acid
  • 64 codons: 61 sense + 3 stop (UAA, UAG, UGA)
  • Genetic code: universal, non-overlapping, degenerate

Evolution

  • Hardy-Weinberg equilibrium: p² + 2pq + q² = 1
  • p + q = 1 (p = dominant allele freq, q = recessive allele freq)
  • Conditions: large population, random mating, no selection/mutation/migration
  • Geological time: Earth ≈ 4.5 Ga ; Life ≈ 3.8 Ga ; Homo sapiens ≈ 200,000 ya
  • Fitness: relative reproductive success of a genotype

Human Health & Disease

  • Immune response: primary (slower) → memory cells ; secondary (faster, stronger)
  • Antibody structure: 2 heavy + 2 light chains; Y-shaped
  • Vaccine: attenuated/killed pathogen or toxoid → active immunity
  • BMI = weight(kg) / height²(m) ; Normal: 18.5–24.9
  • Widal test: for typhoid ; ELISA: for HIV, hepatitis

Biotechnology

  • PCR: denaturation (94°C) → annealing (55–65°C) → extension (72°C)
  • n cycles of PCR → 2ⁿ copies of target DNA
  • Gel electrophoresis: smaller fragments migrate farther in agarose gel
  • Restriction enzymes cut at palindromic sequences; produce sticky/blunt ends
  • Ti plasmid (Agrobacterium) used for plant transformation
  • SCID gene therapy: ADA gene delivered via retroviral vector

Ecology

  • GPP − R = NPP (net primary productivity)
  • Ecological efficiency ≈ 10% (Lindemann's law; energy transfer between trophic levels)
  • Population growth (exponential): dN/dt = rN ; N(t) = N₀eʳᵗ
  • Population growth (logistic): dN/dt = rN[(K−N)/K]
  • r = birth rate − death rate ; K = carrying capacity
  • Simpson's diversity index: D = 1 − Σ(nᵢ/N)²
  • Species-area relationship: log S = log C + z·log A

More for ISC Class 12

Important Derivations →Named Reactions →Last Minute Revision →Revision Checklist →