ISC Class 12 Important Derivations
All must-know derivations for ISC Class 12 Physics and Maths — with chapter, marks, and how often they appear in the board exam.
Physics
Electric field due to a uniformly charged infinite plane sheet using Gauss's Law
Electrostatics
Electric field due to a uniformly charged spherical shell using Gauss's Law (inside and outside)
Electrostatics
Electric potential at a point due to a point charge
Electrostatics
Capacitance of a parallel plate capacitor (with and without dielectric)
Electrostatics
Energy stored in a charged capacitor (U = ½CV²)
Electrostatics
Expression for drift velocity of electrons and hence Ohm's Law
Current Electricity
Principle of Wheatstone bridge using Kirchhoff's laws
Current Electricity
Force per unit length between two parallel current-carrying conductors; definition of ampere
Magnetic Effects of Current and Magnetism
Expression for torque on a rectangular current loop placed in a uniform magnetic field
Magnetic Effects of Current and Magnetism
Working and theory of a moving coil galvanometer
Magnetic Effects of Current and Magnetism
Conversion of galvanometer into ammeter and voltmeter
Magnetic Effects of Current and Magnetism
Expression for EMF induced in a coil rotating uniformly in a magnetic field
Electromagnetic Induction and Alternating Currents
Expression for energy stored in an inductor (U = ½LI²)
Electromagnetic Induction and Alternating Currents
Expression for mutual inductance of two long coaxial solenoids
Electromagnetic Induction and Alternating Currents
Expression for impedance of a series LCR circuit; resonance condition
Alternating Currents
Power dissipated in a series LCR AC circuit; power factor
Alternating Currents
Mirror formula: 1/v + 1/u = 1/f using geometry of reflection
Optics — Ray Optics
Refraction at a single spherical surface: n₂/v − n₁/u = (n₂−n₁)/R
Optics — Ray Optics
Lens formula (1/v − 1/u = 1/f) from two refractions at spherical surfaces
Optics — Ray Optics
Lens maker's formula: 1/f = (n−1)(1/R₁ − 1/R₂)
Optics — Ray Optics
Magnifying power of a compound microscope
Optics — Ray Optics
Magnifying power of an astronomical telescope (normal adjustment)
Optics — Ray Optics
Expression for fringe width in Young's double slit experiment
Optics — Wave Optics
Conditions for constructive and destructive interference of light
Optics — Wave Optics
de Broglie wavelength of an electron accelerated through potential difference V: λ = h/√(2meV)
Dual Nature of Radiation and Matter
Einstein's photoelectric equation: KE_max = hν − φ₀
Dual Nature of Radiation and Matter
Bohr's postulates → expression for radius of nth orbit: rn = n²a₀
Atoms and Nuclei
Expression for energy of electron in nth Bohr orbit: En = −13.6/n² eV
Atoms and Nuclei
Radioactive decay law: N = N₀e^(−λt); expression for half-life
Atoms and Nuclei
Mathematics
Proof: A·(adj A) = (adj A)·A = |A|·I
Matrices & Determinants
Proof of property: det(AB) = det A × det B
Matrices & Determinants
Derivation of A⁻¹ = (adj A)/|A| and conditions for existence
Matrices & Determinants
Derivative from first principles: d/dx(xⁿ) = nxⁿ⁻¹
Differentiation
Derivative from first principles: d/dx(sin x) = cos x
Differentiation
Proof of product rule: d/dx(uv) = u(dv/dx) + v(du/dx)
Differentiation
Rolle's Theorem statement and geometric interpretation
Applications of Derivatives
Lagrange's Mean Value Theorem (LMVT) statement and proof sketch
Applications of Derivatives
Integration by parts: ∫uv dx = u∫v dx − ∫[du/dx·∫v dx]dx (ILATE rule)
Integration
∫dx/(a²+x²) = (1/a)tan⁻¹(x/a)+C and ∫dx/√(a²−x²) = sin⁻¹(x/a)+C
Integration
Proof: ∫[a→b]f(x)dx = ∫[a→b]f(a+b−x)dx
Definite Integrals
Area between two curves: A = ∫[a→b][f(x)−g(x)]dx
Applications of Integrals
Solution of linear DE: dy/dx + Py = Q using integrating factor e^∫P dx
Differential Equations
Reduction of homogeneous DE to separable form using y = vx
Differential Equations
Section formula in 3D; position vector of point dividing PQ in ratio m:n
Vectors
Proof: |a×b|² = |a|²|b|² − (a·b)²
Vectors
Angle between two lines in 3D: cosθ = |l₁l₂+m₁m₂+n₁n₂|
3D Geometry
Equation of plane in normal form and distance from origin
3D Geometry
Perpendicular distance from a point (x₁,y₁,z₁) to plane ax+by+cz+d=0
3D Geometry
Multiplication theorem: P(A∩B) = P(A)·P(B|A); extension to independent events
Probability
Bayes' theorem derivation using conditional probability
Probability
Mean (np) and variance (npq) of a Binomial distribution B(n,p)
Probability