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CBSE Class 12 PCM Core Revision

Essential formulas, reactions, theorems, and definitions across Physics, Chemistry, and Mathematics for CBSE Class 12 Board Exams.

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Coulomb's Law and Dielectric Medium Effect
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1
Coulomb's Law and Dielectric Medium Effect
F = (1 / 4piepsilon_0) (q1q2 / r^2); In a medium of dielectric constant K, the force is reduced to F' = F / K.
2
Gauss's Law in Electrostatics
Total electric flux through any closed Gaussian surface: Phi = oint E . dA = q_enclosed / epsilon_0.
3
Capacitance of Parallel Plate Capacitor and Energy Stored
Capacitance C = (K epsilon_0 A) / d; Electrostatic energy stored U = 1/2 C V^2 = Q^2 / (2C) = 1/2 Q V.
4
Drift Velocity and Current Density Relation
Drift velocity v_d = (e E tau) / m; Current I = n e A v_d; Current density J = I / A = n e v_d = sigma E (microscopic Ohm's law).
5
Biot-Savart Law for Magnetic Field
dB = (mu_0 / 4pi) (I dl x r_hat) / r^2; Center of circular loop of radius R: B = mu_0 I / (2R).
6
Ampere's Circuital Law and Solenoid Field
oint B . dl = mu_0 I_enclosed; Inside an ideal long solenoid of n turns per unit length: B = mu_0 n * I.
7
Faraday's Law of Electromagnetic Induction and Lenz's Law
Induced EMF: epsilon = - dPhi_B / dt; Lenz's law states the polarity of induced EMF opposes the change in magnetic flux producing it (energy conservation).
8
Self-Inductance of Solenoid and Energy Stored in Inductor
L = (mu_0 N^2 A) / l; Magnetic energy stored in an inductor carrying current I: U_B = 1/2 L I^2.
9
Series LCR Resonant Frequency and Quality Factor
Resonant angular frequency omega_0 = 1 / sqrt(L C); Quality factor Q = (omega_0 L) / R = 1/R * sqrt(L / C).
10
Lens Maker's Formula and Thin Lens Formula
Lens maker's: 1/f = (mu - 1) * (1/R1 - 1/R2); Thin lens: 1/f = 1/v - 1/u; Power P = 1 / f (in metres).
11
Einstein's Photoelectric Equation and de Broglie Wavelength
Photoelectric: hnu = Phi_0 + K_max = hnu_0 + eV_0; de Broglie wavelength: lambda = h / p = h / sqrt(2mE) = h / sqrt(2meV).
12
Colligative Properties Summary (Solutions)
Relative lowering of VP: Delta p / p^0 = x_solute; Boiling elevation: Delta T_b = i K_b m; Freezing depression: Delta T_f = i K_f m; Osmotic pressure: pi = i C R * T.
13
Nernst Equation for Electrochemical Cell Potential
E_cell = E^0_cell - (2.303RT / nF) log Q = E^0_cell - (0.0591 / n) log Q at 298 K; Delta G^0 = -n F * E^0_cell.
14
Integrated Rate Equation and Half-Life for First-Order Reactions
k = (2.303 / t) * log([A]_0 / [A]); Half-life: t_1/2 = 0.693 / k (independent of initial concentration).
15
Arrhenius Equation and Temperature Coefficient
k = A exp(-E_a / (RT)); Log form: log(k2 / k1) = (E_a / 2.303R) [(T2 - T1) / (T1 * T2)].
16
Crystal Field Splitting in Octahedral vs Tetrahedral Complexes
Octahedral: d-orbitals split into t_2g (lower, triply degenerate) and e_g (higher, doubly degenerate); Delta_t = 4/9 * Delta_o.
17
Reimer-Tiemann vs Kolbe's Reaction of Phenol
Reimer-Tiemann: Phenol + CHCl3 + NaOH -> Salicylaldehyde (2-hydroxybenzaldehyde); Kolbe: Phenol + CO2 + NaOH -> Salicylic acid (2-hydroxybenzoic acid).
18
Aldol Condensation vs Cannizzaro Reaction
Aldol: Requires alpha-hydrogens, catalyzed by dilute alkali to give beta-hydroxy carbonyls; Cannizzaro: Aldehydes lacking alpha-hydrogens disproportionate in conc. alkali into alcohol + carboxylate.
19
Hoffmann Bromamide Degradation and Carbylamine Test
Hoffmann: RCONH2 + Br2 + 4NaOH -> RNH2 (primary amine with 1 less carbon) + Na2CO3 + 2NaBr + 2H2O; Carbylamine: RNH2 + CHCl3 + 3KOH -> RNC (foul isocyanide).
20
Matrix Invertibility and Adjoint Properties
A adj(A) = adj(A) A = |A| I; Inverse A^-1 = (1 / |A|) adj(A); |adj(A)| = |A|^(n - 1) for n x n matrix.
21
Integration by Parts Formula
integral u v dx = u integral v dx - integral [ (du/dx) * (integral v dx) ] dx; Choice of u follows ILATE order.
22
Properties of Definite Integrals (King's Property and Parity)
King's: integral_0^a f(x) dx = integral_0^a f(a - x) dx; Parity: integral_-a^a f(x) dx = 2 * integral_0^a f(x) dx (even) or 0 (odd).
23
First-Order Linear Differential Equation and Integrating Factor
dy/dx + P(x)y = Q(x); Integrating factor I.F. = exp(integral P dx); Solution: y (I.F.) = integral [ Q * (I.F.) ] dx + C.
24
Shortest Distance Between Two Skew Lines
d = | (a2 - a1) . (b1 x b2) | / | b1 x b2 |, where lines are r = a1 + lambdab1 and r = a2 + mub2; If lines intersect, d = 0.
25
Bayes' Theorem for Conditional Probability
P(E_i | A) = [ P(E_i) P(A | E_i) ] / sum_j [ P(E_j) P(A | E_j) ], where E_1, ..., E_n partition the sample space.

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