JEE ADVANCED PRACTICE

Physics 1: Kinematics

Basics. Displacement is a vector (change in position); distance is a scalar (total path). Average velocity = Δx/Δt; instantaneous v = dx/dt; a = dv/dt = v dv/dx.

Constant acceleration.

• v = u + at
• s = ut + ½at²
• v² = u² + 2as
• s_n (distance in nth second) = u + a(2n − 1)/2
• s = (u + v)t/2

Graphs. Slope of x-t = velocity; slope of v-t = acceleration; area under v-t = displacement; area under a-t = change in velocity. For variable a, always integrate: v = ∫a dt, x = ∫v dt.

Free fall. a = g = 9.8 m/s² downward. Time of flight (up and back) = 2u/g; max height = u²/2g. Body dropped from h: t = √(2h/g), v = √(2gh).

Relative motion. v_AB = v_A − v_B. River-boat: to cross in minimum time, head perpendicular to flow, t = d/v_b, drift = v_rt. To cross by shortest path (v_b > v_r), sinθ = v_r/v_b upstream of the perpendicular, t = d/√(v_b² − v_r²). Rain-man: v_rm = v_r − v_m.

Projectile motion. Horizontal: a = 0, x = u cosθ t. Vertical: y = u sinθ t − ½gt².

• T = 2u sinθ/g
• H = u²sin²θ/2g
• R = u²sin2θ/g (max at 45°; same range for θ and 90°−θ)
• Trajectory: y = x tanθ − gx²/(2u²cos²θ)
• Projectile from a height / on an incline: resolve g along and perpendicular to the plane.

Common JEE traps. Speed vs velocity in averages; sign conventions in free fall; relative acceleration when both bodies accelerate; at highest point velocity is not zero for oblique projectile (only vertical component is).

Physics 2: Laws of Motion and Friction

Newton's laws. (1) A body stays at rest or in uniform motion unless acted on by a net force. (2) F = dp/dt = ma for constant mass. (3) Action and reaction are equal, opposite and act on different bodies.

Impulse. J = ∫F dt = Δp. Area under F-t graph gives impulse.

Constraint relations. For an inextensible string, the length constraint gives relations between accelerations. Pulley: if one block moves down by x, the other moves up by x. Movable pulley: the string velocity is twice the pulley velocity. Always differentiate the string length equation.

Atwood machine. a = (m₁ − m₂)g/(m₁ + m₂); T = 2m₁m₂g/(m₁ + m₂).

Friction. Static friction adjusts up to f_s,max = μ_sN. Kinetic f_k = μ_kN. On an incline, block just slides when tanθ = μ_s; angle of repose = tan^-1μ. Acceleration down incline with friction: a = g(sinθ − μ cosθ).

Pulling at an angle. For a block pulled at angle θ above horizontal, N = mg − F sinθ. Minimum force to move: F = μmg/√(1 + μ²) at tanθ = μ.

Pseudo force. In a frame with acceleration a₀, add −ma₀ to each body. Lift: apparent weight = m(g + a) going up with acceleration a, m(g − a) going down.

Circular motion. a_c = v²/r = ω²r. Banked road (no friction): tanθ = v²/rg. With friction: v_max² = rg(tanθ + μ)/(1 − μ tanθ). Vertical circle: at the bottom, minimum speed to complete the loop is √(5gr); at the top, minimum speed is √(gr). Tension at top: T = mv²/r − mg.

Common traps. Friction direction is opposite to relative motion (or its tendency); a body on a rough surface in a lift or accelerating wedge needs pseudo force before resolving.

Physics 3: Work, Energy, Power and Collisions

Work. W = ∫F · dr = F d cosθ (constant force). Work by gravity = −mgΔh. Work by spring = −½k(x₂² − x₁²). Area under F-x graph = work.

Work-energy theorem. W_net = ΔKE, where KE = ½mv² = p²/2m. Also W_conservative = −ΔU.

Potential energy. U = mgh (near Earth); U = ½kx² (spring); U = −GMm/r (gravity). F = −dU/dx; equilibrium where dU/dx = 0, stable if d²U/dx² > 0.

Conservation. If only conservative forces act, KE + PE = constant. Otherwise ΔE_mech = W_{non-conservative}.

Power. P = dW/dt = F · v. Average power = W/t. Efficiency = output/input.

Momentum and collisions. If no external force, total momentum is conserved. Centre of mass: r_cm = Σm_ir_i/Σm_i; v_cm is unchanged if F_ext = 0.

• Coefficient of restitution e = (v₂ − v₁)/(u₁ − u₂); e = 1 elastic, e = 0 perfectly inelastic.
• Elastic head-on: v₁ = [(m₁ − m₂)u₁ + 2m₂u₂]/(m₁ + m₂). Equal masses exchange velocities.
• Perfectly inelastic: v = (m₁u₁ + m₂u₂)/(m₁ + m₂); loss in KE = ½μ(u₁ − u₂)², μ = reduced mass.
• Ball hitting a wall from height h: after n bounces, height = e²ⁿh.

Variable mass (rocket). m dv/dt = v_rel dm/dt − mg; v = u + v_rel ln(m₀/m) − gt.

Physics 4: Rotational Motion

Kinematics. ω = dθ/dt, α = dω/dt. Equations mirror linear: ω = ω₀ + αt; θ = ω₀t + ½αt²; ω² = ω₀² + 2αθ. Link: v = ωr, a_t = αr.

Torque and angular momentum. τ = r × F; L = r × p = Iω; τ = dL/dt = Iα. If τ_ext = 0 then L is conserved.

Moments of inertia.

• Ring (axis through centre, perpendicular): MR²
• Disc: ½MR²
• Solid sphere: (2/5)MR²; hollow sphere: (2/3)MR²
• Rod about centre: ML²/12; about end: ML²/3
• Solid cylinder: ½MR²

Parallel axis theorem: I = I_cm + Md². Perpendicular axis (planar bodies): I_z = I_x + I_y.

Energy. KE_rot = ½Iω². For rolling without slipping: v = ωR, KE = ½mv²(1 + k²/R²), where k = radius of gyration.

Rolling down an incline. a = g sinθ/(1 + I/mR²). Order of reaching the bottom: solid sphere, disc/solid cylinder, hollow sphere, ring. Friction needed: f = mg sinθ · (I/mR²)/(1 + I/mR²); no slipping requires tanθ ≤ μ(1 + mR²/I).

Equilibrium. ΣF = 0 and Στ = 0 about any point. Ladder problems: take torque about the contact point where unknown forces meet.

Common traps. Angular momentum about a point for a particle moving in a straight line is mvd (constant). In collisions with a pivot, linear momentum is not conserved but angular momentum about the pivot is.

Physics 5: Gravitation, SHM and Waves

Gravitation. F = Gm₁m₂/r²; g = GM/R². Variation: at height h, g' = g(1 + h/R)^-2 ≈ g(1 − 2h/R); at depth d, g' = g(1 − d/R); at the poles g is max, at the equator g' = g − ω²R.

Potential and orbits. V = −GM/r; escape speed v_e = √(2GM/R) = √(2gR) ≈ 11.2 km/s. Orbital speed v = √(GM/r); total energy of a satellite = −GMm/2r; T² ∝ r³ (Kepler's third law). Geostationary orbit: T = 24 h, r ≈ 6.6R.

Simple harmonic motion. a = −ω²x; x = A sin(ωt + φ). v = ω√(A² − x²), v_max = ωA, a_max = ω²A. Energy: E = ½mω²A²; KE and PE each average half of E; the KE/PE frequency is twice that of the motion.

• Spring-mass: T = 2π√(m/k). Springs in series: 1/k = Σ1/k_i; in parallel: k = Σk_i.
• Simple pendulum: T = 2π√(l/g); in a lift, replace g by g_eff.
• Physical pendulum: T = 2π√(I/mgd).
• Torsional: T = 2π√(I/C).

Waves. y = A sin(kx − ωt); v = fλ = ω/k. String wave speed v = √(T/μ). Sound in gas v = √(γP/ρ). Intensity ∝ A².

Standing waves. String fixed at both ends: f_n = nv/2L. Closed pipe: f = (2n−1)v/4L (odd harmonics only). Open pipe: f = nv/2L.

Beats and Doppler. Beat frequency = |f₁ − f₂|. Doppler: f' = f (v ± v_o)/(v ∓ v_s), with signs chosen by direction of approach (approach increases frequency).

Physics 6: Thermal Physics and Thermodynamics

Kinetic theory. P = (1/3)ρv_rms²; v_rms = √(3RT/M); average KE per molecule = (3/2)k_BT. Degrees of freedom f: U = (f/2)nRT; C_v = fR/2; γ = 1 + 2/f (monatomic 5/3, diatomic 7/5).

First law. ΔU = Q − W, W = ∫P dV. Work is the area under the P-V curve; a cyclic process has net W = area enclosed.

• Isothermal: PV = const; W = nRT ln(V₂/V₁); ΔU = 0.
• Isochoric: W = 0; Q = nC_vΔT.
• Isobaric: W = PΔV; Q = nC_pΔT.
• Adiabatic: Q = 0; PV^γ = const; TV^γ−1 = const; W = (P₁V₁ − P₂V₂)/(γ − 1).
• Polytropic: PVⁿ = const; molar heat capacity C = C_v + R/(1 − n).

Mayer's relation: C_p − C_v = R.

Second law. Heat engine efficiency η = W/Q_H = 1 − Q_C/Q_H. Carnot: η = 1 − T_C/T_H. Refrigerator COP = Q_C/W = T_C/(T_H − T_C).

Heat transfer. Conduction: dQ/dt = kA(T₁ − T₂)/L. Thermal resistance R = L/kA; series add resistances, parallel add conductances. Stefan-Boltzmann: P = σAeT⁴. Wien: λ_mT = b. Newton's law of cooling: dT/dt = −k(T − T₀).

Thermal expansion. ΔL = LαΔT; β = 2α; γ = 3α. Calorimetry: heat lost = heat gained; Q = mcΔT or mL during phase change.

Physics 7: Electrostatics and Capacitors

Coulomb's law. F = (1/4πε₀) q₁q₂/r². Electric field E = F/q; for a point charge E = kq/r². Principle of superposition holds for forces and fields.

Gauss's law. Φ = ∮E · dA = q_enc/ε₀.

• Infinite line charge: E = λ/2πε₀r
• Infinite sheet: E = σ/2ε₀; conductor surface: E = σ/ε₀
• Uniformly charged sphere (shell): E = kQ/r² outside; zero inside a shell; E = kQr/R³ inside a solid sphere
• Ring on axis: E = kQx/(R² + x²)^3/2

Potential. V = kq/r; E = −dV/dr; potential energy of two charges U = kq₁q₂/r. Work done in moving q: W = qΔV. Dipole: p = qd; potential on axis kp/r²; torque τ = pE sinθ; energy U = −p · E.

Conductors. Field inside is zero; the surface is an equipotential; charge resides on the surface. Connecting two spheres equalises potential: charges distribute in the ratio of radii. Earthing sets V = 0.

Capacitors. C = Q/V; parallel plate C = ε₀A/d; with dielectric C = Kε₀A/d. Energy U = ½CV² = Q²/2C; energy density = ½ε₀E². Series: 1/C = Σ1/C_i; parallel: C = ΣC_i. Spherical capacitor: C = 4πε₀ab/(b − a).

Battery connected vs disconnected. Battery connected: V fixed; inserting a dielectric increases Q and U. Disconnected: Q fixed; dielectric decreases V and U (energy is pulled in as work). Charge sharing between two capacitors: V = (C₁V₁ + C₂V₂)/(C₁ + C₂), with energy loss ½ C₁C₂(V₁ − V₂)²/(C₁ + C₂).

Physics 8: Current Electricity and Magnetism

Current. I = dQ/dt = nAv_de. Ohm: V = IR; R = ρL/A; temperature dependence R = R₀(1 + αΔT). Power P = VI = I²R = V²/R.

Circuits. Kirchhoff's current law (junction) ΣI = 0 and voltage law (loop) ΣV = 0. Cell with internal resistance: V = ε − Ir; max power transfer when R = r. Wheatstone bridge balanced when P/Q = R/S. Meter bridge: R/S = l/(100 − l). Potentiometer: ε₁/ε₂ = l₁/l₂.

RC circuits. Charging: q = CV(1 − e^−t/RC); discharging: q = q₀e^−t/RC. Time constant τ = RC.

Magnetic force. F = q(v × B); on a wire F = IL × B. Motion in a field: r = mv/qB, T = 2πm/qB. Helical path if v has a component along B; pitch = v_∥T.

Sources. Biot-Savart: dB = (μ₀/4π) I dl × r / r³.

• Long wire: B = μ₀I/2πr
• Circular loop centre: B = μ₀I/2R; axis: μ₀IR²/2(R² + x²)^3/2
• Solenoid: B = μ₀nI
• Force between parallel wires per length: μ₀I₁I₂/2πd (attract if currents are parallel)

Ampere's law: ∮B · dl = μ₀I_enc. Magnetic moment of a loop M = NIA; torque τ = M × B.

Induction. ε = −dΦ/dt (Faraday-Lenz). Motional emf = Blv. Self inductance ε = −L dI/dt; energy = ½LI². LR growth: I = I₀(1 − e^−Rt/L). LC oscillation: ω = 1/√(LC).

AC. V_rms = V₀/√2. X_L = ωL, X_C = 1/ωC. Series LCR: Z = √(R² + (X_L − X_C)²); resonance at ω = 1/√(LC); power factor cosφ = R/Z.

Physics 9: Optics and Modern Physics

Reflection and refraction. Mirror formula 1/v + 1/u = 1/f, f = R/2. Snell: n₁ sinθ₁ = n₂ sinθ₂. Critical angle sinθ_c = n₂/n₁ (for total internal reflection). Apparent depth = real depth/n. Prism: n = sin((A + δ_m)/2)/sin(A/2); small angle δ = (n − 1)A.

Lenses. 1/v − 1/u = 1/f; lens maker: 1/f = (n − 1)(1/R₁ − 1/R₂). Power P = 1/f (dioptre). Combination: 1/f = 1/f₁ + 1/f₂. Magnification m = v/u (lens), −v/u (mirror).

Instruments. Simple microscope m = 1 + D/f. Compound microscope m = (v_o/u_o)(1 + D/f_e). Telescope (normal adjustment) m = f_o/f_e.

Wave optics. Young's double slit: fringe width β = λD/d; path difference = d sinθ = nλ for bright fringes. Intensity I = 4I₀cos²(φ/2). Single slit minima: a sinθ = nλ. Thin film interference: 2μt cos r = (2n ± 1)λ/2 or nλ depending on phase changes. Polarisation: Malus's law I = I₀cos²θ; Brewster angle tanθ_B = n.

Photoelectric effect. E = hf; hf = φ + KE_max; stopping potential eV₀ = KE_max. Threshold frequency f₀ = φ/h. de Broglie wavelength λ = h/p = h/√(2mK).

Atoms. Bohr model: r_n = 0.529 n²/Z Å; E_n = −13.6 Z²/n² eV; v_n = 2.18×10⁶ Z/n m/s. Spectral lines: 1/λ = R Z²(1/n₁² − 1/n₂²). X-rays: Moseley √f = a(Z − b); Bragg 2d sinθ = nλ.

Nuclei. R = R₀A^1/3; binding energy = Δm c² (931.5 MeV per u). Decay: N = N₀e^−λt; t_1/2 = 0.693/λ; activity A = λN; mean life = 1/λ.

Chemistry 1: Physical Chemistry (Part 1)

Mole concept. n = mass/M = N/N_A = V/22.4 L at STP. Molarity M = moles/L solution; molality m = moles/kg solvent; mole fraction x = n_i/n_total. Equivalent weight = M/n-factor; N = n-factor × M. Law of equivalence: equivalents of reactants are equal. Limiting reagent decides the yield.

Atomic structure. Bohr: E_n = −13.6 Z²/n² eV. Number of radial nodes = n − l − 1; angular nodes = l; total nodes = n − 1. Quantum numbers: n, l (0 to n−1), m (−l to +l), s (±½). Aufbau, Pauli, Hund; exceptions Cr (3d⁵4s¹) and Cu (3d¹⁰4s¹). Heisenberg: Δx · Δp ≥ h/4π.

Gaseous state. PV = nRT; Boyle, Charles, Avogadro. Dalton: P = Σp_i. Graham: r₁/r₂ = √(M₂/M₁). Speeds: u_mp = √(2RT/M), u_avg = √(8RT/πM), u_rms = √(3RT/M). Real gas: compressibility Z = PV/nRT; van der Waals (P + an²/V²)(V − nb) = nRT; Boyle temperature T_B = a/Rb.

Thermodynamics and thermochemistry. ΔU = q + w; ΔH = ΔU + Δn_gRT. Hess's law. ΔG = ΔH − TΔS; spontaneous if ΔG < 0. ΔG° = −RT ln K = −nFE°. Kirchhoff: ΔH₂ − ΔH₁ = ΔC_pΔT. Reversible isothermal expansion: w = −nRT ln(V₂/V₁). Entropy ΔS = q_rev/T.

Chemical equilibrium. K_p = K_c(RT)^Δn. Le Chatelier's principle for pressure, temperature and concentration. Reaction quotient Q < K means forward direction. Van't Hoff: ln(K₂/K₁) = (ΔH/R)(1/T₁ − 1/T₂). Degree of dissociation α: for A → 2B, K_p = 4α²P/(1 − α²).

Chemistry 2: Physical Chemistry (Part 2)

Ionic equilibrium. pH = −log[H⁺]; K_w = 10⁻¹⁴ at 25°C. Weak acid: α = √(K_a/C), pH = ½(pK_a − log C). Buffers: pH = pK_a + log([salt]/[acid]). Salt hydrolysis: pH = 7 + ½(pK_a + log C) for a salt of weak acid and strong base. Solubility product: for A_xB_y, K_sp = x^xy^ys^x+y. Common ion effect reduces solubility. Indicator range pH = pK_in ± 1.

Electrochemistry. E_cell = E_cathode − E_anode. Nernst: E = E° − (0.0591/n) log Q at 298 K. ΔG° = −nFE°. Faraday: m = ZIt = (M/nF)It, F = 96500 C. Kohlrausch: Λ_m = Λ_m° − A√c; Λ°_m = λ₊ + λ₋. Conductivity κ = (1/R)(l/A); Λ_m = 1000κ/c. Concentration cells: E = (0.0591/n) log(c₂/c₁).

Chemical kinetics. Rate = k[A]^m[B]ⁿ. Zero order: [A] = [A]₀ − kt, t_1/2 = [A]₀/2k. First order: ln[A] = ln[A]₀ − kt, t_1/2 = 0.693/k. Second order: 1/[A] = 1/[A]₀ + kt. Arrhenius k = Ae^−E_a/RT; ln(k₂/k₁) = (E_a/R)(1/T₁ − 1/T₂). Parallel and consecutive reactions, steady-state approximation, rate-determining step in a mechanism. Pseudo first order, e.g. ester hydrolysis.

Solutions. Raoult: P_A = x_AP_A°. Ideal solutions: ΔH_mix = 0 and ΔV_mix = 0. Colligative properties: relative lowering (P° − P)/P° = x_solute; ΔT_b = iK_bm; ΔT_f = iK_fm; osmotic pressure π = iCRT. Van't Hoff factor i = 1 + (n − 1)α for dissociation; i = 1 − α(1 − 1/n) for association. Henry's law p = K_Hx.

Solid state and surface chemistry. Density ρ = ZM/(N_Aa³). Z: simple cubic 1, bcc 2, fcc 4. Packing efficiency: sc 52%, bcc 68%, fcc/hcp 74%. Radius: bcc 4r = √3a, fcc 4r = √2a. Freundlich adsorption x/m = kp^1/n.

Chemistry 3: Inorganic Chemistry

Periodic trends. Atomic radius decreases across a period, increases down a group. Ionisation energy increases across a period with exceptions (Be > B, N > O). Electron gain enthalpy: Cl > F. Electronegativity increases across a period and decreases down a group. Isoelectronic species: size decreases as nuclear charge increases.

Chemical bonding. Octet rule and exceptions; VSEPR shapes: 2 pairs linear; 3 trigonal planar; 4 tetrahedral; 5 trigonal bipyramidal; 6 octahedral. Hybridisation = (V + M − C + A)/2 for central atom. Bond order from MOT = (N_b − N_a)/2; O₂ is paramagnetic (bond order 2); N₂ is diamagnetic (bond order 3). Fajan's rules: small, highly charged cation and large anion give more covalent character. Dipole moment μ = q × d. Hydrogen bonding raises boiling point (H₂O, HF, NH₃).

s-block. Group 1: reactivity and basicity of hydroxides increase down the group; lattice energy vs hydration energy governs solubility. Group 2: BeO amphoteric; Be and Al show diagonal relationships (Li-Mg too). Sodium carbonate (Solvay process), plaster of Paris is CaSO₄·½H₂O.

p-block. Boron: B₂H₆ has 3-centre 2-electron bonds; boric acid acts as a Lewis acid. Carbon: allotropes; CO is a good ligand. Nitrogen: NH₃ (Haber), HNO₃ (Ostwald); oxides N₂O to N₂O₅. Phosphorus: oxyacids H₃PO₂ (monobasic), H₃PO₃ (dibasic), H₃PO₄ (tribasic). Oxygen group: O₃ is a strong oxidiser; H₂SO₄ by the Contact process. Halogens: acidic strength HF < HCl < HBr < HI; interhalogens, oxoacids HClO < HClO₂ < HClO₃ < HClO₄. Noble gases: XeF₂ (linear), XeF₄ (square planar), XeF₆ (distorted octahedral).

d-block and coordination. Colour from d-d transitions. Magnetic moment μ = √(n(n + 2)) BM. KMnO₄ in acid is reduced to Mn²⁺; K₂Cr₂O₇ gives Cr³⁺. Werner's theory; EAN; CFT: t_2g and e_g splitting; strong-field ligands (CN⁻, CO) give low spin; Δ_t = (4/9)Δ_o. Isomerism: geometrical, optical, linkage, ionisation. IUPAC naming order: ligands alphabetical, then metal with its oxidation state.

Metallurgy. Ore concentration by froth flotation; roasting and calcination; reduction by carbon (Ellingham diagram), electrolytic reduction for Al; refining by zone, van Arkel and Mond processes.

Chemistry 4: Organic Chemistry

Basics. IUPAC nomenclature: longest chain, lowest locants, priority of functional groups (acid > ester > aldehyde > ketone > alcohol > amine). Electronic effects: inductive (−I: NO₂, CN, halogens), resonance (±M), hyperconjugation. Stability: tertiary carbocation > secondary > primary; carbanion order is reversed. Acid strength: carboxylic acid > phenol > alcohol; EWG increases acidity. Basicity of amines depends on solvation and +I (in water 2° > 3° > 1° for ethyl amines).

Stereochemistry. Chiral centre has four different groups; number of stereoisomers = 2ⁿ (no symmetry). R/S by CIP priority rules; E/Z for alkenes. Meso compounds are optically inactive due to internal symmetry. Enantiomers have equal and opposite rotation; diastereomers differ in physical properties.

Hydrocarbons. Alkane: free radical halogenation (reactivity of 3° H > 2° > 1°; Br is more selective than Cl). Alkene: electrophilic addition following Markovnikov; peroxide effect gives anti-Markovnikov for HBr; hydroboration-oxidation gives anti-Markovnikov alcohol; ozonolysis cleaves the double bond. Alkyne: acidic terminal H; Na/NH₃ gives trans-alkene, Lindlar gives cis-alkene; hydration gives a ketone. Benzene: electrophilic substitution (nitration, sulphonation, halogenation, Friedel-Crafts); activating groups are o,p-directing (−OH, −NH₂, −R), deactivating are m-directing (−NO₂, −COOH), halogens are deactivating but o,p-directing.

Halides and alcohols. S_N1: 3°, carbocation intermediate, racemisation, polar protic solvent. S_N2: 1°, backside attack, inversion, polar aprotic solvent. E2 uses a strong base, Saytzeff product. Alcohols: Lucas test (3° fastest), oxidation by PCC stops at aldehyde, dehydration by conc. H₂SO₄. Ethers: Williamson synthesis. Phenol: Reimer-Tiemann (salicylaldehyde), Kolbe (salicylic acid).

Carbonyl compounds. Nucleophilic addition: aldehydes more reactive than ketones. Aldol condensation (needs α-H), Cannizzaro (no α-H), Haloform (methyl ketones), Clemmensen (Zn-Hg/HCl) and Wolff-Kishner (N₂H₄/KOH) reductions, Tollens (aldehyde gives silver mirror), Fehling.

Nitrogen compounds and biomolecules. Hoffmann bromamide (loses one carbon), Gabriel phthalimide (1° amines), diazonium salts (Sandmeyer, coupling to azo dyes), Hinsberg test. Carbohydrates: glucose is an aldohexose with 4 chiral centres; sucrose is non-reducing. Proteins: peptide bond, zwitterion at the isoelectric point.

Maths 1: Algebra

Quadratic equations. ax² + bx + c = 0; roots x = (−b ± √D)/2a, D = b² − 4ac. Sum = −b/a, product = c/a. Nature: D > 0 real distinct; D = 0 equal; D < 0 complex conjugates. For both roots greater than k: D ≥ 0, af(k) > 0, −b/2a > k. Cubic: sum of roots = −b/a, Σαβ = c/a, product = −d/a.

Complex numbers. z = x + iy = r e^iθ; |z| = √(x² + y²). De Moivre: (cosθ + i sinθ)ⁿ = cos nθ + i sin nθ. nth roots of unity: e^2πik/n, sum = 0, product = (−1)ⁿ⁻¹. Properties: |z₁ + z₂| ≤ |z₁| + |z₂|; |z|² = z z̄. Locus |z − z₁| = r is a circle; |z − z₁| = |z − z₂| is the perpendicular bisector. Cube roots of unity: 1 + ω + ω² = 0, ω³ = 1.

Sequences and series. AP: a_n = a + (n−1)d; S_n = n/2[2a + (n−1)d]. GP: a_n = arⁿ⁻¹; S_n = a(rⁿ − 1)/(r − 1); S_∞ = a/(1 − r), |r| < 1. AM ≥ GM ≥ HM; AM = (a + b)/2, GM = √(ab), HM = 2ab/(a + b). Σn = n(n+1)/2; Σn² = n(n+1)(2n+1)/6; Σn³ = [n(n+1)/2]². Method of differences for telescoping series.

Permutations, combinations and binomial. ⁿP_r = n!/(n−r)!; ⁿC_r = n!/(r!(n−r)!). Circular permutation: (n−1)!. Number of derangements D_n = n! Σ(−1)^k/k!. Distributing n identical objects into r groups: ^n+r−1C_r−1. Binomial theorem: (x + y)ⁿ = ΣⁿC_rx^n−ry^r; general term T_r+1 = ⁿC_rx^n−ry^r; sum of coefficients = 2ⁿ; ΣⁿC_r with alternating signs = 0.

Matrices and determinants. |AB| = |A||B|; A · adj A = |A|I; A⁻¹ = adj A/|A|. Cramer's rule; system AX = B has a unique solution if |A| ≠ 0; if |A| = 0, either infinitely many solutions or none. Rank and consistency. Properties: for a skew-symmetric matrix of odd order, determinant = 0.

Maths 2: Trigonometry and Coordinate Geometry

Trigonometric identities. sin²x + cos²x = 1; 1 + tan²x = sec²x. sin(A ± B) = sinA cosB ± cosA sinB; cos(A ± B) = cosA cosB ∓ sinA sinB; tan(A ± B) = (tanA ± tanB)/(1 ∓ tanA tanB). sin2A = 2 sinA cosA; cos2A = cos²A − sin²A = 2cos²A − 1; sin3A = 3sinA − 4sin³A; cos3A = 4cos³A − 3cosA. Sum to product: sinC + sinD = 2 sin((C+D)/2) cos((C−D)/2). Range of a sinx + b cosx is [−√(a²+b²), +√(a²+b²)].

Equations and triangles. sinθ = sinα gives θ = nπ + (−1)ⁿα; cosθ = cosα gives θ = 2nπ ± α; tanθ = tanα gives θ = nπ + α. Sine rule a/sinA = 2R; cosine rule a² = b² + c² − 2bc cosA; area = ½bc sinA = rs = abc/4R; r = (s − a)tan(A/2). Inverse trigonometry: sin⁻¹x + cos⁻¹x = π/2; tan⁻¹x + tan⁻¹y = tan⁻¹[(x + y)/(1 − xy)] (for xy < 1).

Straight lines. Slope form y = mx + c; distance from (x₁, y₁) to ax + by + c = 0 is |ax₁ + by₁ + c|/√(a² + b²). Angle between lines tanθ = |(m₁ − m₂)/(1 + m₁m₂)|. Distance between parallel lines |c₁ − c₂|/√(a² + b²). Pair of lines ax² + 2hxy + by² = 0; angle bisectors; family of lines L₁ + λL₂ = 0.

Circle. (x − h)² + (y − k)² = r²; general x² + y² + 2gx + 2fy + c = 0 with radius √(g² + f² − c). Tangent condition: distance from the centre = r. Radical axis S₁ − S₂ = 0; common tangents; orthogonal circles 2g₁g₂ + 2f₁f₂ = c₁ + c₂.

Conics.

• Parabola y² = 4ax: focus (a, 0); tangent at (at², 2at): ty = x + at²; slope form y = mx + a/m; latus rectum 4a.
• Ellipse x²/a² + y²/b² = 1: e = √(1 − b²/a²); foci (±ae, 0); tangent y = mx ± √(a²m² + b²); sum of focal distances = 2a; area = πab.
• Hyperbola x²/a² − y²/b² = 1: e = √(1 + b²/a²); asymptotes y = ±(b/a)x; tangent y = mx ± √(a²m² − b²).

Maths 3: Calculus and Probability

Limits and continuity. Standard limits: lim sinx/x = 1; lim (1 − cosx)/x² = 1/2; lim (e^x − 1)/x = 1; lim ln(1 + x)/x = 1; lim (1 + 1/x)^x = e. L'Hopital's rule for 0/0 or ∞/∞. A function is continuous if left limit = right limit = f(a). Differentiable implies continuous, but not vice versa (e.g. |x| at 0).

Differentiation. d/dx (xⁿ) = nxⁿ⁻¹; (sinx)' = cosx; (tanx)' = sec²x; (e^x)' = e^x; (ln x)' = 1/x. Product rule (uv)' = u'v + uv'; quotient rule (u/v)' = (u'v − uv')/v²; chain rule dy/dx = (dy/du)(du/dx). Implicit and parametric differentiation; logarithmic differentiation for x^x and similar functions.

Applications. Tangent slope = f'(x). Increasing where f' > 0. Maxima/minima: f' = 0 and f'' < 0 for a maximum. Rolle's and Mean Value theorem: f'(c) = (f(b) − f(a))/(b − a). Rate of change problems and related rates. Optimisation: set derivative of the constraint-substituted function to zero.

Integration. ∫xⁿdx = xⁿ⁺¹/(n+1); ∫dx/x = ln|x|; ∫e^xdx = e^x; ∫sinx dx = −cosx; ∫sec²x dx = tanx; ∫dx/(a² + x²) = (1/a) tan⁻¹(x/a); ∫dx/√(a² − x²) = sin⁻¹(x/a). Integration by parts: ∫u dv = uv − ∫v du (ILATE rule). ∫e^x[f(x) + f'(x)]dx = e^xf(x). Partial fractions for rational functions.

Definite integrals. ∫_a^bf(x)dx = ∫_a^bf(a + b − x)dx. For even f: ∫_−a^a = 2∫₀^a; for odd f it is 0. ∫₀^π/2 sinⁿx dx has the Wallis formula. Leibniz rule for derivative of an integral with variable limits. Area between curves = ∫(y_upper − y_lower)dx. Area of a circle by integration πr².

Differential equations. Variable separable; homogeneous (substitute y = vx); linear dy/dx + Py = Q with integrating factor e^{∫P dx}. Order = highest derivative; degree = its power.

Probability. P(A) = favourable/total; P(A ∪ B) = P(A) + P(B) − P(A ∩ B). Conditional P(A|B) = P(A ∩ B)/P(B). Bayes: P(A_i|B) = P(B|A_i)P(A_i)/ΣP(B|A_j)P(A_j). Independent events: P(A ∩ B) = P(A)P(B). Binomial distribution: P(X = r) = ⁿC_rp^rq^n−r, mean = np, variance = npq. Expected value E(X) = Σx_ip_i.

Vectors and 3D. a · b = |a||b|cosθ; |a × b| = |a||b|sinθ. Scalar triple product [a b c] = volume of the parallelepiped; coplanar if zero. Line in 3D: (r − a) × b = 0. Distance of a point from a plane: |ax₁ + by₁ + cz₁ + d|/√(a² + b² + c²). Angle between two planes is the angle between normals. Shortest distance between skew lines = |(a₂ − a₁) · (b₁ × b₂)|/|b₁ × b₂|.