JEE Main cheat sheet

Projectile: range R = u²sin2θ/g is maximal at θ = 45°; maximum height H = u²sin²θ/(2g); time of flight T = 2u sinθ/g.

Newton's second law in variable-mass form: F = dp/dt; for constant mass F = ma. Friction limit: f_max = μN, with kinetic friction f = μ_k N independent of contact area.

Work–energy theorem: W_net = ΔK. For a conservative force, F = −dU/dx, and mechanical energy is conserved when only conservative forces do work.

Rotational dynamics: τ = Iα, L = Iω; parallel-axis theorem I = I_cm + Md²; rolling without slipping requires v = ωR and static friction does no work.

Simple harmonic motion: a = −ω²x, T = 2π√(m/k) for a spring, T = 2π√(L/g) for a small-angle pendulum; energy E = ½kA².

Thermodynamics: ΔU = Q − W; adiabatic process PV^γ = constant; Carnot efficiency η = 1 − T_c/T_h (temperatures in kelvin).

Electrostatics: Coulomb force F = kq₁q₂/r² with k = 1/(4πε₀); field of a point charge E = kq/r²; potential V = kq/r; capacitor energy U = ½CV².

Current electricity: Ohm's law V = IR; series R = ΣR_i, parallel 1/R = Σ1/R_i; power P = VI = I²R = V²/R; Kirchhoff's junction and loop rules.

Optics: mirror 1/f = 1/v + 1/u, lens 1/f = 1/v − 1/u; Snell's law n₁sinθ₁ = n₂sinθ₂; critical angle sinθ_c = n₂/n₁ for n₁ > n₂; Young's double slit fringe width β = λD/d.

Modern physics: photoelectric equation hν = φ + K_max; Bohr energy E_n = −13.6 Z²/n² eV; de Broglie wavelength λ = h/p; radioactive decay N = N₀e^(−λt), half-life t½ = ln2/λ.

Chemistry - mole concept: n = m/M = N/N_A = V/22.4 L at STP (ideal gas); molarity M = moles solute / litres solution; dilution M₁V₁ = M₂V₂.

Equilibrium and thermodynamics: K_p = K_c(RT)^Δn; ΔG = ΔH − TΔS, spontaneous when ΔG < 0; pH = −log[H₃O⁺]; buffer pH = pKa + log([A⁻]/[HA]).

Organic reaction rules: Markovnikov addition to alkenes, anti-Markovnikov with HBr/peroxide; SN1 favoured by tertiary substrates and polar protic solvents, SN2 by primary substrates and polar aprotic solvents.

Mathematics - calculus: d/dx(xⁿ) = nx^(n−1); ∫xⁿdx = x^(n+1)/(n+1) + C (n ≠ −1); product rule (uv)' = u'v + uv'; L'Hôpital applies to 0/0 and ∞/∞ forms.

Mathematics - coordinate geometry: distance √((x₂−x₁)²+(y₂−y₁)²); slope m = (y₂−y₁)/(x₂−x₁); circle x²+y²+2gx+2fy+c = 0 has centre (−g,−f) and radius √(g²+f²−c).

Mathematics - algebra and probability: quadratic roots sum −b/a and product c/a; discriminant D = b²−4ac; P(A∪B) = P(A)+P(B)−P(A∩B); Bayes' theorem P(A|B) = P(B|A)P(A)/P(B).

Marking scheme rule: each Paper 1 subject paper carries 100 marks with 20 MCQs and 5 numerical-value questions; numerical answers are entered as integers and negative marking applies to MCQ items only.

Paper 2A structure: Mathematics, Aptitude Test, and Drawing; the Drawing section is evaluated on freehand sketching quality and is not a multiple-choice section.

Aptitude objects and surfaces: a cube has 6 faces, 12 edges, 8 vertices; Euler's formula for convex polyhedra V − E + F = 2.

Rotational symmetry rule: a regular n-gon has rotational symmetry of order n; a square has order 4, an equilateral triangle order 3, a circle infinite order.

Orthographic projection: first-angle places the top view below the front view and the left-side view on the right; third-angle places the top view above the front view.

Isometric drawing: all three axes are 120° apart and the isometric scale is √(2/3) ≈ 0.816 times true length; isometric view is not a perspective view.

Scale rule: representative fraction RF = drawing length / actual length; a scale of 1:50 means 1 cm on paper equals 50 cm actual.

Perspective drawing: one-point perspective has one vanishing point; two-point perspective has two, both on the horizon line; the station point is the observer's eye position.

Colour and design principles: primary pigments are red, yellow, blue (additive primaries are red, green, blue); complementary colours lie opposite on the colour wheel; rhythm, balance, and proportion are principles of composition.

Mathematics for architecture: area of a triangle = ½ab sinC; volume of a cylinder = πr²h; surface area of a sphere = 4πr²; total surface area of a cone = πr(l + r).

Architectural awareness: the Taj Mahal is Mughal marble architecture; Le Corbusier's Chandigarh plan applied grid streets and sector zoning; the Lotus Temple in Delhi is a Bahá'í house of worship.

Paper 2B structure: Mathematics, Aptitude Test, and Planning-based questions; there is no drawing section, and planning items test settlement, transport, and land-use reasoning.

Density measures: gross density = total population / total area; net density = population / net residential area; net density is always greater than or equal to gross density for the same population.

Land-use planning rule: residential, commercial, industrial, and recreational uses are separated by zoning; industrial areas are placed downwind and downstream of residential areas.

Neighbourhood unit concept: Clarence Perry's unit is centred on a primary school and defined by a walking radius of about 0.25 mile (400 m), with arterial roads on the periphery.

Transport planning: the gravity model predicts trips T_ij ∝ (P_i A_j)/d_ij²; road hierarchy runs from national highways down to local streets, with access controlled at higher levels.

Census and growth: decadal growth rate = ((P₂ − P₁)/P₁) × 100; doubling time can be estimated by the rule of 70, t ≈ 70/r for growth rate r in percent per year.

Statistics for planning: mean = Σx/n; standard deviation σ = √(Σ(x−x̄)²/n); for grouped data the median lies in the class where cumulative frequency reaches n/2.

Geometry and mensuration: area of a circle πr²; circumference 2πr; volume of a rectangular prism = lbh; 1 hectare = 10,000 m² = 0.01 km².

Probability and data rule: for independent events P(A∩B) = P(A)P(B); a pie chart sector angle = (component/total) × 360°; a line graph is preferred for showing trends over time.