JEE Main Paper 2B - B.Planning cheat sheet

Coordinate geometry: distance AB = √[(x₂−x₁)² + (y₂−y₁)²]; internal section P = ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n)); midpoint is the case m = n.

Straight lines: slope m = (y₂−y₁)/(x₂−x₁); parallel ⇔ m₁ = m₂; perpendicular ⇔ m₁m₂ = −1; point-slope form y − y₁ = m(x − x₁).

Circle: centre (h, k), radius r ⇒ (x−h)² + (y−k)² = r²; general form x² + y² + 2gx + 2fy + c = 0 has centre (−g, −f) and radius √(g² + f² − c).

Series: AP Sₙ = n/2[2a + (n−1)d]; GP Sₙ = a(1−rⁿ)/(1−r) for r ≠ 1 and S∞ = a/(1−r) for |r| < 1; Σn = n(n+1)/2, Σn² = n(n+1)(2n+1)/6, Σn³ = [n(n+1)/2]².

Counting and binomial: ⁿC_r = n!/(r!(n−r)!) = ⁿC_{n−r}; in (a+b)ⁿ, T_{r+1} = ⁿC_r a^{n−r} b^r and the expansion has n+1 terms.

Standard limits: lim(x→0) sin x/x = 1; lim(x→0)(1−cos x)/x² = 1/2; lim(x→0)(1+x)^{1/x} = e; for 0/0 or ∞/∞ forms apply L'Hôpital's rule lim f/g = lim f′/g′.

Differentiation: d/dx(xⁿ) = nxⁿ⁻¹, d/dx(sin x) = cos x, d/dx(cos x) = −sin x, d/dx(eˣ) = eˣ, d/dx(ln x) = 1/x; maxima where f′ = 0 and f″ < 0, minima where f″ > 0.

Integration: ∫xⁿ dx = x^{n+1}/(n+1) + C for n ≠ −1; ∫ₐᵇ f(x)dx = F(b) − F(a); area between two curves = ∫(upper − lower)dx over the intersection interval.

Probability and statistics: P(A∪B) = P(A) + P(B) − P(A∩B); P(A|B) = P(A∩B)/P(B); mean x̄ = Σfx/Σf; variance σ² = Σfx²/Σf − x̄²; σ = √σ².

Painted cube cut into unit cubes (n×n×n): 3 painted faces = 8 (corners); 2 faces = 12(n−2); 1 face = 6(n−2)²; 0 faces = (n−2)³; only valid for n ≥ 2.

Clock: angle between hands at H hours M minutes = |30H − 5.5M| degrees; if greater than 180°, report 360° minus that value; hands coincide and oppose 11 times each in 12 hours.

Calendar odd days: ordinary year 1, leap year 2; 100 years = 5 odd days, 200 = 3, 300 = 1, 400 = 0; the calendar repeats every 400 years.

Speed relations: distance = speed × time; average speed for two equal distances at speeds a and b is 2ab/(a+b); for two equal times it is (a+b)/2; relative speed adds when approaching, subtracts when receding.

Images: a plane mirror reverses left-right (top-bottom unchanged) while a water image reverses top-bottom; the mirror image of a clock at H:M is 11:60 − H:M.

Euler's formula V − E + F = 2 holds for convex polyhedra; a cube has 8 vertices, 12 edges, 6 faces, and a square pyramid has 5, 8, 5 respectively.

Syllogism conversion rules: 'Some A are B' ⇔ 'Some B are A'; 'No A is B' ⇔ 'No B is A'; 'All A are B' does not convert to 'All B are A'. Test conclusions with Venn diagrams.

Series completion order to check: constant first differences (AP), constant ratios (GP), constant second differences (quadratic), then square/cube, alternating and two-interleaved patterns.

Paper folding/cutting: each fold doubles the number of punched holes, and the unfolded hole pattern is the mirror reflection of the punched set across every fold line.

Constitutional framework: 73rd Amendment (1992) added Part IX and the Eleventh Schedule (29 rural subjects); 74th Amendment (1992) added Part IXA and the Twelfth Schedule with 18 municipal functions including urban planning.

Census 2011 urban data: urban population is 31.16% of the total (about 377 million) and there are 53 million-plus urban agglomerations; a census town requires 5,000+ population, 400/km² density and 75% of male main workers in non-agricultural work.

Development controls: FAR (FSI) = total floor area of all storeys ÷ plot area; ground coverage = building footprint ÷ plot area × 100; setbacks are mandatory open margins from every plot boundary.

Plan hierarchy per URDPFI guidelines: perspective plan (about 20–25 years) → development/master plan (10–20 years) → zonal plan → layout or subdivision plan → local area plan for implementation.

Classical models: Von Thünen's land-use rings (1826), Burgess concentric zone model (1925), Hoyt sector model (1939), Harris–Ullman multiple nuclei model (1945), Christaller's central place theory (1933).

Urban form concepts: Howard's Garden City (1898, green belt, about 32,000 people), Le Corbusier's Radiant City, Wright's Broadacre City, Perry's neighbourhood unit (1929, quarter-mile school radius).

Movement: road hierarchy runs arterial → sub-arterial → collector → local, with access increasingly restricted up the hierarchy; the Metro Rail Policy 2017 mandates transit-oriented development and last-mile connectivity.

Institutions and missions: MoHUA (with TCPO) is the national urban ministry; NITI Aayog replaced the Planning Commission on 1 January 2015; Smart Cities Mission, AMRUT and PMAY-U were all launched in 2015.

Sustainability: SDG 11 targets inclusive, safe, resilient and sustainable cities; Habitat III (Quito, 2016) adopted the New Urban Agenda; GRIHA (TERI), IGBC and LEED are the main green-building rating systems in Indian practice.