JEE Main Paper 2B - B.Planning flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Distance formula in 2D

For points A(x₁, y₁) and B(x₂, y₂) in a plane, AB = √[(x₂−x₁)² + (y₂−y₁)²]. It follows from the Pythagoras theorem applied to the horizontal and vertical separations.

Section formula, internal division

If P divides the join of A(x₁, y₁) and B(x₂, y₂) internally in the ratio m:n, then P = ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n)). The midpoint formula is the special case m = n.

Perpendicular and parallel line slopes

Two non-vertical lines are perpendicular when m₁m₂ = −1 and parallel when m₁ = m₂. A line perpendicular to one of slope m has slope −1/m, provided m is not zero.

Sum of squares and cubes

For the first n natural numbers, Σn = n(n+1)/2, Σn² = n(n+1)(2n+1)/6 and Σn³ = [n(n+1)/2]². These identities give standard series sums without term-by-term addition.

General term, binomial expansion

In the expansion of (a + b)ⁿ, the (r+1)th term is T_{r+1} = ⁿC_r · a^{n−r} · b^r. Putting r = 0 gives the first term, and the expansion contains n+1 terms.

Derivatives of sine and cosine

d/dx(sin x) = cos x and d/dx(cos x) = −sin x, valid when x is measured in radians. The chain rule extends these to d/dx sin(f(x)) = cos(f(x))·f′(x).

L'Hôpital's rule

If a limit produces the indeterminate form 0/0 or ∞/∞, then lim f(x)/g(x) = lim f′(x)/g′(x), provided the second limit exists. The rule may be applied repeatedly.

Definite integral as area

For a non-negative function f on [a, b], ∫ₐᵇ f(x)dx equals the area enclosed between the curve y = f(x), the x-axis, and the vertical lines x = a and x = b.

Conditional probability formula

P(A|B) = P(A ∩ B)/P(B), defined for P(B) > 0. Rearranging gives the multiplication rule P(A ∩ B) = P(B)·P(A|B), which underpins tree diagrams and Bayes' theorem.

Inverse of a 2×2 matrix

For A = [[a, b], [c, d]], the determinant is ad − bc and A⁻¹ = (1/det A)·[[d, −b], [−c, a]]. The inverse exists only when the determinant is non-zero.

Dot and cross product

a·b = |a||b|cos θ is a scalar and vanishes for perpendicular vectors; |a × b| = |a||b|sin θ equals the area of the parallelogram spanned by a and b, with direction given by the right-hand rule.

Variance and standard deviation

Variance σ² = (1/n)Σ(xᵢ − x̄)², equivalently (1/n)Σxᵢ² − x̄². Standard deviation σ is its positive square root and is expressed in the same units as the data.

Painted cube: corner cubes

When an n×n×n cube is painted on all faces and cut into unit cubes, exactly 8 small cubes carry three painted faces - the eight corners of the original cube, for any n ≥ 2.

Painted cube: two painted faces

Cubes with exactly two painted faces lie along the edges of the painted n×n×n cube. The 12 edges each contribute (n − 2) such cubes, giving a total of 12(n − 2) for n ≥ 2.

Clock angle formula

The angle between the hour and minute hands at H hours M minutes is |30H − 5.5M| degrees. If the result exceeds 180°, subtract it from 360° to obtain the smaller angle.

Mirror image of clock time

The mirror image of a clock showing H:M is 11:60 − H:M, applied digit-wise to hours and minutes for times between 1:00 and 11:59. For example, 4:30 mirrors to 7:30.

Odd days in 100 years

An ordinary year has 1 odd day and a leap year 2. A span of 100 years has 24 leap and 76 ordinary years, giving 124 odd days, that is 5 odd days after dividing by 7.

Average speed for equal distances

If two equal distances are covered at speeds a and b, the average speed for the whole journey is 2ab/(a + b), the harmonic mean. The arithmetic mean (a + b)/2 overestimates the true value.

Euler's formula for polyhedra

For any convex polyhedron, V − E + F = 2, where V counts vertices, E edges and F faces. A cube satisfies 8 − 12 + 6 = 2, which helps count hidden edges and faces.

Water image reflection

A water image reverses a figure top-to-bottom while preserving left-right order, unlike a mirror image which reverses left-right. Letters such as 'b' and 'p' interchange under a water image.

Syllogism 'some' statement

The statement 'Some A are B' converts validly to 'Some B are A', whereas 'All A are B' does not convert to 'All B are A'. Venn diagrams are used to test whether a conclusion follows.

Number series: difference method

Compute successive differences: constant first differences indicate an arithmetic series, constant second differences a quadratic pattern, and constant ratios a geometric series. Then test alternating, square and cube patterns.

74th Constitutional Amendment Act

Enacted in 1992, it inserted Part IXA and the Twelfth Schedule into the Constitution, granting urban local bodies constitutional status and listing 18 municipal functions including urban planning and land-use regulation.

Floor Area Ratio (FAR)

FAR = total floor area of all storeys ÷ plot area, a dimensionless development control capping built intensity. A 200 m² plot with permissible FAR 2 can accommodate 400 m² of floor space.

Ground coverage

Ground coverage = building footprint area ÷ plot area × 100, expressed as a percentage. It limits how much of the plot the building occupies at ground level, safeguarding open space, light and ventilation.

Neighborhood unit concept

Proposed by Clarence Perry in 1929, it sizes a residential unit around a primary school within roughly a quarter-mile walking radius, bounded by arterial roads and provided with shops and a community centre.

Garden City concept

Ebenezer Howard's 1898 proposal for self-contained towns of about 32,000 people combining the advantages of town and country, encircled by a permanent agricultural green belt that limits outward growth.

Central Place Theory

Walter Christaller (1933) explained the number, size and spacing of settlements as a hexagonal hierarchy of central places serving surrounding hinterlands, ranked by the range and threshold of goods and services.

Concentric zone model

Ernest Burgess (1925) described city growth as five concentric rings outward from the central business district: transition, working-class housing, better residential and commuter zones, driven by invasion and succession.

Transit Oriented Development

TOD concentrates high-density, mixed-use, pedestrian-friendly development within roughly 500–800 m of a transit station to reduce car dependence. India's Metro Rail Policy 2017 makes TOD mandatory for metro projects.

Census town criteria

A census town must have at least 5,000 population, a density above 400 persons per km², and at least 75% of male main workers engaged in non-agricultural work. It is urban in character but has no municipality.

Smart Cities Mission

Launched on 25 June 2015 by the Government of India, it covers 100 cities and funds area-based development plus pan-city smart solutions, delivered through city-level special purpose vehicles with central and state funding.