GATE GE cheat sheet

Beam bending: σ = My/I and τ = VQ/(Ib); maximum shear in a rectangular section is 1.5·V/A, occurring at the neutral axis.

Cantilever with end point load P: δ_max = PL³/(3EI); simply supported with central load: δ_max = PL³/(48EI).

Mohr's circle: centre at (σx+σy)/2, radius R = √[((σx−σy)/2)² + τxy²]; principal stresses = centre ± R.

Torsion: τ = Tr/J, angle of twist θ = TL/(GJ); for a solid circular shaft J = πd⁴/32.

First Law closed system: Q − W = ΔU; for an ideal gas ΔU = m·cv·ΔT regardless of path.

Carnot efficiency η = 1 − T_L/T_H with absolute temperatures; no engine between two reservoirs exceeds it.

Reynolds number Re = ρVD/μ; pipe flow is laminar below ~2000 and fully turbulent above ~4000.

Bode rule: a single pole at ωc gives −20 dB/decade slope and −90° phase; a zero adds +20 dB/decade and +90°.

Binary search on n sorted items takes ⌈log₂n⌉ comparisons worst case; O(log n) time but requires random access and sorted input.

Master theorem for T(n)=aT(n/b)+f(n): compare f(n) with n^(log_b a) to pick Θ(n^(log_b a)), Θ(f(n)), or the regularized case.

TCP uses a three-way handshake (SYN, SYN-ACK, ACK); UDP is connectionless with an 8-byte header and no retransmission.

IPv4 header is 20 bytes minimum; TTL decrements per router hop and reaching zero triggers an ICMP Time Exceeded message.

Normalization: 2NF removes partial dependencies on a composite key; 3NF removes transitive dependencies on non-key attributes.

Cache average access time = hit time + miss rate × miss penalty; AMAT is reduced more by cutting miss rate than hit time when penalties are large.

Deadlock requires all four Coffman conditions: mutual exclusion, hold-and-wait, no preemption, and circular wait.

SHA-256 outputs 256 bits; RSA security rests on integer factorization difficulty while ECC rests on the discrete log problem on elliptic curves.

Floor Area Ratio = total built-up floor area ÷ plot area; it caps developable mass under most municipal zoning bylaws.

Ground coverage = footprint area ÷ plot area; FAR and coverage together fix the average number of storeys.

Setbacks, height limits, and ground coverage are independent bylaw controls - satisfying one does not imply compliance with the others.

Golden ratio φ ≈ 1.618; the Modulor scale and classical proportion systems derive human-scaled dimension series from such ratios.

Daylight factor = (indoor illuminance at a point ÷ simultaneous unobstructed outdoor horizontal illuminance) × 100%.

Fire egress: travel distance to the nearest exit and exit width per occupant are governed by occupant load, not by room area alone.

Seismic design philosophy: structures are detailed for ductility so that damage occurs in planned plastic hinges rather than brittle failure.

Universal design principles include equitable use, flexibility in use, simple intuitive use, and low physical effort.

Newton's second law F = ma; momentum p = mv is conserved in the absence of external net force.

Kinetic energy KE = ½mv²; work-energy theorem states net work equals change in kinetic energy.

Ideal gas law PV = nRT with R = 8.314 J/(mol·K); at STP one mole occupies about 22.4 L.

pH = −log[H⁺]; a buffer resists pH change best when pH is within ±1 of the acid's pKa (Henderson–Hasselbalch).

Le Chatelier's principle: raising pressure shifts equilibrium toward the side with fewer gas moles; raising temperature favors the endothermic direction.

DNA base pairing: adenine pairs with thymine (two hydrogen bonds), guanine with cytosine (three hydrogen bonds).

Mendel's law of segregation: alleles separate during meiosis so each gamete carries one allele per gene.

Snell's law n₁sinθ₁ = n₂sinθ₂; total internal reflection occurs only when light passes from higher to lower refractive index beyond the critical angle.

Accounting equation: Assets = Liabilities + Equity; every transaction preserves this equality.

Double entry: each transaction debits at least one account and credits at least one account by equal amounts.

Break-even quantity = Fixed costs ÷ (Selling price − Variable cost per unit); contribution margin per unit is the denominator.

Net Present Value = Σ CFt/(1+r)^t − initial outlay; accept a project when NPV > 0 at the required rate of return.

IRR is the discount rate making NPV zero; it can give conflicting rankings with NPV for mutually exclusive projects of different scale.

Current ratio = Current assets ÷ Current liabilities; quick ratio excludes inventory from current assets.

Straight-line depreciation = (Cost − Salvage value) ÷ Useful life; written-down-value method applies a fixed rate to the opening book value.

Cost of capital WACC = (E/V)·Re + (D/V)·Rd·(1 − tax rate), using market values of equity and debt.

A sonnet has 14 lines; Shakespearean rhyme scheme is abab cdcd efef gg, Petrarchan is an octave (abbaabba) plus a sestet.

Iambic pentameter is five iambic feet per line, an unstressed syllable followed by a stressed one.

In art history, Renaissance is roughly 14th–16th century, Baroque follows it, and Neoclassicism precedes Romanticism.

Impressionism (from the 1870s) emphasizes visible brushwork and light effects; Cubism (from about 1907) fractures form into multiple viewpoints.

Logical fallacy identification: ad hominem attacks the person, straw man misrepresents the opponent, and post hoc assumes causation from sequence.

Historical periodization: ancient, medieval, early modern, and modern are conventions, not sharp boundaries, and dates differ by region.

Primary sources are contemporary with the event; secondary sources interpret them - the distinction depends on use, not format.

In literary analysis, theme is the underlying idea, motif is a recurring element, and symbol carries meaning beyond its literal sense.