GATE XH cheat sheet

Stress σ = F/A; engineering strain ε = ΔL/L₀; Young's modulus E = σ/ε in the linear elastic region.

For a simply supported beam with central point load W and span L: maximum bending moment M = WL/4 and deflection δ = WL³/(48EI).

Second law for rotation: Στ = Iα; parallel-axis theorem I = I_cm + Md².

Carnot efficiency of a heat engine: η = 1 − T_c/T_h, with temperatures in kelvin; no real engine between the same reservoirs exceeds this.

Reynolds number Re = ρvD/μ; pipe flow is treated as laminar below Re ≈ 2000 and fully turbulent above ≈ 4000.

Bernoulli along a streamline: p + ½ρv² + ρgh = constant for steady, incompressible, inviscid flow.

For a first-order linear ODE y′ + Py = Q, the integrating factor is e^∫P dx; solution y·e^∫P dx = ∫Q·e^∫P dx dx.

Duty cycle of a periodic square wave = (on time / period) × 100%; average value of a periodic waveform equals its integral over one period divided by the period.

Boolean identity: A + A′B = A + B; De Morgan: (AB)′ = A′ + B′ and (A + B)′ = A′B′.

A 2-input XOR outputs 1 only when inputs differ; an n-input XOR outputs 1 when the count of 1s is odd (parity).

Nyquist rate: a signal band-limited to f_max must be sampled at more than 2f_max to allow exact reconstruction.

Shannon capacity: C = B·log₂(1 + S/N); doubling bandwidth doubles capacity, while capacity grows only logarithmically with SNR.

In TCP, the congestion window grows by roughly one MSS per RTT in slow start (exponential) and by one MSS per RTT in congestion avoidance (linear).

Amdahl's law: speedup = 1 / ((1 − p) + p/s), where p is the parallel fraction and s the parallel speedup.

Cache average access time = hit time + miss rate × miss penalty; a two-level cache multiplies the miss rates of successive levels.

Binary search on a sorted array of n elements runs in O(log n) time and requires O(1) extra space iteratively.

Scale factor in architectural drawings: 1:100 means 1 unit on the drawing equals 100 units on site; area scales as the square of the ratio.

Floor Area Ratio (FAR) = total built-up floor area / plot area; ground coverage = footprint area / plot area.

In a plan, a north arrow fixes orientation; sun path diagrams show the sun's altitude and azimuth by latitude and season.

Setback is the minimum mandatory open distance between the building line and the plot boundary, governed by local bye-laws.

Concrete mix designation M25 means characteristic compressive cube strength of 25 N/mm² at 28 days.

Standard brick size (Indian modular) is 190 × 90 × 90 mm; with mortar the nominal module is 200 × 100 × 100 mm.

Thermal comfort depends on air temperature, radiant temperature, humidity, air velocity, and clothing/metabolic rate, not temperature alone.

A load-bearing wall transfers load directly to the foundation along its length; a framed structure transfers loads through beams and columns to footings.

Ideal gas law: PV = nRT; at STP one mole of an ideal gas occupies about 22.4 L.

First law of thermodynamics: ΔU = Q − W, with W the work done by the system; for an adiabatic process Q = 0.

pH = −log₁₀[H⁺]; a buffer resists pH change and is most effective when pH ≈ pKa of its weak acid.

Rate law for a first-order reaction: [A] = [A]₀e^(−kt); half-life t½ = ln2/k, independent of initial concentration.

Newton's law of gravitation F = Gm₁m₂/r²; orbital speed for a circular orbit v = √(GM/r).

Snell's law: n₁sinθ₁ = n₂sinθ₂; total internal reflection occurs only when light travels from denser to rarer medium beyond the critical angle.

DNA base pairing: adenine pairs with thymine (two hydrogen bonds), guanine with cytosine (three); in RNA, uracil replaces thymine.

Michaelis–Menten rate v = V_max[S]/(K_m + [S]); K_m is the substrate concentration at half-maximal velocity.