GATE EY cheat sheet

Stress-strain: σ = F/A, ε = ΔL/L, E = σ/ε; ductile materials show a yield plateau, brittle materials fail near the proportional limit with little plastic strain.

Beam bending: σ = My/I and deflection follows EI·d²y/dx² = M(x); maximum bending stress occurs at the farthest fiber, and I for a rectangle is bd³/12.

First Law of Thermodynamics for a closed system: Q − W = ΔU; for a steady-flow control volume, q − w = Δh + ΔKE + ΔPE.

Carnot efficiency: η = 1 − T_L/T_H (absolute temperatures); no engine operating between two reservoirs can exceed this value.

Thevenin/Norton equivalence: any linear two-terminal network reduces to V_th in series with R_th, where R_th is the resistance seen with all independent sources deactivated (voltage sources shorted, current sources opened).

Op-amp ideal rules: infinite open-loop gain, infinite input impedance, zero output impedance; with negative feedback, the input terminals are at equal potential (virtual short).

Boolean simplification: use K-maps for up to 4-5 variables; SOP from grouping 1s, POS from grouping 0s; don't-care terms may be grouped for larger simplification.

Concrete mix design and workability: lower water-cement ratio raises strength but reduces slump; nominal maximum aggregate size and grading govern paste demand.

Surveying: for a level, backsight − foresight gives the rise/fall; the rise-and-fall and height-of-instrument methods must reconcile to the same reduced levels.

Quantum mechanics: [x̂, p̂] = iħ; stationary states satisfy Ĥψ = Eψ; the infinite square well energies are E_n = n²π²ħ²/(2mL²).

Statistical mechanics: the partition function Z = Σ exp(−E_i/k_BT) generates all thermodynamic quantities via F = −k_BT ln Z.

Thermodynamic identity: dG = −S dT + V dP; at equilibrium at constant T and P, Gibbs free energy is minimized.

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

Molecular biology: transcription proceeds 5'→3' using the template strand read 3'→5'; PCR uses denaturation, primer annealing, and extension steps.

Organic reaction mechanisms: SN1 proceeds via a carbocation (racemization, first-order); SN2 is backside attack with inversion (second-order, favored by polar aprotic solvents).

Spectroscopy: Beer-Lambert A = εcl; IR absorption requires a change in dipole moment; NMR chemical shift reflects electronic shielding.

Genetics: Hardy-Weinberg equilibrium requires no mutation, migration, selection, or drift and random mating; p² + 2pq + q² = 1.

Data analysis: the standard error of the mean is s/√n; a 95% confidence interval uses t or z critical values depending on whether the population variance is known.