GATE GG cheat sheet

Linear algebra: eigenvalues solve det(A − λI) = 0; the trace equals the sum of eigenvalues and the determinant equals their product, so one root can be checked against the other.

Gauss elimination with partial pivoting keeps round-off small; a square system has a unique solution only when the coefficient matrix is nonsingular (det ≠ 0).

First-order ODE dy/dx + P(x)y = Q(x) has integrating factor exp(∫P dx); the solution is y·IF = ∫Q·IF dx + C.

Second-order linear ODE with constant coefficients: roots of the characteristic equation give e^(r1x), e^(r2x) for distinct real roots, e^(αx)(C1 cos βx + C2 sin βx) for complex α ± iβ, and xe^(rx) for a repeated root.

For a particle in plane motion, velocity v = ω × r and acceleration a = α × r + ω × (ω × r); for circular motion at constant speed the acceleration is v²/r directed to the centre.

Newton's second law in a rotating frame adds Coriolis −2m(ω × v) and centrifugal −mω × (ω × r) terms; these vanish in an inertial frame.

Stress tensor σ_ij is symmetric for non-polar continua; principal stresses are its eigenvalues, and maximum shear stress equals (σ1 − σ3)/2 with σ1 ≥ σ2 ≥ σ3.

For a simply supported beam of span L under central point load W, maximum deflection is WL³/(48EI) at midspan and maximum bending moment is WL/4.

For a cantilever of length L with end load W, tip deflection is WL³/(3EI) and maximum bending moment is WL at the fixed end.

Torsion of a solid circular shaft: shear stress τ = Tr/J with J = πd⁴/32, and angle of twist θ = TL/(GJ); stress is maximum at the outer surface.

Thermodynamics: for an ideal gas, ΔU = nC_vΔT always; ΔH = nC_pΔT; and C_p − C_v = R per mole.

Entropy change for an ideal gas: ΔS = nC_v ln(T2/T1) + nR ln(V2/V1), or equivalently nC_p ln(T2/T1) − nR ln(P2/P1).

Carnot efficiency of a heat engine is 1 − T_C/T_H with absolute temperatures; COP of a reversible refrigerator is T_C/(T_H − T_C).

For steady one-dimensional conduction through a plane wall, heat rate q = kAΔT/L; thermal resistances in series add, R = L/(kA).

Thevenin and Norton equivalents: R_th is found by deactivating independent sources (short voltage sources, open current sources); V_th is the open-circuit voltage.

In an ideal transformer, V1/V2 = N1/N2 and I1/I2 = N2/N1; the reflected impedance is Z1 = (N1/N2)²Z2.

Calorific value: higher heating value (HHV) includes latent heat of water vapour in products; lower heating value (LHV) excludes it, so LHV = HHV − m_w·h_fg per unit fuel.

Solid-fuel combustion stoichiometry uses the ultimate analysis (C, H, O, S, N, ash, moisture); theoretical air from mass of O2 required per kg fuel divided by 0.232 (O2 mass fraction of air).

Excess air ratio = actual air / stoichiometric air; flue-gas O2 rises with excess air, and dry flue gas loss is the dominant stack loss in boilers.

Solar geometry: solar declination δ ≈ 23.45° sin[360(284 + n)/365]; at solar noon the altitude angle α = 90° − |φ − δ| for latitude φ.

For a tilted collector surface, the tilt factor R_b = cos θ / cos θ_z (ratio of beam radiation on tilted to horizontal surface); it depends on surface azimuth, tilt, latitude, declination, and hour angle.

Flat-plate collector useful gain: Q_u = A_c[F_R(τα)G − F_R U_L(T_i − T_a)]; efficiency is Q_u/(A_c G), and the collector stagnates when absorbed gain equals losses.

PV cell: short-circuit current I_sc scales with irradiance, open-circuit voltage V_oc falls logarithmically with temperature (~−0.3%/°C to −0.5%/°C for silicon), and fill factor FF = P_max/(V_oc·I_sc).

Wind power in the swept area is P = ½ρAV³; Betz limit caps extractable fraction at 16/27 ≈ 0.593 of the wind's power.

Wind turbine tip-speed ratio λ = ωR/V; power coefficient C_p peaks near λ ≈ 7–8 for modern three-blade horizontal-axis turbines.

Hydro power: P = ρgQH·η_overall; for a Pelton wheel the bucket speed is about half the jet speed for maximum efficiency, and jet velocity v = √(2gH).

Battery metrics: specific energy (Wh/kg), specific power (W/kg), coulombic efficiency = discharge capacity / charge capacity, and C-rate = current / rated capacity in Ah.

Fuel cell thermodynamics: reversible cell voltage E = −ΔG/(nF); the Nernst correction adds (RT/nF)ln(Π reactant/product activities), and activation, ohmic, and concentration overpotentials reduce the operating voltage.

Biomass and waste-to-energy: anaerobic digestion yields biogas (~50–70% CH4), while gasification yields producer gas (CO + H2) whose LHV is roughly 4–6 MJ/Nm³.

Nuclear fission: each U-235 fission releases about 200 MeV (~3.2×10⁻¹¹ J); the multiplication factor k = 1 is critical, k > 1 supercritical, k < 1 subcritical.

Payback and LCOE: simple payback = capital cost / annual savings; LCOE = (sum of discounted costs)/(sum of discounted energy), both evaluated over system lifetime.

Energy storage sizing for a load profile: required capacity = ∫(load − generation) dt over the deficit period divided by round-trip efficiency and depth of discharge.