GATE TF cheat sheet

Linear algebra: for A x = b with n unknowns, a unique solution exists iff det(A) ≠ 0; if det(A) = 0 and rank(A) = rank([A|b]) < n, infinitely many solutions exist; if rank(A) < rank([A|b]), no solution exists.

Eigenvalues satisfy det(A − λI) = 0; trace(A) = Σλi and det(A) = Πλi, so a zero eigenvalue immediately implies singularity.

Calculus: d/dx of the definite integral ∫ from a(x) to b(x) of f(t) dt equals f(b(x))b′(x) − f(a(x))a′(x) (Leibniz rule).

For a function of two variables, a stationary point is a local minimum when fxx > 0 and fxx fyy − fxy² > 0, a local maximum when fxx < 0 and the same determinant is positive, and a saddle point when the determinant is negative.

Ordinary differential equations: the linear first-order equation y′ + P(x)y = Q(x) has integrating factor exp(∫P dx); for constant-coefficient homogeneous equations, substitute y = e^(mx) and solve the characteristic polynomial.

Probability: for independent events P(A∩B) = P(A)P(B); Bayes' rule gives P(A|B) = P(B|A)P(A)/P(B); for a Poisson variable with mean μ, mean and variance both equal μ.

Numerical methods: Newton–Raphson iteration is x(n+1) = x(n) − f(x(n))/f′(x(n)) and converges quadratically near a simple root; the trapezoidal rule error scales as O(h²) and Simpson's 1/3 rule as O(h⁴) for smooth integrands.

Material and energy balances: for steady state, accumulation = 0, so input − output + generation − consumption = 0; use mole fractions with total molar flow and mass fractions with total mass flow, never mix the two bases in one balance.

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, where m_w is mass of water formed per unit fuel.

Coal ultimate analysis reports C, H, O, N, S and ash on mass basis; stoichiometric oxygen demand per kg coal ≈ 2.67C + 8H + S − O (masses in kg), with C, H, S, O as mass fractions.

Combustion flue gas: excess air fraction e gives actual air = (1 + e) × stoichiometric air; dry flue gas analysis excludes water vapour, so O₂ and CO₂ readings must be interpreted on a dry basis.

Solar geometry: solar declination δ ≈ 23.45° sin[360(284 + n)/365]; hour angle ω = 15°(solar time − 12); cos θz = sin φ sin δ + cos φ cos δ cos ω for zenith angle at latitude φ.

Solar collector efficiency: η = (useful heat)/(incident solar energy) = F_R(τα) − F_R U_L (T_i − T_a)/G, so efficiency falls linearly as the fluid inlet temperature rises above ambient.

Wind power: available power in the wind is P = ½ ρ A v³; Betz limit caps extractable fraction at 16/27 ≈ 0.593; tip-speed ratio λ = ωR/v and power coefficient C_p(λ) determine rotor output.

Hydro power: P = ρ g Q H η_overall, where Q is volumetric flow and H is net head; for a Pelton wheel the ideal jet velocity is √(2gH) and bucket speed is about half the jet speed for maximum efficiency.

Energy storage and economics: levelized cost of electricity = (total lifetime cost discounted)/(total lifetime energy discounted); capacity factor = actual annual energy output/(rated power × 8760 h).