GATE ME cheat sheet

Euler's theorem: if f is homogeneous of degree n, then x·∂f/∂x + y·∂f/∂y = n·f; for f(x,y) homogeneous degree n, ∂f/∂x and ∂f/∂y are homogeneous of degree n−1.

Taylor series about a: f(x) = Σ f⁽ⁿ⁾(a)(x−a)ⁿ/n!; for two variables use f(x,y) = f(a,b) + (h∂x + k∂y)f + (1/2!)(h∂x + k∂y)²f + …, h = x−a, k = y−b.

Eigenvalues from characteristic equation |A − λI| = 0; trace = Σλᵢ, determinant = Πλᵢ. A real symmetric matrix has real eigenvalues and orthogonal eigenvectors; Cayley–Hamilton: A satisfies its own characteristic polynomial.

Gauss elimination with partial pivoting reduces round-off; a square system has a unique solution iff rank(A) = rank([A|b]) = n; inconsistent iff rank(A) < rank([A|b]).

Fourier series on (−L, L): a₀ = (1/L)∫f dx, aₙ = (1/L)∫f cos(nπx/L)dx, bₙ = (1/L)∫f sin(nπx/L)dx; even functions give bₙ = 0, odd functions give aₙ = 0.

Laplace transforms: L{e^{at}} = 1/(s−a), L{tⁿ} = n!/s^{n+1}, L{sin ωt} = ω/(s²+ω²), L{f′} = sF(s) − f(0); first shifting theorem L{e^{at}f(t)} = F(s−a).

Probability: for independent events P(A∩B) = P(A)P(B); Bayes' theorem P(Aᵢ|B) = P(B|Aᵢ)P(Aᵢ)/ΣP(B|Aⱼ)P(Aⱼ); binomial mean np, variance np(1−p); Poisson mean = variance = λ.

Numerical integration: Trapezoidal error ∝ h², Simpson's 1/3 rule error ∝ h⁴ and needs an even number of intervals; Newton–Raphson iteration xₙ₊₁ = xₙ − f(xₙ)/f′(xₙ) converges quadratically.

Vector calculus: Green's theorem ∮(P dx + Q dy) = ∬(∂Q/∂x − ∂P/∂y)dA; divergence of curl is zero; a field is conservative iff curl F = 0 on a simply connected domain.

Mohr's circle: centre at (σx+σy)/2, radius R = √[((σx−σy)/2)² + τxy²]; principal stresses σ₁,₂ = centre ± R; maximum shear τmax = R and occurs on planes at 45° to the principal planes.

Thin cylindrical shell under internal pressure p: hoop stress = pr/t, longitudinal stress = pr/2t; thin spherical shell: σ = pr/2t. Thin-wall assumption requires r/t ≥ 10.

Euler buckling load Pcr = π²EI/(Leff)²; effective length Leff = L (both ends pinned), 0.5L (fixed–fixed), 0.7L (fixed–pinned), 2L (fixed–free). Rankine–Gordon formula combines Euler and crushing for intermediate slenderness.

Torsion of a circular shaft: τ = Tr/J, angle of twist θ = TL/GJ; for a hollow shaft J = π(do⁴ − di⁴)/32. Maximum shear theory (Tresca) and von Mises criterion differ by a factor √3 in pure shear.

S-N curve: fatigue strength decreases with cycles; endurance limit applies only to steels below ~10⁶–10⁷ cycles. Goodman line: σa/σe + σm/σu = 1; Soderberg replaces σu with yield strength.

Shafts with combined loading: equivalent twisting moment Te = √(M² + T²) (maximum shear theory); equivalent bending moment Me = ½[M + √(M² + T²)].

Kinematics of a particle: v = u + at, s = ut + ½at², v² = u² + 2as for constant acceleration; projectile range R = u²sin2θ/g on level ground, maximum at θ = 45°.

Vibrations: natural frequency ωn = √(k/m) for a spring-mass system, ωn = √(g/δst) for a beam; damping ratio ζ = c/(2√(km)); logarithmic decrement δ = 2πζ/√(1−ζ²) for small ζ.

Gear trains: velocity ratio = product of teeth on driven gears / product of teeth on driving gears; for a simple gear train ratio = Ndriven/Ndriver; epicyclic trains are solved by the tabular (relative speed) method.

Continuity for steady incompressible flow: A₁V₁ = A₂V₂; Bernoulli along a streamline (inviscid, steady, incompressible): p/ρg + V²/2g + z = constant.

Reynolds number Re = ρVD/μ = VD/ν; pipe flow is laminar below Re ≈ 2000 and turbulent above ≈ 4000. Hagen–Poiseuille laminar flow: Δp = 128μLQ/(πD⁴), friction factor f = 64/Re.

Darcy–Weisbach head loss hf = fLV²/(2gD); in turbulent smooth pipes use Blasius f = 0.316Re^(−1/4) for Re < 10⁵; Moody chart adds relative roughness ε/D.

Boundary layer: Blasius laminar δ ∝ x/√Rex, local Cf = 0.664/√Rex; separation occurs where adverse pressure gradient makes wall shear zero. Drag on a flat plate = ½ρV²A·CD.

Conduction: Fourier law q = −kA dT/dx; plane wall resistance R = L/kA, cylindrical R = ln(r₂/r₁)/(2πkL); critical insulation radius for a cylinder rc = k/h.

Fins: efficiency η = tanh(mL)/(mL) for an adiabatic-tip rectangular fin, m = √(hP/kA); fin effectiveness ε = Qfin/(hAθb).

Convection: Nu = hL/k; laminar flat plate Nu = 0.664Re^0.5Pr^(1/3); Dittus–Boelter for turbulent pipe flow Nu = 0.023Re^0.8Pr^n (n = 0.4 heating, 0.3 cooling).

Radiation: Stefan–Boltzmann Q = εσA(T⁴ − Tsurr⁴), σ = 5.67×10⁻⁸ W/m²K⁴; Wien's law λmaxT = 2898 μm·K; Kirchhoff's law ε = α at a given wavelength and temperature.

Rankine cycle: thermal efficiency = (h₁ − h₂)/(h₁ − hf); boiler pressure raises efficiency but increases moisture at turbine exit; reheat and regeneration reduce this penalty.

Reciprocating compressor: volumetric efficiency ηv = 1 + C − C(p₂/p₁)^(1/n); work for polytropic compression = (n/(n−1))p₁V₁[(p₂/p₁)^((n−1)/n) − 1].

Hooke's law in 3D: εx = (σx − ν(σy + σz))/E; bulk modulus K = E/[3(1−2ν)]; shear modulus G = E/[2(1+ν)]; for an isotropic material only two of E, G, K, ν are independent.

Engineering stress–strain: σ = P/A₀, ε = ΔL/L₀; true stress σt = σ(1+ε) and true strain εt = ln(1+ε) up to necking; Considère criterion dσt/dεt = σt predicts necking.

Hall–Petch relation: yield strength σy = σ₀ + kd^(−1/2), where d is grain diameter; smaller grains raise yield strength and toughness.

Cooling curves: eutectoid plain carbon steel (0.76% C) transforms to pearlite; martensite forms by diffusionless shear on rapid quenching below the Ms temperature and is hard and brittle; tempering restores toughness.

T-T-T diagrams: the nose of the curve sets the critical cooling rate; alloying elements (Cr, Ni, Mo) shift the curve right, improving hardenability, which is measured by the Jominy end-quench test.

Fick's first law: J = −D dC/dx; second law ∂C/∂t = D ∂²C/∂x²; diffusion coefficient D = D₀exp(−Q/RT), so rate rises strongly with temperature.

Fracture mechanics: K = Yσ√(πa); fast fracture occurs when K ≥ KIc. Fatigue crack growth follows Paris law da/dN = C(ΔK)^m.

Creep occurs at T/Tm > 0.4; steady-state creep rate follows Norton's law ε̇ = Aσⁿexp(−Q/RT).

Phase rule F = C − P + 1 (vapour pressure fixed); a eutectic reaction on a binary phase diagram is L → α + β at a single temperature and composition.

Metal cutting: Merchant's relation 2φ + β − α = 90°; shear angle φ, friction angle β, rake angle α. Cutting force Fc = Rcos(β − α) and thrust Ft = Rsin(β − α).

Taylor's tool life: VTⁿ = C, where n ≈ 0.1–0.15 for HSS and 0.4–0.6 for carbide; increasing speed above the optimum shortens tool life sharply.

Machining time for turning = πDL/(f·N); for a milling cut with N teeth, feed per revolution = fz·N; MRR in turning = V·f·d.

Casting: Chvorinov's rule solidification time t = B(V/A)²; a riser must solidify after the casting, so its V/A ratio must be larger.

Welding: heat input per unit length = VI/v (J/mm); in arc welding, arc voltage and current set the heat input; preheating reduces the cooling rate and prevents martensite cracking in steels.

Gating design: sprue profile follows continuity A₁V₁ = A₂V₂ with V = √(2gh); pour time from the choke area and the required fill rate.

Inventory: EOQ = √(2DS/H), with D annual demand, S ordering cost, H holding cost per unit per year; total cost minimum occurs where ordering cost = holding cost.

Forecasting: moving average smooths random variation; exponential smoothing Ft+1 = αDt + (1−α)Ft; larger α tracks recent changes faster.

CPM/PERT: critical path has zero total float; expected activity time te = (a + 4m + b)/6 and variance = ((b − a)/6)².

Work study: standard time = observed time × rating factor + allowances; line balancing efficiency = Σ task times/(number of stations × cycle time).

Percentage change: successive changes of x% and y% give net change x + y + xy/100; a rise of x% followed by a fall of x% leaves a net loss of x²/100 percent.

Ratio and proportion: if a:b = c:d then ad = bc; for mixtures, the alligation rule gives the ratio of two components as (mean − cheaper):(dearer − mean).

Time-speed-distance: distance = speed × time; for a train of length L crossing a platform of length P, time = (L + P)/speed. Boats: downstream speed = u + v, upstream = u − v.

Work: if A alone takes x days and B alone y days, together they take xy/(x + y) days; using the LCM method, total work = LCM(x, y) units.

Permutations and combinations: nPr = n!/(n − r)!, nCr = n!/[r!(n − r)!]; number of arrangements of a word with repeated letters = n!/(p!q!…).

Probability: P(A ∪ B) = P(A) + P(B) − P(A ∩ B); for independent events multiply; for exactly one event, P(A) + P(B) − 2P(A ∩ B).

Data interpretation: mean = Σx/n; median is the middle value of ordered data; for grouped data, mode = L + [(f₁ − f₀)/(2f₁ − f₀ − f₂)]h.

Series: arithmetic sum = (n/2)(2a + (n−1)d); geometric sum = a(1 − rⁿ)/(1 − r) for r ≠ 1; sum of first n squares = n(n+1)(2n+1)/6.

Logical reasoning: syllogisms are tested by Venn diagrams; a statement and its contrapositive are logically equivalent, while the converse is not.

Limit test for convergence: if lim(n→∞) aₙ₊₁/aₙ = L, the series converges when L < 1, diverges when L > 1, and is inconclusive at L = 1 (ratio test).

Partial derivative test for f(x,y): at a critical point, let D = fxx·fyy − (fxy)²; D > 0 and fxx > 0 gives a minimum, D > 0 and fxx < 0 gives a maximum, D < 0 gives a saddle point.

Double integral in polar form: ∬f dA = ∫∫f(r,θ) r dr dθ; the Jacobian r must be included when converting from Cartesian to polar coordinates.

Line integral of a conservative field equals the potential difference between the endpoints and is path independent; the closed-loop integral is zero.

If A is n×n with distinct eigenvalues, its eigenvectors are linearly independent and A is diagonalisable as A = PDP⁻¹, where D holds the eigenvalues.

For a system Ax = 0 with more unknowns than equations, a non-trivial solution always exists because the rank is less than n.

Strain energy: axial U = P²L/(2AE); bending U = ∫M²dx/(2EI); torsion U = T²L/(2GJ); resilience is strain energy per unit volume.

Thermal stress in a fully constrained bar: σ = αEΔT (compressive on heating); if the bar is free to expand, the thermal stress is zero.

Thick cylinders (Lame's equations): σr = A − B/r², σθ = A + B/r²; at the inner radius the hoop stress is maximum and equals p(ro² + ri²)/(ro² − ri²).

Beam deflection by Macaulay's method: integrate EI d²y/dx² = M(x), including the step functions in brackets; boundary conditions at supports fix the constants.

Torsion of a thin-walled closed tube: τ = T/(2Am t), where Am is the area enclosed by the mean centreline and t is the wall thickness (Bredt–Batho formula).

Principal strain: ε₁,₂ = (εx + εy)/2 ± √[((εx − εy)/2)² + (γxy/2)²]; the maximum shear strain equals ε₁ − ε₂.

Heat exchanger LMTD = (ΔT₁ − ΔT₂)/ln(ΔT₁/ΔT₂); for a counter-flow exchanger the LMTD is higher than for parallel flow with the same terminal temperatures.

Effectiveness-NTU: ε = Qactual/Qmax, NTU = UA/Cmin; for a counter-flow exchanger ε = [1 − exp(−NTU(1 − Cr))]/[1 − Cr·exp(−NTU(1 − Cr))].

Second law: the Clausius inequality ∮δQ/T ≤ 0; entropy change for an ideal gas Δs = cp ln(T₂/T₁) − R ln(p₂/p₁).

Carnot efficiency = 1 − Tc/Th (absolute temperatures); for a refrigerator the COP is Tc/(Th − Tc); no real cycle between the same reservoirs can exceed these values.

Nozzle flow: for air, choked flow occurs at the critical pressure ratio p*/p₀ = (2/(γ + 1))^(γ/(γ−1)) ≈ 0.528; the exit velocity is limited by choking.

Psychrometry: relative humidity φ = pv/psat; the dew point is the temperature at which pv equals the saturation pressure; the wet-bulb temperature equals the dry-bulb temperature at saturation.

Hardness: Brinell HB = 2P/[πD(D − √(D² − d²))]; Vickers HV = 1.854P/d², where d is the mean diagonal in mm and P is in kgf.

Impact testing: Charpy V-notch energy drops sharply with decreasing temperature and shows a ductile-to-brittle transition in BCC steels, but not in FCC metals such as austenitic stainless steel.

Fatigue limit (endurance limit) exists for ferrous alloys but not for aluminium alloys, whose S-N curve continues to fall with increasing cycles.

Stress concentration factor Kt = σmax/σnom; for a small notch in a ductile material, the notch sensitivity q = (Kf − 1)/(Kt − 1) reduces the effective factor.

Recrystallisation temperature ≈ 0.4 Tm (absolute); cold work raises strength and dislocation density, while annealing restores ductility by recovery, recrystallisation and grain growth.

Solid solution strengthening: solute atoms of a different size distort the lattice and pin dislocations; precipitation hardening requires solution treatment, quenching and ageing.

Sheet metal: blanking force = τ·t·L, where L is the perimeter cut; the clearance is about 5–10% of sheet thickness and the punch is made smaller than the die for blanking.

Rolling: maximum draft Δhmax = μ²R; the roll force can be reduced by using smaller rolls, reducing friction or applying back tension.

Extrusion: for a perfectly plastic material with ideal deformation, the extrusion pressure is p = σy ln(A₀/Af); the actual pressure is higher because of friction and redundant work.

Forging: the forging load for an open-die compression can be estimated by integrating the flow stress over the contact area; the barrelling effect comes from friction at the die faces.

Welding defects: lack of penetration, porosity, slag inclusion and undercut; in arc welding, electrode polarity and current magnitude control penetration depth.

Quality control: for a control chart, UCL = x̄ + 3σ/√n for the mean; process capability Cp = (USL − LSL)/(6σ), and Cpk accounts for the centring of the process.

Work measurement: allowance factor converts normal time to standard time; the standard time equals normal time/(1 − allowance fraction) when allowances are given as a percentage of the total time.

Simple and compound interest: SI = PRT/100; CI amount = P(1 + R/100)ⁿ; the difference between CI and SI for 2 years equals P(R/100)².

Profit and loss: profit% = (SP − CP)/CP × 100; if two articles are sold at the same price with equal percentage gain and loss, the net result is a loss of x²/100 percent.

Pipes and cisterns: if a pipe fills a tank in x hours and a leak empties it in y hours, the net fill time is xy/(y − x) when y > x.

Clocks and calendars: the minute hand gains 5.5° per minute on the hour hand; the day of the week shifts by 1 for an ordinary year and by 2 for a leap year.

Odd one out and analogy questions rely on the relation between the words (cause–effect, part–whole, synonym/antonym); check the direction of the relation before choosing.

Reading comprehension: the answer must be supported by the passage itself; extreme options with words such as 'always' and 'never' are usually wrong.

First-order linear ODE: dy/dx + Py = Q has integrating factor IF = e^∫P dx and solution y·IF = ∫Q·IF dx + C.

Exact differential equation: M dx + N dy = 0 is exact when ∂M/∂y = ∂N/∂x; the solution is found by integrating M with respect to x and matching the terms.

Second-order linear ODE with constant coefficients: solve the auxiliary equation; for repeated roots the second solution is x·e^(mx); for complex roots α ± iβ the solution is e^(αx)(C₁cos βx + C₂sin βx).

Method of undetermined coefficients: for a forcing term of the form e^(ax), try yp = A e^(ax); if e^(ax) is already a homogeneous solution, multiply the trial by x.

Inverse Laplace transform by partial fractions: 1/[(s + a)(s + b)] = [1/(b − a)]·[1/(s + a) − 1/(s + b)]; use the convolution theorem for products of transforms.

Fourier transform of a Gaussian is Gaussian; the Fourier transform of a real even function is real and even, and the transform of a real odd function is imaginary and odd.

Four-bar Grashof condition: s + l ≤ p + q (s shortest, l longest) guarantees at least one link can fully rotate; equality gives a change-point mechanism.

Instantaneous centre method: the velocity of a point on a link is perpendicular to the line joining the point to the instantaneous centre; Kennedy's theorem applies to three bodies.

Slider-crank: the velocity of the slider is approximately v = ωr(sin θ + (sin 2θ)/(2n)), where n = l/r; the acceleration has a secondary component at twice the crank frequency.

Governor: for a Porter governor, the height h = g/(ω²) for the simple case; the sensitiveness is the ratio of the range of speed to the mean speed.

Flywheel: the energy fluctuation ΔE = Iω²Cs, where Cs is the coefficient of fluctuation of speed; a larger flywheel reduces speed fluctuation.

Gear interference is avoided by undercutting the pinion; the minimum number of teeth on a 20° full-depth pinion with a rack is about 18.

Hydrostatic force on a vertical plane surface: F = ρg·h̄·A, where h̄ is the depth of the centroid; the centre of pressure lies below the centroid at depth h̄ + IG/(A h̄).

A floating body is stable when the metacentre lies above the centre of gravity; the metacentric height GM = I/V − BG, where I is the second moment of the waterplane area.

Manometry: p₁ − p₂ = ρm·g·h for a differential U-tube manometer, where ρm is the manometric fluid density; the fluid in the pipe contributes ρg·h terms that must be included.

Centrifugal pump: head coefficient and specific speed Ns = N√Q/H^(3/4); the best efficiency point fixes the design duty; cavitation is avoided when NPSH available > NPSH required.

Pump and system curves: the operating point is the intersection of the pump characteristic and the system curve; two identical pumps in series double the head, in parallel double the flow at the same head.

Pelton wheel: the bucket speed is about half the jet speed for maximum efficiency; the hydraulic efficiency is η = 2u(V − u)(1 + cos β)/V².

Crystal structures: BCC has 2 atoms per cell and packing factor 0.68; FCC has 4 atoms and 0.74; HCP has 6 atoms and 0.74. FCC and HCP are close-packed; BCC is not.

Miller indices: take reciprocals of the intercepts, clear fractions; for a cubic crystal, the interplanar spacing is d = a/√(h² + k² + l²).

Bragg's law: 2d sin θ = nλ; X-ray diffraction identifies phases by their d-spacings and relative intensities.

Point defects: Schottky defects create vacancies in ionic crystals and preserve charge neutrality; Frenkel defects place an ion in an interstitial site.

The iron–carbon diagram: ferrite is BCC iron with very low carbon; austenite is FCC and can dissolve up to 2.11% C; cementite (Fe₃C) is hard and brittle.

Lever rule on a tie line: fraction of phase α = (Cβ − C₀)/(Cβ − Cα); the fraction of the other phase is the complement.

Micrometer least count = pitch/number of circular scale divisions, typically 0.5 mm/50 = 0.01 mm; vernier calliper least count = 1 MSD − 1 VSD.

Limits and fits: tolerance is the difference between the upper and lower limits; in a hole-basis system the lower deviation of the hole is zero, and clearance fits have the shaft maximum below the hole minimum.

Surface roughness: Ra is the arithmetic mean deviation of the profile; Rz is the maximum height of the profile; a smaller Ra means a smoother surface.

Tool wear: flank wear is measured on the flank face and crater wear on the rake face; tool life ends when the average flank wear reaches the specified criterion, typically 0.3 mm.

Grinding: the specific energy is high because the grits have a large negative rake angle; dressing restores the wheel form, while truing restores the concentricity.

NC and CNC: G-codes define motion (G01 linear interpolation), M-codes handle machine functions (M03 spindle clockwise); the feed rate is specified in mm/min for milling and mm/rev for turning.

Averages: if the average of n numbers is A and one number x is replaced by y, the new average is A + (y − x)/n; the average of consecutive numbers equals the middle term.

Ages and partnership: in a partnership, profit shares are proportional to capital × time; for ages, set up equations from the present age and the given future/past relations.

Permutation with restrictions: the number of arrangements of n items with r specified items always together is (n − r + 1)!·r!; the number with them never together is n! − that value.

Probability with dice and cards: a standard die has 6 equally likely outcomes, and a standard deck has 52 cards with 4 suits of 13; the probability of a face card is 12/52 = 3/13.

Data sufficiency and comparison: compute both quantities fully before comparing; do not assume a figure is drawn to scale or that a marked angle is a right angle.

Verbal ability: subject–verb agreement follows the subject, not the intervening phrase; 'none' and 'each' are usually singular, and 'fewer' applies to countable nouns while 'less' applies to uncountable ones.

Cauchy–Riemann equations: uₓ = vy and uy = −vₓ; a function is analytic where these hold and the partial derivatives are continuous.

Cauchy's integral formula: f(a) = (1/2πi)∮f(z)/(z − a) dz for a simple closed contour enclosing a; the residue theorem gives ∮f dz = 2πi·Σ residues inside.

A singularity is a pole of order n if (z − a)ⁿf(z) is analytic and non-zero at a; the residue at a simple pole is lim(z→a)(z − a)f(z).

Taylor series converges inside the nearest singularity; the Laurent series includes negative powers and its coefficient a₋₁ is the residue.

Newton's forward interpolation: f(x) = f₀ + pΔf₀ + p(p−1)Δ²f₀/2! + …, where p = (x − x₀)/h; use the backward formula near the end of the table.

Runge–Kutta fourth order: k₁ = hf(x₀,y₀), k₂ = hf(x₀ + h/2, y₀ + k₁/2), k₃ = hf(x₀ + h/2, y₀ + k₂/2), k₄ = hf(x₀ + h, y₀ + k₃); y₁ = y₀ + (k₁ + 2k₂ + 2k₃ + k₄)/6.

A block on a horizontal plane: the limiting friction F = μN; on an inclined plane the block slides when tan θ > μ, so the angle of repose equals tan⁻¹μ.

Screw jack: the torque to raise a load W is T = W·rm·tan(α + φ), where α is the helix angle and φ is the friction angle; the jack is self-locking when φ > α.

Band brake: the tension ratio is T₁/T₂ = e^(μθ), where θ is the angle of contact in radians; the braking torque is (T₁ − T₂)r.

Single-plate clutch: the torque capacity is μW·rm, with rm = (r₁ + r₂)/2 for uniform wear and rm = (2/3)(r₁³ − r₂³)/(r₁² − r₂²) for uniform pressure.

Belt drive: the ratio of tensions is T₁/T₂ = e^(μθ/sin β) for a V-belt with groove angle 2β; the centrifugal tension Tc = mv² reduces the power transmitted.

Rolling resistance is much smaller than sliding friction; the coefficient of rolling resistance is the ratio of the resisting force to the normal load.

Orifice meter: the actual discharge Q = Cd·A₀·√(2gh), where Cd ≈ 0.6; the coefficient of contraction Cc = Ac/A₀ and the coefficient of velocity Cv = Cd/Cc.

Venturimeter: Q = Cd·A₁A₂√(2gh)/√(A₁² − A₂²), where h is the manometer reading converted to the equivalent head of the flowing fluid.

A pitot-static tube measures the stagnation pressure; the velocity is V = √(2Δp/ρ); the pitot tube coefficient accounts for imperfect stagnation.

Momentum equation: ΣF = ρQ(V₂ − V₁); the force on a bend carrying a flow Q is obtained by resolving the momentum change and the pressure forces.

Flow over a weir: for a rectangular weir Q = (2/3)Cd·b√(2g)·H^(3/2); for a V-notch Q = (8/15)Cd√(2g)·tan(θ/2)·H^(5/2).

Laminar flow between parallel plates: the velocity profile is parabolic, and the maximum velocity is 1.5 times the average velocity; for a circular pipe it is 2 times the average.

Ductile fracture shows cup-and-cone necking; brittle fracture shows a flat, bright cleavage surface with little plastic deformation.

The Griffith criterion for brittle fracture is σf = √(2Eγs/(πa)) for a through-thickness crack of length a in a plate under plane stress.

Galvanic corrosion occurs when two dissimilar metals are coupled in an electrolyte; the more anodic metal corrodes, so the area of the anode should be large relative to the cathode.

Cathodic protection: a sacrificial anode (Zn, Mg) protects steel; an impressed-current system uses an external DC source.

Dye penetrant testing detects surface-breaking defects; ultrasonic testing detects internal flaws; radiography shows internal voids but is poor for tight planar cracks.

Magnetic particle inspection works only on ferromagnetic materials and detects surface and near-surface defects; eddy current testing suits conductive materials.

Sand casting: the sprue is tapered downward to avoid air entrainment; the pouring basin and sprue act as a reservoir; the gating ratio describes the areas of the sprue, runner and ingate.

Shell moulding and investment casting give better surface finish and dimensional accuracy than green sand casting; die casting suits low-melting-point metals in high volume.

Friction stir welding is a solid-state process that joins aluminium alloys without melting; it avoids porosity and hot cracking found in fusion welding.

Resistance spot welding: heat generated = I²Rt; the current, time and electrode force control the nugget size; the resistance is highest at the faying surfaces.

Brazing and soldering use a filler that melts below the base metal; brazing operates above 450 °C and soldering below it; capillary action fills the joint.

Powder metallurgy: the steps are powder production, blending, compaction and sintering; sintering occurs below the melting point by diffusion and reduces porosity.

Seating arrangement: for a linear row, 'left of' and 'right of' are fixed by the facing direction; for a circular arrangement facing the centre, the left-hand neighbour is clockwise from the person.

Blood relations: draw the family tree and mark gender; a 'maternal uncle' is the mother's brother, and a 'paternal grandfather' is the father's father.

Direction sense: net displacement is found by resolving each move into north–south and east–west components; the final direction uses the resultant vector.

Series completion: check the differences, then the ratios, then the alternating patterns; for letter series, convert letters to positions 1–26 to reveal the rule.

Bar and pie charts: a pie chart shows shares of 360°, so each percentage equals 3.6°; the total value must be known to convert a share into an absolute quantity.

Venn diagrams: the number in exactly one category is found by subtracting the overlaps; the total is the sum of all regions, including the intersection of all sets.