GATE PE cheat sheet

Beam bending: σ = My/I; maximum stress occurs at the largest distance y from the neutral axis, and I for a rectangle is bh³/12.

Euler buckling load: P_cr = π²EI/(KL)²; K = 1 pinned-pinned, 0.5 fixed-fixed, 0.7 fixed-pinned, 2 fixed-free.

Second law for a control volume: ΣF = d/dt ∫ρV dV + ∫ρV(V·n)dA; steady uniform-flow form gives ΣF = ṁ(V_out − V_in).

Carnot efficiency: η = 1 − T_L/T_H with absolute temperatures; no engine operating between two reservoirs can exceed it.

First law, closed system: Q − W = ΔU; sign convention matters, so state it before applying.

Castigliano's theorem: partial derivative of strain energy with respect to a load gives the displacement at that load, in the load's direction.

Reynolds number Re = ρVD/μ; pipe flow is laminar below about 2000 and turbulent above about 4000 for circular pipes.

Mohr's circle: centre at (σ_x + σ_y)/2, radius √[((σ_x − σ_y)/2)² + τ_xy²]; principal stresses are centre ± radius.

For a simply supported beam with central point load W and span L, maximum deflection is WL³/(48EI) at midspan.

Ideal gas law: PV = nRT; at STP one mole occupies 22.4 L, and R = 8.314 J mol⁻¹ K⁻¹.

Arrhenius equation: k = A·exp(−E_a/RT); a plot of ln k versus 1/T is linear with slope −E_a/R.

Gibbs free energy: ΔG = ΔH − TΔS; a reaction is spontaneous at constant T and P when ΔG < 0.

Beer–Lambert law: A = εcl; absorbance is linear in concentration and path length, valid for dilute solutions.

Schrödinger equation for a particle in a 1-D box of length L: E_n = n²h²/(8mL²); energy gaps widen as L shrinks.

Henderson–Hasselbalch: pH = pK_a + log([A⁻]/[HA]); buffer capacity peaks when pH = pK_a.

Newton's second law in rotating frames requires the Coriolis term −2m(ω × v) and centrifugal term −mω × (ω × r).

Hardy–Weinberg equilibrium: p² + 2pq + q² = 1 for genotype frequencies, with p + q = 1 for two alleles.

Net present value: NPV = Σ C_t/(1+r)^t − C_0; accept a project when NPV > 0 at the given discount rate.

Capital asset pricing model: E(R) = R_f + β(E(R_m) − R_f); β measures systematic, not total, risk.

Weighted average cost of capital: WACC = (E/V)·R_e + (D/V)·R_d·(1 − t), using market values.

Break-even quantity: Q = Fixed cost/(Price − Variable cost per unit); contribution margin per unit is the denominator.

Current ratio = current assets/current liabilities; quick ratio excludes inventory and prepaid expenses.

DuPont decomposition: ROE = net margin × asset turnover × equity multiplier.

Economic order quantity: EOQ = √(2DS/H), where D is annual demand, S ordering cost, H holding cost per unit per year.

Fisher effect: (1 + nominal) = (1 + real)(1 + inflation); approximately nominal ≈ real + inflation for low rates.

Aristotle's Poetics defines tragedy by pity and fear producing catharsis, and ranks plot above character.

In Indian aesthetics, rasa is the aesthetic relish evoked in the spectator; the Natyashastra enumerates eight rasas, with shanta added later as the ninth.

Saussure separates langue (the shared system) from parole (individual utterance), and signifier from signified.

Kant's categorical imperative: act only on maxims that could be willed as universal law.

In classical logic, modus ponens is (P → Q, P) ⊢ Q; modus tollens is (P → Q, ¬Q) ⊢ ¬P; affirming the consequent is a fallacy.

Renaissance perspective uses a vanishing point on the horizon; one-point perspective projects orthogonals to a single point.

The syllogism's figure is set by the position of the middle term; a valid mood must not distribute the middle term zero times.

In Marxist criticism, base (relations of production) is held to condition superstructure (law, art, ideology), though later theorists soften this determinism.

Percentage change of a quantity after successive changes a% and b% is a + b + ab/100 percent.

Work rate: if A alone takes x days and B alone y days, together they take xy/(x + y) days.

Compound interest: A = P(1 + r/n)^(nt); simple interest A = P(1 + rt); doubling time under CI is about 72/r years for small r.

Permutations of n distinct objects = n!; combinations C(n, r) = n!/[r!(n − r)!]; arrangements of a word with repeats = n!/(p!q!…).

Speed, time, distance: average speed for equal distances at speeds u and v is the harmonic mean 2uv/(u + v), not the arithmetic mean.

For sets, |A ∪ B| = |A| + |B| − |A ∩ B|; extend to three sets by subtracting pairwise and adding back the triple intersection.

In a number series, check first differences, then ratios, then alternating or second-difference patterns before assuming a rule.

Blood relations and direction sense: draw a diagram and fix one reference orientation; do not rotate the frame mid-problem.

A statement is valid if true in every model where the premises are true; soundness additionally requires the premises to be true.

Contrapositive of P → Q is ¬Q → ¬P and is logically equivalent; the converse Q → P is not.

In data interpretation, percentage of total = (part/total) × 100; percentage point difference is a subtraction of two percentages, not a ratio.

For a pie chart, angle of a sector = (component/total) × 360°; the same fraction gives the percentage.

In seating arrangement problems with circular tables, fix one person to break rotational symmetry before counting arrangements.

For grouped data, median class is where cumulative frequency first reaches N/2; mode class has the highest frequency.

Weighted average = Σ(w_i x_i)/Σw_i; it lies between the minimum and maximum of the x_i.

In syllogism questions, convert all statements to standard form (All A are B, No A is B, Some A are B) and test the conclusion against a Venn diagram.