GATE BM cheat sheet

Michaelis–Menten: v = Vmax[S]/(Km+[S]); Km is the substrate concentration at which v = Vmax/2; Lineweaver–Burk plot is 1/v vs 1/[S] with slope Km/Vmax and y-intercept 1/Vmax.

Competitive inhibition raises apparent Km without changing Vmax; non-competitive inhibition lowers Vmax without changing Km; uncompetitive inhibition lowers both Km and Vmax.

DNA replication: helicase unwinds, primase synthesizes RNA primer, DNA polymerase III extends 5'→3', DNA polymerase I removes primer, ligase seals nicks; leading strand continuous, lagging strand discontinuous (Okazaki fragments).

Central dogma: DNA → RNA (transcription, RNA polymerase) → protein (translation, ribosome); reverse transcriptase converts RNA to DNA; retroviruses use this.

Restriction enzymes: Type II recognize palindromic sequences (e.g., EcoRI GAATTC) and cleave within; sticky ends allow ligation; blunt ends require higher DNA concentration.

PCR cycle: denaturation (~95°C), annealing (~55–65°C, primer Tm dependent), extension (~72°C, Taq polymerase); exponential amplification 2^n after n cycles.

Western blot detects proteins (antibody probe), Southern blot detects DNA, Northern blot detects RNA; ELISA uses enzyme-linked antibody for quantitation.

Monoclonal antibodies: produced by hybridoma (B cell + myeloma) fusion; polyclonal from immunized animal serum; hybridoma selection uses HAT medium.

CRISPR-Cas9: guide RNA directs Cas9 nuclease to complementary DNA; creates double-strand break; repair by NHEJ (knockout) or HDR (knock-in) with template.

BOD: BOD_t = L0(1 − e^(−k t)); ultimate BOD L0; k is deoxygenation constant (base e, day^−1); standard BOD5 at 20°C.

Stoichiometry of complete combustion: C_aH_b + (a + b/4)O2 → a CO2 + (b/2)H2O; air required = O2 × (100/21) by volume.

Soil compaction: Proctor test gives maximum dry unit weight and optimum moisture content; higher compactive effort → higher γ_d,max and lower OMC.

Effective stress: σ' = σ − u; for capillary rise, u is negative; for upward seepage, effective stress reduces and quick condition when i = (G−1)/(1+e).

Beam deflection: simply supported with central point load P: δ_max = PL^3/(48EI); cantilever with end load: δ_max = PL^3/(3EI).

Concrete mix design: IS method uses target mean strength f_ck + 1.65σ; water-cement ratio by exposure condition; slump test for workability.

Surveying: leveling reduces to Δh = BS − FS; reciprocal leveling eliminates refraction and curvature; tacheometry uses stadia constants (f/i and f+d).

Hydrology: rational method Q = C i A (peak runoff); unit hydrograph theory assumes linear response; S-curve converts unit hydrograph durations.

Structural steel: limit state design uses partial safety factors; bolt value in shear = (f_ub/√3)(n_n A_nb + n_s A_sb)/γ_mb; block shear check per IS 800.

Master theorem: T(n) = aT(n/b) + f(n); compare f(n) with n^(log_b a); if f(n) = Θ(n^(log_b a) log^k n), then T(n) = Θ(n^(log_b a) log^(k+1) n).

NP-completeness: a problem is NP-complete if it is in NP and every NP problem reduces to it in polynomial time; SAT is the foundational NP-complete problem (Cook–Levin).

TCP: three-way handshake SYN, SYN-ACK, ACK; congestion control AIMD; slow start doubles cwnd per RTT; fast retransmit on three duplicate ACKs.

Deadlock four necessary conditions: mutual exclusion, hold and wait, no preemption, circular wait; prevention breaks at least one; Banker's algorithm for avoidance.

Page replacement: FIFO suffers Belady's anomaly; LRU does not; optimal (MIN) is theoretical benchmark; working set model for thrashing control.

Relational algebra: selection σ, projection π, join ⋈; SQL GROUP BY with HAVING filters groups; NULL comparisons yield UNKNOWN; COUNT(*) counts rows, COUNT(col) ignores NULLs.

Regular languages closed under union, intersection, complement, concatenation, star; context-free languages not closed under intersection or complement; pumping lemma for non-regularity.

Cache: average access time = hit_time + miss_rate × miss_penalty; write-through vs write-back; direct-mapped, set-associative, fully associative; replacement LRU, FIFO, random.

Floating point IEEE 754 single precision: 1 sign, 8 exponent (bias 127), 23 fraction; double: 1, 11 (bias 1023), 52; normalized, denormalized, infinity, NaN encodings.

Thevenin/Norton: any linear two-terminal network = V_th in series with R_th; Norton current I_N = V_th/R_th; maximum power transfer when R_L = R_th, η = 50%.

Three-phase power: P = √3 V_L I_L cos φ for balanced Y or Δ; apparent power S = √3 V_L I_L; reactive Q = √3 V_L I_L sin φ.

Transformer: EMF equation E = 4.44 f N Φ_m; equivalent circuit referred to primary; efficiency maximum when copper loss = iron loss; regulation = (V_no-load − V_full-load)/V_full-load.

DC machines: back EMF E_b = (PφNZ)/(60A); torque T = (PφI_a Z)/(2πA); separately excited speed control by armature voltage and field flux.

Induction motor: slip s = (N_s − N_r)/N_s; rotor frequency = s f; torque ∝ s V^2 R_2/(R_2^2 + (sX_2)^2); maximum torque at s = R_2/X_2.

Power system: per-unit = actual/base; base impedance Z_base = V_base^2/S_base; symmetrical components (positive, negative, zero sequence) for unbalanced fault analysis.

Protection: overcurrent relay coordination by time-current characteristics; distance relay zones (Zone 1 ~80%, Zone 2 ~120%, Zone 3 backup); CT saturation affects differential protection.

Control systems: second-order ζ, ω_n; overshoot %OS = 100 e^(−ζπ/√(1−ζ^2)); settling time ≈ 4/(ζω_n) for 2% band; Routh–Hurwitz for stability.

Power electronics: single-phase full-bridge rectifier average output V_dc = (2V_m/π) cos α; buck converter V_o = D V_in; boost V_o = V_in/(1−D); buck-boost V_o = −V_in D/(1−D).

First law for steady flow: q − w = Δh + ΔKE + ΔPE; for nozzle and diffuser q=0, w=0; for throttling h constant (ideal gas T constant).

Carnot efficiency η = 1 − T_L/T_H (absolute temperatures); COP of refrigerator = T_L/(T_H − T_L); heat pump COP = T_H/(T_H − T_L).

Rankine cycle: pump work ≈ v_f (p_2 − p_1); boiler heat addition; condenser heat rejection; superheat raises mean temperature of heat addition and cycle efficiency.

Beam bending: σ = My/I; τ = VQ/(Ib); deflection by double integration or superposition; Euler buckling P_cr = π^2 EI/(KL)^2; K depends on end conditions.

Fatigue: S–N curve; endurance limit; Goodman, Gerber, and Soderberg criteria for combined mean and alternating stresses; stress concentration factor K_t.

Machining: Taylor's tool life V T^n = C; Merchant circle: shear angle φ from 2φ + β − α = 90° (approximate); chip thickness ratio r = t_1/t_2 = sin φ / cos(φ−α).

Casting: Chvorinov's rule t = B (V/A)^2; riser design using modulus M = V/A; solidification time proportional to square of modulus.

Welding: heat input H = VI/v (J/mm); arc welding efficiency; residual stresses reduced by preheating and stress-relief annealing.

Refrigeration: vapor-compression cycle components (compressor, condenser, expansion valve, evaporator); subcooling increases COP; superheating protects compressor.

Accounting equation: Assets = Liabilities + Equity; every transaction preserves balance; double-entry bookkeeping requires equal debits and credits.

Depreciation: straight-line = (Cost − Salvage)/Useful life; written-down value = Cost × (1 − r)^n; sum-of-years-digits allocates more in early years.

Break-even point (units) = Fixed cost / (Selling price − Variable cost per unit); margin of safety = Actual sales − Break-even sales; contribution = Sales − Variable cost.

Capital budgeting: NPV = Σ CF_t/(1+r)^t − Initial outlay; accept if NPV > 0; IRR is rate where NPV = 0; payback period ignores time value of money.

Cost of capital: WACC = (E/V) R_e + (D/V) R_d (1 − T); CAPM R_e = R_f + β(R_m − R_f); dividend growth model R_e = D_1/P_0 + g.

Working capital cycle = Inventory days + Receivable days − Payable days; current ratio = Current assets/Current liabilities; quick ratio excludes inventory.

Auditing: materiality, audit risk = inherent risk × control risk × detection risk; true and fair view; going concern assumption; qualified vs unqualified opinion.

Income tax: residential status determines scope of taxable income; capital vs revenue expenditure; depreciation allowance; deductions under specified sections; advance tax and TDS.

Business law: contract essentials - offer, acceptance, consideration, capacity, free consent, lawful object; Sale of Goods Act implied conditions; negotiable instruments.

Economics: elasticity of demand = (%ΔQ)/(%ΔP); price elasticity >1 elastic, <1 inelastic; income elasticity negative for inferior goods; cross elasticity positive for substitutes.

Reasoning: syllogism validity depends on form, not content; modus ponens (P→Q, P ⊢ Q) valid; modus tollens (P→Q, ¬Q ⊢ ¬P) valid; affirming the consequent is invalid.

Statistics: mean, median, mode; variance = Σ(x−μ)^2/N; standard deviation = √variance; normal distribution 68–95–99.7 rule; correlation ≠ causation.

Psychology: classical conditioning (Pavlov) pairs neutral stimulus with unconditioned stimulus; operant conditioning (Skinner) uses reinforcement and punishment; cognitive dissonance (Festinger).

Sociology: social stratification (class, caste, gender); functionalism (Durkheim, Parsons), conflict theory (Marx), symbolic interactionism (Mead, Goffman); anomie = normlessness.

Philosophy: epistemology - justified true belief; skepticism; empiricism (Hume, Locke) vs rationalism (Descartes, Kant); ethics - utilitarianism (Mill), deontology (Kant), virtue ethics (Aristotle).

Linguistics: phoneme is smallest sound unit distinguishing meaning; morpheme is smallest meaningful unit; syntax phrase structure; semantics vs pragmatics; Chomsky's universal grammar.

History: periodization - ancient, medieval, early modern, modern; historiography examines sources and bias; primary vs secondary sources; cause and consequence analysis.