GATE XL flashcards

33 cards. Read the term and the definition here. The deck at the top of this page is optional.

Euler buckling load

Critical axial load for a slender pinned-pinned column: P_cr=π^2EI/L^2, where E is Young's modulus, I is the minimum area moment of inertia, and L is the unsupported length.

Carnot efficiency

Maximum theoretical efficiency of a heat engine between hot reservoir T_H and cold reservoir T_C (absolute temperatures): η=1−T_C/T_H. No real engine between the same reservoirs can exceed it.

Time constant RC

In a first-order RC circuit, τ=RC is the time to reach 63.2% of the final voltage change. After 5τ the transient is within about 1% of its final value.

Mohr's circle principal stresses

Principal stresses are σ_1,2=(σ_x+σ_y)/2 ± √[((σ_x−σ_y)/2)^2+τ_xy^2]. Maximum in-plane shear equals the circle radius and acts on planes 45° from the principal planes.

PID controller equation

u(t)=K_p e(t)+K_i∫e dt+K_d de/dt. Integral action removes steady-state error; derivative action adds damping but amplifies high-frequency noise. Tune to meet phase margin and settling time.

BJT forward-active current

In forward-active mode, collector current is I_C≈I_S e^{V_BE/V_T}, with thermal voltage V_T≈26 mV at 300 K. Base-emitter voltage is about 0.7 V for silicon devices.

Convolution and LTI systems

For a linear time-invariant system, output y(t)=∫x(τ)h(t−τ)dτ, where h is the impulse response. Convolution in time corresponds to multiplication of Laplace transforms.

Capital recovery factor

Factor converting present worth P to an equivalent uniform annual series A over n periods at interest i: A=P[i(1+i)^n]/[(1+i)^n−1]. It is the reciprocal of the uniform-series present-worth factor.

Simply supported beam deflection

For a simply supported beam with central point load W and span L, maximum deflection at midspan is WL^3/(48EI); maximum bending moment is WL/4 at midspan.

Effective length factor

Ratio K of effective column length to actual length, used in P_cr=π^2EI/(KL)^2. Values: pinned-pinned 1.0, fixed-pinned 0.7, fixed-fixed 0.5, fixed-free 2.0.

Thevenin equivalent circuit

Any linear two-terminal network can be replaced by a voltage source V_th in series with R_th. V_th is the open-circuit voltage; R_th is the resistance seen with independent sources zeroed.

Nyquist stability criterion

A closed-loop system is stable if the Nyquist plot of the open-loop transfer function encircles the point −1+j0 exactly P times counterclockwise, where P is the number of open-loop right-half-plane poles.

Reynolds number

Ratio of inertial to viscous forces: Re=ρVD/μ. For pipe flow, Re<2300 is laminar, Re>4000 is turbulent, and 2300–4000 is transitional.

Rankine cycle efficiency

Thermal efficiency of a basic Rankine cycle is η=(W_turbine−W_pump)/Q_in. Superheating steam and lowering condenser pressure raise efficiency; reheating reduces moisture at turbine exit.

Michaelis-Menten equation

Enzyme velocity v=V_max[S]/(K_m+[S]). K_m is the substrate concentration at half-maximal velocity. A Lineweaver-Burk plot of 1/v vs 1/[S] gives slope K_m/V_max and intercept 1/V_max.

Hardy-Weinberg equilibrium

In a large random-mating population with no selection, mutation, migration, or drift, allele frequencies p+q=1 and genotype frequencies are p^2, 2pq, q^2. Deviations indicate evolutionary forces.

First-order reaction half-life

For a first-order reaction, t_1/2=ln2/k≈0.693/k, independent of initial concentration. Integrated rate law: ln([A]_0/[A]_t)=kt.

Particle in a 1D box

Quantum energy levels for a particle confined to length L: E_n=n^2h^2/(8mL^2), n=1,2,3,... The spacing between levels increases with n and decreases with L.

Henderson-Hasselbalch equation

pH=pK_a+log([A−]/[HA]) for a weak acid buffer. At half-neutralization pH=pK_a, and buffer capacity is maximal near that pH.

Gibbs free energy criterion

ΔG=ΔH−TΔS. At constant temperature and pressure, a process is spontaneous if ΔG<0, at equilibrium if ΔG=0, and nonspontaneous if ΔG>0. Also ΔG°=−RT ln K.

Beer-Lambert law

Absorbance A=εcl, where ε is molar absorptivity, c is concentration, and l is path length. Linearity fails at high concentration or with polychromatic light.

Standard error of the mean

SEM=σ/√n, where σ is the population standard deviation and n is sample size. It quantifies uncertainty in the sample mean and decreases as n increases.

Central dogma

Genetic information flows DNA→RNA→protein. Transcription makes RNA from DNA; translation makes protein from mRNA. Reverse transcriptase converts RNA to cDNA.

CRISPR-Cas9 mechanism

A 20-nt single guide RNA directs Cas9 to a complementary DNA sequence adjacent to a PAM. Cas9 creates a double-strand break, repaired by nonhomologous end joining or homology-directed repair.

PCR cycle steps

Polymerase chain reaction repeats denaturation (≈95°C), primer annealing (≈55–65°C), and extension (≈72°C). Each cycle doubles the target DNA, giving 2^n amplification after n cycles.

Normal distribution rule

For a normal distribution, about 68% of values lie within μ±σ, 95% within μ±2σ, and 99.7% within μ±3σ. Standardizing gives z=(x−μ)/σ.

Arrhenius equation

Rate constant k=A e^{−E_a/(RT)}, where A is the pre-exponential factor, E_a is activation energy, R is the gas constant, and T is absolute temperature. A plot of ln k vs 1/T has slope −E_a/R.

Nernst equation

Cell potential E=E°−(RT/nF)ln Q, or at 298 K E=E°−(0.0592/n)log Q. At equilibrium E=0 and Q=K, so E°=(0.0592/n)log K.

Heisenberg uncertainty principle

Position and momentum cannot both be known exactly: ΔxΔp≥ħ/2, where ħ=h/(2π). The product of uncertainties in conjugate variables has a lower bound.

Mendelian segregation ratio

A monohybrid cross of heterozygotes Aa×Aa gives a 3:1 phenotypic ratio and 1:2:1 genotypic ratio in offspring, assuming complete dominance and independent assortment.

Resting membrane potential

Neuron resting potential is about −70 mV, set mainly by K+ permeability and the Na+/K+ ATPase. The Nernst potential for K+ is near −90 mV.

Enzyme inhibition types

Competitive inhibitors raise apparent K_m with unchanged V_max. Noncompetitive inhibitors lower V_max with unchanged K_m. Uncompetitive inhibitors lower both K_m and V_max.

Chi-square goodness of fit

χ^2=Σ(O−E)^2/E, with degrees of freedom = categories − 1 − estimated parameters. Compare to critical value; a large χ^2 rejects the null hypothesis.