GATE XL cheat sheet

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

In a first-order RC circuit, voltage across capacitor v(t)=V(1-e^{-t/RC}); time constant τ=RC; after 5τ the transient is within 1% of final value.

The Carnot efficiency of a heat engine operating between T_H and T_C (absolute) is η=1−T_C/T_H; no real engine between the same reservoirs can exceed this.

For a linear time-invariant system, the output is the convolution of input with impulse response: y(t)=∫x(τ)h(t−τ)dτ; convolution in time equals multiplication in the Laplace domain.

In a BJT biased in forward-active mode, collector current I_C≈I_S e^{V_BE/V_T} with V_T≈26 mV at room temperature; base-emitter drop ≈0.7 V for silicon.

Mohr's circle: principal stresses σ_1,2=(σ_x+σ_y)/2 ± √[((σ_x−σ_y)/2)^2+τ_xy^2]; maximum in-plane shear τ_max=√[((σ_x−σ_y)/2)^2+τ_xy^2].

For a PID controller, output u(t)=K_p e(t)+K_i∫e dt+K_d de/dt; increasing K_i eliminates steady-state error but reduces phase margin; increasing K_d adds damping but amplifies noise.

In engineering economy, present worth of a uniform series A over n periods at interest i is P=A[(1+i)^n−1]/[i(1+i)^n]; capital recovery factor is the reciprocal.

For a column with both ends pinned, Euler buckling load P_cr=π^2EI/L^2; effective length factors: fixed-free 2L, fixed-pinned 0.7L, fixed-fixed 0.5L.

Michaelis-Menten kinetics: v=V_max[S]/(K_m+[S]); K_m is the substrate concentration at half-maximal velocity; Lineweaver-Burk plot is 1/v vs 1/[S] with slope K_m/V_max and intercept 1/V_max.

In molecular biology, the central dogma flows DNA→RNA→protein; reverse transcriptase converts RNA to cDNA, and CRISPR-Cas9 creates double-strand breaks guided by a 20-nt sgRNA.

For a first-order reaction, rate constant k=(1/t)ln([A]_0/[A]_t); half-life t_1/2=ln2/k≈0.693/k, independent of initial concentration.

Hardy-Weinberg equilibrium: allele frequencies p+q=1; genotype frequencies p^2+2pq+q^2=1; deviations indicate selection, drift, migration, mutation, or nonrandom mating.

In quantum mechanics, the time-independent Schrödinger equation is Ĥψ=Eψ; for a particle in a 1D box of length L, E_n=n^2h^2/(8mL^2).

For a weak acid, pH=pK_a+log([A−]/[HA]) (Henderson-Hasselbalch); at half-equivalence, pH=pK_a; buffer capacity is maximal near pH=pK_a.

In thermodynamics, Gibbs free energy ΔG=ΔH−TΔS; spontaneous at constant T and P when ΔG<0; at equilibrium ΔG=0 and ΔG°=−RT ln K.

For a normal distribution, approximately 68% of values lie within μ±σ, 95% within μ±2σ, and 99.7% within μ±3σ; standard error of the mean is σ/√n.

In spectroscopy, Beer-Lambert law A=εcl; absorbance is linear in concentration only at low c; deviations occur at high concentration or with polychromatic light.