GATE AE cheat sheet

Percentage change: ((new − old)/old) × 100; successive changes multiply as (1+a)(1+b), not add.

If a:b = c:d then ad = bc; divide a total T in ratio a:b as T·a/(a+b) and T·b/(a+b).

Combined work rate is the sum of individual rates; A alone in x days and B alone in y days finish together in xy/(x+y) days.

Average = total sum/count; adding a value x changes the mean by (x − old mean)/new count.

Permutations nPr = n!/(n−r)!; combinations nCr = nPr/r!; circular arrangements of n distinct items = (n−1)!.

Probability: P(A∪B)=P(A)+P(B)−P(A∩B); conditional P(A|B)=P(A∩B)/P(B), valid when P(B)>0.

Categorical logic: 'All A are B' does not imply 'All B are A'; 'Some A are B' is equivalent to 'Some B are A'.

For data interpretation, compute required quantity as (part/whole)×100 or use total = sum of category values before percentages.

Eigenvalues: sum = trace(A), product = det(A); Cayley-Hamilton states A satisfies its own characteristic polynomial.

Rank-nullity: rank(A)+nullity(A)=n for an m×n matrix; nullity is dim{x: Ax=0}.

Gradient points normal to level surfaces; divergence measures source density; curl measures local rotation.

Laplace: L{f′}=sF(s)−f(0); L{e^{at}}=1/(s−a); L{sin ωt}=ω/(s²+ω²).

First-order linear ODE y′+P(x)y=Q(x) has integrating factor exp(∫P dx).

Second-order constant-coefficient ODE: roots of characteristic equation decide overdamped, critically damped, or oscillatory response.

Fourier series: even functions use cosine terms only; odd functions use sine terms only; period 2π coefficients integrate over one period.

Numerical integration: composite trapezoidal error O(h²); composite Simpson error O(h⁴) and needs an even number of subintervals.

Lift L = ½ρV²S C_L; dynamic pressure q = ½ρV², so L = qS C_L.

Drag polar C_D = C_D0 + C_L²/(π e AR); induced drag falls as aspect ratio AR rises.

Load factor n = L/W; in a level coordinated turn n = 1/cos φ, so bank angle raises stall speed by sqrt(n).

V_stall = sqrt(2W/(ρS C_Lmax)); for fixed weight and C_Lmax, higher altitude lowers ρ and raises true stall speed.

Jet Breguet range R = (V/c_T)(L/D) ln(W_i/W_f); propeller range uses propeller efficiency and BSFC in the analogous formula.

Rate of climb = excess power/W = V sin γ; maximum climb angle uses maximum excess thrust, maximum rate uses maximum excess power.

Static margin = (x_NP − x_CG)/MAC; positive static margin means CG ahead of neutral point and pitch stiffness.

Short-period mode is a fast angle-of-attack oscillation; phugoid is a slow speed/pitch oscillation with nearly constant angle of attack.

Vis-viva: v² = μ(2/r − 1/a); circular orbit has a=r and v=sqrt(μ/r).

Escape speed v_esc=sqrt(2μ/r), equal to sqrt(2) times circular speed at the same radius.

Specific orbital energy ε=v²/2−μ/r=−μ/(2a); angular momentum h=r v cos γ is constant for two-body motion.

Kepler III: T²=4π²a³/μ; period is independent of eccentricity for a given semi-major axis.

Hohmann transfer between circular radii r1<r2 uses two tangential burns; transfer time is half the ellipse period, π sqrt((r1+r2)³/(8μ)).

Plane change at speed v costs Δv=2v sin(Δi/2); smaller Δi and lower speed reduce the penalty.

Hyperbolic excess speed satisfies v∞²=v²−v_esc²; a hyperbolic trajectory has a<0 and positive specific energy.

Rocket equation Δv=Isp g0 ln(m0/mf); delta-v grows logarithmically with mass ratio.

Closed-loop characteristic equation is 1+G(s)H(s)=0; its roots determine stability.

Routh-Hurwitz: all first-column entries positive means no right-half-plane roots; sign changes count unstable poles.

First-order step response reaches 63.2% of final value in one time constant τ; settling time ≈4τ for 2% band.

Second-order: ω_n and ζ set response; percent overshoot ≈100 exp(−πζ/sqrt(1−ζ²)) for 0<ζ<1.

Gain margin GM_dB=−20 log|G(jω_pc)| at phase −180°; phase margin PM=180°+∠G(jω_gc) at |G|=1.

Bode slope: one pole adds −20 dB/decade and −90°; one zero adds +20 dB/decade and +90°.

PID: proportional acts on present error, integral removes steady-state offset, derivative adds damping and anticipates error.

State model xdot=Ax+Bu is controllable if [B AB ... A^{n−1}B] has rank n; observability uses [C; CA; ...; CA^{n−1}].

Incompressible Bernoulli: p+½ρV²+ρgz=constant along a streamline for steady inviscid flow.

Isentropic relations: T0/T=1+((γ−1)/2)M²; p0/p=(T0/T)^{γ/(γ−1)}; ρ0/ρ=(T0/T)^{1/(γ−1)}.

Area-Mach relation: subsonic flow accelerates in a converging duct; supersonic flow accelerates in a diverging duct.

Reynolds number Re=ρVL/μ sets viscous regime; boundary layer grows downstream and separates under adverse pressure gradient.

Kutta condition fixes circulation at a sharp trailing edge; lift per unit span is L′=ρVΓ.

Thin airfoil in incompressible flow has lift slope 2π per radian; camber shifts the zero-lift angle.

Prandtl-Glauert: subsonic compressible coefficients scale by 1/sqrt(1−M∞²), invalid near sonic speed.

Normal shock: M2<1, static pressure and temperature rise, stagnation pressure falls; entropy increases.

Axial stress σ=F/A; engineering strain ε=ΔL/L; Hooke's law σ=Eε within the elastic limit.

Bending: σ=−My/I; maximum stress at the outer fiber; shear stress τ=VQ/(It).

Beam deflection satisfies EI y″=M(x); boundary conditions at supports determine integration constants.

Circular shaft torsion: τ=Tr/J and twist angle φ=TL/(GJ).

Euler buckling: P_cr=π²EI/(KL)²; K depends on end fixity, and buckling governs slender columns.

Principal stresses from Mohr's circle; von Mises yield occurs when equivalent stress equals yield strength.

Fatigue: cyclic stress below endurance limit may give infinite life in steel; stress concentration raises local stress.

Thin-walled cylindrical pressure vessel: hoop stress σ_h=pr/t, longitudinal stress σ_l=pr/(2t).

Thrust F=mdot v_e+(p_e−p_a)A_e; vacuum thrust is higher because p_a=0.

Specific impulse Isp=F/(mdot g0), in seconds; TSFC is fuel mass flow per unit thrust.

Rocket equation Δv=Isp g0 ln(m0/mf); staging raises effective mass ratio.

Ideal Brayton efficiency η=1−1/r_p^{(γ−1)/γ}; specific work depends on turbine inlet temperature and pressure ratio.

Convergent nozzle chokes at p*/p0=(2/(γ+1))^{γ/(γ−1)}; choked mass flow depends on throat area and chamber conditions.

Compressor pressure ratio π_c=p02/p01; stage work and temperature rise follow isentropic efficiency relations.

Bypass ratio BPR=mdot_bypass/mdot_core; high BPR improves propulsive efficiency for low-speed flight.

Stoichiometric combustion uses the chemically correct fuel-air ratio; actual engines run lean or rich depending on temperature limits and emissions.