GATE XE flashcards

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

Eigenvalue trace identity

For any square matrix, the sum of all eigenvalues equals the trace and their product equals the determinant. This gives a quick consistency check after solving the characteristic equation.

Cayley–Hamilton theorem

Every square matrix satisfies its own characteristic equation. It lets you express high powers of a matrix as lower-degree polynomials and compute inverses without row reduction.

Rank–nullity theorem

For an m×n matrix, rank plus nullity equals n, the number of columns. It tells you the dimension of the solution space of Ax = 0 and the degrees of freedom in Ax = b.

Consistency of linear systems

Ax = b is consistent only if b lies in the column space of A. For square A this means either det(A) ≠ 0 (unique solution) or b orthogonal to the left null space.

Leibniz integral rule

Differentiating an integral with variable limits gives the integrand evaluated at the upper limit times that limit's derivative, minus the same at the lower limit, plus the integral of the partial derivative.

Hessian classification of critical points

At a stationary point, D = f_xx f_yy − f_xy² decides the type: D > 0 and f_xx > 0 is a minimum, D > 0 and f_xx < 0 a maximum, D < 0 a saddle point.

Green's theorem

A line integral around a simple closed plane curve of P dx + Q dy equals the double integral over the enclosed region of (∂Q/∂x − ∂P/∂y), for continuously differentiable P and Q.

Divergence theorem

The outward flux of a vector field through a closed surface equals the volume integral of its divergence over the enclosed region. It converts surface integrals into often simpler volume integrals.

Second-order ODE solution strategy

For constant coefficients, solve the auxiliary equation for the complementary function, then add a particular integral chosen to match the forcing term, avoiding terms already present.

Bayes' theorem

P(A|B) = P(B|A)P(A)/P(B). It updates a prior probability into a posterior using the likelihood of the observed evidence and the total probability of that evidence.

Variance shortcut formula

Var(X) = E[X²] − (E[X])². Computing the second moment first and subtracting the square of the mean is usually faster than summing squared deviations about the mean.

Poisson distribution properties

For a Poisson random variable with rate λ, both the mean and the variance equal λ. Probabilities are P(X = k) = e^(−λ) λ^k / k!, used for counts of rare events.

Newton–Raphson convergence

The iteration x_{n+1} = x_n − f(x_n)/f′(x_n) converges quadratically near a simple root but fails or slows when f′ is near zero or the starting guess is poor.

Trapezoidal and Simpson error orders

The composite trapezoidal rule has error of order h², while Simpson's 1/3 rule has error of order h⁴ and requires an even number of subintervals to apply.

XE sectional paper structure

GATE XE has a compulsory Engineering Mathematics section plus one optional engineering-science section chosen by the candidate, such as Fluid Mechanics, Thermodynamics, or Materials Science.

Bernoulli equation

Along a streamline of steady, inviscid, incompressible flow, p/ρ + V²/2 + gz is constant. It trades pressure, kinetic, and potential energy and fails across shocks or in viscous regions.

Reynolds number

Re = ρVD/μ compares inertial to viscous forces. In pipe flow, Re below about 2300 indicates laminar flow and above about 4000 fully turbulent flow.

Hagen–Poiseuille law

For fully developed laminar pipe flow, Δp = 128μLQ/(πD⁴). The fourth-power dependence on diameter means halving the diameter raises pressure drop sixteen-fold at the same flow rate.

Laminar friction factor

In fully developed laminar pipe flow the Darcy friction factor is f = 64/Re, independent of wall roughness. Roughness matters only in the turbulent regime.

First law for control mass

Q − W = ΔU for a closed system. For an ideal gas, ΔU = m c_v ΔT regardless of path, so internal energy change depends only on the end temperatures.

Ideal gas specific heats

For an ideal gas c_p − c_v = R and the ratio γ = c_p/c_v. Enthalpy change is ΔH = m c_p ΔT and internal energy change is ΔU = m c_v ΔT.

Ideal gas entropy change

Δs = c_p ln(T₂/T₁) − R ln(p₂/p₁) per unit mass. Setting Δs to zero recovers the isentropic relations T₂/T₁ = (p₂/p₁)^((γ−1)/γ).

Bending stress formula

Normal stress in a beam is σ = My/I, where y is distance from the neutral axis and I is the second moment of area; for a rectangle I = bd³/12.

Torsion of circular shafts

Shear stress is τ = Tr/J and angle of twist is θ = TL/GJ. For a solid circular shaft the polar second moment is J = πd⁴/32.

Euler buckling load

The critical load for a slender column is P_cr = π²EI/(KL)², where K depends on end conditions: 0.5 fixed-free, 0.7 pinned-fixed, 1.0 pinned-pinned, 2.0 fixed-fixed.

Fick's first law

Diffusion flux is J = −D dc/dx, proportional to the concentration gradient. The negative sign shows net transport from high to low concentration along the gradient.

Arrhenius diffusion coefficient

Diffusivity follows D = D₀ exp(−Q/RT), where Q is activation energy and R the gas constant. Plotting ln D against 1/T gives a straight line of slope −Q/R.

Hall–Petch relationship

Yield strength increases with finer grain size as σ_y = σ₀ + k d^(−1/2). Grain boundaries block dislocation motion, so smaller grains raise strength at room temperature.

Number-average molar mass

M_n = ΣN_iM_i/ΣN_i, weighting each chain equally by number. It is sensitive to low-molecular-weight chains and controls properties like osmotic pressure and melt viscosity.

Polydispersity index

PDI = M_w/M_n, the ratio of weight-average to number-average molar mass. It equals 1 only for a perfectly monodisperse polymer and grows as the molecular-weight distribution broadens.

Truncation versus round-off error

Truncation error comes from approximating a continuous operation by a discrete formula and shrinks with smaller step size. Round-off error grows as step size shrinks, so total error has an optimum.