GATE XE cheat sheet

Linear algebra: eigenvalues solve det(A − λI) = 0; the trace equals the sum of eigenvalues and the determinant equals their product, so a singular matrix always has λ = 0.

Cayley–Hamilton: every square matrix satisfies its own characteristic equation, so A^n can be reduced to a polynomial of degree n − 1 in A; use it to compute inverses and high powers cheaply.

Rank–nullity: rank(A) + nullity(A) = number of columns; a square system Ax = b has a unique solution iff det(A) ≠ 0, and consistency for singular A requires b orthogonal to the left null space.

Calculus: Leibniz rule for differentiating integrals with parameter-dependent limits; L'Hôpital applies only to 0/0 or ∞/∞ forms, and Taylor expansion with remainder is the safer route for limits.

Multivariable calculus: a critical point is classified by the Hessian determinant D = f_xx f_yy − f_xy²; D > 0 with f_xx > 0 is a minimum, D > 0 with f_xx < 0 a maximum, D < 0 a saddle.

Vector calculus: Green's theorem converts a plane line integral to a double integral of (∂Q/∂x − ∂P/∂y); divergence theorem converts a closed-surface flux to a volume integral of ∇·F.

ODEs: for constant-coefficient linear equations use the auxiliary (characteristic) equation; for non-homogeneous terms use undetermined coefficients or variation of parameters, and check the complementary solution for duplication.

Probability: Bayes' theorem P(A|B) = P(B|A)P(A)/P(B); for continuous variables E[X] = ∫x f(x)dx and Var(X) = E[X²] − (E[X])²; Poisson mean equals variance equals λ.

Numerical methods: Newton–Raphson converges quadratically near a simple root (x_{n+1} = x_n − f/f′); trapezoidal error is O(h²) and Simpson's 1/3 rule error is O(h⁴), valid for an even number of intervals.

XE is a sectional paper: candidates answer Section A (Engineering Mathematics, compulsory) plus one optional section such as Fluid Mechanics, Materials Science, Thermodynamics, Solid Mechanics, Polymer Science, Food Technology, or Computational Science; confirm the current syllabus list before choosing.

Fluid Mechanics: continuity ρAV = constant for steady incompressible flow; Bernoulli p/ρ + V²/2 + gz = constant along a streamline for inviscid flow; Reynolds number Re = ρVD/μ sets laminar–turbulent transition near 2300 in pipes.

Fluid Mechanics: for fully developed laminar pipe flow the Hagen–Poiseuille result gives Δp = 128μLQ/(πD⁴), so pressure drop scales with D⁻⁴; friction factor f = 64/Re in the laminar regime.

Thermodynamics: first law for a control mass Q − W = ΔU; for an ideal gas ΔU = m c_v ΔT and ΔH = m c_p ΔT with c_p − c_v = R; entropy change for an ideal gas is Δs = c_p ln(T₂/T₁) − R ln(p₂/p₁).

Solid Mechanics: bending stress σ = My/I with I = bd³/12 for a rectangle; torsion τ = Tr/J with J = πd⁴/32 for a solid shaft; Euler buckling load P_cr = π²EI/(KL)² with K depending on end conditions.

Materials Science: Fick's first law J = −D dc/dx; diffusion obeys Arrhenius D = D₀ exp(−Q/RT); the Hall–Petch relation σ_y = σ₀ + k d^(−1/2) links yield strength to grain size.

Polymer Science: number-average and weight-average molar masses M_n = ΣN_iM_i/ΣN_i and M_w = ΣN_iM_i²/ΣN_iM_i; polydispersity index M_w/M_n ≥ 1, equalling 1 only for a monodisperse sample.

Computational Science: distinguish truncation error (from the discretization, controlled by step size and order) from round-off error (from finite precision); total error is minimized at an intermediate step size, not the smallest one.

Exam strategy fact: the XE paper is 100 marks over 3 hours with negative marking for wrong MCQ answers, so unanswered questions score zero while a guessed wrong answer costs marks; numerical-answer questions carry no negative marking.