Engineering Mathematics
Engineering Mathematics carries roughly 13–15 marks in GATE EE and is the highest-leverage section because the questions are formulaic: recognition of the standard form usually finishes the problem. The syllabus spans linear algebra, calculus, differential equations, complex variables, probability and statistics, and numerical methods.
Linear algebra
- Rank decides consistency of Ax = b; eigenvalue from |A − λI| = 0, with trace = sum of eigenvalues and det = product.
- Symmetric real matrices have real eigenvalues and orthogonal eigenvectors; use Cayley–Hamilton for high powers and inverses.
Calculus and ODEs
- Taylor expansion, partial derivatives, maxima/minima and Lagrange multipliers.
- Second-order linear ODE: characteristic roots give the complementary function; particular integral by undetermined coefficients or variation of parameters. Euler–Cauchy equations are linearised with x = e^t.
Complex variables
- Simple pole residue = lim (z − z0)f(z); a pole of order m uses the (m−1)-th derivative formula.
Worked example: ∮ e^z/(z²+1) dz over |z| = 2. The poles ±i both lie inside. Residue at i is e^i/(2i); at −i it is e^(−i)/(−2i). Their sum is (e^i − e^(−i))/(2i) = sin 1, so the integral equals 2πi sin 1. The reasoning hinges on identifying every enclosed pole before summing.
Traps that cost marks
- Contour orientation: a clockwise contour flips the sign of the answer.
- Forgetting that undetermined coefficients fail when the forcing term duplicates a homogeneous solution - multiply by t instead.
- Probability: P(A ∪ B) = P(A) + P(B) − P(A ∩ B); Bayes' theorem is not the same as a conditional probability.
- Numerical methods: Newton–Raphson converges quadratically but diverges near a zero derivative.