Engineering Mathematics (XE Section A, compulsory)
Engineering Mathematics is Section A of GATE XE and is compulsory for every candidate regardless of the sectional paper chosen. It rewards computational accuracy, not proof-writing: short questions on linear algebra, calculus and vector calculus, complex variables, ordinary and partial differential equations, probability and statistics, and numerical methods.
Rules that decide most questions
- Consistency of Ax = b requires rank(A) = rank([A|b]); for square systems a nonzero determinant is the quick sufficient check.
- For a 2×2 matrix, λ² − (trace)λ + det = 0. Symmetric matrices have real eigenvalues and orthogonal eigenvectors; non-symmetric ones generally do not.
- f is analytic in a domain iff the Cauchy–Riemann equations hold there with continuous partials. CR at a single point proves nothing.
- Closed-contour integrals: the integral equals 2πi times the sum of residues at poles strictly inside C. Simple pole: Res = lim(z−a)f(z) as z→a.
- Constant-coefficient ODEs: solve the auxiliary equation, then add a particular integral. For Cauchy–Euler equations x²y″ + …, substitute x = eᵗ.
- Probability: binomial mean np, variance npq; Poisson mean = variance = λ; exponential mean 1/λ, variance 1/λ². Regression slope b(yx) = r·σy/σx.
- Simpson's 1/3 rule needs an even number of subintervals; Newton–Raphson converges quadratically once the iterate is close.
Worked example
Evaluate ∮ (3z+1)/(z(z−1)) dz counter-clockwise around |z| = 2. Poles z = 0 and z = 1 both lie strictly inside. Residue at 0 is (0+1)/(0−1) = −1; at 1 it is (3+1)/1 = 4. Sum = 3, so the integral is 2πi(3) = 6πi. Confirm the location of each pole before summing.
Traps
Using Simpson's rule with an odd interval count; confusing determinant with rank for rectangular systems; treating gauge pressure as absolute in gas-law items; dropping the Jacobian r in a polar double integral; assuming a non-symmetric matrix has real eigenvalues.