Engineering Mathematics
Engineering Mathematics in GATE MN is tested as a tool for mining problems, not as pure theory: matrices for linear systems and rock-mass computations, calculus for optimisation, Laplace transforms for ventilation and heat-flow ODEs, probability for reliability and reserve risk, and numerical methods for equations with no closed form. Expect roughly one-sixth of the paper here, mostly single-step questions where careless arithmetic, not insight, costs the mark.
What is actually examined
- Linear algebra: eigenvalues of small matrices, rank and consistency of systems, Cayley–Hamilton.
- Calculus: partial derivatives, maxima–minima of f(x,y), gradient, divergence, curl, Gauss and Stokes theorems.
- ODEs: second-order linear with constant coefficients, Cauchy–Euler form; Laplace transforms are the workhorse.
- Probability: binomial, Poisson, normal, mean/median/mode/SD, correlation and regression.
- Numerical methods: Newton–Raphson, trapezoidal versus Simpson's rule, Runge–Kutta.
Worked example
Solve y'' + y = 1 with y(0) = 0, y'(0) = 0. Taking Laplace transforms: (s^2 + 1)Y = 1/s, so Y = 1/[s(s^2 + 1)]. Partial fractions: 1/[s(s^2 + 1)] = 1/s - s/(s^2 + 1). Inverse transform: y = 1 - cos t. Check: y(0) = 0 correct; y' = sin t so y'(0) = 0 correct; y'' = cos t and y'' + y = cos t + 1 - cos t = 1, confirming the particular solution.
Traps that cost marks
- Simpson's 1/3 rule needs an even number of subintervals; applying it to nine strips invalidates the answer.
- For f(x,y), classify a stationary point with D = f_xx f_yy - f_xy^2; positive D with positive f_xx means a minimum, not a maximum.
- Poisson has mean equal to variance; do not substitute the binomial variance np(1-p).
- The Laplace transform of 1 is 1/s, of t is 1/s^2; swapping these ruins every downstream step.