Engineering Mathematics
Engineering Mathematics in GATE ME is a computation section: linear algebra, calculus, differential equations, complex analysis, probability and numerical methods. Questions are short, so the marks sit in picking the right formula and in bookkeeping of signs, limits and units rather than in long algebra.
Linear algebra
- Rank is the count of nonzero rows after row reduction. Ax = b is consistent only when rank(A) = rank([A|b]), and the solution is unique when that common rank equals the number of unknowns.
- Eigenvalues come from det(A − λI) = 0, but check with trace = Σλ and det = Πλ.
- Real symmetric matrices have real eigenvalues and mutually orthogonal eigenvectors.
Calculus and complex analysis
- For f(x, y), set both first partials to zero, then test D = f_xx·f_yy − f_xy²: D > 0 with f_xx > 0 is a minimum, f_xx < 0 a maximum, D < 0 a saddle.
- Analyticity requires Cauchy–Riemann (u_x = v_y, u_y = −v_x). A residue at a simple pole is lim(z−z₀)f(z), and ∮f dz = 2πi Σ residues.
Differential equations and numerics
- First-order linear: multiply by the integrating factor e^{∫P dx}. Second-order constant coefficient: complementary function from characteristic roots, particular integral by undetermined coefficients.
- Newton–Raphson uses x_{n+1} = x_n − f(x_n)/f′(x_n) with quadratic convergence. Simpson's 1/3 rule needs an even number of intervals and is O(h⁴); trapezoidal is O(h²).
Worked example
For A = [[4,1],[1,4]]: trace = 8, det = 15, so λ² − 8λ + 15 = 0 gives λ = 5 and λ = 3. The product check (5×3 = 15) confirms the pair without expanding the determinant.
Traps
Assuming every square matrix is invertible; running Simpson's rule on an odd number of intervals; forgetting that a Poisson variable has mean equal to variance; ignoring branch cuts when evaluating a closed-contour integral.