Engineering Mathematics
Engineering Mathematics in GATE NM supplies the analytical tools used in hydrostatics, structures and dynamics: linear algebra, calculus, differential equations, numerical methods and probability. Questions are usually short and computational, so the marks come from setting up the standard form correctly and checking signs, units and limiting cases.
Core rules
- Eigenvalues follow det(A − λI) = 0; eigenvectors follow (A − λI)x = 0. For a real symmetric matrix, eigenvalues are real and eigenvectors corresponding to distinct eigenvalues are orthogonal.
- First-order linear ODE y′ + P(x)y = Q(x) is solved with the integrating factor exp(∫P dx). Constant-coefficient second-order ODEs use the characteristic equation.
- Trapezoidal rule has error O(h²); Simpson’s 1/3 rule has error O(h⁴) and needs an even number of intervals.
- Newton–Raphson uses x_{n+1} = x_n − f(x_n)/f′(x_n); convergence is fast near a simple root but can fail at a zero derivative.
- For probability, use total probability and Bayes’ theorem; check whether events are independent before multiplying probabilities.
Worked example
For A = [[2, 1], [1, 2]], det(A − λI) = (2 − λ)² − 1 = λ² − 4λ + 3 = (λ − 1)(λ − 3). Eigenvalues are 1 and 3. For λ = 3, (A − 3I)x = 0 gives −x1 + x2 = 0, so x = (1, 1). For λ = 1, x1 + x2 = 0, so x = (1, −1). Check: trace 4 = 1 + 3; determinant 3 = 1 × 3.
Traps
A sign slip in the characteristic equation changes both eigenvalues. Do not use degrees in calculus or in Newton–Raphson. Simpson’s rule requires evenly spaced ordinates and an even number of intervals. In ODEs, apply the initial condition after finding the general solution, not before. Keep radians for trigonometric terms in differential equations.