Engineering Mathematics and Numerical Methods
ISRO uses this domain as a filter: items are short, computational and unforgiving of sign or unit slips. Expect rank and eigenvalue problems, partial differentiation, first-order ODEs, standard probability distributions, and numerical methods judged on order of error and convergence.
Core rules
- Sum of eigenvalues equals the trace; the product equals the determinant. A matrix is singular exactly when an eigenvalue is zero, so its rank falls below n.
- Newton–Raphson: xₙ₊₁ = xₙ − f(xₙ)/f′(xₙ); quadratic convergence, but it fails at f′ = 0 or where iterates enter a cycle.
- Bisection halves the bracket each step; error after n steps ≤ (b − a)/2ⁿ. It converges whenever f changes sign on the bracket.
- The trapezoidal rule is exact for polynomials up to degree 1; Simpson's 1/3 rule is exact up to degree 3 and needs an even number of intervals.
- For y′ + Py = Q, the integrating factor is e^(∫P dx).
- For the Poisson distribution the mean equals the variance; for the normal distribution about 68% of area lies within ±1σ.
Worked example
Find √2 by Newton–Raphson from x₀ = 1, with f(x) = x² − 2 and f′(x) = 2x. x₁ = 1 − (1 − 2)/2 = 1.5; x₂ = 1.5 − (2.25 − 2)/3 = 1.41667; x₃ = 1.41667 − (2.00694 − 2)/2.83333 = 1.41422. Three iterations give four correct decimals - the signature of second-order convergence, where each step roughly doubles the number of correct digits.
Traps
Applying Simpson's rule to an odd number of ordinates; starting Newton–Raphson far from the root; mixing degrees and radians in trigonometric derivatives; reporting variance when standard deviation is asked; and using the error term of the trapezoidal rule with the wrong sign.