Engineering Mathematics and Basic Sciences
Engineering Mathematics and Basic Sciences tests linear algebra, calculus, differential equations, probability, numerical methods and the physics and chemistry that underpin civil engineering.
Core rules
- For a 2×2 matrix, trace equals the sum of eigenvalues and determinant equals their product; this gives the characteristic equation λ² − (trace)λ + det = 0.
- A linear system Ax = b is consistent only if rank(A) = rank([A|b]); if this common rank equals the number of unknowns, the solution is unique.
- First-order linear ODE dy/dx + P(x)y = Q(x) has integrating factor e^(∫P dx).
- Probability: for independent events P(A∩B) = P(A)P(B); Bayes theorem updates prior probabilities with likelihoods.
- Numerical integration: Simpson 1/3 rule needs an even number of intervals; the trapezoidal rule works with any number.
Worked example
Find the eigenvalues of A = [[4, 1], [2, 3]]. Trace = 4 + 3 = 7; determinant = 4×3 − 1×2 = 10. Characteristic equation: λ² − 7λ + 10 = 0 = (λ − 5)(λ − 2). Eigenvalues are 5 and 2. Check: sum 7, product 10.
Traps
Using degrees in calculus limits and derivatives; dropping the constant of integration; treating a 3×3 matrix as if trace and determinant alone determine all eigenvalues; applying Simpson rule with an odd number of intervals; confusing P(A|B) with P(B|A) in Bayes problems.