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04-BS-1: May 2015

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

  1. Question 1 Three First- and Second-Order ODEs — Integrating Factor, Separable/Bernoulli, and Complex Characteristic Roots
  2. Question 2 Cauchy–Euler Equation with a Resonant Power-Law Forcing
  3. Question 3 Verifying an Eigenvector and an Eigenvalue, and Building Two Solutions of a Linear System
  4. Question 4 Work Done by a Vector Field Along a Helical Path — Conservative-Field Shortcut
  5. Question 5 Tangent Plane to an Implicitly Defined Surface
  6. Question 6 Plane Through Three Points, and Its Intersection Line With Another Plane
  7. Question 7 Classifying and Diagonalizing a Quadratic Form (Conic Section)
  8. Question 8 Closed-Surface Flux Through a Paraboloid Cap via the Divergence Theorem

Start with Question 1 →

National Exams — May 2015 — 04-BS-1 Mathematics. Three-hour, closed-book exam (one double-sided aid sheet permitted, approved calculator allowed). Format: eight questions offered; any five constitute a complete paper and only the first five appearing in the answer book are marked. All eight are solved below for completeness.

Reference texts: Kreyszig, Advanced Engineering Mathematics (10th ed., Wiley) — first-order and constant-coefficient ODEs, Cauchy–Euler equations, eigenvalues and linear systems, line/surface integrals, the divergence theorem; Stewart, Calculus: Early Transcendentals (9th ed., Cengage) — lines and planes in space, tangent-plane linear approximation, line integrals of vector fields; Strang, Introduction to Linear Algebra (6th ed.) — eigenvalues/eigenvectors, systems of linear ODEs, quadratic forms and principal axes.