04-BS-1
Worked solutions to 14 past sittings (2013–2019), 111 questions. Pick a sitting, or start from a topic below.
- December 2019
- December 2018
- May 2018
- December 2017
- May 2017
- December 2016
- May 2016
- December 2015
- May 2015
- December 2014
- May 2014
- December 2013
- May 2013
- Undated paper
Topics across the sittings
Topics that come up in more than one sitting, taken from the headings of our worked solutions. A topic counts once per sitting.
- Line Integral via Stokes' Theorem (10 sittings): December 2018 Q8 · May 2018 Q6 · December 2017 Q5 · May 2017 Q6 · May 2016 Q8 · December 2015 Q7 · December 2014 Q5 · May 2014 Q7 · December 2013 Q6 · May 2013 Q7
- Above a Cone (5 sittings): December 2019 Q6 · December 2017 Q4 · December 2015 Q8 · May 2013 Q8 · Undated paper Q5
- Tangent Plane (5 sittings): December 2019 Q5 · May 2018 Q3 · December 2017 Q7 · December 2016 Q5 · May 2013 Q4
- Intersection (4 sittings): December 2018 Q4 · December 2015 Q3 · December 2014 Q3 · May 2013 Q6
- Linear Approximation (4 sittings): December 2019 Q5 · December 2017 Q7 · December 2014 Q8 · May 2013 Q4
- Second-Order ODEs (4 sittings): May 2017 Q1 · May 2016 Q1 · December 2015 Q1 · May 2015 Q1
- Two Lines in Space (4 sittings): December 2018 Q4 · December 2015 Q3 · December 2014 Q3 · May 2013 Q6
- Clockwise Orientation (3 sittings): May 2017 Q6 · December 2014 Q5 · May 2014 Q7
- Common Plane (3 sittings): December 2018 Q4 · December 2014 Q3 · May 2013 Q6
- Constrained Minimum via Lagrange Multipliers (3 sittings): December 2019 Q3 · May 2016 Q4 · December 2013 Q3
- Orthogonal Line (3 sittings): December 2018 Q4 · December 2014 Q3 · May 2013 Q6
- Plane Through Three Points (3 sittings): May 2017 Q5 · May 2015 Q6 · December 2013 Q4
- Two Linear ODEs (3 sittings): December 2014 Q1 · May 2014 Q1 · December 2013 Q1
- Volume Inside an Ellipsoid (3 sittings): December 2019 Q6 · December 2015 Q8 · May 2013 Q8
- Angle Between a Line (2 sittings): December 2018 Q5 · May 2018 Q5
Questions by sitting
December 2019
- Question 1: Forced Second-Order IVP with Complex Roots
- Question 2: General Solution of a Linear First-Order ODE via Integrating Factor
- Question 3: Constrained Minimum via Lagrange Multipliers
- Question 4: General Solution of a Resonant Linear System
- Question 5: Tangent Plane and Linear Approximation
- Question 6: Volume Inside an Ellipsoid and Above a Cone
- Question 7: Work Done by a Conservative Field Along a Parametrized Path
- Question 8: Surface Area of a Cone in the First Octant
December 2018
- Question 1: General Solutions of Two First–/Second–Order ODEs
- Question 2: General Solution of a Doubly-Forced Oscillator (Resonant + Non-Resonant)
- Question 3: Constrained Extrema on an Ellipsoid (Lagrange Multipliers)
- Question 4: Two Lines in Space — Intersection, Orthogonal Line, Common Plane
- Question 5: Angle Between a Line and a Surface
- Question 6: Flux Through a Closed Surface (Divergence Theorem)
- Question 7: Work Done by a Conservative Field Along a Curved Path
- Question 8: Line Integral via Stokes' Theorem (Cylinder ∩ Plane)
May 2018
- Question 1: Nonhomogeneous Euler–Cauchy Equation
- Question 2: Two Initial Value Problems — Separable and Constant-Coefficient
- Question 3: Tangent Plane and Tangent Line to a Surface Intersection
- Question 4: General Solution of a Resonantly-Forced Linear System
- Question 5: Angle Between a Line and a Surface
- Question 6: Line Integral via Stokes' Theorem (Cylinder ∩ Tilted Plane)
- Question 7: Closed-Surface Flux Integral via the Divergence Theorem (Paraboloid Cap)
- Question 8: Lagrange Multipliers — Minimum on an Ellipsoid
December 2017
- Question 1: Resonant Initial Value Problem for a Forced Harmonic Oscillator
- Question 2: Euler–Cauchy Equation with Non-Resonant Exponential–Polynomial Forcing
- Question 3: Eigenvalues/Eigenvectors and a Linear System Initial Value Problem
- Question 4: Volume Inside a Sphere and Above a Cone (Spherical Coordinates)
- Question 5: Line Integral via Stokes' Theorem (Cylinder ∩ Tilted Plane)
- Question 6: Laplace-Transform Response of a Damped Mass-Spring System to a Square-Wave Force
- Question 7: Tangent Plane and Linear Approximation
- Question 8: Closed-Surface Flux Integral via the Divergence Theorem (Quarter-Elliptical Cylinder)
May 2017
- Question 1: Three First- and Second-Order ODEs
- Question 2: Euler–Cauchy Equation with a Non-Resonant Power Forcing
- Question 3: Extrema of a Linear Function on a Sphere via Lagrange Multipliers
- Question 4: Initial Value Problem for a 2×2 Linear System with Complex Eigenvalues
- Question 5: Plane Through Three Points and Its Intersection with a Second Plane
- Question 6: Line Integral via Stokes' Theorem, Clockwise Orientation
- Question 7: Closed-Surface Flux Through a Paraboloid Cap via the Divergence Theorem
- Question 8: Damped Mass–Spring Response to a Square-Wave Force
December 2016
- Question 1: Euler–Cauchy Equation with Exponential–Polynomial Forcing
- Question 2: Forced Oscillator — Resonant and Non-Resonant Forcing Together
- Question 3: Constrained Extrema on an Ellipsoid via Lagrange Multipliers
- Question 4: Verifying an Eigenvector and Finding a Second Eigenpair
- Question 5: Tangent Plane and Tangent Line for Intersecting Surfaces
- Question 6: Closed-Surface Flux Through a Paraboloid Cap via the Divergence Theorem
- Question 7: Work Done by a Field Along a Helical Path — Conservative-Field Shortcut
- Question 8: Surface Integral Over a Half-Cylinder
May 2016
- Question 1: Three First- and Second-Order ODEs
- Question 2: Forced Second-Order IVP
- Question 3: Linear System of ODEs (Eigenvalue Method with Resonant Forcing)
- Question 4: Constrained Minimum via Lagrange Multipliers
- Question 5: Tangent Line to the Intersection of Two Surfaces
- Question 6: Volume Between a Paraboloid and a Plane, Outside a Cone
- Question 7: Surface Area of a Cone in the First Octant
- Question 8: Line Integral via Stokes' Theorem
December 2015
- Question 1: Two First- and Second-Order ODEs
- Question 2: Two Initial Value Problems
- Question 3: Two Lines in Space — Intersection, Common Perpendicular, and Containing Plane
- Question 4: Constrained Max/Min via Lagrange Multipliers (Ellipsoid Constraint)
- Question 5: Closed-Surface Flux Integral via the Divergence Theorem (Paraboloid Capped by a Disk)
- Question 6: Angle of Intersection Between a Line and a Hyperboloid
- Question 7: Line Integral via Stokes' Theorem (Cylinder ∩ Tilted Plane)
- Question 8: Volume Inside an Ellipsoid and Above a Cone
May 2015
- Question 1: Three First- and Second-Order ODEs — Integrating Factor, Separable/Bernoulli, and Complex Characteristic Roots
- Question 2: Cauchy–Euler Equation with a Resonant Power-Law Forcing
- Question 3: Verifying an Eigenvector and an Eigenvalue, and Building Two Solutions of a Linear System
- Question 4: Work Done by a Vector Field Along a Helical Path — Conservative-Field Shortcut
- Question 5: Tangent Plane to an Implicitly Defined Surface
- Question 6: Plane Through Three Points, and Its Intersection Line With Another Plane
- Question 7: Classifying and Diagonalizing a Quadratic Form (Conic Section)
- Question 8: Closed-Surface Flux Through a Paraboloid Cap via the Divergence Theorem
December 2014
- Question 1: Two Linear ODEs — Variation of Parameters with a Secant Forcing, and a Resonant Polynomial+Exponential Forcing
- Question 2: Cauchy–Euler Equation with a Resonant Power-Law Forcing
- Question 3: Two Lines in Space — Intersection, Orthogonal Line, and Common Plane
- Question 4: Closed-Surface Flux Through a Cone via the Divergence Theorem
- Question 5: Line Integral via Stokes' Theorem, Clockwise Orientation
- Question 6: Volume Outside a Cylinder, Inside an Ellipsoid
- Question 7: Eigenvalues/Eigenvectors of a $3\times3$ Matrix, and the Associated Linear System
- Question 8: Tangent Plane to a Logarithmic Surface and Linear Approximation
May 2014
- Question 1: Two Linear ODEs — Variation of Parameters with a Secant Forcing, and a Resonant Polynomial+Exponential Forcing
- Question 2: Extrema of a Linear Function on a Sphere
- Question 3: Tangent Line to the Intersection of Two Surfaces
- Question 4: Eigenvalues/Eigenvectors of a $2\times2$ Matrix, and the Associated Linear IVP System
- Question 5: Closed-Surface Flux Through a Paraboloid Cap via the Divergence Theorem
- Question 6: Volume Between a Paraboloid and a Plane, Outside a Cone
- Question 7: Line Integral via Stokes' Theorem, Clockwise Orientation
- Question 8: Damped Mass-Spring System Driven by a Square Pulse (Laplace Transform)
December 2013
- Question 1: Two Linear ODEs — Resonant IVP and a Reducible Second-Order Equation
- Question 2: Classifying and Diagonalizing a Quadratic Form (Conic Section)
- Question 3: Constrained Minimum via Lagrange Multipliers
- Question 4: Plane Through Three Points, and Its Intersection Line With Another Plane
- Question 5: Closed-Surface Flux Integral via the Divergence Theorem
- Question 6: Line Integral via Stokes' Theorem
- Question 7: Angle of Intersection Between a Line and a Hyperboloid
- Question 8: Driven Damped Mass-Spring System (2nd-Order IVP)
May 2013
- Question 1: Second-Order IVP by Undetermined Coefficients
- Question 2: General Solutions of Three First/Second-Order ODEs
- Question 3: Eigenvalues/Eigenvectors and a Nonhomogeneous Linear System
- Question 4: Tangent Plane and Linear Approximation
- Question 5: Surface Area of a Cone in the First Octant
- Question 6: Two Lines in Space — Intersection, Orthogonal Line, Common Plane
- Question 7: Line Integral via Stokes' Theorem
- Question 8: Volume Inside an Ellipsoid and Above a Cone
Undated paper
- Question 1: General Solutions of Three First–/Second–Order ODEs
- Question 2: Critically-Damped Forced Oscillator (Initial Value Problem)
- Question 3: Initial Value Problem with Real Distinct Roots and Linear Forcing
- Question 4: Tangent Line to the Intersection of Two Quadric Surfaces
- Question 5: Volume Inside a Sphere and Above a Cone (Spherical Coordinates)
- Question 6: Line Integral Along the Equatorial Circle of a Hemisphere
- Question 7: Steady-State Response of a Damped Mass-Spring System to a Periodic Square Wave