FE Other Disciplines Mathematics

Knowledge area 1 of the FE Other Disciplines exam: 8–12 of the 110 questions. The subtopics NCEES lists, how they are tested and worked practice problems with solutions.

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110 questions on the real 5 hours 20 minutes clock, weighted to the NCEES specifications, with a score by knowledge area and a worked solution for every answer.

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What the specification lists

  • Analytic geometry and trigonometry
  • Differential equations
  • Numerical methods (e.g., algebraic equations, roots of equations, approximations, precision limits, convergence)
  • Linear algebra (e.g., matrix operations)
  • Single-variable calculus

From the NCEES FE Other Disciplines CBT exam specifications, knowledge area 1.

Line drawing of graph paper with a plotted curve, a compass, a protractor and a calculator

How Mathematics is tested

Mathematics questions are short and calculator-friendly: a slope, an intersection, a derivative or integral, a vector product. Most of the formulas sit in the Reference Handbook section on mathematics, so the skill being tested is recognizing the setup quickly and not spending ten minutes on a two-minute question.

On exam day the relationships you need are in the Reference Handbook section on Mathematics. The handbook is the only reference allowed, so practice finding each formula in it rather than memorizing a separate sheet.

Start the free Mathematics drill

FE Other Disciplines Mathematics practice problems

Work each problem with your calculator and the handbook, then open the solution.

Problem 1 · Numerical methods

Select all that apply.

Select all statements that are true of Newton's method for finding a root of f(x) = 0.

  1. A.Near a simple root it converges quadratically.
  2. B.It converges from any starting point.
  3. C.It can fail or diverge when f′(xₙ) is close to zero.
  4. D.It requires a starting interval over which f changes sign.
  5. E.Each iteration requires evaluating the derivative f′(xₙ).
Show the answer and solution

Answer: A: Near a simple root it converges quadratically.; C: It can fail or diverge when f′(xₙ) is close to zero.; E: Each iteration requires evaluating the derivative f′(xₙ).

Newton's update xₙ₊₁ = xₙ − f(xₙ)/f′(xₙ) needs the derivative at every step, roughly doubles the number of correct digits per iteration near a simple root, and takes huge or undefined steps when f′ is near zero. It starts from a single guess, and a poor guess can make it diverge or converge to a different root.

Numerical methods · Reference Handbook: Mathematics

Problem 2 · Linear algebra

The eigenvalues of the matrix with rows [4, 1] and [2, 3] are:

  1. A.7 and 10
  2. B.−2 and −5
  3. C.2 and 5
  4. D.3 and 4
Show the answer and solution

Answer: C: 2 and 5

det(A − λI) = (4 − λ)(3 − λ) − 2 = λ² − 7λ + 10 = (λ − 2)(λ − 5) = 0, so λ = 2 and 5. As checks, their sum equals the trace (7) and their product equals the determinant (10); the diagonal entries are eigenvalues only for a triangular matrix.

Linear algebra · Reference Handbook: Mathematics

Problem 3 · Single-variable calculus

The displacement of a vibrating sensor is x(t) = 5 sin(3t²), with x in mm and t in seconds (the argument is in radians). The velocity at t = 0.5 s is most nearly:

  1. A.3.66 mm/s
  2. B.11.0 mm/s
  3. C.15.0 mm/s
  4. D.22.0 mm/s
Show the answer and solution

Answer: B: 11.0 mm/s

By the chain rule, v = dx/dt = 5 cos(3t²)(6t) = 30t cos(3t²). At t = 0.5 s: v = 15 cos(0.75 rad) = 15(0.7317) = 11.0 mm/s. Evaluating cos 0.75 in degree mode gives about 15.0 mm/s.

Single-variable calculus · Reference Handbook: Mathematics

Problem 4 · Ordinary differential equations

The general solution of y″ + 4y′ + 13y = 0 is:

  1. A.y = e^(−3t)(C₁ cos 2t + C₂ sin 2t)
  2. B.y = e^(−2t)(C₁ cos 3t + C₂ sin 3t)
  3. C.y = e^(2t)(C₁ cos 3t + C₂ sin 3t)
  4. D.y = e^(−4t)(C₁ cos √13 t + C₂ sin √13 t)
Show the answer and solution

Answer: B: y = e^(−2t)(C₁ cos 3t + C₂ sin 3t)

The characteristic equation r² + 4r + 13 = 0 gives r = [−4 ± √(16 − 52)]/2 = −2 ± 3i. Complex roots α ± βi give y = e^(αt)(C₁ cos βt + C₂ sin βt) with α = −2 and β = 3. A positive exponent would describe a growing response, which the positive damping term rules out.

Ordinary differential equations · Reference Handbook: Mathematics

More FE Other Disciplines practice problems from every knowledge area

Sit the full exam under the real clock

Take a full timed FE Other Disciplines practice exam

110 questions on the real 5 hours 20 minutes clock, weighted to the NCEES specifications, with a score by knowledge area and a worked solution for every answer.

Or start the free Mathematics drill

Common questions

How many Mathematics questions are on the FE Other Disciplines exam?

The NCEES FE Other Disciplines specification lists 8–12 questions on Mathematics, out of 110. Every examinee gets the same number per knowledge area.

What does Mathematics cover on the FE Other Disciplines exam?

Analytic geometry and trigonometry; Differential equations; Numerical methods (e.g., algebraic equations, roots of equations, approximations, precision limits, convergence); Linear algebra (e.g., matrix operations); Single-variable calculus.

Where are the Mathematics formulas on exam day?

In the NCEES FE Reference Handbook, which appears on screen during the exam: the Mathematics section. Practice with your free copy from MyNCEES so you know where each relationship sits.

Original independent practice questions written to the published NCEES exam specifications. Not affiliated with or endorsed by NCEES (National Council of Examiners for Engineering and Surveying).