FE Mechanical Mathematics

Knowledge area 1 of the FE Mechanical exam: 6–9 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
  • Calculus (e.g., differential, integral, single-variable, multivariable)
  • Ordinary differential equations (e.g., homogeneous, nonhomogeneous, Laplace transforms)
  • Linear algebra (e.g., matrix operations, vector analysis)
  • Numerical methods (e.g., approximations, precision limits, error propagation, Taylor’s series, Newton’s method)
  • Algorithm and logic development (e.g., flowcharts, pseudocode)

From the NCEES FE Mechanical 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 Mechanical Mathematics practice problems

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

Problem 1 · Differential equations

A casting cools in a 20°C room according to dT/dt = −k(T − 20), with k = 0.05 min⁻¹ and T in °C. Its temperature at t = 0 is 90°C. Enter the temperature at t = 30 min, in °C, to one decimal place.

Fill in the blank: answer in °C.

Show the answer and solution

Answer: 35.6 °C (accepted 35.2 to 36)

Separating variables gives T(t) = 20 + (90 − 20)e^(−kt). At t = 30 min: T = 20 + 70e^(−1.5) = 20 + 70(0.2231) = 35.6°C. Applying the decay to 90°C instead of to the 70°C excess over ambient gives 20.1°C.

Differential equations · Reference Handbook: Mathematics

Problem 2 · Analytic geometry

On a site grid with coordinates in meters, a straight property line follows 3x + 4y − 12 = 0. A survey monument sits at (4, −2) m. The perpendicular distance from the monument to the property line is most nearly:

  1. A.0.32 m
  2. B.1.14 m
  3. C.1.60 m
  4. D.8.00 m
Show the answer and solution

Answer: C: 1.60 m

d = |Ax₀ + By₀ + C|/√(A² + B²) = |3(4) + 4(−2) − 12|/√(9 + 16) = |−8|/5 = 1.60 m. Leaving out the denominator gives 8.00 m, and dividing by 25 instead of √25 gives 0.32 m.

Analytic geometry · Reference Handbook: Mathematics

Problem 3 · Numerical methods

Newton's method is used to find the root of f(x) = x³ − 30, starting from x₀ = 4. The estimate after two iterations, x₂, is most nearly:

  1. A.1.17
  2. B.3.12
  3. C.3.29
  4. D.4.71
Show the answer and solution

Answer: B: 3.12

x₁ = 4 − (64 − 30)/(3·16) = 4 − 0.7083 = 3.2917. x₂ = 3.2917 − (35.666 − 30)/(3·10.835) = 3.2917 − 0.1743 = 3.1174 ≈ 3.12. For comparison, the true root is ∛30 = 3.107. Stopping after one step leaves the estimate at 3.29.

Numerical methods · Reference Handbook: Mathematics

Problem 4 · Vector operations

A force F = −400j + 300k N acts at a point whose position vector from origin O is r = 2i + 1.5k m. The magnitude of the moment of F about O (M = r × F) is most nearly:

  1. A.450 N·m
  2. B.1,166 N·m
  3. C.1,250 N·m
  4. D.2,000 N·m
Show the answer and solution

Answer: B: 1,166 N·m

r × F = (0·300 − 1.5·(−400))i − (2·300 − 1.5·0)j + (2·(−400) − 0·0)k = 600i − 600j − 800k N·m. |M| = √(600² + 600² + 800²) = 1,166 N·m. Multiplying |r| = 2.5 m by |F| = 500 N gives 1,250 N·m, which assumes a 90° angle between them.

Vector operations · Reference Handbook: Mathematics

More FE Mechanical practice problems from every knowledge area

Sit the full exam under the real clock

Take a full timed FE Mechanical 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 Mechanical exam?

The NCEES FE Mechanical specification lists 6–9 questions on Mathematics, out of 110. Every examinee gets the same number per knowledge area.

What does Mathematics cover on the FE Mechanical exam?

Analytic geometry; Calculus (e.g., differential, integral, single-variable, multivariable); Ordinary differential equations (e.g., homogeneous, nonhomogeneous, Laplace transforms); Linear algebra (e.g., matrix operations, vector analysis); Numerical methods (e.g., approximations, precision limits, error propagation, Taylor’s series, Newton’s method); Algorithm and logic development (e.g., flowcharts, pseudocode).

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).