FE math practice problems with solutions
Below are six FE math and statistics practice problems in exam style: a quadratic root, a derivative, a definite integral, the law of cosines, a binomial probability and a sample standard deviation. Each has a step-by-step solution and the FE Reference Handbook 10.6 page. Most should take 1 to 2 minutes.
Last reviewed October 10, 2026 by Engineer Exam Lab. Exam facts link to NCEES. Problems are from our practice packs, written by a PhD environmental engineer, and every numeric answer was checked by a second, independent calculation.
Where math shows up
- FE Civil: Mathematics and Statistics, 8–12 questions. FE Environmental: Mathematics, 5–8 questions, plus Probability and Statistics, 4–6 (NCEES specifications).
- The handbook's Mathematics and Engineering Probability and Statistics chapters hold every formula used below.
- These are the fastest points on the exam if you can do them in 1 to 2 minutes, which leaves more time for long discipline problems.
Six FE math and statistics practice problems
Problem 1 · FE Civil · Mathematics and Statistics · Multiple choice
The larger root of 3x² - 18x - 16 = 0 is most nearly:
Show the worked solution
Answer: D) 6.79
Equation: x = [-b ± √(b² - 4ac)]/(2a)
- Discriminant: b² - 4ac = (-18)² - 4(3)(-16) = 516
- x = [-b + √(b² - 4ac)]/(2a) = [18 + √516]/(2 × 3) = [18 + 22.72]/6 = 6.79
Why the wrong choices are wrong:
- A used b² + 4ac under the root instead of b² - 4ac.
- B divided only -b by 2a and left the square root undivided.
- C divided by a instead of 2a.
Handbook: FE Reference Handbook 10.6, Mathematics, Quadratic Equation, p. 37.
Problem 2 · FE Environmental · Mathematics · Multiple choice
For f(x) = 2x³ - 7x² + 7x - 8, the slope of the curve at x = 3 is most nearly:
Show the worked solution
Answer: D) 19.0
Equation: d(axⁿ)/dx = n·a·xⁿ⁻¹; the slope is f′(x0)
- f′(x) = 6x² - 14x + 7
- f′(3) = 3(2)(3)² + 2(-7)(3) + (7) = 19
Why the wrong choices are wrong:
- A dropped the derivative of the linear term.
- B lowered each power but did not multiply by it.
- C multiplied by each power but did not lower it.
Handbook: FE Reference Handbook 10.6, Mathematics, Differential Calculus (the derivative), p. 47.
Problem 3 · FE Environmental · Mathematics · Multiple choice
The area under the curve y = 4x² - 6x + 16 between x = 0 and x = 5 (the curve is above the x-axis there) is most nearly:
Show the worked solution
Answer: D) 172
Equation: Area = ∫ f(x) dx from x1 to x2 = F(x2) - F(x1)
- F(x) = ∫(4x² - 6x + 16) dx = (4/3)x³ + (-6/2)x² + 16x
- F(5) = 171.7; F(0) = 0
- Area = F(5) - F(0) = 172
Why the wrong choices are wrong:
- A used one trapezoid (average of the end heights times the width), not the integral.
- B differentiated instead of integrating.
- C raised each power but did not divide by the new power.
Handbook: FE Reference Handbook 10.6, Mathematics, Integral Calculus (definite integral), p. 49.
Problem 4 · FE Civil · Mathematics and Statistics · Multiple choice
Two sides of a triangular lot are b = 57.5 m and c = 10.0 m, and the angle between them is A = 152°. The third side a is most nearly:
Show the worked solution
Answer: C) 66.5 m
Equation: a² = b² + c² - 2bc cos A
- a² = b² + c² - 2bc cos A = (57.5)² + (10.0)² - 2(57.5)(10.0) cos 152° = 4,422 m²
- a = √4,422 = 66.5 m
Why the wrong choices are wrong:
- A added the 2bc cos A term instead of subtracting it.
- B took the cosine with the calculator in radians.
- D left out the 2 in 2bc cos A.
Handbook: FE Reference Handbook 10.6, Mathematics, Trigonometry, Law of Cosines, p. 39.
Problem 5 · FE Civil · Mathematics and Statistics · Multiple choice
Each water sample from a distribution system has a probability of 0.4 of failing a coliform test, independently. For 8 samples, the probability that exactly 2 fail is most nearly:
Show the worked solution
Answer: C) 0.209
Equation: P(x) = C(n, x) pˣ qⁿ⁻ˣ, with q = 1 - p
- C(8, 2) = 8!/[2!(6)!] = 28
- P(2) = C(n, x) pˣ qⁿ⁻ˣ = 28 × (0.4)^2 × (0.60)^6 = 0.209
Why the wrong choices are wrong:
- A left out the (1 - p)^(n - x) term.
- B is P(X ≥ 2), not exactly 2.
- D left out the number of combinations C(n, x).
Handbook: FE Reference Handbook 10.6, Engineering Probability and Statistics, Binomial Distribution, p. 67.
Problem 6 · FE Civil · Mathematics and Statistics · Multiple choice
A lab reports the tensile strength of rebar samples: 594, 427, 489, 542, 612 MPa. The sample standard deviation is most nearly:
Show the worked solution
Answer: C) 76.2 MPa
Equation: s = √[Σ(xi - x̄)²/(n - 1)]
- Mean: x̄ = Σx/n = 2,664/5 = 532.8 MPa
- Σ(x - x̄)² = 23,210 MPa²
- s = √[Σ(x - x̄)²/(n - 1)] = √(23,210/4) = 76.2 MPa
Why the wrong choices are wrong:
- A is half the range, not a standard deviation.
- B is the mean absolute deviation, not the standard deviation.
- D divided by n (population) instead of n - 1 for a sample.
Handbook: FE Reference Handbook 10.6, Engineering Probability and Statistics, Dispersion, Mean, Median, and Mode Values, p. 64.
The slips the wrong choices are built from
- Sign errors in the quadratic formula: −b, not b.
- Degrees vs radians. The law of cosines with the calculator in the wrong mode gives a wrong side length that can still look plausible.
- Sample vs population standard deviation. Divide by n − 1 for a sample. On a calculator that is Sx or sx, not σx.
- Exactly vs at most in binomial questions: one term, or a sum of terms.
Want 100 more like these, plus a timed 110-question mock? Each pack is $39, one time, with every solution tied to the handbook page.
What this page doesn't cover
Six problems sample a wide area. Vectors, matrices, differential equations, numerical methods and other distributions also appear. For calculator shortcuts on these problems, see FE calculator functions.
Questions
How many math questions are on the FE exam?
FE Civil has 8 to 12 Mathematics and Statistics questions. FE Environmental has 5 to 8 Mathematics questions plus 4 to 6 Probability and Statistics questions, according to NCEES.
Is calculus on the FE exam?
Yes. Derivatives and definite integrals of simple functions appear, and the handbook lists the common derivative and integral forms.
Can my calculator do these?
Most approved models compute integrals, derivatives, roots and statistics, but the steps differ by model. Practice on the exact model you will bring.
Are these real FE exam questions?
No. They are original practice problems from our packs. Each numeric answer was computed by code and recomputed independently.
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Engineer Exam Lab is not affiliated with or endorsed by NCEES. The problems here are original practice problems, not questions from a real exam.