What is the value of the integral of (3x^2 + 2) dx from x = 0 to x = 2?
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About this test
This free FE exam practice test has 15 problems written by the StudyPDF team: math, statics, dynamics, mechanics of materials, fluid mechanics and ethics, the subjects that show up on most FE exams. The real FE has 110 questions in 6 hours, so this is a short sampler that shows where you are strong, not a full-length exam. Work each problem on paper first, then check your score and read the worked solution for every one.
All 15 questions at a glance
Prefer to read before you play? Every question in the sampler is listed below. Answers and explanations stay collapsed until you open them.
1.What is the value of the integral of (3x^2 + 2) dx from x = 0 to x = 2?
- A. 8
- B. 10
- C. 12
- D. 16
Answer: C. 12
The antiderivative of 3x^2 + 2 is x^3 + 2x. At x = 2 that is 8 + 4 = 12, and at x = 0 it is 0, so the definite integral is 12. A common slip is to forget the 2x term and stop at 8, so integrate every term before you plug in the limits.
2.Which of these is the general solution of the differential equation y'' - 4y = 0?
- A. y = C1 e^(2x) + C2 e^(-2x)
- B. y = C1 cos(2x) + C2 sin(2x)
- C. y = C1 e^(4x) + C2 e^(-4x)
- D. y = C1 e^(2x) + C2 x e^(2x)
Answer: A. y = C1 e^(2x) + C2 e^(-2x)
Try y = e^(rx). The characteristic equation is r^2 - 4 = 0, so r = 2 or r = -2, and the two real roots give e^(2x) and e^(-2x). The cosine and sine form would come from y'' + 4y = 0, the term x e^(2x) appears only for a repeated root, and e^(4x) mistakes r^2 = 4 for r = 4.
3.Complete the sentence.
For the matrix A with rows [2, 1] and [4, 3], the determinant of A is _____ and the trace of A is _____.
Options for blank 1: 2, 6, 10, 14
Options for blank 2: 3, 5, 7, 10
Answer: 2, 5
For a 2 by 2 matrix, the determinant is ad - bc = (2)(3) - (1)(4) = 6 - 4 = 2. The trace is the sum of the diagonal entries, 2 + 3 = 5. Mixing up the two is a common error: the determinant subtracts the cross products, and the trace only adds the diagonal.
4.A simply supported beam is 6 m long with a support at each end. A single downward load of 12 kN acts 2 m from the left support. What is the vertical reaction at the left support?
- A. 4 kN
- B. 6 kN
- C. 10 kN
- D. 8 kN
Answer: D. 8 kN
Take moments about the right support: the left reaction times 6 m equals 12 kN times 4 m, the distance from the load to the right support. That gives 48 / 6 = 8 kN. The right reaction is then 12 - 8 = 4 kN. The load sits closer to the left support, so the left support carries more of it.
5.A 50 kg block rests on a horizontal surface. The coefficient of static friction is 0.40. Use g = 9.81 m/s^2. What is the largest horizontal force that can be applied before the block starts to slide?
- A. 20 N
- B. 196 N
- C. 490 N
- D. 1,226 N
Answer: B. 196 N
The normal force on a horizontal surface equals the weight: 50 x 9.81 = 490.5 N. Maximum static friction is the coefficient times the normal force, 0.40 x 490.5 = 196.2 N. Choosing 490 N forgets the coefficient, and 1,226 N comes from dividing by 0.40 instead of multiplying.
6.A truss member is loaded only at its two pinned ends, and its own weight is neglected. Select all statements that are true of this two-force member.
Select all that apply.
- A. The two end forces are equal in magnitude and opposite in direction.
- B. The forces act along the line joining the two pins.
- C. The member carries shear force and a bending moment along its length.
- D. The member can be in tension or in compression.
- E. The two end forces must be perpendicular to each other.
Answer: A. The two end forces are equal in magnitude and opposite in direction., B. The forces act along the line joining the two pins., D. The member can be in tension or in compression.
For a body loaded at two points to stay in equilibrium, the two forces must be equal, opposite and collinear, which means they act along the line between the pins. Whether they pull the ends apart or push them together decides tension or compression. With no loads along the length there is no shear or bending in the member, and the forces are collinear, not perpendicular.
7.A 1,500 kg car is slowed by a constant net braking force of 6,000 N. How long does it take to stop from a speed of 24 m/s?
- A. 4 s
- B. 9 s
- C. 6 s
- D. 16 s
Answer: C. 6 s
Newton's second law gives the deceleration: a = F / m = 6,000 / 1,500 = 4 m/s^2. With constant deceleration, the time to stop is t = v / a = 24 / 4 = 6 s. The choice of 4 s takes the value of the deceleration as the time, so check that your final number has the units the question asks for.
8.Complete the sentence.
A 2 kg mass moves at 6 m/s. Its kinetic energy is _____ J and its linear momentum is _____ kg*m/s.
Options for blank 1: 12, 18, 36, 72
Options for blank 2: 6, 12, 24, 36
Answer: 36, 12
Kinetic energy is (1/2) m v^2 = 0.5 x 2 x 36 = 36 J. Momentum is m v = 2 x 6 = 12 kg*m/s. Energy depends on the square of the speed and momentum on the speed itself, which is why 72 is the typical wrong answer when the one-half is forgotten.
9.A solid round steel rod with a diameter of 20 mm carries an axial tensile load of 30 kN. What is the average normal stress in the rod?
- A. 23.9 MPa
- B. 95.5 MPa
- C. 47.7 MPa
- D. 191 MPa
Answer: B. 95.5 MPa
Stress is force divided by area. The area is pi x (10 mm)^2 = 314.2 mm^2, so the stress is 30,000 N / 314.2 mm^2 = 95.5 N/mm^2, which is 95.5 MPa. If you use the 20 mm diameter as the radius, the area is four times too big and you get 23.9 MPa. Always check whether a formula needs diameter or radius.
10.A simply supported beam is 4 m long and carries a uniformly distributed load of 10 kN/m over its full length. What is the maximum bending moment?
- A. 20 kN*m
- B. 10 kN*m
- C. 40 kN*m
- D. 80 kN*m
Answer: A. 20 kN*m
For a simply supported beam with a uniform load, the maximum moment is at mid-span and equals w L^2 / 8 = 10 x 16 / 8 = 20 kN*m. The choice of 80 kN*m uses w L^2 / 2, which is the formula for a cantilever's fixed end, and 40 kN*m uses w L^2 / 4. Match the formula to the support conditions before you plug in numbers.
11.A ductile metal is tested in uniaxial tension. Select all statements that are true.
Select all that apply.
- A. In the linear elastic region, stress is proportional to strain.
- B. Young's modulus is the slope of the linear part of the stress-strain curve.
- C. The yield strength is always equal to the ultimate tensile strength.
- D. Necking begins at about the ultimate tensile strength.
- E. Stress stays proportional to strain after the metal yields.
Answer: A. In the linear elastic region, stress is proportional to strain., B. Young's modulus is the slope of the linear part of the stress-strain curve., D. Necking begins at about the ultimate tensile strength.
Hooke's law holds in the linear elastic region, and Young's modulus is the slope there. On the engineering stress-strain curve, the ultimate tensile strength is the peak, and necking, the local thinning of the specimen, starts about there. Yield strength is lower than ultimate strength in a ductile metal, and once the metal yields, stress is no longer proportional to strain.
12.A storage tank is filled to a depth of 6 m with oil that has a density of 800 kg/m^3. Use g = 9.81 m/s^2. What is the gauge pressure at the bottom of the tank?
- A. 4.7 kPa
- B. 58.9 kPa
- C. 470.9 kPa
- D. 47.1 kPa
Answer: D. 47.1 kPa
Hydrostatic gauge pressure is rho x g x h = 800 x 9.81 x 6 = 47,088 Pa, or about 47.1 kPa. The 58.9 kPa choice uses the density of water, 1,000 kg/m^3, and the others are off by a factor of ten. The pressure depends only on the density and the depth, not on the shape or width of the tank.
13.Complete the sentence.
Water flows at 2 m/s in a pipe with a 100 mm diameter that narrows to a 50 mm diameter. The velocity in the narrow section is _____ m/s, and the flow rate is about _____ m^3/s.
Options for blank 1: 4, 6, 8, 16
Options for blank 2: 0.0039, 0.0157, 0.0628, 0.1571
Answer: 8, 0.0157
By continuity the flow rate stays the same, so area times velocity is constant. Halving the diameter cuts the area to one quarter, so the velocity is four times larger: 8 m/s. The flow rate is area times velocity: pi x (0.05 m)^2 x 2 m/s = 0.0157 m^3/s. Doubling the speed is the common mistake, since area depends on the square of the diameter.
14.An engineer finds that a design her employer wants to release could endanger the public, and the employer dismisses her concern. According to the codes of ethics used by engineers, what should she do?
- A. Release the design, since the employer has the final say.
- B. Document her concern, raise it through the proper channels, and notify the appropriate authorities if the danger remains.
- C. Quietly resign and say nothing further about the design.
- D. Post her concerns about the design publicly right away.
Answer: B. Document her concern, raise it through the proper channels, and notify the appropriate authorities if the danger remains.
The codes of ethics for engineers put the safety, health and welfare of the public first, above loyalty to an employer. That means writing down the concern, using the proper channels and, if the danger remains, informing the proper authorities. Staying silent or simply resigning leaves the danger in place, and going public first skips the channels the code expects you to use.
15.Select all actions that are consistent with the codes of professional conduct for licensed engineers.
Select all that apply.
- A. Accept work only in areas where you are competent.
- B. Make public statements only in an objective and truthful way.
- C. Accept gifts from a contractor to speed up an approval.
- D. Disclose a conflict of interest to the affected parties.
- E. Sign and seal plans prepared by others that you have not reviewed.
Answer: A. Accept work only in areas where you are competent., B. Make public statements only in an objective and truthful way., D. Disclose a conflict of interest to the affected parties.
Professional codes call for working within your competence, speaking honestly in public and disclosing conflicts of interest. Taking gifts to influence a decision is a conflict and can amount to bribery. Sealing work you have not reviewed is wrong because the seal means you take responsibility for it, so you must have direct control over the work.
What the FE exam looks like and what this page practices
The Fundamentals of Engineering (FE) exam is the first exam on the path to a professional engineering license in the United States, and it is made by NCEES. According to NCEES, it is a computer-based test with 110 questions in 6 hours, and the 6 hours include a tutorial and an optional scheduled break. It is closed book, with an electronic reference on screen. Questions use both SI and U.S. Customary units, and besides single-answer questions there are other item types, such as select-all and fill-in answers.
There are seven FE exams, one for each discipline, such as civil, mechanical, electrical and computer, and Other Disciplines. They share a lot of ground. The Other Disciplines specification lists mathematics at 8 to 12 questions and then probability and statistics, chemistry, ethics, safety, engineering economics, statics, dynamics, mechanics of materials, materials, fluid mechanics, basic electrical engineering and thermodynamics. These 15 problems stay on the shared base: math, statics, dynamics, mechanics of materials, fluids and ethics, so they are useful for most disciplines. Your own discipline's specification on the NCEES site lists the exact topics and question counts for your exam.
Every problem here gives you the numbers it needs, so you do not need the NCEES FE Reference Handbook open to solve them. On the real exam you will use the handbook, so after you finish, look up the matching formulas in your own copy and practice finding them fast.
How to work FE problems and where the time goes
On the real exam you have an average of about 3 minutes per question, so the goal is a fast, reliable method, not a clever one. Read the question, write down what is given and what is asked, pick the one formula that connects them, and check the units before you calculate.
Statics and dynamics. Start with a free-body diagram and write the equilibrium equations: the forces in each direction and the moments about a point. For motion problems, list the quantities you know before you choose among the constant-acceleration equations.
Mechanics of materials and fluids. Most of these are one-formula problems with a units trap, such as using a diameter where the formula needs a radius, or millimeters where it needs meters. Convert first, then calculate. Fluids problems often come down to continuity (flow rate equals area times velocity) or hydrostatic pressure.
Ethics. These questions test judgment against the codes of ethics for engineers, and they reward one idea: the safety, health and welfare of the public come first. Pick the answer that protects the public and follows proper channels.
No one section decides the exam, so do not sink ten minutes into a single problem. Flag it, move on and come back. NCEES also sells interactive practice exams for each FE exam, with 50 questions each, if you want problems from the test maker. An honest note: the StudyPDF team wrote every problem here to the public NCEES specification. None of them are real FE questions or come from a prep company's bank, and StudyPDF is not affiliated with NCEES.
How to build more practice tests from your own materials
Fifteen problems can show you which topics need work. They cannot get you ready on their own, and a generic set does not know which review book or course notes you are using.
StudyPDF works from your material instead. Upload your FE review book, your course notes, your class slides or the problem sets you already own. Then press New material, pick a practice test, choose those files as the sources and set how long it should be. You get a test written from those exact pages, and every explanation cites where the answer comes from. Narrow one to a single topic, such as beam bending or fluid statics, and run it again after you review.
Bo is the tutor for the rest. Ask him why a method works and he explains it from your own pages, with a hint first if you want one.
You do not have to upload anything to start. Name a topic instead, for example Mohr's circle or Bernoulli's equation, and you get practice problems on it. A free account is enough to start, and Free covers about four new materials a week, with Pro and Ultra covering more.
Written by the StudyPDF team. Last updated August 19, 2026.
Good to know
How many questions are on the FE exam, and how long is it?
According to NCEES, the FE exam has 110 questions, and you have 6 hours, which include a tutorial and an optional scheduled break. It is computer based and closed book, with an electronic reference on screen. That works out to about 3 minutes per question on average.
Which FE exam does this practice test match?
It stays on topics that the exams share, mainly math, statics, dynamics, mechanics of materials, fluid mechanics and ethics, so it is a fair warm-up for most disciplines. Each of the seven FE exams has its own specification on the NCEES site, which lists the exact topics and question counts for yours.
Do I need the FE Reference Handbook for these problems?
No. Every problem gives the numbers it needs. On the real exam, though, you work from the electronic FE Reference Handbook, so practice looking up formulas in it. NCEES offers the handbook as a free download on its site.
Are these real FE questions?
No. The StudyPDF team wrote all 15 to the public NCEES exam specification, and StudyPDF is not affiliated with NCEES. The practice test is free to play in your browser, and NCEES sells its own interactive FE practice exams for each discipline.
Need more than a sampler?
Upload your own prep book, slides or notes and Bo builds full-length practice tests from your exact materials, tracks which concepts you keep missing and targets your weak spots. Free to start.