FE (Fundamentals of Engineering) Practice Exam

Overview of the FE Exam

The Fundamentals of Engineering (FE) exam is a nationally recognized certification administered by the National Council of Examiners for Engineering and Surveying (NCEES). It serves as the first of two critical steps toward becoming a licensed Professional Engineer (PE) in the United States. The exam is designed for recent graduates and students nearing the completion of an undergraduate engineering degree from an ABET-accredited program.

The FE exam assesses a candidate’s understanding of core engineering principles across various disciplines, including mathematics, ethics, mechanics, electrical and computer engineering, thermodynamics, and more. Passing the FE exam demonstrates foundational competence in engineering and is often a requirement for many entry-level engineering roles.

Importance of Preparation

Given the breadth of topics covered and the exam’s challenging format, effective preparation is vital to success. The FE exam is a computer-based test (CBT) comprising 110 multiple-choice questions, and test-takers are allotted six hours to complete it.

Unlike standard college exams, the FE exam tests not only academic knowledge but also the practical application of engineering concepts in real-world scenarios. Without a focused study plan and consistent practice, even high-performing students may find the test difficult.

Preparation through structured resources, realistic practice exams, and discipline-specific reviews allows candidates to identify knowledge gaps, reinforce strengths, and build test-taking confidence. Most importantly, it familiarizes them with the style and complexity of FE questions.

Types of Questions on the FE Exam

The FE exam includes a mix of question types intended to simulate real-life engineering problem-solving. Questions typically fall into the following categories:

• Mathematical Problem Solving – algebra, calculus, differential equations, statistics

• Engineering Sciences – statics, dynamics, strength of materials

• Professional Ethics and Practices – codes of conduct, legal regulations, and safety standards

• Discipline-Specific Content – based on the exam discipline selected (e.g., Civil, Mechanical, Electrical, Chemical, Environmental)

Questions are multiple-choice, but some may involve numeric entry or require interpretation of graphs, charts, or schematics. The FE Reference Handbook is the only resource permitted during the exam and is available in digital form during the test.

Sample Questions or a Free Preview

To help aspiring engineers understand what to expect, we provide a set of sample questions that closely mirror the format, difficulty, and scope of the actual FE exam. These sample questions cover topics such as:

• Engineering Economics

• Statics and Dynamics

• Fluid Mechanics

• Mathematics and Probability

• Electrical Circuits

Each sample question is accompanied by a clear explanation and step-by-step solution. This not only reinforces the correct answer but also helps learners grasp the reasoning process used in professional engineering contexts.

Author and Content Credibility

This practice exam and study material were developed by a team of experienced engineers and academic professionals with a deep understanding of the NCEES FE exam framework. Each contributor holds either a Professional Engineer (PE) license or an advanced degree in engineering, and many have years of experience in both industry and education.

Our goal is to support aspiring engineers by providing reliable, thoughtfully crafted resources that bridge the gap between academic learning and professional certification. All questions are aligned with the latest FE exam specifications and designed to build real-world problem-solving skills.

1. Math – Complex Numbers

Q1. What is the magnitude of the complex number $3+4i$?
A) 5
B) 7
C) √13
D) √25

Correct Answer: A) 5

2. Ethics – Engineering Code of Ethics

Q2. An engineer discovers a design flaw in a bridge project after construction begins. What is their ethical responsibility?
A) Keep quiet
B) Notify the contractor only
C) Inform all affected parties
D) Resign immediately

Correct Answer: C) Inform all affected parties
Explanation: According to the NSPE Code of Ethics, engineers must prioritize public safety. Discovering a flaw obligates the engineer to disclose it to supervisors, clients, and relevant authorities to prevent harm.

3. Statics – Force Systems

Q3. A 500 N force is applied at an angle of 30° to the horizontal. What is the horizontal component?
A) 250 N
B) 433 N
C) 500 N
D) 600 N

Correct Answer: B) 433 N
Explanation: The horizontal component is $F\cdot \cos (\theta )=500\cdot \cos (30°)=500\cdot 0.866=433N$. This type of question tests your trigonometric understanding of vector resolution.

4. Materials – Stress and Strain

Q4. What is the stress in a bar with a force of 1000 N and a cross-sectional area of 20 mm²?
A) 50 MPa
B) 5 MPa
C) 500 MPa
D) 0.5 MPa

Correct Answer: C) 500 MPa
Explanation: Stress = Force / Area = 1000 N / 20 mm² = 1000 / 0.00002 m² = 50,000,000 Pa = 500 MPa. Make sure units are consistent when calculating stress.

5. Thermodynamics – First Law

Q5. A system receives 200 kJ of heat and does 80 kJ of work. What is the change in internal energy?
A) 280 kJ
B) 120 kJ
C) –120 kJ
D) –280 kJ

Correct Answer: B) 120 kJ
Explanation: According to the first law of thermodynamics, $\Delta U=Q-W=200-80=120$ kJ. Internal energy increases when more heat is added than work done.

6. Dynamics – Newton’s Second Law

Q6. A 10 kg mass accelerates at 3 m/s². What is the net force?
A) 30 N
B) 13 N
C) 3 N
D) 300 N

Correct Answer: A) 30 N
Explanation: From Newton’s second law: $F=ma=10\cdot 3=30$ N. This concept is frequently tested in the FE exam’s dynamics section.

7. Math – Integration

Q7. What is ∫02(3×2) dx\int_0^2 (3x^2) \, dx∫02​(3x2)dx?
A) 4
B) 8
C) 12
D) 16

Correct Answer: C) 12
Explanation: The integral of 3x23x^23x2 is x3x^3x3. So ∫023×2 dx=[x3]02=8−0=8\int_0^2 3x^2 \, dx = [x^3]_0^2 = 8 – 0 = 8∫02​3x2dx=[x3]02​=8−0=8, then multiply by 3 gives 24. Correction: The integral is 3⋅∫02x2dx=3⋅(8/3)=83 \cdot \int_0^2 x^2 dx = 3 \cdot (8/3) = 83⋅∫02​x2dx=3⋅(8/3)=8. Correct answer: B) 8 (Updated)

8. Electrical – Ohm’s Law

Q8. A circuit has 10 V across a 5 Ω resistor. What is the current?
A) 0.5 A
B) 2 A
C) 5 A
D) 50 A

Correct Answer: B) 2 A
Explanation: Ohm’s Law: $I=V/R=10/5=2$ A. Basic electrical circuit analysis is an important FE topic.

9. Chemistry – Ideal Gas Law

Q9. How many moles are in a 10 L container at 1 atm and 298 K? Use $R=0.0821$ L·atm/mol·K
A) 0.41
B) 1.23
C) 0.51
D) 0.33

Correct Answer: C) 0.41
Explanation: $PV=nRT\to n=PV/RT=(1)(10)/(0.0821)(298)\approx 0.41$ mol. The ideal gas law is commonly tested in chemical and general engineering FE exams.

10. Economics – Break-even Analysis

Q10. If fixed cost is $10,000, variable cost is $2/unit, and price is $5/unit, what is the break-even quantity?
A) 1000
B) 2000
C) 2500
D) 3333

Correct Answer: C) 3333
Explanation: Break-even quantity = Fixed cost / (Price – Variable cost) = 10000 / (5 – 2) = 3333.33 units. Choose closest whole number. Engineering economics is essential for FE prep.

11. Math – Differential Equations

Q11. What is the general solution of the differential equation dydx=3×2\frac{dy}{dx} = 3x^2dxdy​=3x2?
A) $y=3x+C$
B) y=x3+Cy = x^3 + Cy=x3+C
C) y=x2+Cy = x^2 + Cy=x2+C
D) y=3×3+Cy = 3x^3 + Cy=3x3+C

Correct Answer: B) y=x3+Cy = x^3 + Cy=x3+C
Explanation: To solve dydx=3×2\frac{dy}{dx} = 3x^2dxdy​=3x2, integrate both sides: y=∫3x2dx=x3+Cy = \int 3x^2 dx = x^3 + Cy=∫3x2dx=x3+C. Recognizing common derivatives and integrals is essential for the FE’s mathematics section.

12. Thermodynamics – Heat Transfer

Q12. In conduction, heat transfer rate increases with:
A) Lower thermal conductivity
B) Decreasing temperature difference
C) Larger cross-sectional area
D) Shorter time duration

Correct Answer: C) Larger cross-sectional area
Explanation: According to Fourier’s Law, heat transfer by conduction is q=−kAdTdxq = -kA \frac{dT}{dx}q=−kAdxdT​. Larger area (A) allows more heat to flow. Time duration does not directly affect steady-state conduction.

13. Statics – Moments

Q13. A 200 N force acts at a distance of 4 m from a pivot. What is the moment about the pivot?
A) 400 Nm
B) 800 Nm
C) 50 Nm
D) 600 Nm

Correct Answer: B) 800 Nm
Explanation: Moment = Force × Distance = 200 N × 4 m = 800 Nm. Understanding moment calculations is key in the statics portion of the FE exam.

14. Dynamics – Kinematics

Q14. A car accelerates from rest at 4 m/s² for 5 seconds. What is its final velocity?
A) 10 m/s
B) 15 m/s
C) 20 m/s
D) 25 m/s

Correct Answer: C) 20 m/s
Explanation: Using $v=u+at$, where $u=0$, $a=4$, $t=5$:
$v=0+4\times 5=20$ m/s. Kinematics questions often appear in the dynamics section.

15. Materials – Young’s Modulus

Q15. If a material has a stress of 200 MPa and a strain of 0.01, what is its Young’s Modulus?
A) 2 MPa
B) 20 MPa
C) 2000 MPa
D) 20,000 MPa

Correct Answer: C) 2000 MPa
Explanation: Young’s Modulus $E=\sigma /\epsilon =200/0.01=20000$ MPa. Young’s Modulus reflects material stiffness, a common topic in materials questions.

16. Math – Probability

Q16. What is the probability of getting a 3 or a 4 when rolling a fair 6-sided die?
A) 1/3
B) 1/2
C) 2/3
D) 1/6

Correct Answer: A) 1/3
Explanation: There are 2 favorable outcomes (3 or 4) out of 6 total outcomes. So, $P=2/6=1/3$. Probability questions are common in the FE mathematics section.

17. Ethics – Conflict of Interest

Q17. An engineer is evaluating bids and notices their cousin’s company submitted one. What should the engineer do?
A) Ignore the relationship
B) Accept the bid
C) Inform a supervisor
D) Favor the cousin’s bid

Correct Answer: C) Inform a supervisor
Explanation: Engineering ethics requires transparency and avoidance of conflicts of interest. The proper course of action is disclosure to ensure fairness and uphold ethical standards.

18. Chemistry – pH Calculation

Q18. What is the pH of a solution with [H+]=1×10−4[H^+] = 1 \times 10^{-4}[H+]=1×10−4 M?
A) 3
B) 4
C) 5
D) 10

Correct Answer: B) 4
Explanation: pH = –logH+H⁺H+ = –log(10⁻⁴) = 4. Knowledge of pH and basic chemistry is expected on the FE exam.

19. Engineering Economics – Interest

Q19. If $1,000 is invested at 10% annual interest compounded yearly, what will it be worth in 2 years?
A) $1,100
B) $1,210
C) $1,200
D) $1,300

Correct Answer: B) $1,210
Explanation: FV=PV(1+i)n=1000(1.1)2=1000×1.21=1210FV = PV(1 + i)^n = 1000(1.1)^2 = 1000 × 1.21 = 1210FV=PV(1+i)n=1000(1.1)2=1000×1.21=1210. Compound interest questions are very frequent in engineering economics.

20. Fluid Mechanics – Bernoulli’s Principle

Q20. Bernoulli’s equation applies to:
A) Viscous flows
B) Compressible flows
C) Steady, incompressible flows
D) All turbulent flows

Correct Answer: C) Steady, incompressible flows
Explanation: Bernoulli’s equation is valid only for steady, incompressible, and inviscid flows without significant energy losses. It’s a staple of fluid mechanics in the FE exam.

Set 2

Mathematics

1. What is the Laplace transform of f(t)=e−2tsin⁡(3t)f(t) = e^{-2t}\sin(3t)f(t)=e−2tsin(3t)?
A. 3(s+2)2+9\frac{3}{(s+2)^2 + 9}(s+2)2+93​
B. 3s2+9\frac{3}{s^2 + 9}s2+93​
C. s+2(s+2)2+9\frac{s+2}{(s+2)^2 + 9}(s+2)2+9s+2​
D. 3(s+2)(s+2)2+9\frac{3(s+2)}{(s+2)^2 + 9}(s+2)2+93(s+2)​

Answer: A
Explanation:
The Laplace transform of e−atsin⁡(bt)e^{-at}\sin(bt)e−atsin(bt) is b(s+a)2+b2\frac{b}{(s+a)^2 + b^2}(s+a)2+b2b​.
Here, $a=2$, $b=3$.
So, the answer is 3(s+2)2+9\frac{3}{(s+2)^2 + 9}(s+2)2+93​.

2. Evaluate the integral ∫0πxsin⁡(x) dx\int_0^{\pi} x\sin(x)\,dx∫0π​xsin(x)dx.
A. $2\pi$
B. $\pi$
C. $-2$
D. $\pi$

Answer: B
Explanation:
Use integration by parts:
Let $u=x$, $dv=\sin (x)dx$ → $du=dx$, $v=-\cos (x)$
Then:
$\int x\sin (x)dx=-x\cos (x)+\int \cos (x)dx=-x\cos (x)+\sin (x)$
Evaluate from 0 to $\pi$:
$[-\pi \cos (\pi )+\sin (\pi )]-[0\cdot \cos (0)+\sin (0)]=-\pi (-1)=\pi$

Ethics and Professional Practice

4. An engineer is asked to approve a design outside their area of expertise. What is the ethical action?
A. Approve if confident
B. Delegate and sign later
C. Decline and refer to a qualified engineer
D. Estimate and sign off to save time

Answer: C
Explanation:
According to NSPE Code of Ethics, engineers shall perform services only in areas of their competence.

5. Accepting a gift from a supplier can be considered unethical if:
A. It costs over $25
B. It is not reported
C. It influences decision-making
D. All of the above

Answer: D
Explanation:
Gifts must not influence professional judgment or create a conflict of interest.

Engineering Economics

6. What is the future value of $5,000 invested at 6% annually for 5 years (compounded annually)?
A. $6,690
B. $6,593.85
C. $6,500
D. $7,000

Answer: B
Explanation:
Use formula: F=P(1+i)n=5000(1.06)5≈6593.85F = P(1 + i)^n = 5000(1.06)^5 \approx 6593.85F=P(1+i)n=5000(1.06)5≈6593.85

7. If a project costs $100,000 and yields $25,000 annually for 5 years, what is the simple payback period?
A. 3 years
B. 4 years
C. 5 years
D. Cannot be determined

Answer: C
Explanation:
Total inflow in 5 years = $25,000 × 5 = $125,000
Payback occurs in exactly 4 years with $100,000 recovered.

Statics

8. A simply supported beam has a 10 kN load at midspan. What’s the reaction at each support?
A. 10 kN each
B. 5 kN each
C. 0 kN
D. 20 kN each

Answer: B
Explanation:
Symmetrical loading: each support carries half the load → 102=5\frac{10}{2} = 5210​=5 kN

9. Which of the following is a two-force member?
A. Beam
B. Cable
C. Truss element
D. Rigid frame

Answer: C
Explanation:
Truss members carry force only at two points, hence two-force members.

10. What is the moment about point A due to a 100 N force applied 2 m from A at 30° angle?
A. 100 Nm
B. 200 Nm
C. 100 sin(30°) × 2 = 100 Nm
D. 100 cos(30°) × 2 = 173.2 Nm

Answer: C
Explanation:
Moment = Force × Perpendicular distance = $100\sin (30°)\times 2=100$ Nm

Dynamics

11. A car accelerates uniformly from rest to 20 m/s in 5 seconds. What’s the acceleration?
A. 5 m/s²
B. 4 m/s²
C. 2 m/s²
D. 3 m/s²

Answer: B
Explanation:
a=Δvt=205=4 m/s2a = \frac{\Delta v}{t} = \frac{20}{5} = 4 \, \text{m/s}^2a=tΔv​=520​=4m/s2

12. What is the centripetal force on a 2 kg mass moving in a circle of radius 3 m at 4 m/s?
A. 10.7 N
B. 2.5 N
C. 5.3 N
D. 10.7 Nm

Answer: A
Explanation:
F=mv2r=2×163=10.67≈10.7 NF = \frac{mv^2}{r} = \frac{2 \times 16}{3} = 10.67 \approx 10.7 \, \text{N}F=rmv2​=32×16​=10.67≈10.7N

Materials

13. Which material property describes resistance to permanent deformation?
A. Toughness
B. Yield Strength
C. Ductility
D. Hardness

Answer: B
Explanation:
Yield strength defines the stress at which a material begins to deform plastically.

14. Which material is typically most brittle?
A. Steel
B. Rubber
C. Concrete
D. Aluminum

Answer: C
Explanation:
Concrete fails with little deformation → very brittle.

Thermodynamics

15. A heat engine absorbs 1000 J and does 300 J of work. What’s its efficiency?
A. 70%
B. 30%
C. 50%
D. 60%

Answer: B
Explanation:
Efficiency = WQin=3001000=0.3=30%\frac{W}{Q_{in}} = \frac{300}{1000} = 0.3 = 30\%Qin​W​=1000300​=0.3=30%

16. What does the first law of thermodynamics state?
A. Energy cannot be created nor destroyed
B. Entropy always increases
C. All processes are irreversible
D. Work equals pressure times volume

Answer: A
Explanation:
It is the principle of conservation of energy.

Electrical Engineering

17. A 10 Ω resistor has 2 A current. What’s the power dissipated?
A. 5 W
B. 10 W
C. 20 W
D. 40 W

Answer: D
Explanation:
Power = I2R=4×10=40 WI^2R = 4 \times 10 = 40 \, \text{W}I2R=4×10=40W

18. What is the equivalent resistance of two 10 Ω resistors in parallel?
A. 5 Ω
B. 10 Ω
C. 20 Ω
D. 2 Ω

Answer: A
Explanation:
Req=10×1010+10=5 ΩR_{eq} = \frac{10 \times 10}{10 + 10} = 5 \, \OmegaReq​=10+1010×10​=5Ω

Fluid Mechanics

19. Bernoulli’s equation assumes:
A. Viscous fluid
B. Compressible flow
C. Irrotational flow
D. Unsteady flow

Answer: C
Explanation:
Bernoulli’s equation is valid for incompressible, inviscid, steady, irrotational flow.

20. What is the specific gravity of a fluid with density 800 kg/m³?
A. 1
B. 0.8
C. 0.6
D. 1.2

Answer: B
Explanation:
SG=density1000=8001000=0.8SG = \frac{\text{density}}{1000} = \frac{800}{1000} = 0.8SG=1000density​=1000800​=0.8

More Practice Questions

21. Which law states that strain is proportional to stress?
A. Newton’s Law
B. Hooke’s Law
C. Bernoulli’s Law
D. Pascal’s Law

Answer: B
Explanation:
Hooke’s Law: $\sigma =E\epsilon$, where stress is proportional to strain.

22. The moment of inertia for a rectangle (base b, height h) about its base is:
A. 112bh3\frac{1}{12}bh^3121​bh3
B. 13bh3\frac{1}{3}bh^331​bh3
C. 12bh2\frac{1}{2}bh^221​bh2
D. 14bh2\frac{1}{4}bh^241​bh2

Answer: B
Explanation:
About its base: I=13bh3I = \frac{1}{3}bh^3I=31​bh3

24. What is the efficiency of a pump that requires 1.5 kW input and delivers 1.2 kW output?
A. 80%
B. 90%
C. 60%
D. 70%

Answer: A
Explanation:
Efficiency = 1.21.5=0.8=80%\frac{1.2}{1.5} = 0.8 = 80\%1.51.2​=0.8=80%

25. In project management, what does the critical path represent?
A. Tasks with least cost
B. Shortest project duration
C. Longest path with zero slack
D. Activities that can be delayed

Answer: C
Explanation:
Critical path = longest duration path; no slack.

26. Which function represents a unit impulse?
A. δ(t)
B. u(t)
C. t
D. 1

Answer: A
Explanation:
δ(t) is the Dirac delta function → unit impulse.

27. Which method is used for solving linear differential equations?
A. Laplace transform
B. Newton-Raphson
C. Simpson’s rule
D. Euler’s formula

Answer: A
Explanation:
Laplace transforms simplify differential equations into algebraic ones.

28. What is a common failure mode in brittle materials?
A. Creep
B. Yielding
C. Fracture
D. Buckling

Answer: C
Explanation:
Brittle materials fail by fracture without yielding.

29. What is the modulus of elasticity a measure of?
A. Energy
B. Resistance to heat
C. Stiffness
D. Plasticity

Answer: C
Explanation:
Modulus of elasticity quantifies stiffness of a material.

30. A structure with more unknown forces than equilibrium equations is:
A. Statically determinate
B. Kinematically unstable
C. Statically indeterminate
D. Over-constrained

Answer: C
Explanation:
More unknowns than equations → statically indeterminate.

31. A simply supported beam has a point load $P$ applied at its midpoint. What is the maximum bending moment in the beam?

A. PL2\frac{PL}{2}2PL​
B. PL4\frac{PL}{4}4PL​
C. PL6\frac{PL}{6}6PL​
D. PL8\frac{PL}{8}8PL​

Answer: B. PL4\frac{PL}{4}4PL​

Explanation:
For a simply supported beam with a point load at the midpoint:
Mmax=PL4M_{\text{max}} = \frac{PL}{4}Mmax​=4PL​

32. Which of the following is a correct unit for dynamic viscosity?

A. Pascal
B. Newton
C. Pascal-second
D. Newton-second

Answer: C. Pascal-second

Explanation:
Dynamic viscosity $\mu$ has units of Pa·s (Pascal-second) in SI units.

33. What is the Laplace transform of f(t)=t2f(t) = t^2f(t)=t2?

A. 1s2\frac{1}{s^2}s21​
B. 2s3\frac{2}{s^3}s32​
C. 6s3\frac{6}{s^3}s36​
D. 1s3\frac{1}{s^3}s31​

Answer: C. 2!s3=6s3\frac{2!}{s^{3}} = \frac{6}{s^3}s32!​=s36​

Explanation:
The Laplace transform of tnt^ntn is n!sn+1\frac{n!}{s^{n+1}}sn+1n!​. Here, $n=2$.

34. The modulus of elasticity (E) relates which two material properties?

A. Stress and strain
B. Force and displacement
C. Moment and curvature
D. Load and pressure

Answer: A. Stress and strain

Explanation:
Hooke’s Law: $\sigma =E\cdot \epsilon$, where $\sigma$ is stress, and $\epsilon$ is strain.

35. A capacitor has a capacitance of 10 μF and is charged to 5 V. How much energy is stored?

A. 0.125 J
B. 1.25 mJ
C. 25 μJ
D. 50 μJ

Answer: B. 1.25 mJ

Explanation:
Energy stored: E=12CV2=0.5⋅10×10−6⋅25=1.25×10−3E = \frac{1}{2} C V^2 = 0.5 \cdot 10 \times 10^{-6} \cdot 25 = 1.25 \times 10^{-3}E=21​CV2=0.5⋅10×10−6⋅25=1.25×10−3 J = 1.25 mJ

36. Which of the following equations is used to calculate the moment of inertia of a rectangular area?

A. 12bh2\frac{1}{2}bh^221​bh2
B. 13bh3\frac{1}{3}bh^331​bh3
C. 112bh3\frac{1}{12}bh^3121​bh3
D. 16bh3\frac{1}{6}bh^361​bh3

Answer: C. 112bh3\frac{1}{12}bh^3121​bh3

Explanation:
Moment of inertia for a rectangle about its base: I=112bh3I = \frac{1}{12}bh^3I=121​bh3

37. In a thermodynamic system, which law states that energy cannot be created or destroyed?

A. Zeroth Law
B. First Law
C. Second Law
D. Third Law

Answer: B. First Law

Explanation:
The First Law of Thermodynamics is the conservation of energy: $\Delta U=Q-W$

38. What is the determinant of the matrix
[1234]\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}[13​24​]?

A. -2
B. -5
C. -4
D. -1

Answer: B. -2

Explanation:
Determinant = $1\ast 4-2\ast 3=4-6=-2$

39. The eigenvalues of a matrix are found by solving which equation?

A. $Ax=b$
B. $|A-\lambda I|=0$
C. A−1x=λxA^{-1}x = \lambda xA−1x=λx
D. $A+\lambda I=0$

Answer: B. $|A-\lambda I|=0$

Explanation:
Eigenvalues are roots of the characteristic polynomial from $|A-\lambda I|=0$

40. What is the break-even point in engineering economics?

A. When total revenue is less than total cost
B. When net present value is maximum
C. When total revenue equals total cost
D. When internal rate of return equals inflation

Answer: C. When total revenue equals total cost

Explanation:
Break-even occurs when no profit or loss exists, i.e., revenue = cost.

41. Which of the following is a valid assumption for Bernoulli’s Equation?

A. Viscous flow
B. Compressible flow
C. Irrotational flow
D. Turbulent flow

Answer: C. Irrotational flow

Explanation:
Bernoulli’s equation assumes incompressible, inviscid, steady, and irrotational flow.

42. In structural analysis, a truss is statically determinate if:

A. $m>2j-3$
B. $m<2j-3$
C. $m=2j-3$
D. $m+r=j$

Answer: C. $m=2j-3$

Explanation:
Where $m$ is members and $j$ is joints. This is the determinacy condition for a planar truss.

43. The unit of thermal conductivity is:

A. W/m·K
B. J/kg·K
C. W/K
D. W·m

Answer: A. W/m·K

Explanation:
Thermal conductivity $k$ has units of watts per meter per Kelvin.

44. The equation $F=q(E+v\times B)$ is:

A. Newton’s Law
B. Gauss’s Law
C. Faraday’s Law
D. Lorentz Force Law

Answer: D. Lorentz Force Law

Explanation:
The Lorentz Force Law describes the force on a charged particle in electric and magnetic fields.

45. In a binary tree, the maximum number of nodes at level $l$ is:

A. 2l−12^{l-1}2l−1
B. 2l2^l2l
C. $l$
D. log⁡2(l)\log_2(l)log2​(l)

Answer: A. 2l−12^{l-1}2l−1

Explanation:
In a binary tree, level $l$ has at most 2l−12^{l-1}2l−1 nodes.

46. Which of the following functions is not continuous at $x=0$?

A. $\sin (x)$
B. exe^xex
C. 1x\frac{1}{x}x1​
D. $\cos (x)$

Answer: C. 1x\frac{1}{x}x1​

Explanation:
1x\frac{1}{x}x1​ is undefined at $x=0$, hence not continuous there.

47. In electrical circuits, Kirchhoff’s Voltage Law states:

A. Current in = Current out
B. Voltage across resistor = 0
C. Sum of voltages in a loop = 0
D. Total power = total resistance

Answer: C. Sum of voltages in a loop = 0

Explanation:
KVL is based on conservation of energy.

48. In project management, the critical path is:

A. The shortest path
B. The longest path
C. The least expensive path
D. The safest path

Answer: B. The longest path

Explanation:
Critical path determines the minimum project duration.

49. For a normal distribution, about 68% of data falls within:

A. ±1σ
B. ±2σ
C. ±3σ
D. ±4σ

Answer: A. ±1σ

Explanation:
This is the empirical rule for normal distributions.

50. If the determinant of a 3×3 matrix is zero, the matrix is:

A. Invertible
B. Diagonalizable
C. Singular
D. Symmetric

Answer: C. Singular

Explanation:
A matrix with a zero determinant is singular and non-invertible.

51. Which material property defines resistance to shape change?

A. Bulk modulus
B. Shear modulus
C. Poisson’s ratio
D. Thermal conductivity

Answer: B. Shear modulus

Explanation:
Shear modulus relates shear stress to shear strain.

52. A root locus plot is typically used in:

A. Circuit design
B. Control systems
C. Fluid dynamics
D. Structural analysis

Answer: B. Control systems

Explanation:
Root locus shows pole locations as a system parameter varies.

53. The limit lim⁡x→0sin⁡(x)x\lim_{x \to 0} \frac{\sin(x)}{x}limx→0​xsin(x)​ equals:

A. 0
B. 1
C. Infinity
D. Undefined

Answer: B. 1

Explanation:
This is a standard limit in calculus.

54. A gear train with 3 gears (20T, 40T, 80T) gives a gear ratio of:

A. 2
B. 4
C. 8
D. 16

Answer: C. 8

Explanation:
Gear ratio = output/input = $80/10=8$

55. Which component stores mechanical energy in rotational systems?

A. Damper
B. Spring
C. Mass
D. Flywheel

Answer: D. Flywheel

Explanation:
Flywheels store energy using rotational inertia.

56. The smallest number that satisfies x2−5x+6=0x^2 – 5x + 6 = 0x2−5x+6=0 is:

A. 3
B. 2
C. 6
D. 1

Answer: B. 2

Explanation:
Factoring: $(x-2)(x-3)=0$ → Roots: 2, 3 → Smallest is 2

57. A Newton is equivalent to:

A. kg·m/s
B. kg·m/s²
C. N·m
D. J/s

Answer: B. kg·m/s²

Explanation:
Newton’s second law: $F=ma$

58. Which logic gate returns TRUE only when inputs are different?

A. AND
B. OR
C. XOR
D. NAND

Answer: C. XOR

Explanation:
XOR (exclusive OR) = 1 if inputs differ.

59. The rank of a matrix represents:

A. The number of columns
B. The number of non-zero entries
C. The number of independent rows or columns
D. The trace of the matrix

Answer: C. The number of independent rows or columns

Explanation:
Rank = maximum number of linearly independent rows or columns.

60. What is the derivative of f(x)=ln⁡(x2+1)f(x) = \ln(x^2 + 1)f(x)=ln(x2+1)?

A. 1×2+1\frac{1}{x^2 + 1}x2+11​
B. 2xx2+1\frac{2x}{x^2 + 1}x2+12x​
C. $\ln (x)$
D. $2x\ln (x)$

Answer: B. 2xx2+1\frac{2x}{x^2 + 1}x2+12x​

Explanation:
Use chain rule: derivative of $\ln (u)$ = u′u\frac{u’}{u}uu′​

61. What is the ethical responsibility of an engineer if they find a major flaw in a project after it has been approved?

A. Keep quiet and let the contractor handle it
B. Notify their superior only if asked
C. Ignore it if they are not the lead engineer
D. Report the flaw to the responsible authority immediately

Answer: D. Report the flaw to the responsible authority immediately
Explanation: Engineers are ethically bound to protect public safety and must report any serious issues that could lead to failure or harm.

62. A 10-ohm resistor has a current of 2 A passing through it. What is the voltage drop?

A. 5 V
B. 10 V
C. 20 V
D. 200 V

Answer: C. 20 V
Explanation: Ohm’s Law: $V=IR=2\times 10=20\text{ V}$

63. Which of the following quantities is not a vector?

A. Force
B. Acceleration
C. Displacement
D. Temperature

Answer: D. Temperature
Explanation: Temperature is a scalar quantity; it has magnitude but no direction.

64. What is the Fourier Transform primarily used for?

A. Finding eigenvalues
B. Solving differential equations
C. Converting time-domain to frequency-domain
D. Modeling fluid flow

Answer: C. Converting time-domain to frequency-domain
Explanation: Fourier Transforms help analyze signal behavior in the frequency domain.

65. In thermodynamics, entropy is a measure of:

A. Pressure
B. Energy
C. Disorder
D. Heat capacity

Answer: C. Disorder
Explanation: Entropy measures the randomness or disorder in a thermodynamic system.

66. What is the moment of a 50 N force applied 3 meters from a pivot point?

A. 150 Nm
B. 50 Nm
C. 3 Nm
D. 150 N

Answer: A. 150 Nm
Explanation: Moment = Force × Distance = $50\times 3=150\text{ Nm}$

67. Which of the following processes is reversible and adiabatic?

A. Isothermal
B. Isobaric
C. Isochoric
D. Isentropic

Answer: D. Isentropic
Explanation: Isentropic = reversible + adiabatic; entropy remains constant.

68. In probability, the expected value of a fair six-sided die roll is:

A. 3
B. 3.5
C. 4
D. 6

Answer: B. 3.5
Explanation: E=1+2+3+4+5+66=3.5E = \frac{1 + 2 + 3 + 4 + 5 + 6}{6} = 3.5E=61+2+3+4+5+6​=3.5

69. A process with constant pressure is called:

A. Isobaric
B. Isochoric
C. Isothermal
D. Adiabatic

Answer: A. Isobaric
Explanation: “Iso” means same, “baric” refers to pressure.

70. The velocity of a particle is given by v(t)=3t2+2tv(t) = 3t^2 + 2tv(t)=3t2+2t. What is its acceleration at $t=2$ s?

A. 12 m/s²
B. 14 m/s²
C. 16 m/s²
D. 18 m/s²

Answer: C. 16 m/s²
Explanation: Acceleration is the derivative of velocity: $a(t)=dv/dt=6t+2$. At $t=2$: $6(2)+2=14$

(Correction: that yields 14, so correct answer is:)

Answer: B. 14 m/s²

71. Which of the following is not a mode of heat transfer?

A. Conduction
B. Convection
C. Radiation
D. Compression

Answer: D. Compression
Explanation: Heat is transferred via conduction, convection, and radiation. Compression relates to mechanical work.

72. In control systems, a PID controller includes:

A. Power, Impedance, Delay
B. Proportional, Integral, Derivative
C. Phase, Input, Damping
D. Pulse, Isolation, Distance

Answer: B. Proportional, Integral, Derivative
Explanation: PID controllers improve system stability and response.

73. The area under a force-displacement graph represents:

A. Acceleration
B. Velocity
C. Energy
D. Pressure

Answer: C. Energy
Explanation: Work (energy) = Force × Displacement, which is the area under the curve.

74. Which of the following numbers is not a prime?

A. 2
B. 3
C. 9
D. 11

Answer: C. 9
Explanation: 9 is divisible by 3; hence, not a prime.

75. In statistics, a Type I error refers to:

A. Accepting a false null hypothesis
B. Rejecting a true null hypothesis
C. Accepting a true alternative hypothesis
D. Failing to reject a false null hypothesis

Answer: B. Rejecting a true null hypothesis
Explanation: Type I error is a false positive.

76. A transformer operates based on:

A. Newton’s law
B. Ohm’s law
C. Faraday’s law of induction
D. Kirchhoff’s laws

Answer: C. Faraday’s law of induction
Explanation: Transformers work via electromagnetic induction.

77. A system with an impulse response h(t)=e−tu(t)h(t) = e^{-t}u(t)h(t)=e−tu(t) is:

A. Linear and time-invariant
B. Nonlinear and time-variant
C. Linear but time-variant
D. Nonlinear but time-invariant

Answer: A. Linear and time-invariant
Explanation: The response is exponential and causal, satisfying LTI properties.

78. A bar has an original length of 2 m and elongates by 1 mm under load. What is the strain?

A. 0.0005
B. 0.001
C. 0.005
D. 0.00005

Answer: B. 0.0005
Explanation: Strain = ΔLL=0.0012=0.0005\frac{\Delta L}{L} = \frac{0.001}{2} = 0.0005LΔL​=20.001​=0.0005

79. Which programming language is most associated with numerical computing?

A. HTML
B. JavaScript
C. Fortran
D. CSS

Answer: C. Fortran
Explanation: Fortran is still widely used in scientific and numerical computing.

80. The phase angle between voltage and current in a pure resistor is:

A. 0°
B. 90°
C. -90°
D. 180°

Answer: A. 0°
Explanation: In a pure resistor, voltage and current are in phase.

81. In fluid mechanics, Reynolds number is used to:

A. Predict pressure drop
B. Identify turbulent or laminar flow
C. Calculate flow velocity
D. Determine pump power

Answer: B. Identify turbulent or laminar flow
Explanation: Re = ρvDμ\frac{\rho v D}{\mu}μρvD​, a dimensionless number.

82. In project management, a Gantt chart is used to:

A. Analyze risk
B. Track scheduling
C. Determine stress levels
D. Estimate cost

Answer: B. Track scheduling
Explanation: Gantt charts display project timelines and task durations.

83. The SI unit of angular velocity is:

A. m/s²
B. degrees/s
C. radians/s
D. revolutions/min

Answer: C. radians/s
Explanation: SI uses radians for angular measures.

84. The Laplace transform of a constant $a$ is:

A. $a$
B. as\frac{a}{s}sa​
C. 1a\frac{1}{a}a1​
D. $as$

Answer: B. as\frac{a}{s}sa​
Explanation: L{a}=as\mathcal{L}\{a\} = \frac{a}{s}L{a}=sa​

85. In materials science, ductility refers to:

A. Hardness
B. Ability to return to original shape
C. Ability to absorb energy
D. Ability to deform plastically without breaking

Answer: D. Ability to deform plastically without breaking
Explanation: Ductile materials can stretch significantly before failure.

86. Which of these is a second-order differential equation?

A. y′′+5y=0y” + 5y = 0y′′+5y=0
B. y′+3y=0y’ + 3y = 0y′+3y=0
C. $y+y=0$
D. $y=3x+1$

Answer: A. y′′+5y=0y” + 5y = 0y′′+5y=0
Explanation: A second-order equation includes y′′y”y′′, the second derivative.

87. In DC circuits, power is given by:

A. $P=IV$
B. P=IR2P = IR^2P=IR2
C. P=V2/RP = V^2/RP=V2/R
D. All of the above

Answer: D. All of the above
Explanation: All expressions are derived from Ohm’s law and power formulas.

88. A vector with components (3, 4) has a magnitude of:

A. 5
B. 6
C. 7
D. 25

Answer: A. 5