FE Exam Prep 2026Updated Practice Questions,
Comprehensive Fundamentals of Engineering Review,
Detailed Explanations, Verified Answers & Complete
Success Workbook
This practice exam is designed as a comprehensive cross-discipline
review covering the core topics common to all FE disciplines —
Mathematics, Probability & Statistics, Ethics & Professional Practice,
Engineering Economics, Statics, Dynamics, Mechanics of Materials, Fluid
Mechanics, Thermodynamics, and more. Questions are written in the
NCEES CBT format with four answer choices, detailed step-by-step
solutions, and rationales referencing the FE Reference Handbook where
applicable.
SECTION I: MATHEMATICS (Questions 1–12)
Question 1
What is the derivative of 𝑓(𝑥) = 3𝑥 4 − 2𝑥 3 + 5𝑥 − 7?
A. 12𝑥 3 − 6𝑥 2 + 5
B. 12𝑥 3 − 6𝑥 2 + 5𝑥
C. 3𝑥 3 − 2𝑥 2 + 5
D. 12𝑥 4 − 6𝑥 3 + 5
Correct Answer: A
𝑑
Rationale: Using the power rule [𝑥 𝑛 ] = 𝑛𝑥 𝑛−1 :
𝑑𝑥
𝑑
• [3𝑥 4 ] = 12𝑥 3
𝑑𝑥
𝑑
• [−2𝑥 3 ] = −6𝑥 2
𝑑𝑥
𝑑
• [5𝑥] = 5
𝑑𝑥
𝑑
• [−7] = 0
𝑑𝑥
,Therefore, 𝑓 ′ (𝑥) = 12𝑥 3 − 6𝑥 2 + 5. Option B incorrectly includes an
extra 𝑥 term. Option C has incorrect coefficients. Option D has incorrect
exponents. FE Reference Handbook: Derivative rules are found in the
Mathematics section.
Question 2
2
Evaluate the definite integral: ∫0 ( 3𝑥 2 − 4𝑥 + 1) 𝑑𝑥
A. 2
B. 4
C. 6
D. 0
Correct Answer: A
Rationale: First find the antiderivative:
∫( 3𝑥 2 − 4𝑥 + 1)𝑑𝑥 = 𝑥 3 − 2𝑥 2 + 𝑥 + 𝐶
Evaluate from 0 to 2:
[𝑥 3 − 2𝑥 2 + 𝑥]20 = (8 − 8 + 2) − (0 − 0 + 0) = 2 − 0 = 2
Question 3
1 2 5 6
Given the matrices 𝐴 = [ ] and 𝐵 = [ ], what is 𝐴 + 𝐵?
3 4 7 8
6 8
A. [ ]
10 12
6 8
B. [ ]
10 12
5 12
C. [ ]
21 32
1 2
D. [ ]
3 4
Correct Answer: A
Rationale: Matrix addition is performed element-wise:
, 1+5 2+6 6 8
𝐴+𝐵 =[ ]=[ ]
3+7 4+8 10 12
19 22
Option C shows matrix multiplication (which would be [ ]), not
43 50
addition.
Question 4
2 3
What is the determinant of matrix 𝐴 = [ ]?
4 5
A. -2
B. 2
C. 22
D. -22
Correct Answer: A
𝑎 𝑏
Rationale: For a 2 × 2 matrix [ ], determinant = 𝑎𝑑 − 𝑏𝑐.
𝑐 𝑑
det(𝐴) = (2)(5) − (3)(4) = 10 − 12 = −2
Question 5
Solve the system of equations:
2𝑥 + 3𝑦 = 8
4𝑥 − 𝑦 = 2
A. 𝑥 = 1, 𝑦 = 2
B. 𝑥 = 2, 𝑦 = 1
C. 𝑥 = 1, 𝑦 = 1
D. 𝑥 = 0, 𝑦 = 2
Correct Answer: A
Rationale: Using substitution or elimination. From the second equation: 𝑦 =
4𝑥 − 2. Substitute into the first:
2𝑥 + 3(4𝑥 − 2) = 8
2𝑥 + 12𝑥 − 6 = 8
14𝑥 = 14
, 𝑥=1
Then 𝑦 = 4(1) − 2 = 2. Solution: 𝑥 = 1, 𝑦 = 2.
Question 6
What is the Laplace transform of 𝑓(𝑡) = 𝑒 𝑎𝑡 ?
1
A.
𝑠−𝑎
1
B.
𝑠+𝑎
𝑠
C.
𝑠 2 +𝑎2
𝑎
D.
𝑠 2 +𝑎2
Correct Answer: A
1
Rationale: The Laplace transform of 𝑒 𝑎𝑡 is ℒ{𝑒 𝑎𝑡 } = , for 𝑠 > 𝑎. Option B is
𝑠−𝑎
the transform of 𝑒 −𝑎𝑡 . Option C is the transform of cos(𝑎𝑡). Option D is the
transform of sin(𝑎𝑡).
Question 7
𝑑𝑦
What is the solution to the differential equation = 2𝑦 with initial
𝑑𝑥
condition 𝑦(0) = 3?
A. 𝑦 = 3𝑒 2𝑥
B. 𝑦 = 3𝑒 −2𝑥
C. 𝑦 = 2𝑒 3𝑥
D. 𝑦 = 𝑒 2𝑥 + 3
Correct Answer: A
𝑑𝑦
Rationale: This is a separable differential equation. = 2𝑑𝑥. Integrate both
𝑦
sides: ln ∣ 𝑦 ∣= 2𝑥 + 𝐶. Thus 𝑦 = 𝐶𝑒 2𝑥 . Using 𝑦(0) = 3: 3 = 𝐶 ⋅ 𝑒 0 = 𝐶.
Therefore, 𝑦 = 3𝑒 2𝑥 .
Question 8
Comprehensive Fundamentals of Engineering Review,
Detailed Explanations, Verified Answers & Complete
Success Workbook
This practice exam is designed as a comprehensive cross-discipline
review covering the core topics common to all FE disciplines —
Mathematics, Probability & Statistics, Ethics & Professional Practice,
Engineering Economics, Statics, Dynamics, Mechanics of Materials, Fluid
Mechanics, Thermodynamics, and more. Questions are written in the
NCEES CBT format with four answer choices, detailed step-by-step
solutions, and rationales referencing the FE Reference Handbook where
applicable.
SECTION I: MATHEMATICS (Questions 1–12)
Question 1
What is the derivative of 𝑓(𝑥) = 3𝑥 4 − 2𝑥 3 + 5𝑥 − 7?
A. 12𝑥 3 − 6𝑥 2 + 5
B. 12𝑥 3 − 6𝑥 2 + 5𝑥
C. 3𝑥 3 − 2𝑥 2 + 5
D. 12𝑥 4 − 6𝑥 3 + 5
Correct Answer: A
𝑑
Rationale: Using the power rule [𝑥 𝑛 ] = 𝑛𝑥 𝑛−1 :
𝑑𝑥
𝑑
• [3𝑥 4 ] = 12𝑥 3
𝑑𝑥
𝑑
• [−2𝑥 3 ] = −6𝑥 2
𝑑𝑥
𝑑
• [5𝑥] = 5
𝑑𝑥
𝑑
• [−7] = 0
𝑑𝑥
,Therefore, 𝑓 ′ (𝑥) = 12𝑥 3 − 6𝑥 2 + 5. Option B incorrectly includes an
extra 𝑥 term. Option C has incorrect coefficients. Option D has incorrect
exponents. FE Reference Handbook: Derivative rules are found in the
Mathematics section.
Question 2
2
Evaluate the definite integral: ∫0 ( 3𝑥 2 − 4𝑥 + 1) 𝑑𝑥
A. 2
B. 4
C. 6
D. 0
Correct Answer: A
Rationale: First find the antiderivative:
∫( 3𝑥 2 − 4𝑥 + 1)𝑑𝑥 = 𝑥 3 − 2𝑥 2 + 𝑥 + 𝐶
Evaluate from 0 to 2:
[𝑥 3 − 2𝑥 2 + 𝑥]20 = (8 − 8 + 2) − (0 − 0 + 0) = 2 − 0 = 2
Question 3
1 2 5 6
Given the matrices 𝐴 = [ ] and 𝐵 = [ ], what is 𝐴 + 𝐵?
3 4 7 8
6 8
A. [ ]
10 12
6 8
B. [ ]
10 12
5 12
C. [ ]
21 32
1 2
D. [ ]
3 4
Correct Answer: A
Rationale: Matrix addition is performed element-wise:
, 1+5 2+6 6 8
𝐴+𝐵 =[ ]=[ ]
3+7 4+8 10 12
19 22
Option C shows matrix multiplication (which would be [ ]), not
43 50
addition.
Question 4
2 3
What is the determinant of matrix 𝐴 = [ ]?
4 5
A. -2
B. 2
C. 22
D. -22
Correct Answer: A
𝑎 𝑏
Rationale: For a 2 × 2 matrix [ ], determinant = 𝑎𝑑 − 𝑏𝑐.
𝑐 𝑑
det(𝐴) = (2)(5) − (3)(4) = 10 − 12 = −2
Question 5
Solve the system of equations:
2𝑥 + 3𝑦 = 8
4𝑥 − 𝑦 = 2
A. 𝑥 = 1, 𝑦 = 2
B. 𝑥 = 2, 𝑦 = 1
C. 𝑥 = 1, 𝑦 = 1
D. 𝑥 = 0, 𝑦 = 2
Correct Answer: A
Rationale: Using substitution or elimination. From the second equation: 𝑦 =
4𝑥 − 2. Substitute into the first:
2𝑥 + 3(4𝑥 − 2) = 8
2𝑥 + 12𝑥 − 6 = 8
14𝑥 = 14
, 𝑥=1
Then 𝑦 = 4(1) − 2 = 2. Solution: 𝑥 = 1, 𝑦 = 2.
Question 6
What is the Laplace transform of 𝑓(𝑡) = 𝑒 𝑎𝑡 ?
1
A.
𝑠−𝑎
1
B.
𝑠+𝑎
𝑠
C.
𝑠 2 +𝑎2
𝑎
D.
𝑠 2 +𝑎2
Correct Answer: A
1
Rationale: The Laplace transform of 𝑒 𝑎𝑡 is ℒ{𝑒 𝑎𝑡 } = , for 𝑠 > 𝑎. Option B is
𝑠−𝑎
the transform of 𝑒 −𝑎𝑡 . Option C is the transform of cos(𝑎𝑡). Option D is the
transform of sin(𝑎𝑡).
Question 7
𝑑𝑦
What is the solution to the differential equation = 2𝑦 with initial
𝑑𝑥
condition 𝑦(0) = 3?
A. 𝑦 = 3𝑒 2𝑥
B. 𝑦 = 3𝑒 −2𝑥
C. 𝑦 = 2𝑒 3𝑥
D. 𝑦 = 𝑒 2𝑥 + 3
Correct Answer: A
𝑑𝑦
Rationale: This is a separable differential equation. = 2𝑑𝑥. Integrate both
𝑦
sides: ln ∣ 𝑦 ∣= 2𝑥 + 𝐶. Thus 𝑦 = 𝐶𝑒 2𝑥 . Using 𝑦(0) = 3: 3 = 𝐶 ⋅ 𝑒 0 = 𝐶.
Therefore, 𝑦 = 3𝑒 2𝑥 .
Question 8
