Fundamentals of Engineering (FE Exam - Core Disciplines: Math, Statics, Dynamics, Mechanics
of Materials, Fluid Mechanics, Thermodynamics, Electrical Circuits, Materials Science,
Economics)
1. What is the derivative of f(x) = ln(sin(x))?
A) cot(x)
B) tan(x)
C) sec(x)csc(x)
D) 1/sin(x)
Answer: A) Using the chain rule: d/dx[ln(u)] = (1/u) * du/dx. u = sin(x), du/dx = cos(x). So
derivative = (1/sin(x)) * cos(x) = cos(x)/sin(x) = cot(x).
2. Evaluate the integral: ∫₀¹ x² dx
A) 1/3
B) 1/2
C) 1
D) 1/4
Answer: A) ∫ x² dx = x³/3. Evaluate from 0 to 1: (1³/3) - (0³/3) = 1/3.
3. What is the determinant of the 2x2 matrix [[2, 3], [4, 5]]?
A) -2
B) 22
C) 10
D) -22
Answer: A) For a 2x2 matrix [[a, b], [c, d]], the determinant is ad - bc. (2*5) - (3*4) = 10 - 12 = -2.
4. In a simply supported beam with a central point load P, what is the reaction at each
support?
A) P/2
B) P
C) 2P
D) P/4
Answer: A) By symmetry, the load is equally shared between the two supports. Each reaction =
P/2.
5. A force vector F = 3i + 4j N acts on a particle. What is the magnitude of the force?
A) 5 N
B) 7 N
C) 1 N
,D) 12 N
Answer: A) Magnitude = √(3² + 4²) = √(9 + 16) = √25 = 5 N.
6. What is the moment of inertia of a solid cylinder of mass M and radius R about its central
axis?
A) ½MR²
B) MR²
C) ¼MR²
D) ⅔MR²
Answer: A) The mass moment of inertia of a solid cylinder about its central longitudinal axis is I
= (1/2)MR².
7. What is the definition of stress in a material?
A) Force per unit area
B) Deformation per unit length
C) Force per unit volume
D) Energy per unit volume
Answer: A) Stress (σ) is defined as the internal resisting force per unit area. σ = F/A. Strain is
deformation per unit length.
8. What is the modulus of rigidity (shear modulus) G related to Young's modulus E and
Poisson's ratio ν by?
A) G = E / [2(1 + ν)]
B) G = E / [3(1 - 2ν)]
C) G = E(1 + ν)
D) G = E(1 - ν)
Answer: A) The relationship is G = E / [2(1 + ν)]. This is derived from elastic constants.
9. What is the specific weight of water at standard conditions (approx)?
A) 62.4 lb/ft³
B) 1000 kg/m³
C) 9.81 kN/m³
D) Both A and C
Answer: D) Specific weight (γ) = ρg. In SI: 9.81 kN/m³. In US units: 62.4 lb/ft³. Density of water is
1000 kg/m³ (B).
10. For a fluid at rest, the pressure at a point is:
A) Dependent on direction
B) Independent of direction (Pascal's Law)
C) Always zero
, D) Directly proportional to velocity
Answer: B) Pascal's Law states that pressure at a point in a static fluid is the same in all
directions.
11. The ideal gas law is expressed as:
A) PV = nRT
B) PV = mRT
C) P = ρRT
D) All of the above
Answer: D) PV = nRT (with n = number of moles, R = universal gas constant). PV = mRT (with m =
mass, R = specific gas constant). P = ρRT (with ρ = density). All are equivalent forms.
12. In a Carnot cycle, which process is isentropic?
A) Isothermal expansion
B) Isothermal compression
C) Adiabatic expansion and compression
D) All processes are isentropic
Answer: C) A Carnot cycle consists of two isothermal (reversible, constant temperature)
processes and two adiabatic (reversible, constant entropy - isentropic) processes.
13. What is the efficiency of a Carnot heat engine operating between 500 K and 300 K?
A) 60%
B) 40%
C) 50%
D) 30%
Answer: B) η = 1 - (T_c / T_h) = 1 - (300/500) = 1 - 0.6 = 0.4 or 40%.
14. Kirchhoff's Voltage Law (KVL) states that:
A) The sum of currents entering a node is zero
B) The sum of voltages around a closed loop is zero
C) The voltage across a resistor is IR
D) Power is conserved in a circuit
Answer: B) KVL is based on conservation of energy. The algebraic sum of all voltages around any
closed loop in a circuit is zero.
15. Kirchhoff's Current Law (KCL) states that:
A) The sum of currents entering a node equals the sum of currents leaving the node
B) The voltage across a node is constant
C) Current is proportional to voltage
D) Charge is conserved in a circuit
of Materials, Fluid Mechanics, Thermodynamics, Electrical Circuits, Materials Science,
Economics)
1. What is the derivative of f(x) = ln(sin(x))?
A) cot(x)
B) tan(x)
C) sec(x)csc(x)
D) 1/sin(x)
Answer: A) Using the chain rule: d/dx[ln(u)] = (1/u) * du/dx. u = sin(x), du/dx = cos(x). So
derivative = (1/sin(x)) * cos(x) = cos(x)/sin(x) = cot(x).
2. Evaluate the integral: ∫₀¹ x² dx
A) 1/3
B) 1/2
C) 1
D) 1/4
Answer: A) ∫ x² dx = x³/3. Evaluate from 0 to 1: (1³/3) - (0³/3) = 1/3.
3. What is the determinant of the 2x2 matrix [[2, 3], [4, 5]]?
A) -2
B) 22
C) 10
D) -22
Answer: A) For a 2x2 matrix [[a, b], [c, d]], the determinant is ad - bc. (2*5) - (3*4) = 10 - 12 = -2.
4. In a simply supported beam with a central point load P, what is the reaction at each
support?
A) P/2
B) P
C) 2P
D) P/4
Answer: A) By symmetry, the load is equally shared between the two supports. Each reaction =
P/2.
5. A force vector F = 3i + 4j N acts on a particle. What is the magnitude of the force?
A) 5 N
B) 7 N
C) 1 N
,D) 12 N
Answer: A) Magnitude = √(3² + 4²) = √(9 + 16) = √25 = 5 N.
6. What is the moment of inertia of a solid cylinder of mass M and radius R about its central
axis?
A) ½MR²
B) MR²
C) ¼MR²
D) ⅔MR²
Answer: A) The mass moment of inertia of a solid cylinder about its central longitudinal axis is I
= (1/2)MR².
7. What is the definition of stress in a material?
A) Force per unit area
B) Deformation per unit length
C) Force per unit volume
D) Energy per unit volume
Answer: A) Stress (σ) is defined as the internal resisting force per unit area. σ = F/A. Strain is
deformation per unit length.
8. What is the modulus of rigidity (shear modulus) G related to Young's modulus E and
Poisson's ratio ν by?
A) G = E / [2(1 + ν)]
B) G = E / [3(1 - 2ν)]
C) G = E(1 + ν)
D) G = E(1 - ν)
Answer: A) The relationship is G = E / [2(1 + ν)]. This is derived from elastic constants.
9. What is the specific weight of water at standard conditions (approx)?
A) 62.4 lb/ft³
B) 1000 kg/m³
C) 9.81 kN/m³
D) Both A and C
Answer: D) Specific weight (γ) = ρg. In SI: 9.81 kN/m³. In US units: 62.4 lb/ft³. Density of water is
1000 kg/m³ (B).
10. For a fluid at rest, the pressure at a point is:
A) Dependent on direction
B) Independent of direction (Pascal's Law)
C) Always zero
, D) Directly proportional to velocity
Answer: B) Pascal's Law states that pressure at a point in a static fluid is the same in all
directions.
11. The ideal gas law is expressed as:
A) PV = nRT
B) PV = mRT
C) P = ρRT
D) All of the above
Answer: D) PV = nRT (with n = number of moles, R = universal gas constant). PV = mRT (with m =
mass, R = specific gas constant). P = ρRT (with ρ = density). All are equivalent forms.
12. In a Carnot cycle, which process is isentropic?
A) Isothermal expansion
B) Isothermal compression
C) Adiabatic expansion and compression
D) All processes are isentropic
Answer: C) A Carnot cycle consists of two isothermal (reversible, constant temperature)
processes and two adiabatic (reversible, constant entropy - isentropic) processes.
13. What is the efficiency of a Carnot heat engine operating between 500 K and 300 K?
A) 60%
B) 40%
C) 50%
D) 30%
Answer: B) η = 1 - (T_c / T_h) = 1 - (300/500) = 1 - 0.6 = 0.4 or 40%.
14. Kirchhoff's Voltage Law (KVL) states that:
A) The sum of currents entering a node is zero
B) The sum of voltages around a closed loop is zero
C) The voltage across a resistor is IR
D) Power is conserved in a circuit
Answer: B) KVL is based on conservation of energy. The algebraic sum of all voltages around any
closed loop in a circuit is zero.
15. Kirchhoff's Current Law (KCL) states that:
A) The sum of currents entering a node equals the sum of currents leaving the node
B) The voltage across a node is constant
C) Current is proportional to voltage
D) Charge is conserved in a circuit
