FE Other Disciplines (NCEES Fundamentals of Engineering) — All Questions

What is the derivative of f(x) = 3x^3 - 5x evaluated at x = 2?

• a.31✓
• b.27
• c.36
• d.19

Differentiate term by term: f'(x) = 9x^2 - 5. At x = 2, f'(2) = 9(4) - 5 = 36 - 5 = 31. The power rule d/dx(x^n) = n·x^(n-1) is the core single-variable calculus tool tested on the FE.

What is the determinant of the 2x2 matrix [[4, 3], [2, 5]]?

• a.26
• b.7
• c.20
• d.14✓

For a 2x2 matrix [[a, b], [c, d]], the determinant is ad - bc. Here (4)(5) - (3)(2) = 20 - 6 = 14. Determinants indicate whether a linear system has a unique solution (nonzero determinant).

Evaluate the definite integral of sin(x) from 0 to pi.

• a.1
• b.2✓
• c.pi
• d.0

The antiderivative of sin(x) is -cos(x). Evaluating from 0 to pi gives [-cos(pi)] - [-cos(0)] = -(-1) - (-(1)) = 1 + 1 = 2. This is a standard application of the Fundamental Theorem of Calculus.

Two fair six-sided dice are rolled. What is the probability that the sum equals 7?

• a.1/12
• b.1/9
• c.1/8
• d.1/6✓

There are 36 equally likely outcomes. The combinations summing to 7 are (1,6),(2,5),(3,4),(4,3),(5,2),(6,1) = 6 outcomes. Probability = 6/36 = 1/6. Seven is the most likely sum for two dice.

Find the sample standard deviation of the data set 2, 4, 6, 8, 10.

• a.3.16✓
• b.6.32
• c.2.83
• d.10

The mean is 6. Deviations are -4, -2, 0, 2, 4; squared they are 16, 4, 0, 4, 16 summing to 40. Sample variance divides by (n - 1) = 4, giving 10; the standard deviation is sqrt(10) = 3.16. Dividing by n-1 (not n) gives the unbiased SAMPLE estimate.

What is the pH of a 0.001 M solution of the strong acid HCl?

• a.3✓
• b.2
• c.11
• d.1

HCl fully dissociates, so [H+] = 0.001 = 1x10^-3 M. pH = -log10[H+] = -log10(10^-3) = 3. A pH below 7 is acidic, consistent with an acid solution.

What is the oxidation state of sulfur in sulfuric acid, H2SO4?

• a.+2
• b.+4
• c.+6✓
• d.-2

Each H is +1 (total +2) and each O is -2 (total -8). For a neutral molecule the sum is zero: 2 + S + (-8) = 0, so S = +6. Assigning oxidation states this way underlies redox and corrosion analysis.

By the Nyquist sampling theorem, what is the minimum sampling rate needed to reconstruct a signal whose highest frequency is 1 kHz?

• a.500 Hz
• b.1 kHz
• c.4 kHz
• d.2 kHz✓

The Nyquist criterion requires a sampling rate greater than twice the highest signal frequency to avoid aliasing: f_s > 2 x 1 kHz = 2 kHz. Sampling below this rate causes higher frequencies to masquerade as lower ones (aliasing).

Under the fundamental canons of engineering ethics (e.g., the NSPE Code), what must an engineer hold paramount?

• a.The employer's schedule
• b.The client's cost savings
• c.The safety, health, and welfare of the public✓
• d.The engineer's professional reputation

The first fundamental canon states that engineers shall hold paramount the safety, health, and welfare of the public. This obligation overrides schedule, cost, and personal interest and is the basis of licensure to protect the public.

After three half-lives have elapsed, what fraction of a radioactive substance remains?

• a.50%
• b.12.5%✓
• c.25%
• d.33%

Each half-life halves the remaining amount: (1/2)^3 = 1/8 = 0.125 = 12.5%. The general relation is N/N0 = (1/2)^(t/T_half), central to radiation-exposure and toxicology calculations.

An atmosphere is generally classified as oxygen-deficient (a confined-space hazard) when the oxygen concentration falls below what value?

• a.16%
• b.20.9%
• c.19.5%✓
• d.23.5%

Normal air is about 20.9% oxygen. OSHA classifies an atmosphere as oxygen-deficient below 19.5% and oxygen-enriched above 23.5%. Confined-space entry requires atmospheric testing before entry.

What is the future worth of $1,000 invested for 5 years at 6% annual compound interest?

• a.$1,060
• b.$1,500
• c.$1,300
• d.$1,338✓

Future worth F = P(1 + i)^n = 1000(1.06)^5. Since 1.06^5 = 1.3382, F = $1,338. This single-payment compound-amount factor is a core time-value-of-money relation.

What is the present worth of $5,000 to be received 10 years from now at an 8% annual discount rate?

• a.$2,316✓
• b.$2,000
• c.$2,500
• d.$4,630

Present worth P = F / (1 + i)^n = 5000 / (1.08)^10. Since 1.08^10 = 2.159, P = 5000 / 2.159 = $2,316. Discounting converts a future cash flow to its value today.

Two forces, 30 N and 40 N, act at the same point and are perpendicular to each other. What is the magnitude of their resultant?

• a.35 N
• b.10 N
• c.50 N✓
• d.70 N

For perpendicular forces the resultant is the vector sum magnitude sqrt(30^2 + 40^2) = sqrt(900 + 1600) = sqrt(2500) = 50 N. This is the 3-4-5 right-triangle relationship. Simply adding to 70 N would only be valid if the forces were collinear.

A 100 N force is applied perpendicular to a wrench handle at a distance of 0.5 m from the bolt. What moment does it produce about the bolt?

• a.200 N·m
• b.100 N·m
• c.150 N·m
• d.50 N·m✓

Moment M = force x perpendicular distance = 100 N x 0.5 m = 50 N·m. Because the force is already perpendicular to the moment arm, no angle factor is needed.

A block rests on a horizontal surface with a normal force of 200 N and a coefficient of static friction of 0.3. What is the maximum static friction force available?

• a.600 N
• b.60 N✓
• c.200 N
• d.0.3 N

Maximum static friction F = mu x N = 0.3 x 200 = 60 N. Friction is proportional to the normal force, not the contact area, and cannot exceed this value before sliding begins.

A particle starts from rest and accelerates uniformly at 3 m/s^2. What is its velocity after 4 seconds?

• a.0.75 m/s
• b.12 m/s✓
• c.7 m/s
• d.24 m/s

With constant acceleration, v = v0 + a·t = 0 + (3)(4) = 12 m/s. This is the basic kinematic equation for linear motion starting from rest.

What is the kinetic energy of a 10 kg mass moving at 4 m/s?

• a.40 J
• b.80 J✓
• c.160 J
• d.20 J

Kinetic energy KE = (1/2) m v^2 = 0.5 x 10 x 4^2 = 0.5 x 10 x 16 = 80 J. Note the velocity is squared, so KE grows rapidly with speed.

What is the linear momentum of a 5 kg object moving at 10 m/s?

• a.100 kg·m/s
• b.2 kg·m/s
• c.50 kg·m/s✓
• d.15 kg·m/s

Linear momentum p = m·v = 5 kg x 10 m/s = 50 kg·m/s. Momentum is conserved in the absence of external impulse, the basis of impact and collision analysis.

An axial tensile load of 10,000 N acts on a bar with cross-sectional area 0.002 m^2. What is the normal stress?

• a.20 MPa
• b.5 MPa✓
• c.0.5 MPa
• d.50 MPa

Normal stress sigma = P / A = 10,000 N / 0.002 m^2 = 5,000,000 Pa = 5 MPa. Stress is force per unit area and does not depend on the bar's length.

A simply supported beam of span 4 m carries a single concentrated load of 8 kN at midspan. What is the maximum bending moment?

• a.32 kN·m
• b.16 kN·m
• c.4 kN·m
• d.8 kN·m✓

For a central point load on a simply supported beam, the maximum moment occurs at midspan and equals P·L / 4 = (8 kN)(4 m) / 4 = 8 kN·m. The moment diagram is triangular, peaking under the load.

A steel bar (E = 200 GPa) 2 m long with cross-sectional area 1x10^-4 m^2 carries an axial load of 10 kN. What is its elongation?

• a.1 mm✓
• b.2 mm
• c.0.5 mm
• d.0.1 mm

Axial elongation delta = P·L / (A·E) = (10,000 x 2) / (1x10^-4 x 200x10^9) = 20,000 / 2x10^7 = 0.001 m = 1 mm. Stiffer materials (higher E) and larger areas reduce elongation.

What is the gauge pressure at a depth of 10 m in water (density 1000 kg/m^3, g = 9.81 m/s^2)?

• a.100 kPa
• b.981 kPa
• c.9.81 kPa
• d.98.1 kPa✓

Hydrostatic pressure p = rho·g·h = 1000 x 9.81 x 10 = 98,100 Pa = 98.1 kPa. Pressure increases linearly with depth and is independent of the container's shape.

Incompressible water flows through a pipe. If the cross-sectional area is reduced to half, what happens to the flow velocity?

• a.It doubles✓
• b.It halves
• c.It quadruples
• d.It stays the same

By the continuity equation A1·V1 = A2·V2, velocity is inversely proportional to area for incompressible flow. Halving the area doubles the velocity so that volumetric flow rate is conserved.

Water flows at 3 m/s through a circular pipe of 0.1 m diameter. What is the volumetric flow rate?

• a.0.03 m^3/s
• b.0.0236 m^3/s✓
• c.0.236 m^3/s
• d.0.0079 m^3/s

Flow rate Q = A·V. The area A = (pi/4)·D^2 = (pi/4)(0.1)^2 = 0.00785 m^2. Then Q = 0.00785 x 3 = 0.0236 m^3/s. Continuity ties cross-sectional area and velocity to discharge.

In pipe flow, flow is generally considered laminar when the Reynolds number is below approximately what value?

• a.4,000
• b.2,100✓
• c.500,000
• d.10

For flow in a circular pipe, laminar flow generally persists below Re of about 2,100; transitional and turbulent flow occur at higher values. The Reynolds number Re = rho·V·D/mu compares inertial to viscous forces.

The modulus of elasticity (Young's modulus) of structural steel is approximately what value?

• a.20 GPa
• b.1,000 GPa
• c.70 GPa
• d.200 GPa✓

Structural steel has a Young's modulus of about 200 GPa (29,000 ksi). Aluminum, by contrast, is near 70 GPa. Modulus measures stiffness (stress-to-strain ratio in the elastic region), not strength.

A 10 m steel bar (coefficient of thermal expansion 12x10^-6 /degC) is heated by 50 degC. How much does it expand?

• a.60 mm
• b.1.2 mm
• c.6 mm✓
• d.0.6 mm

Thermal expansion delta = alpha·L·(delta T) = 12x10^-6 x 10 x 50 = 0.006 m = 6 mm. If the bar were restrained instead of free to expand, this would generate thermal stress.

A resistor of 10 ohms carries a current of 2 A. What is the voltage across it?

• a.5 V
• b.12 V
• c.20 V✓
• d.0.2 V

Ohm's law gives V = I·R = 2 A x 10 ohm = 20 V. Voltage, current, and resistance are directly linked; doubling current at fixed resistance doubles the voltage drop.

Two 10-ohm resistors are connected in parallel. What is their equivalent resistance?

• a.20 ohms
• b.0.05 ohms
• c.10 ohms
• d.5 ohms✓

For two equal resistors in parallel, R_eq = R/2 = 10/2 = 5 ohms. In general 1/R_eq = 1/R1 + 1/R2; parallel combinations always yield a resistance smaller than the smallest branch.

A Carnot engine operates between a hot reservoir at 600 K and a cold reservoir at 300 K. What is its maximum thermal efficiency?

• a.0.25
• b.0.50✓
• c.2.0
• d.0.75

Carnot efficiency eta = 1 - T_cold/T_hot = 1 - 300/600 = 0.50, or 50%. Temperatures must be absolute (kelvin). No real engine between the same reservoirs can exceed this limit.

How much heat is required to raise the temperature of 2 kg of water by 10 degC (specific heat 4,186 J/kg·degC)?

• a.837 kJ
• b.8.37 kJ
• c.41.9 kJ
• d.83.7 kJ✓

Sensible heat Q = m·c·(delta T) = 2 x 4,186 x 10 = 83,720 J = 83.7 kJ. Water's high specific heat is why it is an effective coolant and thermal-storage medium.

For an ideal gas held at constant temperature, if the absolute pressure is doubled, the volume becomes:

• a.Doubled
• b.Unchanged
• c.Halved✓
• d.Quadrupled

Boyle's law (a special case of the ideal gas law at constant temperature) states P·V = constant, so pressure and volume are inversely proportional. Doubling pressure halves the volume.

What is the specific weight of water at standard conditions (density 1000 kg/m^3, g = 9.81 m/s^2)?

• a.1000 N/m^3
• b.981 N/m^3
• c.9.81 kN/m^3✓
• d.98.1 kN/m^3

Specific weight gamma = rho·g = 1000 x 9.81 = 9810 N/m^3 = 9.81 kN/m^3. Specific weight is weight per unit volume, distinct from density (mass per unit volume).

What is the gauge pressure at the bottom of a 5 m deep tank of oil with specific gravity 0.8 (g = 9.81 m/s^2)?

• a.4.9 kPa
• b.39.2 kPa✓
• c.392 kPa
• d.49.1 kPa

Hydrostatic pressure p = SG·rho_water·g·h = 0.8 x 1000 x 9.81 x 5 = 39,240 Pa = 39.2 kPa. Using water (SG = 1) would wrongly give 49.1 kPa; the fluid's specific gravity scales the result.

A manometer shows a mercury column height of 0.2 m (mercury SG = 13.6). What pressure does this represent (g = 9.81 m/s^2)?

• a.26.7 kPa✓
• b.1.96 kPa
• c.2.72 kPa
• d.267 kPa

p = rho·g·h = (13.6 x 1000) x 9.81 x 0.2 = 26,683 Pa = 26.7 kPa. Mercury's high density (SG 13.6) makes it compact for measuring large pressures; using water would give only 1.96 kPa.

Water discharges from a small opening 5 m below the free surface of a large tank. What is the ideal exit velocity (g = 9.81 m/s^2)?

• a.9.9 m/s✓
• b.98.1 m/s
• c.49.1 m/s
• d.19.8 m/s

By Torricelli's theorem (from Bernoulli), v = sqrt(2·g·h) = sqrt(2 x 9.81 x 5) = sqrt(98.1) = 9.9 m/s. Forgetting the square root gives the incorrect 98.1.

Water (density 1000 kg/m^3) flows at 2 m/s through a duct of cross-sectional area 0.01 m^2. What is the mass flow rate?

• a.20 kg/s✓
• b.0.02 kg/s
• c.2 kg/s
• d.200 kg/s

Mass flow rate m_dot = rho·A·V = 1000 x 0.01 x 2 = 20 kg/s. This is the continuity equation expressed in mass terms; leaving out density yields the volumetric rate 0.02 m^3/s instead.

Water (density 1000 kg/m^3, viscosity 1x10^-3 Pa·s) flows at 2 m/s in a 0.05 m diameter pipe. What is the Reynolds number?

• a.100,000✓
• b.100
• c.10,000
• d.1,000

Re = rho·V·D/mu = (1000 x 2 x 0.05)/(1x10^-3) = 100/0.001 = 100,000. This far exceeds ~2100, so the flow is turbulent. Re is the ratio of inertial to viscous forces.

An object of volume 0.02 m^3 is fully submerged in water (density 1000 kg/m^3, g = 9.81 m/s^2). What is the buoyant force on it?

• a.196.2 N✓
• b.1,962 N
• c.19.6 N
• d.98.1 N

By Archimedes' principle, buoyant force F_b = rho·g·V_displaced = 1000 x 9.81 x 0.02 = 196.2 N. It equals the weight of the displaced fluid and is independent of the object's own weight.