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Question 1Mathematics and numerical methods

Evaluate the limit lim(x→0) sin(3x)/x.

  1. 1
  2. 1/3
  3. 3 Correct answer.
  4. 0

Explanation

Rewrite the ratio as 3·sin(3x)/(3x) and use the standard result sin(u)/u → 1 as u → 0. The limit therefore equals 3·1 = 3. The coefficient inside the argument scales the limit, so the answer differs from the familiar sin(x)/x case.

Question 2Mathematics and numerical methods

Find the determinant of the 2×2 matrix whose first row is (4, −2) and second row is (3, 5).

  1. 26 Correct answer.
  2. 22
  3. 14
  4. 20

Explanation

For a 2×2 matrix with entries a, b in the first row and c, d in the second, the determinant is ad − bc. Here ad = 4·5 = 20 and bc = (−2)·3 = −6, so the determinant is 20 − (−6) = 26. The sign of the off-diagonal product is the most common source of error.

Question 3Mathematics and numerical methods

What is the modulus of the complex number z = 3 − 4i?

  1. 1
  2. 7
  3. 25
  4. 5 Correct answer.

Explanation

The modulus of a + bi is √(a² + b²), giving √(3² + (−4)²) = √(9 + 16) = √25 = 5. The imaginary part is squared, so its sign has no effect on the result.

Question 4Mathematics and numerical methods

Apply one iteration of the Newton–Raphson method to f(x) = x² − 2 starting from x₀ = 1. What is x₁?

  1. 1.25
  2. 1.75
  3. 1.414 (approximately)
  4. 1.5 Correct answer.

Explanation

The update is x₁ = x₀ − f(x₀)/f′(x₀), with f′(x) = 2x. Since f(1) = −1 and f′(1) = 2, we get x₁ = 1 − (−1)/2 = 1.5. The correction is added because f is negative at the starting point.

Question 5Mathematics and numerical methods

Use the composite trapezoidal rule with two equal subintervals to approximate the integral of f(x) = x² over the interval [0, 2].

  1. 2.5
  2. 3.0 Correct answer.
  3. 4.0
  4. 2.6667

Explanation

With h = (2 − 0)/2 = 1, the rule gives (h/2)[f(0) + 2f(1) + f(2)] = 0.5[0 + 2 + 4] = 3.0. The interior node carries weight 2, and the endpoints carry weight 1.

Question 6Mathematics and numerical methods

Determine the eigenvalues of the symmetric matrix whose rows are (2, 1) and (1, 2).

  1. 0 and 4
  2. 2 and 2
  3. 1 and 3 Correct answer.
  4. −1 and 3

Explanation

Eigenvalues satisfy det(A − λI) = (2 − λ)² − 1 = 0, so 2 − λ = ±1 and λ = 1 or λ = 3. As a check, the trace is 4 (the sum) and the determinant is 3 (the product).

Question 7Mathematics and numerical methods

A random variable follows a binomial distribution with n = 10 independent trials and success probability p = 0.3. What is its variance?

  1. 3.0
  2. 7.0
  3. 1.45
  4. 2.1 Correct answer.

Explanation

The variance of a binomial random variable is np(1 − p). Substituting gives 10 × 0.3 × 0.7 = 2.1, while the mean is np = 3.0.

Question 8Mathematics and numerical methods

Using Euler's method with step size h = 0.1, estimate y(0.1) for the initial value problem dy/dx = x + y with y(0) = 1.

  1. 1.20
  2. 1.00
  3. 1.10 Correct answer.
  4. 1.05

Explanation

Euler's method advances by y₁ = y₀ + h·f(x₀, y₀). At the initial point f(0, 1) = 0 + 1 = 1, so y₁ = 1 + 0.1(1) = 1.10.

Question 9Mathematics and numerical methods

A linear system Ax = b has a coefficient matrix with condition number κ(A) = 1.0 × 10⁶, solved in double precision. Which statement best describes the numerical behaviour of the computed solution?

  1. Relative errors in b can be amplified by about 10⁶ in x. Correct answer.
  2. The system is only badly scaled and stays exact.
  3. The coefficient matrix is singular and elimination fails.
  4. The relative error in x is always 10⁻⁶ times that in b.

Explanation

The condition number bounds how much a relative perturbation in the right-hand side can be magnified in the solution: the relative error in x is at most about κ(A) times the relative error in b. A value of 10⁶ signals ill-conditioning and a likely loss of roughly six significant digits in double precision.

Question 10Mathematics and numerical methods

For the composite Simpson's 1/3 rule applied over an interval divided into n equal subintervals, which combination of requirement and global error order is correct?

  1. n may be any positive integer; global error is O(h⁴).
  2. n must be even; global error is O(h⁴). Correct answer.
  3. n must be odd; global error is O(h²).
  4. n must be even; global error is O(h³).

Explanation

The composite Simpson's 1/3 rule fits a quadratic over each pair of adjacent subintervals, so the subinterval count must be even. Because the rule integrates cubics exactly, the global error is O(h⁴), one order better than the O(h²) of the composite trapezoidal rule.

Question 11Mathematics and numerical methods

Given vectors u = (2, −1, 3) and v = (1, 4, −2), what is the dot product u · v?

  1. 8
  2. −8 Correct answer.
  3. −4
  4. 12

Explanation

The dot product is the sum of the products of corresponding components: (2)(1) + (−1)(4) + (3)(−2) = 2 − 4 − 6 = −8. This signed scalar measures alignment and is negative when the angle between the vectors exceeds 90 degrees.

Question 12Mathematics and numerical methods

For f(x) = x ln x, what is the value of f′(1)?

  1. e
  2. ln 2
  3. 0
  4. 1 Correct answer.

Explanation

Using the product rule, f′(x) = 1·ln x + x·(1/x) = ln x + 1. At x = 1, ln 1 = 0, so f′(1) = 1.

Question 13Mathematics and numerical methods

If A is a 3×4 matrix and B is a 4×2 matrix, what are the dimensions of the product AB?

  1. 4×4
  2. 3×4
  3. 3×2 Correct answer.
  4. 2×3

Explanation

The product AB is defined because the number of columns of A, 4, equals the number of rows of B, 4. The result has the row count of A and the column count of B, so its dimensions are 3×2.

Question 14Mathematics and numerical methods

Use the first three nonzero terms of the Maclaurin series for e^x to approximate e^{0.5}. Which value results?

  1. 1.750
  2. 1.500
  3. 1.648
  4. 1.625 Correct answer.

Explanation

The Maclaurin series begins e^x = 1 + x + x²/2 + x³/6 + …, so the first three nonzero terms at x = 0.5 give 1 + 0.5 + 0.125 = 1.625. The omitted cubic term is about 0.0208, making this approximation slightly low.

Question 15Mathematics and numerical methods

For f(x) = x³ − x − 1 on the interval [1, 2], after one bisection step which subinterval is guaranteed to contain a root?

  1. [1.25, 1.5]
  2. [1.0, 1.25]
  3. [1.0, 1.5] Correct answer.
  4. [1.5, 2.0]

Explanation

Bisection retains a subinterval whose endpoints have opposite signs. Here f(1) = −1 and f(1.5) = 1.5³ − 1.5 − 1 = 0.875, so a sign change occurs on [1.0, 1.5] and that interval must contain a root.

Question 16Mathematics and numerical methods

Which property defines a real square matrix Q as orthogonal?

  1. det(Q) = 0
  2. Q = Qᵀ
  3. Q² = I
  4. QᵀQ = I Correct answer.

Explanation

A real square matrix is orthogonal when its transpose is its inverse, equivalently QᵀQ = I. This means its columns form an orthonormal basis with respect to the standard inner product.

Question 17Mathematics and numerical methods

A simple random sample of size n = 25 is drawn from a population with standard deviation σ = 10. What is the standard error of the sample mean?

  1. 2.0 Correct answer.
  2. 5.0
  3. 0.4
  4. 10.0

Explanation

The standard error of the sample mean is σ divided by the square root of n. With σ = 10 and n = 25, the standard error is 10/5 = 2.0. It describes the sampling variability of the mean rather than the spread of individual observations.

Question 18Mathematics and numerical methods

Which expression gives the polar form of z = 1 + i√3?

  1. 4(cos π/6 + i sin π/6)
  2. 2(cos π/3 + i sin π/3) Correct answer.
  3. √3(cos π/3 + i sin π/3)
  4. 2(cos π/6 + i sin π/6)

Explanation

For z = a + bi, the modulus is r = √(a² + b²) = √(1 + 3) = 2, and the argument is θ = arctan(b/a) = arctan(√3) = π/3. Therefore z = 2(cos π/3 + i sin π/3).

Question 19Mathematics and numerical methods

For the matrix A = [[1, 2], [3, 4]], what is the infinity norm ‖A‖∞?

  1. 7 Correct answer.
  2. 6
  3. 10
  4. 4

Explanation

The infinity norm of a matrix is the maximum absolute row sum. The row sums are |1| + |2| = 3 and |3| + |4| = 7, so the infinity norm is 7.

Question 20Mathematics and numerical methods

Apply the secant method to f(x) = x² − 2 using initial points x₀ = 1 and x₁ = 2. What is x₂?

  1. 1.250
  2. 1.500
  3. 1.414
  4. 1.333 Correct answer.

Explanation

The secant update is x₂ = x₁ − f(x₁)(x₁ − x₀)/(f(x₁) − f(x₀)). Here f(1) = −1 and f(2) = 2, so x₂ = 2 − 2(1)/(2 − (−1)) = 2 − 2/3 ≈ 1.333.

Question 21Mathematics and numerical methods

In floating-point arithmetic, what does machine epsilon represent?

  1. The largest finite representable floating-point number
  2. The gap between 1.0 and the next larger representable floating-point number Correct answer.
  3. The smallest positive subnormal floating-point number
  4. The smallest positive normalized floating-point number

Explanation

Machine epsilon is defined by the spacing of representable numbers at 1.0: it equals the distance from 1.0 to the next larger floating-point value. It is a relative precision measure, not the smallest normalized or subnormal number, and it has nothing to do with the overflow limit. In binary64 arithmetic it is 2^-52, approximately 2.22e-16.

Question 22Mathematics and numerical methods

A linear system Ax = b has coefficient matrix A = [[4, 1], [2, 3]]. Which statement about the Jacobi iteration for this system is correct?

  1. It converges only if the initial vector is chosen close enough to the exact solution
  2. It is not applicable because A is not a symmetric matrix
  3. It diverges because the spectral radius of the iteration matrix exceeds 1
  4. It converges for any initial vector because A is strictly diagonally dominant Correct answer.

Explanation

A matrix is strictly diagonally dominant when each diagonal entry exceeds the sum of the off-diagonal magnitudes in its row; here 4 > 1 and 3 > 2. For such a matrix the Jacobi iteration matrix has spectral radius below 1, so the iteration converges for every starting vector. Symmetry is not required for the Jacobi method to be defined.

Question 23Mathematics and numerical methods

The classical fourth-order Runge–Kutta method is used to solve an initial value problem. How many derivative evaluations per step does it use, and what is the order of its local truncation error?

  1. Three evaluations; local truncation error O(h^4)
  2. Two evaluations; local truncation error O(h^3)
  3. Four evaluations; local truncation error O(h^5) Correct answer.
  4. Five evaluations; local truncation error O(h^6)

Explanation

The classical fourth-order Runge–Kutta method evaluates the derivative at four stage points per step. Its global error is O(h^4), while the local truncation error accumulated in one step is O(h^5). The other choices confuse the number of stages with the order of the error term.

Question 24Mathematics and numerical methods

A two-point Gaussian quadrature rule on the interval [-1, 1] uses nodes at ±1/√3. What is the highest degree of polynomial for which this rule is exact for every polynomial of that degree?

  1. 4
  2. 1
  3. 2
  4. 3 Correct answer.

Explanation

An n-point Gaussian quadrature rule is constructed to be exact for polynomials of degree up to 2n − 1. For n = 2 this gives exactness through degree 3, so the two-point rule integrates every cubic exactly. A general quartic is not integrated exactly by a two-point Gaussian rule.

Question 25Probability, statistics, and data analysis

For a unimodal data set, the mean is 50 and the median is 62. Which description of the distribution's shape is best supported?

  1. symmetric, because the mean and median are both measures of center
  2. right-skewed, because the mean is greater than the median
  3. bimodal, because the mean and median differ
  4. left-skewed, because the mean is less than the median Correct answer.

Explanation

In a unimodal distribution, extreme values pull the mean toward the longer tail. When the mean is less than the median, the tail extends to the left, which is called left-skewed or negatively skewed. The median is more resistant to extreme low values.

Question 26Probability, statistics, and data analysis

Two independent safety components have failure probabilities 0.10 and 0.20. What is the probability that at least one component works?

  1. 0.72
  2. 0.10
  3. 0.98 Correct answer.
  4. 0.80

Explanation

The complement of at least one working is that both components fail. Because the failures are independent, P(both fail) = 0.10 × 0.20 = 0.02, so P(at least one works) = 1 − 0.02 = 0.98.

Question 27Probability, statistics, and data analysis

When computing an unbiased sample variance from n independent observations, which denominator is used?

  1. n - 1 Correct answer.
  2. n
  3. n + 1
  4. sqrt(n)

Explanation

The sample variance uses n - 1 in the denominator, a correction known as Bessel's correction. Dividing by n - 1 rather than n removes the downward bias in estimating the population variance from a sample.

Question 28Probability, statistics, and data analysis

Ten independent Bernoulli trials each have success probability 0.20. What is the probability of exactly two successes?

  1. 0.678
  2. 0.268
  3. 0.107
  4. 0.302 Correct answer.

Explanation

The binomial probability is C(10,2)(0.20)^2(0.80)^8. This equals 45 × 0.04 × 0.16777216, which rounds to 0.302.

Question 29Probability, statistics, and data analysis

Times to failure of a component are normally distributed with mean 100 hours and standard deviation 15 hours. What proportion of failures occurs after 130 hours?

  1. 0.0228 Correct answer.
  2. 0.1587
  3. 0.0455
  4. 0.9772

Explanation

The z-score is (130 − 100) / 15 = 2.0. The upper-tail area beyond z = 2.0 for a standard normal distribution is approximately 0.0228, so about 2.28% of failures occur after 130 hours.

Question 30Probability, statistics, and data analysis

Defects occur independently at an average rate of 3 per inspection interval. Using a Poisson model, what is the probability of exactly 4 defects in one interval?

  1. 0.0498
  2. 0.168 Correct answer.
  3. 0.224
  4. 0.100

Explanation

The Poisson probability for exactly k events is e^(−λ) λ^k / k!. With λ = 3 and k = 4, this is e^(−3) × 3^4 / 4! = 0.049787 × 81 / 24 ≈ 0.168.

Question 31Probability, statistics, and data analysis

A 95% confidence interval for a population mean is reported as [10.2, 12.8]. Which interpretation is correct?

  1. 95% of the population values lie between 10.2 and 12.8.
  2. 95% of the sample means fall between 10.2 and 12.8.
  3. About 95% of similarly built intervals contain the true mean. Correct answer.
  4. There is a 95% probability that the true mean lies in this interval.

Explanation

A confidence interval is based on a procedure whose long-run coverage is stated by the confidence level. After the interval is computed, the true mean is either inside it or not; the 95% describes the reliability of repeated interval construction, not a probability assigned to this fixed interval.

Question 32Probability, statistics, and data analysis

A data set gives Pearson correlation r = -0.85 between ambient temperature and heating energy use. Which conclusion is best supported?

  1. Heating energy use causes ambient temperature to decrease.
  2. There is no linear association because the correlation is negative.
  3. About 85% of the variation in energy use is explained by temperature.
  4. Strong negative linear association: warmer temperature, less energy use. Correct answer.

Explanation

The sign of r gives the direction of the linear association, and |r| = 0.85 indicates a strong linear relationship. A negative r means that higher temperature tends to accompany lower heating energy use. Correlation alone does not establish causation.

Question 33Probability, statistics, and data analysis

A hypothesis test of H0: mu = 50 versus Ha: mu > 50 has significance level 0.05. The true mean is 55, and the test's power at that value is 0.80. What is the probability of a Type II error when the true mean is 55?

  1. 0.20 Correct answer.
  2. 0.95
  3. 0.80
  4. 0.05

Explanation

Power is the probability of rejecting H0 when the alternative value is true, so power = 1 − P(Type II error). If power is 0.80 at mu = 55, then the Type II error probability is 1 − 0.80 = 0.20.

Question 34Probability, statistics, and data analysis

In a simple linear regression of tensile strength on curing time, the fitted slope is 2.5 MPa per day and r^2 = 0.64. Which statement is correct?

  1. The model explains 64% of the variability in curing time.
  2. Each extra MPa of strength raises curing time by 2.5 days.
  3. The correlation between strength and curing time is negative.
  4. Each extra curing day raises predicted strength by 2.5 MPa. Correct answer.

Explanation

The slope gives the change in the predicted response for a one-unit increase in the predictor, so the predicted strength rises by 2.5 MPa per additional curing day. The value r^2 = 0.64 means 64% of the variability in strength is explained, and the positive slope implies r = +0.80.

Question 35Probability, statistics, and data analysis

A qualification test is set up with H0: mean life = 1,000 hours versus Ha: mean life > 1,000 hours, run at a significance level of alpha = 0.01. What does that 0.01 significance level represent?

  1. The probability that H0 is true given that the test did not reject it
  2. The probability of failing to reject H0 when Ha is true
  3. The probability of accepting H0 when H0 is actually true
  4. The probability of rejecting H0 when H0 is actually true Correct answer.

Explanation

The significance level is the test's Type I error rate: the probability, computed under the assumption that H0 holds, that the sample statistic falls in the rejection region. Setting alpha = 0.01 therefore caps the false-alarm rate at one percent; the chance of correctly rejecting a false null is the power, 1 - beta, which depends on sample size and effect size.

Question 36Probability, statistics, and data analysis

A machined dimension has a population standard deviation of 4.0 mm. Independent random samples of 64 parts are drawn repeatedly and the sample mean is recorded each time. What is the standard deviation of the sampling distribution of the sample mean?

  1. 0.25 mm
  2. 0.0625 mm
  3. 4.00 mm
  4. 0.50 mm Correct answer.

Explanation

The standard error of the mean equals sigma divided by the square root of n, so 4.0 / sqrt(64) = 0.50 mm. Because the divisor grows with the square root of sample size, quadrupling n from 64 to 256 would halve the standard error to 0.25 mm.

Question 37Probability, statistics, and data analysis

A functional test flags faulty units with sensitivity 0.90, meaning P(flag | faulty) = 0.90, and specificity 0.95, meaning P(no flag | good) = 0.95. In the lot being tested, 4% of units are faulty. A unit is flagged. What is the probability that it is actually faulty?

  1. 0.04
  2. 0.57
  3. 0.90
  4. 0.43 Correct answer.

Explanation

Bayes' theorem gives P(faulty | flag) = (0.90 x 0.04) / (0.90 x 0.04 + 0.05 x 0.96) = 0.036 / 0.084, which is about 0.43. Because faulty units are rare, the 5% false-flag rate applied to the large good population produces more flags than the true faults do.

Question 38Probability, statistics, and data analysis

Two independent groups are compared on a strength measurement. Group 1 has n1 = 36 and s1 = 6.0; group 2 has n2 = 64 and s2 = 8.0. Using the separate-variance formula, what is the standard error of the difference in sample means?

  1. 0.71
  2. 10.0
  3. 1.41 Correct answer.
  4. 2.00

Explanation

The variance of the difference between two independent means is the sum of the individual variances of the means: 36/36 + 64/64 = 2.00. The standard error is the square root of that sum, about 1.41, which is why the two contributions are combined before taking the root.

Question 39Probability, statistics, and data analysis

An engineer tests H0: mean coating thickness = 40 micrometres against Ha: mean coating thickness not equal to 40 micrometres and obtains p = 0.03. Which interpretation of this p-value is correct?

  1. There is a 3% chance that the null hypothesis is true.
  2. There is a 3% chance that a Type II error was committed in this test.
  3. If the true mean were 40 micrometres, a result at least as far from 40 as the one observed would occur about 3% of the time. Correct answer.
  4. The probability that the observed sample mean equals exactly 40 micrometres is 0.03.

Explanation

A p-value is calculated assuming H0 is true: it is the probability of a test statistic at least as extreme as the observed one, in the direction(s) specified by Ha. It is not the probability that H0 is true, nor the probability of a Type II error, and it is a tail area rather than the probability of any single value.

Question 40Probability, statistics, and data analysis

A plant tracks defect rate by shift for two lines. Line A: day shift 400 units at 6% defective, night shift 100 units at 8% defective. Line B: day shift 100 units at 5% defective, night shift 400 units at 7% defective. Line A's overall rate is 6.4% and Line B's is 6.6%, even though Line A is worse on each shift. Which explanation is correct?

  1. Line B's night shift is inspected more strictly than its day shift, which inflates its recorded night-shift defect count and therefore its overall rate.
  2. Line A's aggregate figure is a rate rather than a count, and rates from lines with different shift schedules are not comparable in any circumstances.
  3. Shift mix confounds the comparison: Line A puts most of its units on the day shift, which has the lower rate for both lines. Correct answer.
  4. The comparison is invalid because the two lines have equal total output, whereas a Simpson reversal can only arise when the aggregated groups differ in total size.

Explanation

This is an aggregation, or Simpson's, reversal produced by confounding with shift: the overall rate is a weighted average, and Line A places 80% of its units on the day shift, where both lines have lower rates. Line A's night shift holds only 100 units, so its higher 8% rate receives little weight and barely moves the aggregate. Stratifying by shift removes the confounding and shows Line B is the better performer on both shifts.

Question 41Ethics, professional practice, and engineering economics

An engineer is retained by a client who asks for a design change that could reduce the margin of safety in a public structure. Under the NCEES Rules of Professional Conduct, which obligation takes precedence?

  1. Maximizing the client's financial return
  2. Keeping the client's trade secrets confidential
  3. Following the client's instruction without review
  4. Protecting public health, safety, and welfare Correct answer.

Explanation

The NCEES Rules require engineers to hold paramount the safety, health, and welfare of the public. When a client request conflicts with that duty, the engineer must refuse or modify the request rather than defer to the client. Confidentiality remains important but does not override public safety.

Question 42Ethics, professional practice, and engineering economics

Which interest factor converts a single future amount to its equivalent present worth at a stated interest rate and time period?

  1. (P/F, i, n) Correct answer.
  2. (F/A, i, n)
  3. (A/P, i, n)
  4. (F/P, i, n)

Explanation

The single-payment present-worth factor (P/F, i, n) equals 1/(1+i)^n and discounts one future amount to time zero. It is used when only a single future cash flow is known and the equivalent present value is required.

Question 43Ethics, professional practice, and engineering economics

A licensed civil engineer is asked to seal mechanical-system drawings for a project, but the engineer has no mechanical design experience. What is the most appropriate action?

  1. Have a drafting technician sign the drawings instead
  2. Decline to seal and refer the work to a qualified professional Correct answer.
  3. Seal the drawings because a professional license authorizes all disciplines
  4. Seal the drawings after a brief review of the cover sheet

Explanation

Engineers may seal only work for which they are competent and in which they have responsible charge. Accepting work outside one's competence risks public safety and violates professional conduct rules. The engineer should decline and direct the client to a qualified professional.

Question 44Ethics, professional practice, and engineering economics

A consulting engineer owns a significant block of shares in a supplier that is bidding on a project the engineer is evaluating for a client. What action best addresses the conflict of interest?

  1. Bid the supplier's competitors against each other
  2. Disclose the ownership and withdraw from the selection process Correct answer.
  3. Keep the ownership private to avoid influencing the client's decision
  4. Accept the lowest bid regardless of the supplier's identity

Explanation

A financial interest in a bidding supplier creates a conflict of interest that could impair, or appear to impair, professional judgment. The engineer should disclose the interest to the client and recuse from evaluating or selecting that supplier. Concealment or merely choosing the lowest bid does not remove the conflict.

Question 45Ethics, professional practice, and engineering economics

A machine costs $10,000 and has no salvage value. What is its equivalent uniform annual cost over 5 years at an interest rate of 8% per year?

  1. Approximately $2,505 Correct answer.
  2. Approximately $1,850
  3. Approximately $2,000
  4. Approximately $2,800

Explanation

The annual cost is P(A/P, i, n), where (A/P, 8%, 5) equals 0.08(1.08)^5/[(1.08)^5 - 1], or about 0.2505. Multiplying $10,000 by 0.2505 gives approximately $2,505 per year. Simple division by 5 would ignore the time value of money.

Question 46Ethics, professional practice, and engineering economics

An engineer learns that a required safety test failed, but the company's report to a regulator omits that result. What should the engineer do first?

  1. Raise the issue through internal company channels and document the concern Correct answer.
  2. Ignore the omission because the report was signed by management
  3. Post the test results on social media to force disclosure
  4. Alter the test record so the report is technically accurate

Explanation

The engineer's paramount duty is to protect public health and safety, and internal escalation is normally the first step. The engineer should document the concern and follow the employer's procedure for correcting a false or incomplete regulatory report. If internal channels fail and a serious risk remains, further reporting to the appropriate authority may be required.

Question 47Ethics, professional practice, and engineering economics

In a negligence claim against an engineering firm, the applicable standard of care is best described as:

  1. The client's stated expectations in the contract, even if unreasonable
  2. The highest level of skill attainable by any engineer in the country
  3. The care a reasonably prudent engineer would exercise under similar circumstances Correct answer.
  4. The firm's own written procedures, regardless of industry practice

Explanation

Negligence is judged against the conduct of a reasonably prudent professional with similar training and experience in comparable circumstances. A firm is not ordinarily held to a standard of perfection or to every client expectation. Written procedures and contracts may inform the analysis but do not replace the professional standard.

Question 48Ethics, professional practice, and engineering economics

A public project has a present worth of benefits equal to $500,000, a present worth of costs equal to $400,000, and a present worth of disbenefits equal to $50,000. What is the conventional benefit-cost ratio?

  1. 0.80
  2. 1.125 Correct answer.
  3. 1.25
  4. 12.5

Explanation

The conventional benefit-cost ratio is (benefits minus disbenefits) divided by costs. Here, ($500,000 - $50,000)/$400,000 equals 1.125. A ratio above 1.0 indicates that discounted benefits exceed discounted costs under this convention.

Question 49Ethics, professional practice, and engineering economics

A licensed engineer directly supervises an unlicensed designer who prepares calculations for a foundation. The engineer checks the assumptions, reviews the calculations in detail, and accepts professional responsibility for the design. May the engineer seal the drawings?

  1. No, because every calculation must be prepared and signed by a licensed engineer
  2. Yes, if the engineer has responsible charge and the review supports acceptance Correct answer.
  3. Yes, but only if the client waives the licensing requirement in writing
  4. No, because sealing work by an unlicensed designer is always misconduct

Explanation

Licensing laws allow an engineer to seal work prepared by others when the engineer exercises responsible charge and has sufficient knowledge to review and accept it. Direct supervision plus a detailed technical review supports that responsibility. The seal certifies the engineer's professional acceptance, not who performed each calculation.

Question 50Ethics, professional practice, and engineering economics

A project requires an initial investment of $50,000 and returns $20,000 at the end of each year for 3 years. What is the approximate internal rate of return?

  1. 12.0% per year
  2. 20.0% per year
  3. 9.7% per year Correct answer.
  4. 6.7% per year

Explanation

The IRR makes the present worth of the three annual returns equal the $50,000 investment. At about 9.7%, the factor (P/A, i, 3) is approximately 2.50, so $20,000 times 2.50 equals $50,000. Higher rates would produce a present worth below the initial cost.