Tricky Engineering Math interview questions

75 questions with answers (MAT-426 to MAT-500). Try each one aloud before reading the answer.

Linear Algebra

MAT-426 · Can a matrix have a zero eigenvalue?

Trap: Assuming eigenvalues must be non-zero. Yes, they can be zero: λ = 0 means Av = 0 for some v ≠ 0, so the matrix is singular (det = 0) and its null space is non-trivial.

MAT-427 · Is AB = BA for square matrices?

Trap: Carrying over commutativity from scalars. Matrix multiplication is generally not commutative; even the shapes may differ for non-square factors. It holds in special cases, such as diagonal matrices or a matrix and its own powers.

MAT-428 · If AB = 0, must A = 0 or B = 0?

Trap: Assuming there are no zero divisors. A = [[1, 0], [0, 0]] and B = [[0, 0], [0, 1]] are both non-zero, yet AB = 0. You can only conclude that at least one of them is singular.

MAT-429 · Is every square matrix diagonalizable?

Trap: Thinking every n×n matrix has n independent eigenvectors. [[1, 1], [0, 1]] has the repeated eigenvalue 1 but only one eigenvector, so it is defective. Its best form is the Jordan form.

MAT-430 · Is det(A + B) = det A + det B?

Trap: Treating the determinant as linear. It is not: for A = B = I (2×2), det(2I) = 4 but det A + det B = 2. The determinant is multiplicative: det(AB) = det A·det B.

MAT-431 · Does a real matrix always have real eigenvalues?

Trap: Generalising from symmetric matrices. The 90° rotation [[0, −1], [1, 0]] has eigenvalues ±i, since no real vector keeps its direction. Only real symmetric (more generally Hermitian) matrices are guaranteed real eigenvalues.

MAT-432 · Do similar matrices have the same eigenvectors?

Trap: Confusing shared eigenvalues with shared eigenvectors. B = P⁻¹AP has the same eigenvalues, trace and determinant as A, but its eigenvectors are P⁻¹v for each eigenvector v of A.

Differential Calculus

MAT-433 · Is |x| differentiable everywhere?

Trap: Assuming a continuous function must be differentiable. |x| is continuous at x = 0, but the left derivative is −1 and the right is +1 (a sharp corner), so it is not differentiable there.

MAT-434 · Is 0.999… equal to 1?

Trap: Thinking it is "just less than" 1. They are exactly equal: if x = 0.999…, then 10x − x = 9, so x = 1. The geometric series 9/10 + 9/100 + … also sums to 1.

MAT-435 · Does f′(c) = 0 guarantee a maximum or minimum at c?

Trap: Treating every stationary point as an extremum. f(x) = x³ has f′(0) = 0, but 0 is an inflection point with no sign change of f′. An extremum also needs f′ to change sign (or a higher-order test).

MAT-436 · Is a function that is continuous everywhere differentiable at least somewhere?

Trap: Thinking continuity implies some smoothness. The Weierstrass function Σaⁿcos(bⁿπx) (suitable a, b) is continuous everywhere and differentiable nowhere.

MAT-437 · If both partial derivatives exist at a point, is f differentiable (or continuous) there?

Trap: Extending one-variable intuition to several variables. f = xy/(x² + y²) with f(0, 0) = 0 has f_x = f_y = 0 at the origin, but it equals 1/2 along y = x, so it is not even continuous. Continuous partials are needed.

MAT-438 · Does lim(x→0) x·sin(1/x) fail to exist because sin(1/x) oscillates?

Trap: Thinking oscillation always prevents a limit. Since |x·sin(1/x)| ≤ |x| → 0, the squeeze theorem gives a limit of 0. (Without the x factor, sin(1/x) alone has no limit.)

MAT-439 · Is 1^∞ equal to 1?

Trap: Treating it as a fixed number. 1^∞ is indeterminate: (1 + 1/n)ⁿ → e and (1 + 2/n)ⁿ → e². Evaluate it as exp(lim g·ln f).

Integral Calculus

MAT-440 · Is ∫₋₁¹ dx/x² = [−1/x] from −1 to 1 = −2?

Trap: Applying the Fundamental Theorem across a singularity. The integrand is positive, so a negative answer is impossible. It is unbounded at x = 0, and each half diverges, so the integral diverges.

MAT-441 · Is ∫ dx/x = ln x + C?

Trap: Forgetting the domain. The correct antiderivative is ln|x| + C (with separate constants on x > 0 and x < 0), since ln x is undefined for negative x.

MAT-442 · Does every continuous function have an elementary antiderivative?

Trap: Thinking every integral has a closed form. e^(−x²), sin x/x and √(1 + x³) have antiderivatives that are not elementary. They define special functions (erf, Si) or need numerical integration.

MAT-443 · Does Σ1/n converge because its terms tend to 0?

Trap: Treating aₙ → 0 as sufficient. It is only necessary. The harmonic series diverges, like ∫₁^∞ dx/x = ∞ by the integral test, while Σ1/n² converges.

MAT-444 · Is ∫₋∞^∞ x dx = 0 by symmetry?

Trap: Using odd symmetry on a divergent integral. The improper integral requires both ∫₀^∞ and ∫₋∞⁰ to converge, and both diverge. Only the Cauchy principal value, lim ∫₋ᴿᴿ, is 0.

MAT-445 · Is ∫₀^(2π) sin x dx the area between sin x and the x-axis?

Trap: Equating the definite integral with area. The integral is 0 because the positive and negative lobes cancel. The geometric area is ∫|sin x|dx = 4.

Vector Calculus

MAT-446 · If ∇×F = 0 wherever F is defined, is F conservative?

Trap: Ignoring the topology of the domain. F = (−y, x)/(x² + y²) has zero curl away from the origin, yet its circulation around the unit circle is 2π. The implication holds only on simply connected domains.

MAT-447 · Is the flux of r̂/r² through a sphere zero because its divergence is zero?

Trap: Applying the divergence theorem across a singularity. ∇·(r̂/r²) = 0 except at the origin, where it is singular. The flux through any enclosing sphere is 4π; in distribution terms, ∇·(r̂/r²) = 4πδ(r).

MAT-448 · Is the cross product associative?

Trap: Assuming vector products behave like scalar products. In general a×(b×c) ≠ (a×b)×c: i×(i×j) = i×k = −j, but (i×i)×j = 0.

MAT-449 · Is the gradient tangent to the level curves?

Trap: Mixing up tangent and normal. ∇f is perpendicular to level curves and surfaces, because f does not change along them. The tangent direction gives a zero directional derivative.

MAT-450 · Does |∇f| give the rate of change of f in every direction?

Trap: Ignoring direction. |∇f| is only the maximum rate, along ∇f. In direction û the rate is |∇f|cos θ, which is zero along level curves and −|∇f| opposite the gradient.

MAT-451 · Since closed-loop line integrals are zero, is the net work done by friction around a closed path zero?

Trap: Assuming every force is conservative. Only conservative fields have zero circulation. Friction always opposes motion, so its work around a closed path is negative and depends on path length.

Ordinary Differential Equations

MAT-452 · Does every first-order IVP have a unique solution?

Trap: Assuming uniqueness always holds. y′ = √y, y(0) = 0 has both y = 0 and y = x²/4 (x ≥ 0), because ∂f/∂y = 1/(2√y) is unbounded at y = 0, breaking the Lipschitz condition.

MAT-453 · Does a smooth-looking ODE always have a solution valid for all x?

Trap: Expecting nice equations to give global solutions. y′ = y², y(0) = 1 gives y = 1/(1 − x), which blows up at x = 1 even though f(y) = y² is smooth. Nonlinear ODEs can have finite-time blow-up.

MAT-454 · Does y″ + y = 0 with y(0) = 0 and y(π) = 0 have a unique solution?

Trap: Carrying IVP uniqueness over to BVPs. Every y = c·sin x satisfies both conditions, giving infinitely many solutions; this is an eigenvalue case. With y(π) = 1 instead, there would be no solution.

MAT-455 · For y″ − y = eˣ, is the trial particular integral Aeˣ?

Trap: Ignoring overlap with the complementary function. eˣ already solves y″ − y = 0, so Aeˣ gives 0 = eˣ. Use Axeˣ, which gives A = 1/2, so y_p = xeˣ/2.

MAT-456 · Does a damped system oscillate at its natural frequency?

Trap: Confusing ωₙ with the damped frequency. The free oscillation is at ω_d = ωₙ√(1 − ζ²), which is lower. For forced response, peak displacement occurs at ωₙ√(1 − 2ζ²), lower still.

MAT-457 · If y₁ and y₂ solve y′ = y², does y₁ + y₂ also solve it?

Trap: Applying superposition to a nonlinear equation. (y₁ + y₂)′ = y₁² + y₂², but (y₁ + y₂)² has the extra 2y₁y₂ term. Superposition holds only for linear homogeneous equations.

Partial Differential Equations

MAT-458 · Can you run the heat equation backwards in time to recover a past temperature profile?

Trap: Assuming time-reversibility. Backward heat conduction is ill-posed: mode n grows as e^(+α²n²π²t/L²), so tiny measurement noise in high modes explodes. It needs regularization.

MAT-459 · Do higher modes of an ideal vibrating string decay faster, as in the heat equation?

Trap: Mixing up parabolic and hyperbolic behaviour. The ideal wave equation conserves energy: every mode oscillates as cos(nπct/L) with constant amplitude. Decay comes only from added damping terms.

MAT-460 · Can a non-constant solution of Laplace's equation have a maximum inside the domain?

Trap: Picturing a "hot spot" in steady state. By the maximum principle, a non-constant harmonic function attains its maximum and minimum only on the boundary, so steady temperature without sources peaks on the boundary.

MAT-461 · Does separation of variables work for any linear PDE?

Trap: Treating it as universal. It needs a separable equation and coordinate system, a domain matching those coordinates (rectangle, disk, sphere), and usually homogeneous boundary conditions. Otherwise use transforms, Green's functions or numerics.

MAT-462 · In an explicit heat-equation scheme, does refining Δx alone always improve accuracy?

Trap: Ignoring the stability limit. r = α²Δt/Δx² grows as Δx shrinks; once r > 1/2 the solution blows up. Δt must be reduced in proportion to Δx².

MAT-463 · Does the Neumann problem for Laplace's equation have a unique solution?

Trap: Expecting it to behave like the Dirichlet problem. A solution exists only if the net boundary flux ∮∂u/∂n ds = 0, and then it is unique only up to an additive constant.

Complex Variables

MAT-464 · Is iⁱ imaginary?

Trap: Assuming an imaginary base and exponent give an imaginary result. iⁱ = e^(i·log i) = e^(i·i(π/2 + 2πk)) = e^(−π/2 − 2πk), which is real. The principal value is e^(−π/2) ≈ 0.208.

MAT-465 · Is |sin z| ≤ 1 for all complex z?

Trap: Carrying over the real bound. sin(iy) = i·sinh y, which is unbounded; sin(3i) ≈ 10.02i. By Liouville's theorem, a non-constant entire function cannot be bounded.

MAT-466 · Is √(ab) = √a·√b always true?

Trap: Using a rule that holds only for non-negative reals. √(−1)·√(−1) = i·i = −1, but √((−1)(−1)) = √1 = 1. Complex square roots are multivalued and branch choices break the identity.

MAT-467 · Is f(z) = |z|² analytic because it is smooth?

Trap: Equating real smoothness with complex differentiability. With u = x² + y², v = 0, the Cauchy–Riemann equations hold only at z = 0. It is complex-differentiable only there and analytic nowhere.

MAT-468 · Can a bounded entire function be non-constant?

Trap: Generalising from bounded real functions like sin x. Liouville's theorem says every bounded entire function is constant. This leads directly to a proof of the fundamental theorem of algebra.

MAT-469 · Is i > 0?

Trap: Assuming complex numbers can be ordered. ℂ cannot be an ordered field: if i > 0 then i·i = −1 > 0, a contradiction; i < 0 fails similarly. Only magnitudes |z| can be compared.

Probability

MAT-470 · In the Monty Hall problem, should you switch doors?

Trap: Thinking the two remaining doors are equally likely. Your first pick wins with 1/3. The host, who knows where the car is, always opens a losing door, so switching wins with probability 2/3.

MAT-471 · How many people are needed for a 50% chance that two share a birthday?

Trap: Guessing about 183 (half of 365). Only 23 are needed (≈50.7%), because there are C(23, 2) = 253 pairs. With 70 people the probability exceeds 99.9%.

MAT-472 · After five heads in a row with a fair coin, is tails more likely next?

Trap: The gambler's fallacy. Tosses are independent, so P(tails) is still 1/2. The probability of the full six-toss sequence was small beforehand, but that does not change the next toss.

MAT-473 · Are mutually exclusive events independent?

Trap: Treating "exclusive" and "independent" as the same. If P(A), P(B) > 0 and A∩B = ∅, then P(A∩B) = 0 ≠ P(A)P(B). Knowing A occurred rules B out, so they are strongly dependent.

MAT-474 · Does a probability of zero mean the event is impossible?

Trap: Equating zero probability with impossibility. For a continuous variable, P(X = 2.5) = 0 for every exact value, yet some value always occurs. Such events are "almost never", not impossible.

MAT-475 · Does zero correlation imply independence?

Trap: Thinking zero covariance means unrelated. With X ~ N(0, 1) and Y = X², Cov(X, Y) = E[X³] = 0, yet Y is fully determined by X. Zero correlation implies independence only in special cases, such as jointly Gaussian variables.

MAT-476 · A family has two children, and at least one is a boy. What is the probability that both are boys?

Trap: Answering 1/2. The equally likely cases with at least one boy are BB, BG and GB, so the probability is 1/3. If you know the older child is a boy, it becomes 1/2.

Statistics

MAT-477 · If a treatment works better in every subgroup, must it work better overall?

Trap: Ignoring Simpson's paradox. Unequal group sizes and a confounder can reverse an aggregated trend; the classic kidney-stone data show this. Stratify, or use causal reasoning, before pooling.

MAT-478 · Is a p-value of 0.03 the probability that the null hypothesis is true?

Trap: Inverting the conditional. The p-value is P(data at least this extreme | H₀), not P(H₀ | data). The probability of H₀ needs a prior and Bayes' theorem.

MAT-479 · Does a 95% confidence interval mean a 95% probability that the true mean is in this interval?

Trap: Giving a frequentist interval a Bayesian meaning. The true mean is fixed; a computed interval either contains it or does not. 95% refers to the long-run success rate of the procedure. A Bayesian credible interval supports the probability reading.

MAT-480 · Does a high correlation coefficient mean a steep regression slope?

Trap: Confusing strength with steepness. r measures how tightly points follow a line; the slope is b = r·s_y/s_x and depends on units. r = 0.99 can have a nearly flat slope, and r = 0.3 a steep one.

MAT-481 · Is the average of group averages equal to the overall average?

Trap: Ignoring group sizes. A group of 10 averaging 80 and a group of 90 averaging 60 have an overall mean of (800 + 5400)/100 = 62, not 70. Use a weighted mean.

MAT-482 · Does a larger sample remove bias?

Trap: Confusing random error with systematic error. More data shrinks the standard error σ/√n, but a biased sampling method (selection or non-response bias, or miscalibration) gives a precise wrong answer. Bias is fixed by design, not size.

Numerical Methods

MAT-483 · Does a smaller step size always make a finite-difference derivative more accurate?

Trap: Ignoring round-off. Truncation error falls with h, but the subtraction f(x + h) − f(x) loses digits, so round-off grows like ε/h. For forward differences in double precision the best h is about √ε ≈ 10⁻⁸; smaller is worse.

MAT-484 · Is Newton–Raphson always faster than bisection?

Trap: Assuming quadratic convergence is guaranteed. Newton can diverge, cycle, or slow to linear convergence at multiple roots. For f = x^(1/3) it gives xₙ₊₁ = −2xₙ and diverges from any non-zero start. Bisection is slow but guaranteed for a bracketed root.

MAT-485 · In floating point, is 0.1 + 0.2 == 0.3?

Trap: Assuming decimal fractions are stored exactly. 0.1 and 0.2 have no exact binary representation; in IEEE double, 0.1 + 0.2 = 0.30000000000000004. Compare with a tolerance, |a − b| < ε, not ==.

MAT-486 · If the residual ‖b − Ax̂‖ is tiny, is the solution x̂ accurate?

Trap: Equating a small residual with a small error. For an ill-conditioned A, the relative error can be up to κ(A) times the relative residual. With κ = 10¹⁰, a residual near machine precision can still leave large errors in x.

MAT-487 · Is Simpson's 1/3 rule exact only for quadratics?

Trap: Thinking fitting parabolas means exactness only up to degree 2. By symmetry, the cubic error term cancels, so Simpson's rule is exact for cubics. Its error depends on the fourth derivative.

MAT-488 · Are implicit methods more accurate than explicit ones?

Trap: Confusing stability with accuracy. Backward Euler is first order, like forward Euler. Implicit methods are chosen for large stable steps on stiff problems, not for accuracy, and each step costs more because it needs a solve.

Transforms (Laplace, Fourier, Z)

MAT-489 · Does the final value theorem always give the steady-state value?

Trap: Applying it without checking poles. For F(s) = 1/(s² + 1) it gives lim sF(s) = 0, but f(t) = sin t never settles. It is valid only if all poles of sF(s) are in the open left half-plane.

MAT-490 · Does every function have a Laplace transform?

Trap: Assuming the integral always converges. The function must be piecewise continuous and of exponential order. e^(t²) grows faster than any e^(ct), so ∫e^(t²)e^(−st)dt diverges for every s.

MAT-491 · Does sin ω₀t have a Fourier transform in the ordinary sense?

Trap: Ignoring the convergence condition. sin ω₀t is not absolutely integrable, so the classical integral fails. In the generalized sense its transform is (π/i)[δ(ω − ω₀) − δ(ω + ω₀)], two impulses.

MAT-492 · Does a Z-transform expression uniquely determine the sequence?

Trap: Forgetting the ROC. z/(z − a) corresponds to aⁿu[n] for |z| > |a| and to −aⁿu[−n − 1] for |z| < |a|. The ROC must be specified, as with the Laplace transform.

MAT-493 · Does a Fourier series converge to f(x) at a jump discontinuity?

Trap: Expecting pointwise equality everywhere. At a jump, the series converges to the average [f(x⁺) + f(x⁻)]/2 (Dirichlet conditions). Partial sums also overshoot near the jump (Gibbs phenomenon).

MAT-494 · Is sampling at exactly twice the maximum frequency enough?

Trap: Treating the Nyquist rate as sufficient. The condition is f_s > 2f_max, strictly. A sinusoid at exactly f_s/2 can be sampled at its zero crossings and vanish. Practical systems sample well above 2f_max because anti-aliasing filters are not ideal.

Discrete Math & Optimization

MAT-495 · If p → q is true, is q → p also true?

Trap: Affirming the consequent. The converse is not equivalent: "if it rains, the road is wet" does not mean a wet road implies rain. The contrapositive ¬q → ¬p is equivalent.

MAT-496 · Is p → q false when p is false?

Trap: Reading implication as causation. A conditional is false only when p is true and q is false. With a false premise it is vacuously true, so "if 2 + 2 = 5 then the moon is cheese" is true.

MAT-497 · Does a greedy algorithm always give the optimal answer?

Trap: Trusting the locally best choice. With coins {1, 3, 4} and target 6, greedy picks 4 + 1 + 1 (three coins), but 3 + 3 uses two. Greedy is optimal only when the problem has the greedy-choice property and optimal substructure.

MAT-498 · Are there fewer even numbers than integers?

Trap: Applying finite intuition to infinite sets. n ↦ 2n is a bijection between the integers and the even integers, so they have the same cardinality (countably infinite). An infinite set can be equinumerous with a proper subset.

MAT-499 · Does every LP with a non-empty feasible region have an optimal solution?

Trap: Assuming feasibility guarantees an optimum. If the region is unbounded in an improving direction, the objective is unbounded (maximize x with x ≥ 0). An optimum is guaranteed if the region is bounded.

MAT-500 · Does rounding an LP relaxation give the integer optimum?

Trap: Treating integer programs as rounded LPs. The rounded point may be infeasible or far from optimal, and the true integer optimum can lie away from the LP vertex. Use branch and bound or cutting planes; the LP optimum is only a bound.

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