Advanced Engineering Math interview questions

125 questions with answers (MAT-301 to MAT-425). Try each one aloud before reading the answer.

Linear Algebra

MAT-301 · What is SVD, and where is it used?

Singular value decomposition: A = UΣVᵀ, with orthogonal U and V and non-negative singular values on the diagonal of Σ. It exists for any m×n matrix and is used in PCA, data compression, recommender systems, pseudo-inverses, rank estimation and noise filtering.

MAT-302 · How does PCA use eigendecomposition?

Centre the data and form the covariance matrix. Its eigenvectors are the principal directions and its eigenvalues are the variances along them. Projecting onto the top k eigenvectors keeps the most variance in k dimensions.

MAT-303 · What is the condition number and why does it matter?

κ(A) = ‖A‖·‖A⁻¹‖, which equals σ_max/σ_min in the 2-norm. Relative errors in b can be amplified by up to κ in x, so you can lose roughly log₁₀κ significant digits.

MAT-304 · You must solve Ku = F for a sparse symmetric positive definite FEM stiffness matrix with a million unknowns. Which method do you use?

Never form K⁻¹. Use a sparse Cholesky factorization with fill-reducing reordering if memory allows, or the preconditioned conjugate gradient method (e.g. incomplete Cholesky or multigrid preconditioner), which needs only matrix-vector products.

MAT-305 · What is the Moore–Penrose pseudo-inverse?

A⁺ = VΣ⁺Uᵀ, where Σ⁺ inverts the non-zero singular values. x = A⁺b gives the minimum-norm least-squares solution, valid for rectangular or rank-deficient A.

MAT-306 · How does the power method find the dominant eigenvalue?

Iterate x_(k+1) = Ax_k/‖Ax_k‖. The vector converges to the dominant eigenvector at a rate set by |λ₂/λ₁|, and the Rayleigh quotient xᵀAx/xᵀx gives λ₁. Inverse iteration finds the smallest eigenvalue.

MAT-307 · Find the eigenvalues and eigenvectors of A = [[4, 1], [2, 3]].

λ² − 7λ + 10 = 0, so λ = 5 and 2. For λ = 5: −x + y = 0, v = (1, 1). For λ = 2: 2x + y = 0, v = (1, −2). Check: trace 7 = 5 + 2, det 10 = 5×2.

MAT-308 · How do eigenvalues determine the behaviour of x′ = Ax?

The solution is x(t) = e^(At)x₀ = Σcᵢvᵢe^(λᵢt). The system is asymptotically stable if all eigenvalues have negative real parts; imaginary parts give oscillation.

MAT-309 · Why is QR decomposition preferred for least squares and eigenvalue problems?

With orthogonal Q, solving Rx = Qᵀb avoids forming AᵀA, so the condition number is not squared. The QR algorithm, iterating A = QR → RQ, is the standard method for computing all eigenvalues.

MAT-310 · How much storage does a rank-50 SVD approximation of a 1000×1000 image need?

k(m + n + 1) = 50×2001 = 100,050 numbers versus 10⁶ for the full image, about 10%. By the Eckart–Young theorem, the truncated SVD is the best rank-k approximation in the 2-norm and Frobenius norm.

MAT-311 · Why is the stiffness matrix of an unsupported structure singular?

Rigid-body motions (translations and rotations) produce no strain energy, so they are null-space vectors with zero eigenvalues. Applying enough supports removes them and makes K positive definite.

Differential Calculus

MAT-312 · What is a Jacobian, and where is it used?

The matrix of first-order partial derivatives J_ij = ∂fᵢ/∂xⱼ of a vector function. It is used in coordinate transforms (|det J| scales areas and volumes), robot kinematics (joint to end-effector velocity) and Newton's method for systems.

MAT-313 · When is a two-link planar robot arm singular?

Its Jacobian determinant is l₁l₂ sin θ₂, so it is singular when θ₂ = 0 or π (arm fully stretched or folded). There the arm loses a direction of motion and inverse kinematics needs unbounded joint speeds.

MAT-314 · Resistance R = V/I with V uncertain by 1% and I by 2%. What is the uncertainty in R?

Relative errors add for products and quotients: the worst case is 1% + 2% = 3%. For independent random errors, combine in quadrature: √(1² + 2²) ≈ 2.24%.

MAT-315 · Design an open-top square-base box of volume 32 m³ with minimum material.

S = x² + 4xh with x²h = 32, so S = x² + 128/x. dS/dx = 2x − 128/x² = 0 gives x = 4 m, h = 2 m, S = 48 m². S″ > 0 confirms a minimum.

MAT-316 · How do you linearize a nonlinear system about an operating point?

Expand x′ = f(x, u) in a Taylor series about equilibrium (x₀, u₀) and keep first-order terms: Δx′ = AΔx + BΔu, with A = ∂f/∂x and B = ∂f/∂u evaluated there. It is valid only for small deviations.

MAT-317 · How many terms of the series for e are needed for an error below 10⁻⁶?

The remainder after the 1/n! term is less than 3/(n + 1)!. 3/10! ≈ 8.3×10⁻⁷ < 10⁻⁶ but 3/9! ≈ 8.3×10⁻⁶ is too large, so sum terms through 1/9! (10 terms).

MAT-318 · The second-derivative test gives D = 0 at a critical point. What next?

The test is inconclusive. Examine higher-order terms or behaviour along paths: f = x⁴ + y⁴ has a minimum at the origin, while f = x⁴ − y⁴ has a saddle, though both give D = 0.

MAT-319 · How accurate is sin θ ≈ θ at 10°?

θ = 0.1745 rad and sin θ = 0.1736. The relative error is about θ²/6 ≈ 0.5%, from the next Taylor term θ³/6.

MAT-320 · What is normalized sensitivity, and what is it for f₀ = 1/(2π√(LC))?

S = (p/y)(∂y/∂p), the percentage change in output per percentage change in parameter. Here S_L = S_C = −1/2, so a 2% increase in L lowers f₀ by about 1%.

MAT-321 · Use calculus to derive the maximum power transfer condition.

P = V²R_L/(R_s + R_L)². Setting dP/dR_L = 0 gives (R_s + R_L) − 2R_L = 0, so R_L = R_s, with P_max = V²/(4R_s) and efficiency 50%.

Integral Calculus

MAT-322 · Derive the second moment of area of a rectangle about its centroidal axis.

I = ∫y² dA = ∫ from −h/2 to h/2 of y²·b dy = b[y³/3] = bh³/12. About the base it is bh³/3 (parallel-axis theorem: bh³/12 + bh·(h/2)²).

MAT-323 · Find the centroid of a semicircular area of radius r.

By symmetry x̄ = 0. ȳ = ∬y dA/A = (2r³/3)/(πr²/2) = 4r/(3π) ≈ 0.424r from the diameter.

MAT-324 · Show that the RMS value of v = V_m sin ωt is V_m/√2.

V_rms² = (1/T)∫₀ᵀ V_m²sin²ωt dt = V_m²/2, because sin² averages to 1/2 over a period. So V_rms = V_m/√2 ≈ 0.707V_m.

MAT-325 · What is ∫₋∞^∞ e^(−ax²) dx, and where is it used?

√(π/a) for a > 0. It normalizes the Gaussian PDF: with a = 1/(2σ²) the integral is σ√(2π), which gives the 1/(σ√(2π)) factor.

MAT-326 · Evaluate ∫₀¹ x⁴(1 − x)⁵ dx.

It is B(5, 6) = Γ(5)Γ(6)/Γ(11) = 4!·5!/10! = 2880/3,628,800 = 1/1260.

MAT-327 · What work is needed to pump a full cylindrical tank (radius R, height H) of water out over the top?

A slice at height y lifts a distance H − y: W = ∫₀ᴴ ρgπR²(H − y)dy = ρgπR²H²/2. For R = 1 m, H = 2 m: 1000×9.81×π×4/2 ≈ 61.6 kJ.

MAT-328 · Find the mass of the unit cube [0,1]³ with density ρ = x + y + z.

m = ∭(x + y + z)dV = 1/2 + 1/2 + 1/2 = 3/2 units, since ∭x dV over the cube is 1/2.

MAT-329 · For which p does ∫₀¹ dx/xᵖ converge, and what about ∫₀^∞ dx/xᵖ?

∫₀¹ converges for p < 1, to 1/(1 − p). ∫₁^∞ needs p > 1. No single p satisfies both, so ∫₀^∞ dx/xᵖ diverges for every p.

MAT-330 · Find the hydrostatic force on a vertical rectangular gate of width b submerged from the surface to depth d.

F = ∫₀ᵈ ρgh·b dh = ρgbd²/2, acting at 2d/3 below the surface. For b = 2 m, d = 3 m: 1000×9.81×2×9/2 ≈ 88.3 kN.

MAT-331 · Find the length of a hanging cable y = a cosh(x/a) from x = −L to L.

y′ = sinh(x/a), so √(1 + y′²) = cosh(x/a) and the length is ∫ cosh(x/a)dx = 2a sinh(L/a).

Vector Calculus

MAT-332 · Derive the continuity equation using the divergence theorem.

The rate of mass decrease in V equals the outflow: −d/dt∭ρ dV = ∯ρv·dS = ∭∇·(ρv)dV. Since V is arbitrary, ∂ρ/∂t + ∇·(ρv) = 0; for incompressible flow, ∇·v = 0.

MAT-333 · How do Gauss's and Stokes' theorems convert Maxwell's equations from integral to differential form?

Gauss: ∯E·dA = Q/ε₀ becomes ∇·E = ρ/ε₀. Stokes: ∮E·dl = −dΦ_B/dt becomes ∇×E = −∂B/∂t, because the surfaces and volumes are arbitrary.

MAT-334 · Verify Stokes' theorem for F = (−y, x, 0) over the unit disk in the xy-plane.

∇×F = (0, 0, 2), so the flux through the disk is 2×π = 2π. On the circle r = (cos t, sin t), F·dr = (sin²t + cos²t)dt, integrating to 2π. They agree.

MAT-335 · Find the outward flux of F = (x, y, z) through the surface of the unit cube [0,1]³.

∇·F = 3, so by the divergence theorem the flux is 3×1 = 3. Directly, only the three faces at x, y, z = 1 contribute, 1 each.

MAT-336 · Show F = (2xy + z³, x², 3xz²) is conservative and find its potential.

∇×F = (0 − 0, 3z² − 3z², 2x − 2x) = 0. Integrating gives φ = x²y + xz³, and ∇φ reproduces each component.

MAT-337 · What is the Laplacian in cylindrical coordinates?

∇²f = (1/r)∂/∂r(r∂f/∂r) + (1/r²)∂²f/∂θ² + ∂²f/∂z². It is used for heat conduction in pipes and fields around wires.

MAT-338 · What is the Helmholtz decomposition?

A sufficiently smooth field that decays at infinity can be written F = −∇φ + ∇×A, the sum of an irrotational and a solenoidal part. It underlies the scalar and vector potentials in electromagnetics and fluid dynamics.

MAT-339 · What is the vorticity of a fluid in rigid-body rotation?

With v = Ω×r, the vorticity ∇×v = 2Ω, twice the angular velocity. Vorticity measures local spin of fluid elements.

MAT-340 · Find the work done by F = (y², 2xy) from (0, 0) to (1, 1) along any path.

∂Q/∂x = 2y = ∂P/∂y, so F is conservative with φ = xy². W = φ(1, 1) − φ(0, 0) = 1, independent of path.

MAT-341 · How do Fourier's law and the divergence theorem give the heat equation?

Heat flux q = −k∇T. Energy balance on any volume gives ρc∂T/∂t = −∇·q = k∇²T (constant k), so ∂T/∂t = α∇²T with α = k/(ρc).

Ordinary Differential Equations

MAT-342 · A spring–mass system has m = 1 kg, k = 100 N/m, c = 4 N·s/m. Find ωₙ, ζ and ω_d.

ωₙ = √(k/m) = 10 rad/s. ζ = c/(2√(km)) = 4/20 = 0.2. ω_d = ωₙ√(1 − ζ²) = 10√0.96 ≈ 9.80 rad/s. It is underdamped.

MAT-343 · What damping coefficient gives critical damping for m = 2 kg, k = 800 N/m?

c_c = 2√(km) = 2√1600 = 80 N·s/m. Below 80 the system oscillates; above it, the return is slower without overshoot.

MAT-344 · A series RLC circuit has L = 1 mH and C = 1 µF. Find the resonant frequency and the critical resistance.

ω₀ = 1/√(LC) = 1/√(10⁻⁹) ≈ 31,623 rad/s, so f₀ ≈ 5.03 kHz. Critical damping when R = 2√(L/C) = 2√1000 ≈ 63.2 Ω.

MAT-345 · How do eigenvalues classify the phase portrait of a 2D linear system?

Real, same sign: node (stable if negative). Real, opposite signs: saddle (unstable). Complex with non-zero real part: spiral, stable if negative. Purely imaginary: centre.

MAT-346 · How do you analyse the stability of an equilibrium of a nonlinear ODE system?

Linearize with the Jacobian at the equilibrium. If all eigenvalues have Re < 0 it is stable; if any has Re > 0 it is unstable. If some have Re = 0, linearization is inconclusive and a Lyapunov function is needed.

MAT-347 · When is the Frobenius method needed instead of a power series?

At a regular singular point (e.g. x = 0 for Bessel's equation), an ordinary power series may fail. Frobenius assumes y = Σaₙxⁿ⁺ʳ and solves an indicial equation for r.

MAT-348 · Where do Bessel and Legendre equations arise in engineering?

Bessel's equation, x²y″ + xy′ + (x² − n²)y = 0, comes from separating PDEs in cylindrical coordinates: drum vibration, heat in rods, waveguides. Legendre's equation comes from spherical problems such as potential around a sphere.

MAT-349 · What is a stiff ODE, and how should it be solved?

A system with widely separated time scales, such as fast and slow chemical reactions. Explicit methods need tiny steps for stability, not accuracy. Implicit methods (backward Euler, BDF, implicit Runge–Kutta) allow much larger steps.

MAT-350 · What is the logistic growth model?

dP/dt = rP(1 − P/K), with solution P = K/(1 + Ae^(−rt)), A = (K − P₀)/P₀. Growth is fastest at P = K/2 (the inflection point) and levels off at the carrying capacity K.

MAT-351 · A 100 L tank holds 10 kg of dissolved salt. Pure water enters and mixture leaves at 5 L/min. When is half the salt gone?

dQ/dt = −5Q/100, so Q = 10e^(−t/20). Half remains when t = 20 ln 2 ≈ 13.9 min.

MAT-352 · What is the tip deflection of a cantilever with end load P?

From EI·y″ = M(x) = −P(L − x), integrating twice with y(0) = y′(0) = 0 gives δ = PL³/(3EI) at the tip, with slope PL²/(2EI).

Partial Differential Equations

MAT-353 · How do you classify second-order PDEs?

For Au_xx + Bu_xy + Cu_yy + … = 0, use the discriminant B² − 4AC: < 0 elliptic (Laplace), = 0 parabolic (heat), > 0 hyperbolic (wave). The type determines the boundary and initial conditions and the numerical method.

MAT-354 · Classify the Tricomi equation u_xx + x·u_yy = 0.

B² − 4AC = −4x. It is elliptic for x > 0, parabolic on x = 0 and hyperbolic for x < 0. This mixed type models transonic flow.

MAT-355 · A 1 m rod with ends at 0 °C starts at u = sin(πx) °C, with α² = 10⁻⁴ m²/s. When does the peak halve?

u = sin(πx)e^(−π²α²t). Halving needs π²α²t = ln 2, so t = 0.693/(9.87×10⁻⁴) ≈ 702 s, about 11.7 minutes.

MAT-356 · What is the stability condition for the explicit FTCS scheme for the heat equation?

r = α²Δt/Δx² ≤ 1/2. Halving Δx forces Δt to drop fourfold, which makes explicit schemes expensive on fine grids.

MAT-357 · What is the CFL condition for the wave equation?

For explicit schemes, the Courant number C = cΔt/Δx ≤ 1. The numerical domain of dependence must contain the physical one, so information cannot travel more than one cell per step.

MAT-358 · What are the properties of the Crank–Nicolson scheme?

It averages the explicit and implicit schemes at time levels n and n + 1. It is unconditionally stable and second-order accurate in time and space, but large r can cause non-physical oscillations near sharp gradients.

MAT-359 · What is the series solution for a plucked string released from rest?

u(x, t) = Σ bₙ sin(nπx/L)cos(nπct/L), with bₙ = (2/L)∫₀ᴸ f(x)sin(nπx/L)dx from the initial shape f(x). Zero initial velocity removes the sine-in-time terms.

MAT-360 · How is Laplace's equation solved by finite differences?

The 5-point stencil on a uniform grid gives u_(i,j) = (u_(i+1,j) + u_(i−1,j) + u_(i,j+1) + u_(i,j−1))/4. The resulting sparse system is solved by Gauss–Seidel, SOR, conjugate gradient or multigrid.

MAT-361 · What boundary conditions would you use for a convectively cooled fin?

Fixed base temperature (Dirichlet) at the root, and a Robin condition −k∂T/∂n = h(T − T∞) at convecting surfaces. If tip losses are negligible, use an insulated tip, ∂T/∂x = 0 (Neumann).

MAT-362 · Why are circular drum overtones not harmonic?

Separating the wave equation in polar coordinates gives Bessel functions. The modal frequencies are proportional to the zeros of Jₙ, which are not integer multiples of the fundamental, unlike the sines of a string.

MAT-363 · What is the temperature in a semi-infinite solid after its surface is suddenly set to T_s?

(T − T_s)/(Tᵢ − T_s) = erf(x/(2√(αt))). The thermal penetration depth grows as √(αt), so heating twice as deep takes four times as long.

Complex Variables

MAT-364 · What do the Cauchy–Riemann equations say about analytic functions?

f = u + iv is analytic in a region if u_x = v_y and u_y = −v_x there, with continuous partial derivatives. In polar form: u_r = (1/r)v_θ and v_r = −(1/r)u_θ. They make u and v harmonic with orthogonal level curves.

MAT-365 · Evaluate ∫₋∞^∞ dx/(1 + x²) using residues.

Close the contour in the upper half-plane. The only enclosed pole is z = i, with residue 1/(2i). The integral is 2πi·1/(2i) = π, matching [arctan x] = π.

MAT-366 · Evaluate ∫₀^(2π) dθ/(2 + cos θ).

Substitute z = e^(iθ) to get a contour integral on |z| = 1, or use the standard result 2π/√(a² − b²) for a > |b|. With a = 2, b = 1 the integral is 2π/√3 ≈ 3.628.

MAT-367 · Evaluate ∫₋∞^∞ cos x/(x² + 1) dx.

Take the real part of ∮e^(iz)/(z² + 1)dz over the upper half-plane; Jordan's lemma makes the arc vanish. The residue at z = i is e^(−1)/(2i), so the integral is π/e ≈ 1.156.

MAT-368 · How are residues used to find inverse Laplace transforms?

The Bromwich integral gives f(t) as the sum of residues of F(s)e^(st) at all poles of F(s). Example: F = 1/(s + a)² has a double pole with residue d/ds[e^(st)] at s = −a, so f(t) = t·e^(−at).

MAT-369 · How does the Nyquist stability criterion use complex analysis?

It applies the argument principle to 1 + G(s)H(s) along a contour enclosing the right half-plane: N = Z − P, where N is the number of clockwise encirclements of −1. The closed loop is stable if Z = N + P = 0.

MAT-370 · State the argument principle.

For f meromorphic inside and on C with no zeros or poles on C, (1/2πi)∮f′(z)/f(z)dz = Z − P, the number of zeros minus poles inside, counted with multiplicity. It equals the winding number of f(C) around the origin.

MAT-371 · How is the Joukowski transform used in aerodynamics?

w = z + c²/z maps circles in the z-plane to airfoil-like shapes. Since the potential flow around a cylinder is known, the conformal map gives the flow and lift around the airfoil (Kutta–Joukowski).

MAT-372 · What are the complex potentials for uniform flow and a source?

Uniform flow: w = Uz. Source of strength m: w = (m/2π)ln z. In w = φ + iψ, φ is the velocity potential and ψ the stream function, and the velocity is u − iv = dw/dz.

MAT-373 · How is Rouché's theorem used?

If |g| < |f| on a closed contour C, then f and f + g have the same number of zeros inside C. Example: z⁵ + 3z + 1 has all 5 zeros in |z| < 2 because |3z + 1| ≤ 7 < 32 = |z⁵| on |z| = 2.

MAT-374 · Find the residue of eᶻ/(z − 1)² at z = 1.

It is a pole of order 2, so the residue is d/dz[eᶻ] at z = 1, which is e.

Probability

MAT-375 · Compare the reliability of two 0.9-reliable components in series and in parallel.

Series: R = 0.9 × 0.9 = 0.81, since both must work. Parallel (redundant): R = 1 − (0.1)² = 0.99. Redundancy cuts failure probability tenfold.

MAT-376 · A component has a constant failure rate λ = 10⁻⁴ per hour. Find its MTBF and its reliability at 1000 h.

MTBF = 1/λ = 10,000 h. R(t) = e^(−λt) = e^(−0.1) ≈ 0.905. Only about 36.8% of units survive to the MTBF itself.

MAT-377 · An M/M/1 queue has arrival rate 8/h and service rate 10/h. Find utilization, the mean number in the system and the mean time in the system.

ρ = λ/μ = 0.8. L = ρ/(1 − ρ) = 4 customers. W = 1/(μ − λ) = 0.5 h, consistent with Little's law L = λW.

MAT-378 · Find the stationary distribution of the Markov chain P = [[0.9, 0.1], [0.5, 0.5]].

Solve πP = π with π₁ + π₂ = 1. Balance gives 0.1π₁ = 0.5π₂, so π₁ = 5π₂ and π = (5/6, 1/6).

MAT-379 · Requests arrive as a Poisson process at 3 per minute. What is the probability of no arrival in one minute?

P(0) = e^(−3) ≈ 0.0498, about 5%. Equivalently, the inter-arrival time is exponential with mean 20 s.

MAT-380 · Each item has a 10% defect probability. In a sample of 10, what is P(no defects)?

P(0) = 0.9¹⁰ ≈ 0.349, so P(at least one defect) ≈ 0.651. The expected number of defects is np = 1.

MAT-381 · Three parts each have tolerance ±0.1 mm at 3σ. What is the stack-up tolerance?

Worst case: ±0.3 mm. Statistically, independent variances add: ±√(3×0.1²) ≈ ±0.173 mm at 3σ. The root-sum-square method permits looser part tolerances for the same assembly yield.

MAT-382 · What are the mean and variance of the number of die rolls until the first six?

Geometric with p = 1/6: mean 1/p = 6 rolls, variance (1 − p)/p² = (5/6)×36 = 30.

MAT-383 · What is the multivariate normal distribution?

f(x) = (2π)^(−n/2)|Σ|^(−1/2)exp(−½(x − μ)ᵀΣ⁻¹(x − μ)), with mean vector μ and covariance matrix Σ, which must be symmetric positive definite. Contours are ellipsoids along Σ's eigenvectors.

MAT-384 · How does Monte Carlo error scale with the number of samples?

The standard error is σ/√N, independent of dimension. Halving the error needs four times the samples; variance reduction (importance or stratified sampling) improves the constant.

MAT-385 · Shafts have diameter μ = 25.00 mm and σ = 0.02 mm. The tolerance is 25.00 ± 0.04 mm. What fraction is rejected?

The limits are ±2σ, so about 95.45% conform and about 4.55% are rejected. Halving σ to 0.01 mm makes the limits ±4σ, with nearly zero rejects.

Statistics

MAT-386 · How large a sample is needed to estimate a mean within ±1 at 95% confidence if σ = 5?

n = (zσ/E)² = (1.96×5/1)² = 96.04, rounded up to 97.

MAT-387 · H₀: μ = 50. A sample of n = 36 gives x̄ = 52 with σ = 6. Test at 5% (two-tailed).

z = (52 − 50)/(6/√36) = 2/1 = 2.0. Since 2.0 > 1.96, reject H₀ (p ≈ 0.046). The result is just significant.

MAT-388 · What is maximum likelihood estimation?

Choose parameters θ that maximize L(θ) = Π f(xᵢ|θ), usually through the log-likelihood. For Poisson data λ̂ = x̄; for normal data μ̂ = x̄ and σ̂² = (1/n)Σ(xᵢ − x̄)², which is slightly biased.

MAT-389 · How are control limits set in Shewhart x̄ charts?

Centre line x̄̄ with limits at ±3σ/√n (often estimated from ranges). For an in-control normal process, a point falls outside by chance with probability ≈ 0.27%, about 1 in 370.

MAT-390 · What is multicollinearity in regression, and how do you handle it?

When predictors are highly correlated, coefficient estimates become unstable with inflated variances, though predictions may be fine. Detect it with the variance inflation factor (VIF); remedies include dropping or combining variables, ridge regression or PCA.

MAT-391 · What is the bias–variance trade-off?

Expected squared error = bias² + variance + irreducible noise. Simple models underfit (high bias); complex models overfit (high variance). Cross-validation or regularization picks the balance.

MAT-392 · What are Cp and Cpk? Compute them for μ = 25.00, σ = 0.02 and limits 24.96–25.04 mm.

Cp = (USL − LSL)/(6σ) = 0.08/0.12 ≈ 0.67. Cpk = min(USL − μ, μ − LSL)/(3σ) ≈ 0.67, equal because the process is centred. Values below 1 mean the process is not capable.

MAT-393 · When should you use a paired t-test instead of a two-sample t-test?

When the same units are measured twice (before and after) or are naturally matched. Testing the differences removes unit-to-unit variability and gives more power.

MAT-394 · What is a 2ᵏ factorial design?

k factors, each at two levels, in 2ᵏ runs, estimating all main effects and interactions. Randomization, replication and blocking control bias and noise; fractional factorials reduce runs at the cost of confounding interactions.

MAT-395 · When would you choose a nonparametric test?

When data are ordinal, clearly non-normal, contain outliers, or samples are too small to check assumptions. Examples: Mann–Whitney U instead of the two-sample t, Wilcoxon signed-rank instead of the paired t, and Kruskal–Wallis instead of one-way ANOVA.

Numerical Methods

MAT-396 · When does Newton–Raphson converge, and when does it fail?

It converges quadratically near a simple root when the start is close enough. It fails or slows when f′(x) ≈ 0 (a huge jump), when the guess is poor, near oscillating or inflection regions (cycling), or at multiple roots (only linear convergence).

MAT-397 · Apply Newton–Raphson to f(x) = x³ − 2x − 5 starting at x₀ = 2.

f(2) = −1, f′(2) = 10, so x₁ = 2.1. f(2.1) = 0.061, f′(2.1) = 11.23, so x₂ ≈ 2.0946. The root is 2.09455.

MAT-398 · How do adaptive step-size ODE solvers work?

Embedded Runge–Kutta pairs (e.g. Dormand–Prince 4(5)) compute two solutions of different order each step; their difference estimates local error. The step is rejected and reduced if the error exceeds tolerance, and enlarged when it is well below.

MAT-399 · What is Richardson extrapolation?

If A(h) = A + Chᵖ + …, combine two step sizes to cancel the leading error: A ≈ [2ᵖA(h/2) − A(h)]/(2ᵖ − 1). Applied repeatedly to the trapezoidal rule it becomes Romberg integration.

MAT-400 · Why is Gaussian quadrature efficient?

An n-point Gauss–Legendre rule is exact for polynomials up to degree 2n − 1 by choosing both nodes and weights optimally. The 2-point rule on [−1, 1] uses nodes ±1/√3, both with weight 1.

MAT-401 · Estimate ∫₀¹ x² dx with h = 0.5 by the trapezoidal and Simpson's rules.

Trapezoidal: (0.5/2)[0 + 2(0.25) + 1] = 0.375, an error of 0.0417. Simpson: (0.5/3)[0 + 4(0.25) + 1] = 0.3333, exact, because Simpson integrates cubics exactly.

MAT-402 · What step size keeps explicit Euler stable for y′ = −λy (λ > 0)?

yₙ₊₁ = (1 − hλ)yₙ, which is stable when |1 − hλ| ≤ 1, i.e. h ≤ 2/λ. With λ = 1000 s⁻¹, h must be at most 2 ms even if the solution is smooth.

MAT-403 · Why is pivoting used in Gaussian elimination?

Dividing by a small pivot amplifies round-off. Partial pivoting swaps rows so that the largest-magnitude entry in the column becomes the pivot, which keeps multipliers ≤ 1 and the algorithm numerically stable in practice.

MAT-404 · What is Runge's phenomenon, and how is it avoided?

High-degree polynomial interpolation at equally spaced nodes oscillates wildly near the interval ends, e.g. for 1/(1 + 25x²). Use Chebyshev nodes, piecewise interpolation or splines.

MAT-405 · What is a cubic spline?

A piecewise cubic through the data with continuous first and second derivatives at the knots. A natural spline sets y″ = 0 at the ends. Splines avoid the oscillation of high-degree polynomials.

Transforms (Laplace, Fourier, Z)

MAT-406 · Find the transfer function and unit-step response of y″ + 3y′ + 2y = u with zero initial conditions.

G(s) = 1/[(s + 1)(s + 2)]. For a unit step, Y = 1/[s(s + 1)(s + 2)] = (1/2)/s − 1/(s + 1) + (1/2)/(s + 2), so y(t) = 1/2 − e^(−t) + (1/2)e^(−2t).

MAT-407 · What is the steady-state output of G(s) = 10/(s + 5) to a unit step?

The final value theorem gives lim(s→0) s·(1/s)·10/(s + 5) = 2, valid because the only pole, s = −5, is stable. The DC gain is 2 and the time constant is 0.2 s.

MAT-408 · A 1 kHz tone is sampled at 1.5 kHz. What frequency appears?

The Nyquist rate is 2 kHz, so the tone aliases to |1.0 − 1.5| = 0.5 kHz. An anti-aliasing low-pass filter before the ADC prevents this.

MAT-409 · What do the DFT and FFT compute, and what sets frequency resolution?

The N-point DFT gives N spectral samples of a finite sequence; direct computation is O(N²), and the FFT reduces it to O(N log N). Frequency resolution is Δf = f_s/N = 1/(record length).

MAT-410 · What is spectral leakage, and how is it reduced?

When the record does not contain an integer number of cycles, energy spreads into neighbouring bins. Windowing (Hann, Hamming, Blackman) tapers the ends to lower sidelobes, at the cost of a wider main lobe.

MAT-411 · How do you judge the stability of a discrete-time system from its Z-transform?

A causal LTI system is BIBO stable if all poles of H(z) lie strictly inside the unit circle, which means the ROC includes |z| = 1.

MAT-412 · What is the bilinear transform used for?

Converting analog filters to digital by s = (2/T)(z − 1)/(z + 1). It maps the left half-plane inside the unit circle, preserving stability, but it warps frequency, so critical frequencies are pre-warped with ω_a = (2/T)tan(ω_dT/2).

MAT-413 · Find the impulse response of y[n] − 0.5y[n − 1] = x[n].

H(z) = 1/(1 − 0.5z⁻¹) = z/(z − 0.5), so h[n] = (0.5)ⁿu[n]. The pole at 0.5 is inside the unit circle, so the system is stable.

MAT-414 · How is the convolution theorem used in filtering?

Time-domain convolution y = h*x becomes multiplication Y = H·X. Long FIR filters are often run as FFT, multiply, inverse FFT (overlap-add or overlap-save), which is faster than direct convolution.

MAT-415 · What is the Fourier transform of a rectangular pulse of unit height and width T?

F(ω) = T·sin(ωT/2)/(ωT/2), a sinc with first zeros at ω = ±2π/T. A narrower pulse has a wider spectrum: the time–bandwidth trade-off.

Discrete Math & Optimization

MAT-416 · How does Dijkstra's algorithm work, and what are its limits?

It grows a set of settled vertices, repeatedly taking the unsettled vertex with the smallest tentative distance and relaxing its edges. With a binary heap it runs in O((V + E)log V). It requires non-negative weights; use Bellman–Ford otherwise.

MAT-417 · How do Kruskal's and Prim's algorithms build a minimum spanning tree?

Kruskal sorts edges and adds the cheapest one that forms no cycle (union–find), O(E log E). Prim grows one tree, adding the cheapest edge leaving it. Both are greedy and optimal by the cut property.

MAT-418 · How is the assignment problem solved?

Assigning n workers to n jobs at minimum cost is solved exactly by the Hungarian method in O(n³). It is a special LP whose optimal solutions are integral.

MAT-419 · Maximize Z = 3x + 5y subject to x ≤ 4, 2y ≤ 12, 3x + 2y ≤ 18, x, y ≥ 0.

The vertices are (0,0), (4,0), (4,3), (2,6) and (0,6). Z values: 0, 12, 27, 36 and 30. The optimum is x = 2, y = 6, Z = 36.

MAT-420 · What is LP duality, and what are shadow prices?

Every LP has a dual; for a maximization primal, the dual is a minimization with any feasible dual value ≥ any primal value. Strong duality gives equal optimal values. Dual variables are shadow prices: the objective gain per unit increase of each resource.

MAT-421 · Why is convex optimization preferred in engineering design?

Any local minimum is global, and efficient algorithms (interior-point methods) solve large problems reliably with certificates of optimality. Many design problems are formulated or relaxed into convex form for this reason.

MAT-422 · How does integer programming differ from LP?

Integer constraints make the problem NP-hard in general. Branch and bound solves LP relaxations and splits on fractional variables; cutting planes tighten the relaxation. The LP relaxation gives a bound on the integer optimum.

MAT-423 · Solve T(n) = 2T(n/2) + n with the master theorem.

a = 2, b = 2, f(n) = n = n^(log_b a), which is case 2, so T(n) = Θ(n log n), as for merge sort.

MAT-424 · How is graph colouring used for scheduling?

Make tasks vertices and join conflicting tasks (e.g. exams sharing students) with an edge. The chromatic number is the minimum number of time slots. Planar graphs need at most 4 colours.

MAT-425 · What is the max-flow min-cut theorem?

In a flow network, the maximum flow from source to sink equals the minimum total capacity of edges whose removal separates them. Ford–Fulkerson or Edmonds–Karp (O(VE²)) find it using augmenting paths.

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