Intermediate Engineering Math interview questions

150 questions with answers (MAT-151 to MAT-300). Try each one aloud before reading the answer.

Linear Algebra

MAT-151 · How do you check whether Ax = b is consistent?

Compare rank(A) with rank of the augmented matrix [A|b]. If they differ, there is no solution. If equal to n (number of unknowns), the solution is unique; if less than n, there are infinitely many.

MAT-152 · Why are the eigenvalues of a triangular matrix its diagonal entries?

For a triangular matrix, det(A − λI) is the product of the diagonal terms (aᵢᵢ − λ), so the characteristic roots are exactly the aᵢᵢ.

MAT-153 · What is the Cayley–Hamilton theorem and how is it used?

Every square matrix satisfies its own characteristic equation. It is used to express A⁻¹ and high powers Aⁿ as polynomials in A of degree less than n.

MAT-154 · How does diagonalization work, and what does it require?

A = PDP⁻¹, where P holds eigenvectors as columns and D holds eigenvalues. It needs n linearly independent eigenvectors, and it makes powers easy: Aᵏ = PDᵏP⁻¹.

MAT-155 · Find the eigenvalues of A = [[2, 1], [1, 2]].

det(A − λI) = (2 − λ)² − 1 = λ² − 4λ + 3 = 0, so λ = 1 and 3. Eigenvectors are (1, −1) and (1, 1).

MAT-156 · How are the eigenvalues related to the trace and determinant?

The sum of the eigenvalues equals the trace, and their product equals the determinant. For a 2×2 matrix: λ² − (tr A)λ + det A = 0.

MAT-157 · If A has eigenvalue λ, what are the eigenvalues of A⁻¹, Aᵏ and A + cI?

1/λ (for invertible A), λᵏ and λ + c respectively, all with the same eigenvectors. For a polynomial p(A), the eigenvalue is p(λ).

MAT-158 · What is a positive definite matrix?

A symmetric matrix with xᵀAx > 0 for every non-zero x. Equivalently, all eigenvalues are positive or all leading principal minors are positive (Sylvester's criterion).

MAT-159 · What is LU decomposition and why is it useful?

Factoring A = LU into lower and upper triangular matrices. After one O(n³) factorization, each new right-hand side b is solved by forward and back substitution in O(n²).

MAT-160 · What is Gram–Schmidt orthogonalization?

A procedure that converts linearly independent vectors into an orthonormal set by subtracting projections onto the earlier vectors and normalising. It is the basis of QR factorization.

MAT-161 · How do you find the least-squares solution of an overdetermined system?

Minimise ‖Ax − b‖² by solving the normal equations AᵀAx = Aᵀb. In practice QR or SVD is preferred because forming AᵀA squares the condition number.

MAT-162 · What is the rank of the outer product uvᵀ of two non-zero vectors?

• Every column is a multiple of u, so the column space is one-dimensional.

MAT-163 · How many solutions does a homogeneous system with more unknowns than equations have?

Infinitely many. With m < n, rank ≤ m < n, so the nullity n − rank ≥ 1 and non-trivial solutions exist.

Differential Calculus

MAT-164 · Evaluate lim(x→0) sin x / x.

• It follows from the squeeze theorem or L'Hôpital's rule (cos x/1 → 1), and it justifies sin x ≈ x for small angles in radians.

MAT-165 · What is lim(n→∞) (1 + 1/n)ⁿ?

e ≈ 2.71828. More generally, lim (1 + x/n)ⁿ = eˣ, which is the basis of continuous compounding.

MAT-166 · How do you find and classify extrema of f(x, y)?

Solve f_x = f_y = 0. Compute D = f_xx·f_yy − f_xy². D > 0 with f_xx > 0 is a minimum; D > 0 with f_xx < 0 is a maximum; D < 0 is a saddle; D = 0 is inconclusive.

MAT-167 · What is the method of Lagrange multipliers?

To optimise f subject to g = 0, solve ∇f = λ∇g together with g = 0. At a constrained extremum the gradients are parallel, and λ gives the sensitivity of the optimum to the constraint.

MAT-168 · State Euler's theorem on homogeneous functions.

If f(x, y) is homogeneous of degree n, i.e. f(tx, ty) = tⁿf(x, y), then x·∂f/∂x + y·∂f/∂y = n·f.

MAT-169 · If the radius of a circle has a 1% error, what is the error in its area?

A = πr², so dA/A = 2·dr/r = 2%. In general, the relative error in xⁿ is about n times the relative error in x.

MAT-170 · Find dy/dx for x² + y² = 25 by implicit differentiation.

2x + 2y·dy/dx = 0, so dy/dx = −x/y. At (3, 4) the slope is −3/4.

MAT-171 · What is the derivative of xˣ?

Take logs: ln y = x ln x, so y′/y = ln x + 1, giving d/dx(xˣ) = xˣ(1 + ln x).

MAT-172 · What is the formula for radius of curvature?

R = [1 + (y′)²]^(3/2)/|y″|. Curvature κ = 1/R. For a beam with small slopes, κ ≈ |y″|.

MAT-173 · What is the Hessian matrix?

The square matrix of second partial derivatives, H_ij = ∂²f/∂xᵢ∂xⱼ. At a critical point, positive definite H means a minimum, negative definite a maximum, and indefinite a saddle.

MAT-174 · What is the Leibniz rule for differentiating under the integral sign?

d/dx ∫_(a(x))^(b(x)) f(x, t)dt = f(x, b)·b′(x) − f(x, a)·a′(x) + ∫ₐᵇ ∂f/∂x dt.

MAT-175 · What are the Maclaurin series of sin x and cos x?

sin x = x − x³/3! + x⁵/5! − …, and cos x = 1 − x²/2! + x⁴/4! − …, both valid for all x in radians.

MAT-176 · What are the nth derivatives of e^(ax) and sin(ax)?

dⁿ/dxⁿ e^(ax) = aⁿe^(ax), and dⁿ/dxⁿ sin(ax) = aⁿ sin(ax + nπ/2).

Integral Calculus

MAT-177 · Evaluate ∫₀^(π/2) sin²x dx.

Using sin²x = (1 − cos 2x)/2, the integral is [x/2 − sin 2x/4] from 0 to π/2 = π/4.

MAT-178 · What are Wallis' formulas?

∫₀^(π/2) sinⁿx dx = ∫₀^(π/2) cosⁿx dx = [(n − 1)(n − 3)…]/[n(n − 2)…], multiplied by π/2 if n is even and by 1 if n is odd. Example: n = 4 gives 3π/16.

MAT-179 · What is ∫₋∞^∞ e^(−x²) dx?

√π. Square the integral, convert the double integral to polar coordinates, and get ∫₀^(2π)∫₀^∞ e^(−r²)r dr dθ = π.

MAT-180 · What are Γ(5), Γ(1/2) and Γ(3/2)?

Γ(5) = 4! = 24, Γ(1/2) = √π and Γ(3/2) = (1/2)Γ(1/2) = √π/2.

MAT-181 · Why change the order of integration in a double integral?

An inner integral may be impossible or tedious in one order but easy in the other. You sketch the region and rewrite the limits; for example, ∫₀¹∫ₓ¹ e^(y²)dy dx becomes ∫₀¹∫₀ʸ e^(y²)dx dy = (e − 1)/2.

MAT-182 · Find the area of a circle of radius a using polar coordinates.

A = ∫₀^(2π)∫₀ᵃ r dr dθ = 2π·a²/2 = πa².

MAT-183 · How do you find the centroid of a plane region?

x̄ = ∬x dA/∬dA and ȳ = ∬y dA/∬dA. For a triangle this gives the intersection of the medians, at one-third of the height from the base.

MAT-184 · How is the volume of a sphere found with a triple integral?

In spherical coordinates dV = r² sin φ dr dφ dθ, so V = ∫₀^(2π)∫₀^π∫₀ᵃ r² sin φ dr dφ dθ = 2π·2·a³/3 = 4πa³/3.

MAT-185 · For which p does ∫₁^∞ dx/xᵖ converge?

For p > 1, where it equals 1/(p − 1). For p ≤ 1 it diverges; p = 1 gives ln x → ∞.

MAT-186 · Evaluate ∫ dx/(x² − 1) using partial fractions.

1/(x² − 1) = (1/2)[1/(x − 1) − 1/(x + 1)], so the integral is (1/2)ln|(x − 1)/(x + 1)| + C.

MAT-187 · What is the arc-length formula?

For y = f(x) on [a, b], L = ∫ₐᵇ √(1 + (dy/dx)²) dx. For a parametric curve, L = ∫√((dx/dt)² + (dy/dt)²) dt.

MAT-188 · What is ∫₀^∞ (sin x/x) dx?

π/2 (the Dirichlet integral). It converges conditionally but not absolutely, and it can be evaluated by Laplace transforms or contour integration.

MAT-189 · What is the formula for surface area of revolution?

Rotating y = f(x) about the x-axis gives S = 2π∫ₐᵇ y√(1 + (y′)²) dx.

Vector Calculus

MAT-190 · Find ∇φ for φ = x²y + yz at (1, 1, 1).

∇φ = (2xy, x² + z, y) = (2, 2, 1) at that point, with magnitude 3.

MAT-191 · Find the directional derivative of φ = x²y + yz at (1, 1, 1) along (1, 2, 2).

The unit vector is (1, 2, 2)/3, so D = (2, 2, 1)·(1, 2, 2)/3 = (2 + 4 + 2)/3 = 8/3.

MAT-192 · Why is div(curl F) always zero?

Expanding ∇·(∇×F) gives pairs of mixed second partial derivatives with opposite signs, which cancel when the partials are continuous (Clairaut's theorem).

MAT-193 · Why is curl(grad φ) always zero?

Each component of ∇×∇φ is a difference such as ∂²φ/∂y∂z − ∂²φ/∂z∂y, which is zero for a twice continuously differentiable φ. Gradient fields are irrotational.

MAT-194 · How do you find the unit normal to a surface φ(x, y, z) = c?

n̂ = ∇φ/|∇φ|, because the gradient is perpendicular to level surfaces.

MAT-195 · Is F = (y, x, 0) conservative? If so, find its potential.

∇×F = (0, 0, 1 − 1) = 0, so it is conservative on ℝ³. The potential is φ = xy, since ∇(xy) = (y, x, 0).

MAT-196 · How do you compute the work done by a force along a path?

W = ∫_C F·dr = ∫(F₁dx + F₂dy + F₃dz), parameterising the curve r(t) and integrating F(r(t))·r′(t) dt.

MAT-197 · What are the divergence and curl of the position vector r = (x, y, z)?

∇·r = 3 and ∇×r = 0. Also ∇|r| = r/|r| = r̂.

MAT-198 · Find the outward flux of F = r through a sphere of radius a.

By the divergence theorem, flux = ∭3 dV = 3·(4/3)πa³ = 4πa³. Directly: F·n̂ = a on the surface, times area 4πa², gives the same result.

MAT-199 · How can Green's theorem compute the area of a region?

Choose P, Q with ∂Q/∂x − ∂P/∂y = 1: A = (1/2)∮(x dy − y dx). This is the principle of the planimeter.

MAT-200 · State the vector triple product identity.

a×(b×c) = b(a·c) − c(a·b), the "BAC-CAB" rule. The result lies in the plane of b and c.

MAT-201 · What is ∇·(φF) in terms of simpler quantities?

∇·(φF) = φ(∇·F) + F·∇φ, the product rule for divergence.

Ordinary Differential Equations

MAT-202 · What makes an ODE linear?

The dependent variable and its derivatives appear only to the first power, are not multiplied together and are not inside nonlinear functions. The coefficients may depend on x. Example: y″ + x²y = sin x is linear; y″ + y² = 0 is not.

MAT-203 · Solve y″ − 3y′ + 2y = 0.

The auxiliary equation m² − 3m + 2 = 0 gives m = 1, 2, so y = c₁eˣ + c₂e^(2x).

MAT-204 · Solve y″ − 2y′ + y = 0.

m² − 2m + 1 = 0 has the repeated root m = 1, so y = (c₁ + c₂x)eˣ.

MAT-205 · Solve y″ + 4y = 0.

m² + 4 = 0 gives m = ±2i, so y = c₁cos 2x + c₂sin 2x, an undamped oscillation at ω = 2 rad/s.

MAT-206 · What happens in y″ + ω²y = cos ωt?

Resonance: the forcing frequency matches the natural frequency, so the particular solution is y_p = t·sin(ωt)/(2ω), whose amplitude grows linearly without bound.

MAT-207 · When can the method of undetermined coefficients be used?

For linear constant-coefficient ODEs whose forcing term is a polynomial, exponential, sine/cosine, or a product of these. If the trial term duplicates the CF, multiply it by x.

MAT-208 · What is variation of parameters?

A general method for y″ + p y′ + q y = g(x). With CF solutions y₁, y₂ and Wronskian W: y_p = −y₁∫(y₂g/W)dx + y₂∫(y₁g/W)dx. It works for any continuous g.

MAT-209 · What is a Cauchy–Euler equation and how is it solved?

x²y″ + axy′ + by = 0. Substitute y = xᵐ to get m(m − 1) + am + b = 0, or use x = eᵗ to turn it into a constant-coefficient ODE.

MAT-210 · Solve y′ + y = eˣ.

The integrating factor is eˣ: (y eˣ)′ = e^(2x), so y eˣ = e^(2x)/2 + C, giving y = eˣ/2 + Ce^(−x).

MAT-211 · How is damping classified for m x″ + c x′ + k x = 0?

By the damping ratio ζ = c/(2√(km)). ζ < 1 is underdamped (oscillatory decay), ζ = 1 critically damped (fastest non-oscillatory return), ζ > 1 overdamped.

MAT-212 · What ODE describes Newton's law of cooling?

dT/dt = −k(T − Tₐ), where Tₐ is ambient temperature. The solution is T = Tₐ + (T₀ − Tₐ)e^(−kt).

MAT-213 · What is the voltage on a charging capacitor in an RC circuit?

RC·dv/dt + v = V gives v(t) = V(1 − e^(−t/RC)). The time constant is τ = RC; v reaches about 63.2% of V at t = τ.

MAT-214 · When does an IVP y′ = f(x, y), y(x₀) = y₀ have a unique solution?

By the Picard–Lindelöf theorem, if f and ∂f/∂y are continuous (more generally, f is Lipschitz in y) in a region around (x₀, y₀), a unique solution exists locally.

Partial Differential Equations

MAT-215 · What is the solution of the heat equation on 0 < x < L with both ends at zero?

u(x, t) = Σ Bₙ sin(nπx/L)·e^(−α²n²π²t/L²), with Bₙ = (2/L)∫₀ᴸ f(x)sin(nπx/L)dx from the initial profile f(x).

MAT-216 · Why do higher modes decay faster in the heat equation?

The decay rate of mode n is α²n²π²/L², proportional to n². Sharp spatial variations smooth out quickly, leaving the fundamental mode at long times.

MAT-217 · What is d'Alembert's solution of the wave equation?

u(x, t) = (1/2)[φ(x − ct) + φ(x + ct)] + (1/(2c))∫ψ(s)ds from x − ct to x + ct, for initial displacement φ and velocity ψ. It is two waves travelling in opposite directions at speed c.

MAT-218 · What is the fundamental frequency of a string fixed at both ends?

f₁ = c/(2L), where c = √(T/μ) for tension T and linear density μ. The harmonics are fₙ = n·f₁.

MAT-219 · Why must the separation constant be negative for the heat equation with zero end temperatures?

A positive or zero constant gives X(x) solutions that cannot vanish at both ends except trivially. A negative constant −λ² gives sines that satisfy the boundary conditions and decaying exponentials in time.

MAT-220 · What is Laplace's equation in polar coordinates?

∂²u/∂r² + (1/r)∂u/∂r + (1/r²)∂²u/∂θ² = 0. Separation gives solutions rⁿcos nθ, rⁿsin nθ (and r⁻ⁿ, ln r terms if the origin is excluded).

MAT-221 · What is the steady-state temperature in a rod with ends held at T₁ and T₂?

Setting ∂u/∂t = 0 gives u″ = 0, so u is linear: u(x) = T₁ + (T₂ − T₁)x/L.

MAT-222 · What makes a PDE problem well-posed (Hadamard)?

A solution exists, it is unique, and it depends continuously on the initial and boundary data, so small data errors cause small solution changes.

MAT-223 · How is Lagrange's linear PDE Pp + Qq = R solved?

Here p = ∂z/∂x and q = ∂z/∂y. Solve the auxiliary equations dx/P = dy/Q = dz/R to get two independent integrals u = c₁ and v = c₂; the general solution is φ(u, v) = 0.

MAT-224 · What are characteristic curves of a first-order PDE?

Curves along which the PDE reduces to an ODE. For the advection equation u_t + c·u_x = 0 they are lines x − ct = constant, so u(x, t) = f(x − ct).

MAT-225 · What is the mean value property of harmonic functions?

The value of a harmonic function at a point equals its average over any circle (or sphere) centred there that lies within the domain.

MAT-226 · Why do Fourier series appear when solving PDEs?

Separation of variables yields eigenfunctions such as sin(nπx/L) that are orthogonal and complete. Expanding the initial condition in them gives the coefficients of the series solution.

Complex Variables

MAT-227 · State Cauchy's integral formula.

If f is analytic inside and on C and z₀ is inside, f(z₀) = (1/2πi)∮ f(z)/(z − z₀) dz. For derivatives, f⁽ⁿ⁾(z₀) = (n!/2πi)∮ f(z)/(z − z₀)ⁿ⁺¹ dz.

MAT-228 · State the residue theorem.

If f is analytic inside and on C except at isolated singularities inside C, then ∮_C f(z)dz = 2πi × (sum of residues inside C).

MAT-229 · How do you find the residue at a simple pole?

Res = lim(z→z₀) (z − z₀)f(z). If f = p/q with q(z₀) = 0 and q′(z₀) ≠ 0, Res = p(z₀)/q′(z₀).

MAT-230 · How do you find the residue at a pole of order m?

Res = (1/(m − 1)!)·lim(z→z₀) dᵐ⁻¹/dzᵐ⁻¹[(z − z₀)ᵐf(z)].

MAT-231 · Evaluate ∮ dz/(z − 1) around |z| = 2.

The simple pole z = 1 lies inside with residue 1, so the integral is 2πi.

MAT-232 · Is f(z) = z̄ analytic?

No. With u = x, v = −y, u_x = 1 but v_y = −1, so the Cauchy–Riemann equations fail everywhere. It is continuous but nowhere complex-differentiable.

MAT-233 · What is a removable singularity?

A point where f is undefined but the limit exists and the Laurent series has no negative powers. Example: sin z/z at z = 0; defining f(0) = 1 makes it analytic.

MAT-234 · What is an essential singularity?

An isolated singularity whose Laurent series has infinitely many negative powers, e.g. e^(1/z) at z = 0. Near it the function takes nearly every complex value (Picard's theorem).

MAT-235 · Find the harmonic conjugate of u = x² − y².

From v_y = u_x = 2x and v_x = −u_y = 2y, v = 2xy + C. Then f = u + iv = z².

MAT-236 · What is a conformal mapping?

A mapping w = f(z) that preserves angles and their orientation locally. Any analytic f is conformal wherever f′(z) ≠ 0.

MAT-237 · What is a bilinear (Möbius) transformation?

w = (az + b)/(cz + d) with ad − bc ≠ 0. It maps circles and lines to circles and lines, and it is fixed uniquely by the images of three points.

MAT-238 · What is the logarithm of a complex number?

log z = ln|z| + i(Arg z + 2πk), k ∈ ℤ, which is multivalued. The principal value Log z uses −π < Arg z ≤ π, e.g. Log(−1) = iπ.

Probability

MAT-239 · State Bayes' theorem.

P(A|B) = P(B|A)·P(A)/P(B), where P(B) = Σ P(B|Aᵢ)P(Aᵢ). It updates a prior belief P(A) into a posterior P(A|B) using evidence B.

MAT-240 · What is the law of total probability?

If events A₁…Aₙ partition the sample space, P(B) = Σ P(B|Aᵢ)P(Aᵢ). It supplies the denominator in Bayes' theorem.

MAT-241 · A disease affects 1% of people. A test has 99% sensitivity and a 5% false-positive rate. What is P(disease | positive)?

P(+) = 0.99×0.01 + 0.05×0.99 = 0.0099 + 0.0495 = 0.0594. P(D|+) = 0.0099/0.0594 ≈ 16.7%, low because the disease is rare.

MAT-242 · What is the probability of a sum of 7 with two fair dice?

6 favourable outcomes (1-6, 2-5, 3-4, 4-3, 5-2, 6-1) out of 36, so 1/6. Seven is the most likely sum.

MAT-243 · What is the probability of at least one head in three coin tosses?

Use the complement: 1 − P(no heads) = 1 − (1/2)³ = 7/8.

MAT-244 · When can the Poisson distribution approximate the binomial?

When n is large and p is small with λ = np moderate (a common rule of thumb is n ≥ 20, p ≤ 0.05). Then C(n, k)pᵏ(1 − p)ⁿ⁻ᵏ ≈ e^(−λ)λᵏ/k!.

MAT-245 · What is the memoryless property of the exponential distribution?

P(X > s + t | X > s) = P(X > t). A component with exponential life that has survived s hours has the same remaining-life distribution as a new one.

MAT-246 · What is Var(aX + b)?

a²·Var(X). Adding a constant shifts the distribution but does not change spread; scaling by a scales the standard deviation by |a|.

MAT-247 · What is Var(X + Y)?

Var(X) + Var(Y) + 2Cov(X, Y). For independent (or uncorrelated) variables the covariance is zero, so the variances simply add.

MAT-248 · What is the geometric distribution?

The number of trials until the first success: P(X = k) = (1 − p)ᵏ⁻¹p. Mean 1/p, variance (1 − p)/p². It is the discrete memoryless distribution.

MAT-249 · How do you check independence of two continuous random variables?

X and Y are independent if the joint PDF factors into the marginals, f(x, y) = f_X(x)·f_Y(y), for all x and y.

MAT-250 · What is the Central Limit Theorem?

The mean of n independent, identically distributed variables with finite variance σ² tends to a normal distribution with mean μ and standard deviation σ/√n as n grows, regardless of the original distribution.

MAT-251 · What is a moment generating function?

M(t) = E[e^(tX)]. Its nth derivative at t = 0 gives E[Xⁿ]. The MGF of a sum of independent variables is the product of their MGFs.

Statistics

MAT-252 · What is the 68-95-99.7 rule for the normal distribution?

About 68% of data lie within ±1σ of the mean, about 95% within ±2σ, and about 99.7% within ±3σ.

MAT-253 · Why does sample variance divide by n − 1?

Deviations are measured from x̄, which is itself fitted to the data and uses one degree of freedom. Dividing by n − 1 (Bessel's correction) makes s² an unbiased estimator of σ².

MAT-254 · What is the standard error of the mean?

SE = σ/√n (estimated by s/√n). It is the standard deviation of the sampling distribution of x̄; quadrupling n halves it.

MAT-255 · What are Type I and Type II errors?

Type I: rejecting a true H₀ (false positive), with probability α. Type II: failing to reject a false H₀ (false negative), with probability β. Power = 1 − β.

MAT-256 · What is a 95% confidence interval for a mean with known σ?

x̄ ± 1.96·σ/√n. If the sampling were repeated many times, about 95% of intervals built this way would contain the true mean.

MAT-257 · When do you use a t-test instead of a z-test?

When the population σ is unknown and estimated by s, especially for small samples from a roughly normal population. The t-distribution has n − 1 degrees of freedom and heavier tails.

MAT-258 · What is the chi-square test used for?

Testing goodness of fit to a distribution and independence in contingency tables, using χ² = Σ(O − E)²/E compared with the χ² distribution at the appropriate degrees of freedom.

MAT-259 · What is the least-squares slope of a regression line?

b = Σ(x − x̄)(y − ȳ)/Σ(x − x̄)² = r·(s_y/s_x), with intercept a = ȳ − b·x̄. The line passes through (x̄, ȳ).

MAT-260 · What is R², the coefficient of determination?

The fraction of the variance in y explained by the model: R² = 1 − SS_res/SS_tot. For simple linear regression R² = r².

MAT-261 · What is the difference between covariance and correlation?

Covariance, E[(X − μ_X)(Y − μ_Y)], has units and depends on scale. Correlation divides by σ_Xσ_Y, making it unitless and bounded by −1 and 1.

MAT-262 · What is ANOVA?

Analysis of variance tests whether three or more group means are equal by comparing between-group to within-group variance through F = MS_between/MS_within.

MAT-263 · What is the product of the two regression coefficients?

b_yx × b_xy = r². Both coefficients share the sign of r, and r is their geometric mean.

MAT-264 · What is the empirical relation between mean, median and mode?

For moderately skewed distributions, Mode ≈ 3·Median − 2·Mean. It is an approximation, not an identity.

Numerical Methods

MAT-265 · Compare the convergence orders of bisection, secant and Newton–Raphson.

Bisection is linear (error halves each step). Secant is superlinear with order ≈ 1.618. Newton–Raphson is quadratic for simple roots, roughly doubling correct digits per step.

MAT-266 · How many bisection iterations are needed to reach tolerance ε?

The interval halves each step, so n ≥ log₂((b − a)/ε). Starting from length 1 with ε = 10⁻⁶ needs n ≥ 19.93, i.e. 20 iterations.

MAT-267 · What are the error orders of the composite trapezoidal and Simpson's rules?

Trapezoidal error is O(h²) and is exact for straight lines. Simpson's 1/3 error is O(h⁴), and it is exact for polynomials up to degree 3.

MAT-268 · What is the classical fourth-order Runge–Kutta method?

y_(n+1) = yₙ + (h/6)(k₁ + 2k₂ + 2k₃ + k₄), with k₁ = f(xₙ, yₙ), k₂ = f(xₙ + h/2, yₙ + hk₁/2), k₃ = f(xₙ + h/2, yₙ + hk₂/2), k₄ = f(xₙ + h, yₙ + hk₃). Global error is O(h⁴).

MAT-269 · Take one Euler step for y′ = x + y, y(0) = 1, h = 0.1.

y₁ = y₀ + h·f(0, 1) = 1 + 0.1×(0 + 1) = 1.1. The exact value is 2e^(0.1) − 1.1 ≈ 1.1103.

MAT-270 · What is Simpson's 3/8 rule?

∫ f dx ≈ (3h/8)[y₀ + 3y₁ + 3y₂ + 2y₃ + 3y₄ + 3y₅ + 2y₆ + … + yₙ]. It fits cubics over groups of three intervals, so n must be a multiple of 3.

MAT-271 · When do the Jacobi and Gauss–Seidel methods converge?

A sufficient condition is strict diagonal dominance (|aᵢᵢ| > Σ|aᵢⱼ| for j ≠ i). Gauss–Seidel also converges for symmetric positive definite matrices. The general condition is spectral radius of the iteration matrix < 1.

MAT-272 · What is Lagrange interpolation?

P(x) = Σ yᵢLᵢ(x), where Lᵢ(x) = Π (x − xⱼ)/(xᵢ − xⱼ) over j ≠ i. It gives the unique polynomial of degree ≤ n through n + 1 points and works for unequal spacing.

MAT-273 · When is Newton's forward difference formula used?

For equally spaced data when interpolating near the start of the table. Backward differences are used near the end, and central differences in the middle.

MAT-274 · What is the central difference formula for f′(x), and why is it preferred?

f′(x) ≈ [f(x + h) − f(x − h)]/(2h). Its truncation error is O(h²), versus O(h) for forward or backward differences.

MAT-275 · What is the secant method?

x_(n+1) = xₙ − f(xₙ)(xₙ − xₙ₋₁)/[f(xₙ) − f(xₙ₋₁)]. It replaces the derivative in Newton–Raphson with a finite-difference slope, so f′ is not needed.

MAT-276 · How does Newton–Raphson compute √N?

Apply it to f(x) = x² − N to get x_(n+1) = (xₙ + N/xₙ)/2. For √2 from x₀ = 1: 1.5, 1.41667, 1.414216.

Transforms (Laplace, Fourier, Z)

MAT-277 · Why use the Laplace transform?

It turns linear constant-coefficient ODEs into algebraic equations in s and builds in initial conditions automatically. It handles discontinuous and impulsive inputs and gives transfer functions for control systems.

MAT-278 · What is the difference between a Fourier series and a Fourier transform?

A Fourier series represents a periodic signal with discrete harmonic frequencies nω₀. The Fourier transform handles aperiodic signals and gives a continuous spectrum F(ω).

MAT-279 · What is the first shifting theorem of the Laplace transform?

L{e^(at)f(t)} = F(s − a). Example: L{e^(−2t)sin 3t} = 3/((s + 2)² + 9).

MAT-280 · What are the Laplace transforms of derivatives?

L{f′} = sF(s) − f(0), and L{f″} = s²F(s) − s·f(0) − f′(0). This is how initial conditions enter the algebra.

MAT-281 · State the initial and final value theorems.

f(0⁺) = lim(s→∞) sF(s). f(∞) = lim(s→0) sF(s), valid only if all poles of sF(s) lie in the left half-plane.

MAT-282 · Solve y′ + 2y = 0, y(0) = 3, using the Laplace transform.

sY − 3 + 2Y = 0, so Y = 3/(s + 2) and y(t) = 3e^(−2t).

MAT-283 · Find the inverse Laplace transform of 1/[s(s + 1)].

Partial fractions give 1/s − 1/(s + 1), so f(t) = 1 − e^(−t): the unit-step response of a first-order system with τ = 1.

MAT-284 · What is Parseval's theorem?

Energy is the same in time and frequency domains: ∫|f(t)|²dt = (1/2π)∫|F(ω)|²dω. For Fourier series, the mean square value equals the sum of squared harmonic amplitudes (with appropriate factors).

MAT-285 · What is the Z-transform of aⁿu[n]?

Σ aⁿz⁻ⁿ = z/(z − a), with ROC |z| > |a|. With a = 1 it gives the unit step, z/(z − 1).

MAT-286 · How is the Z-transform related to the DTFT?

The DTFT is the Z-transform evaluated on the unit circle, z = e^(jω). It exists only if the ROC includes the unit circle.

MAT-287 · What is L{t·f(t)}?

−dF(s)/ds. Example: L{t sin ωt} = 2ωs/(s² + ω²)², which appears in resonance problems.

MAT-288 · What is the Gibbs phenomenon?

Near a jump discontinuity, a truncated Fourier series overshoots by about 9% of the jump. Adding terms narrows the overshoot but does not reduce its height.

Discrete Math & Optimization

MAT-289 · How many edges does the complete graph Kₙ have?

n(n − 1)/2, one for each pair of vertices. K₅ has 10 edges.

MAT-290 · What is the handshaking lemma?

The sum of all vertex degrees equals 2|E|, since each edge adds 2. So the number of odd-degree vertices is always even.

MAT-291 · When does a graph have an Euler circuit or Euler path?

A connected graph has an Euler circuit if every vertex has even degree, and an Euler path (not closed) if exactly two vertices have odd degree.

MAT-292 · What is the inclusion–exclusion principle for three sets?

|A∪B∪C| = |A| + |B| + |C| − |A∩B| − |B∩C| − |A∩C| + |A∩B∩C|.

MAT-293 · How many subsets does an n-element set have?

2ⁿ, because each element is either in or out. Of these, C(n, k) have exactly k elements.

MAT-294 · State De Morgan's laws.

¬(p ∧ q) ≡ ¬p ∨ ¬q and ¬(p ∨ q) ≡ ¬p ∧ ¬q. For sets: (A∪B)ᶜ = Aᶜ∩Bᶜ and (A∩B)ᶜ = Aᶜ∪Bᶜ. They are used to build logic from NAND or NOR gates.

MAT-295 · How are linear homogeneous recurrences solved?

Substitute aₙ = rⁿ to get the characteristic equation. For Fibonacci, aₙ = aₙ₋₁ + aₙ₋₂ gives r² = r + 1, so aₙ = Aφⁿ + Bψⁿ with φ, ψ = (1 ± √5)/2.

MAT-296 · How does the simplex method work?

It starts at a vertex of the feasible polytope and moves along edges to adjacent vertices that improve the objective, stopping when no improving direction remains.

MAT-297 · Why does a linear program's optimum occur at a vertex?

The feasible region is a convex polyhedron and the objective is linear, so its level sets are hyperplanes. The extreme value is reached at a corner point, or along an edge or face when there are ties.

MAT-298 · What are the KKT conditions?

For min f subject to gᵢ ≤ 0 and hⱼ = 0: stationarity (∇f + Σμᵢ∇gᵢ + Σλⱼ∇hⱼ = 0), primal feasibility, dual feasibility (μᵢ ≥ 0) and complementary slackness (μᵢgᵢ = 0).

MAT-299 · What is the gradient descent update rule?

x_(k+1) = x_k − η∇f(x_k), where η is the step size (learning rate). Too large an η diverges or oscillates; too small converges slowly.

MAT-300 · What is a convex function?

f(θx + (1 − θ)y) ≤ θf(x) + (1 − θ)f(y) for 0 ≤ θ ≤ 1: the chord lies above the graph. For twice-differentiable f this means the Hessian is positive semidefinite.

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